Theoretical, Modelling and Numerical Simulations Toward Industry 4.0 [1st ed.] 9789811589867, 9789811589874

Table of contents :
Front Matter ....Pages i-xiii
Rainfall Scattered Data Interpolation Using Rational Quartic Triangular Patches (Nur Nabilah Che Draman, Samsul Ariffin Abdul Karim, Ishak Hashim, Yeo Wee Ping)....Pages 1-19
A Hybrid Metaheuristic Algorithm for Truss Structure Domain’s Optimization Problem (Kallol Biswas, Pandian Vasant, Jose Antonio Gamez Vintaned, Junzo Watada, Arunava Roy, Rajalingam Sokkalingam)....Pages 21-34
Electric Field Behavior in Time Domain for Semicircle Antenna in Homogeneous Multi Layered Media (Elsayed Dahy, Afza Shafie, Noorhana Yahya)....Pages 35-51
Surface Interpolation Using Partially Blended Rational Bi-Quartic Spline (Samsul Ariffin Abdul Karim, Lila Iznita Izhar, Mahmod Othman, Nooraini Zainuddin)....Pages 53-70
Cost-Benefit Analysis of Sustainable Solar-Powered Workplace Electric Vehicle Charging Station (Kameswara Satya Prakash Oruganti, Chockalingam Aravind Vaithilingam, Gowthamraj Rajendran, Agileswari Ramasamy)....Pages 71-86
Positivity-Preserving Interpolation Using Rational Quartic Spline Functions (Samsul Ariffin Abdul Karim, Van Thien Nguyen)....Pages 87-98
Decision Support Method for Agricultural Irrigation Scenarios Performance Using WEAP Model (Saiful Azmi Husain, Nor Hamizah Mohd Rhyme)....Pages 99-106
Bayesian Variable Selection for Linear Models Using I-Priors (Haziq Jamil, Wicher Bergsma)....Pages 107-132
On the Space of m-Subharmonic Functions (Samsul Ariffin Abdul Karim, Van Thien Nguyen)....Pages 133-166
Back Matter ....Pages 167-169

Citation preview

Studies in Systems, Decision and Control 319

Samsul Ariffin Abdul Karim   Editor

Theoretical, Modelling and Numerical Simulations Toward Industry 4.0

Studies in Systems, Decision and Control Volume 319

Series Editor Janusz Kacprzyk, Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland

The series “Studies in Systems, Decision and Control” (SSDC) covers both new developments and advances, as well as the state of the art, in the various areas of broadly perceived systems, decision making and control–quickly, up to date and with a high quality. The intent is to cover the theory, applications, and perspectives on the state of the art and future developments relevant to systems, decision making, control, complex processes and related areas, as embedded in the fields of engineering, computer science, physics, economics, social and life sciences, as well as the paradigms and methodologies behind them. The series contains monographs, textbooks, lecture notes and edited volumes in systems, decision making and control spanning the areas of Cyber-Physical Systems, Autonomous Systems, Sensor Networks, Control Systems, Energy Systems, Automotive Systems, Biological Systems, Vehicular Networking and Connected Vehicles, Aerospace Systems, Automation, Manufacturing, Smart Grids, Nonlinear Systems, Power Systems, Robotics, Social Systems, Economic Systems and other. Of particular value to both the contributors and the readership are the short publication timeframe and the world-wide distribution and exposure which enable both a wide and rapid dissemination of research output. Indexed by SCOPUS, DBLP, WTI Frankfurt eG, zbMATH, SCImago. All books published in the series are submitted for consideration in Web of Science.

More information about this series at http://www.springer.com/series/13304

Samsul Ariffin Abdul Karim Editor

Theoretical, Modelling and Numerical Simulations Toward Industry 4.0

123

Editor Samsul Ariffin Abdul Karim Fundamental and Applied Sciences Department and Centre for Systems Engineering (CSE) Institute of Autonomous System Universiti Teknologi PETRONAS Seri Iskandar, Perak Darul Ridzuan, Malaysia

ISSN 2198-4182 ISSN 2198-4190 (electronic) Studies in Systems, Decision and Control ISBN 978-981-15-8986-7 ISBN 978-981-15-8987-4 (eBook) https://doi.org/10.1007/978-981-15-8987-4 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

Mathematics and statistics have been evolved throughout the years from manual calculation to the use of sophisticated tools such as software and hardware in solving complicated problems. Besides, in the era of Industrial Revolution 4.0 (IR 4.0), mathematics and statistics play an important role in all nine pillars of IR4.0. This includes autonomous robots, cybersecurity, Internet of things (IOT), additive manufacturing, augmented reality, system integration, cloud computing, simulations and big data. This book is trying to cater one aspect of IR 4.0, i.e., modeling and simulations. This book is originated from the collaboration between Universiti Teknologi PETRONAS (UTP), Malaysia, with Universiti Brunei Darussalam (UBD), Brunei Darussalam, and FPT University, Vietnam. There are nine contributed chapters in this book spanning all three universities. Chapter “Rainfall Scattered Data Interpolation Using Rational Quartic Triangular Patches” presents scattered data interpolation scheme to visualize rainfall data by using rational quartic spline defined on triangle domain. The scheme has three free parameters for shape modification. The resulting interpolating rainfall surfaces have C1 continuity as well as visually pleasing. Furthermore, the proposed scheme with free parameters has the capability to produce a surface that is near to the positivity. Chapter “A Hybrid Metaheuristic Algorithm for Truss Structure Domain’s Optimization Problem” presents a self-adaptive penalty function approach to eliminate the need for frequent trials for setting up penalty parameters. This approach is hybridized with particle swarm optimization (PSO) and colliding bodies algorithm (CBO). Later, this approach is tested with truss structure problems and compared with previous works. The effect of the behavior of penalty parameter, penalty function and constrained violation analyzed and discussed with the advantages over other algorithms. Chapter “Electric Field Behavior in Time Domain for Semicircle Antenna in Homogeneous Multi Layered Media” presents the simulation of the electric field for semicircle antenna used to detect the presence of hydrocarbon based on resistivity contrast. The simulation results are based on seawater depth; thickness of hydrocarbon shows that the semicircle antenna has the potential to be used in seabed logging. Chapter “Surface Interpolation Using Partially Blended Rational Bi-Quartic Spline” discussed the construction of new rational bi-quartic spline interpolation by using v

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partially blended scheme. There are twelve parameters in the description of the proposed scheme, and it can be used for shape modification. Numerical results suggest that the proposed rational bi-quartic spline interpolation is the best for surface interpolating compared with existing schemes. The interpolating surfaces are very smooth and have C1 continuity everywhere. Chapter “Cost-Benefit Analysis of Sustainable Solar-Powered Workplace Electric Vehicle Charging Station” considers deployment of solar integrated EVs are feasible through embedding the IR 4.0 components, including the information communication technology (ICT), sensors and intelligent networks. This makes the low-voltage distribution network free from overloading of distribution transformers, overloading of network feeders. It avoids various impacts caused by the charging of EVs. In this chapter, the design and sizing of SPEVCS for workplace charging system along with performance and cost analysis of the SPECS are carried out. Chapter “Positivity-Preserving Interpolation Using Rational Quartic Spline Functions” considers the construction of new rational quartic spline of the form quartic numerator and linear denominator. This rational spline is used for positivity-preserving interpolation. Data-dependent sufficient condition for the positivity of the rational spline is derived on the parameter. Numerical results indicate that the proposed scheme has the capability to preserve the positivity of the data. The interpolating positive curve is smooth as well as visually pleasing. Chapter “Decision Support Method for Agricultural Irrigation Scenarios Performance Using WEAP Model” presents a suitable decision support method to look at the performance of the effect of different agricultural irrigation scenarios on special type of rice variety, i.e., MRQ76, planted in Wasan padi field, situated at Brunei-Muara District, Brunei Darussalam. Specifically, the WEAP-MABIA model (based on soil-water balance approach) is applied to perform evaluation of irrigation scheduling for three different rice growing seasons. We found that the WEAP model is a good model for providing the best irrigation scheduling strategy for optimum rice yield and efficient water management. Chapter “Bayesian Variable Selection for Linear Models Using I-Priors” presents a Bayesian methodology for variable selection using a novel, information-theoretic, objective prior choice called the I-prior (Bergsma, 2019). In this work, we show that using the I-prior in Kuo and Mallick’s (1998) variable selection model, we are able to achieve better results (improved parsimony, R2 values and predictive abilities), especially in data with strong multicollinearity. This is shown by way of a simulation study and in real data examples. Chapter “On the Space of m-Subharmonic Functions” presents radially symmetric m-subharmonic functions on the unit ball and their convexity as well as their relationship with a solution of the complex Hessian equations. Furthermore, the ordered vector space of delta radially symmetric m-subharmonic functions which is a Riesz space is investigated. Moreover, we propose a space of m-subharmonic functions along with the Mabuchi metric and introduce a geodesic between two points in this space. The editor would like to express his gratitude to all contributing authors for their great effort and full commitment in preparing their final manuscripts. Furthermore, we would like to thank you all reviewers for their constructive comments. This project is fully supported by Universiti Teknologi PETRONAS (UTP) and the

Preface

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Ministry of Education, Malaysia, through a research grant FRGS/1/2018/STG06/ UTP/03/1015MA0-020 (Rational quartic spline interpolation for image refinement) and YUTP: 0153AA-H24 (Spline Triangulation for Spatial Interpolation of Geophysical Data). This book is suitable for all mathematics and statistics postgraduate students and researchers. Seri Iskandar, Malaysia July 2020

Samsul Ariffin Abdul Karim

Contents

Rainfall Scattered Data Interpolation Using Rational Quartic Triangular Patches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nur Nabilah Che Draman, Samsul Ariffin Abdul Karim, Ishak Hashim, and Yeo Wee Ping A Hybrid Metaheuristic Algorithm for Truss Structure Domain’s Optimization Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kallol Biswas, Pandian Vasant, Jose Antonio Gamez Vintaned, Junzo Watada, Arunava Roy, and Rajalingam Sokkalingam Electric Field Behavior in Time Domain for Semicircle Antenna in Homogeneous Multi Layered Media . . . . . . . . . . . . . . . . . . . . . . . . . Elsayed Dahy, Afza Shafie, and Noorhana Yahya Surface Interpolation Using Partially Blended Rational Bi-Quartic Spline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Samsul Ariffin Abdul Karim, Lila Iznita Izhar, Mahmod Othman, and Nooraini Zainuddin Cost-Benefit Analysis of Sustainable Solar-Powered Workplace Electric Vehicle Charging Station . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kameswara Satya Prakash Oruganti, Chockalingam Aravind Vaithilingam, Gowthamraj Rajendran, and Agileswari Ramasamy

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35

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Positivity-Preserving Interpolation Using Rational Quartic Spline Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Samsul Ariffin Abdul Karim and Van Thien Nguyen

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Decision Support Method for Agricultural Irrigation Scenarios Performance Using WEAP Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Saiful Azmi Husain and Nor Hamizah Mohd Rhyme

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Bayesian Variable Selection for Linear Models Using I-Priors . . . . . . . . 107 Haziq Jamil and Wicher Bergsma On the Space of m-Subharmonic Functions . . . . . . . . . . . . . . . . . . . . . . 133 Samsul Ariffin Abdul Karim and Van Thien Nguyen Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

Editor and Contributors

About the Editor Samsul Ariffin Abdul Karim is a senior lecturer at Fundamental and Applied Sciences Department, Universiti Teknologi PETRONAS (UTP), Malaysia. He has been in the department for more than eleven years. He obtained his B.App.Sc., M.Sc. and Ph.D. in Computational Mathematics and Computer Aided Geometric Design (CAGD) from Universiti Sains Malaysia (USM). He had 20 years of experience using Mathematica and MATLAB software for teaching and research activities. His research interests include curves and surfaces designing, geometric modeling and wavelets applications in image compression and statistics. He has published more than 120 papers in Journal and Conferences as well as seven books including two research monographs and three Edited Conferences Volume and 25 book chapters. He is the recipient of Effective Education Delivery Award and Publication Award (Journal and Conference Paper), UTP Quality Day 2010, 2011 and 2012 respectively. He was Certified WOLFRAM Technology Associate, Mathematica Student Level. He has published four books with Springer.

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Contributors Wicher Bergsma Department of Statistics, London School of Economics and Political Science, London, United Kingdom Kallol Biswas Fundamental and Applied Sciences Department, Universiti Teknologi PETRONAS, Seri Iskandar, Perak Darul Ridzuan, Malaysia Elsayed Dahy Department of Mathematics, Al-Azhar University, Assuit, Egypt Nur Nabilah Che Draman Department of Fundamental and Applied Sciences, Universiti Teknologi PETRONAS, Bandar Seri Iskandar, Perak Darul Ridzuan, Malaysia Ishak Hashim Department of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, UKM Bangi, Selangor, Malaysia Saiful Azmi Husain Faculty of Science, Universiti Brunei Darussalam, Gadong, Brunei Darussalam Lila Iznita Izhar Department of Electrical and Electronic Engineering, Universiti Teknologi PETRONAS, Seri Iskandar, Perak Darul Ridzuan, Malaysia Haziq Jamil Department of Statistics, London School of Economics and Political Science, London, United Kingdom Samsul Ariffin Abdul Karim Fundamental and Applied Sciences Department and Centre for Systems Engineering (CSE), Institute of Autonomous System, Universiti Teknologi PETRONAS, Seri Iskandar, Perak Darul Ridzuan, Malaysia Van Thien Nguyen Department of Mathematics, FPT University, Hanoi, Vietnam Kameswara Satya Prakash Oruganti Faculty of Innovation and Technology, Taylor’s University, Selangor, Malaysia Mahmod Othman Fundamental and Applied Sciences Department, Universiti Teknologi PETRONAS, Seri Iskandar, Perak Darul Ridzuan, Malaysia Yeo Wee Ping Faculty of Science, University Brunei Darussalam, Bandar Seri Begawan, Brunei Darussalam Gowthamraj Rajendran Faculty of Innovation and Technology, Taylor’s University, Selangor, Malaysia Agileswari Ramasamy Institute of Power Engineering (IPE), Universiti Tenaga Nasional, Selangor, Malaysia Nor Hamizah Mohd Rhyme Faculty of Science, Universiti Brunei Darussalam, Gadong, Brunei Darussalam

Editor and Contributors

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Arunava Roy Department of Computer and Information Sciences, Universiti Teknologi PETRONAS, Seri Iskandar, Perak Darul Ridzuan, Malaysia Afza Shafie Fundamental and Applied Sciences Department, Universiti Teknologi PETRONAS, Seri Iskandar, Perak Darul Ridzuan, Malaysia Rajalingam Sokkalingam Fundamental and Applied Sciences Department, Universiti Teknologi PETRONAS, Seri Iskandar, Perak Darul Ridzuan, Malaysia Chockalingam Aravind Vaithilingam Faculty of Innovation and Technology, Taylor’s University, Selangor, Malaysia Pandian Vasant Fundamental and Applied Sciences Department, Universiti Teknologi PETRONAS, Seri Iskandar, Perak Darul Ridzuan, Malaysia Jose Antonio Gamez Vintaned Department of Petroleum Geosciences, Universiti Teknologi PETRONAS, Seri Iskandar, Perak Darul Ridzuan, Malaysia Junzo Watada Department of Computer and Information Sciences, Universiti Teknologi PETRONAS, Seri Iskandar, Perak Darul Ridzuan, Malaysia Noorhana Yahya Fundamental and Applied Sciences Department, Universiti Teknologi PETRONAS, Seri Iskandar, Perak Darul Ridzuan, Malaysia Nooraini Zainuddin Fundamental and Applied Sciences Department, Universiti Teknologi PETRONAS, Seri Iskandar, Perak Darul Ridzuan, Malaysia

Rainfall Scattered Data Interpolation Using Rational Quartic Triangular Patches Nur Nabilah Che Draman, Samsul Ariffin Abdul Karim , Ishak Hashim, and Yeo Wee Ping

Abstract Scattered data interpolation refers to the problems of constructing a smooth surface through non-uniform and uniform set of data points. Besides that, this subject is very vital in various fields such as medical, sciences, engineering and others. There are many researchers who have done their studies in interpolation methods with different approaches and has been reported in many papers. There are two different approaches to the interpolation of scattered data where it can be clarified as global and local methods. The global method is influenced by all data while the local method is not influenced all the data where it effected at nearby points. This chapter discusses the rainfall distributions in several areas in Malaysia by using the rational quartic triangular patches. Keywords Scattered data · Interpolation · Rational quartic triangular patches · C 1 continuity · Visualization · Cubic Ball · Cubic Bézier

N. N. C. Draman Department of Fundamental and Applied Sciences, Universiti Teknologi PETRONAS, 32610 Seri Iskandar, Bandar Seri Iskandar, Perak Darul Ridzuan, Malaysia e-mail: [email protected] S. A. A. Karim (B) Fundamental and Applied Sciences Department and Centre for Systems Engineering (CSE), Institute of Autonomous System, Universiti Teknologi PETRONAS, Bandar Seri Iskandar, 32610 Seri Iskandar, Perak Darul Ridzuan, Malaysia e-mail: [email protected] I. Hashim Department of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor, Malaysia e-mail: [email protected] Y. W. Ping Faculty of Science, University Brunei Darussalam, Jalan Tungku LinkGadong, Bandar Seri Begawan BE1410, Brunei Darussalam e-mail: [email protected]

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 S. A. Abdul Karim (ed.), Theoretical, Modelling and Numerical Simulations Toward Industry 4.0, Studies in Systems, Decision and Control 319, https://doi.org/10.1007/978-981-15-8987-4_1

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1 Introduction Scattered data interpolation has been widely used in the research field. The earliest interpolation schemes have been found by Shepard [1] where interpolations were based on the inverse distance weighting of data. It is knowns as Shepard’s methods. This scheme is defined as meshfree interpolant as a weighted average of the data. After a few years later, Franke and Nielson [2] introduced C 1 interpolant scheme by modifying Shepard’s interpolation and known as Modified Shepard’s interpolation. Another way to solve scattered data interpolation is to triangulate the domain data points on the plane. This method is the main objective of this research. This is because this method is famous and has been widely used in solving scattered data interpolation nowadays. The construction of scattered data interpolation using Bezier triangular patches can be described as: (a) Triangulate the domain by using Delaunay triangulation; (b) Specify the derivatives at the data points, and then assign Bezier ordinates values at each triangular patches; (c) Generate the triangular patches of the surfaces and (d) Apply spatial data interpolation to estimate the missing value [3–7]. Chan and Ong [8] proposed the range restricted scattered data interpolation using cubic Bezier triangular patches. The sufficient nonnegativity conditions are derived on the Bezier ordinates to ensure the positivity of a Cubic Bezier triangular patches. Besides that, the gradients also modified if necessaries to ensure the positivity conditions are fulfilled. Awang and Rahmat [9] proposed the derivative estimation of triangular patches by using cubic Least square method. Every smooth surface that built from Delaunay triangulation needs partial derivates at every vertex. Usually, partial derivatives at vertices and midpoints of each side are unavailable. Thus, we need to approximate the derivatives at vertices and midpoints of each side. This scheme is focused on the estimation derivatives by using the cubic approximation of least square method and comparisons the surface between quadratic and cubic approximation. Sulaiman and Jaafar [10] discussed the construction of a smooth surface of scattered data using Delaunay triangulation for six different test functions. This objective in this scheme is to test the exactness of Delaunay triangulation by generating different surfaces when the points are removed. The points are removed according to the percentage of points and new surface generating after removing the points of scattered data. The result shows the surface by removing the points is not consistent in shape and needs more points to produce a smooth surface. Saaban et al. [11] describe rainfall data distribution using quintic triangular Bezier patches where it uses C 2 interpolant and positive everywhere. The sufficient conditions are derived on Bezier points and will be used to ensure the surface are always positive. Besides that, the first and second derivatives are calculated and need to derive if essential in order these conditions are satisfied. Another researcher, Karim et al. [12] discussed the shape preserving such as positivity by using a rational bicubic spline. This scheme is an extension to the C 1 rational cubic spline interpolant from Karim and Pang [13, 14] to the bivariate cases. The sufficient conditions derive on every four boundary curves network on the

Rainfall Scattered Data Interpolation …

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rectangular patches. The partially blended rational bicubic spline has 12 parameters and 8 of them are free parameters. Unfortunately, this scheme gives the result is on par with the established methods. Hussain and Hussain [15] proposed C 1 construction positive scattered data interpolation arranged over a triangular grid. Each boundary of the triangle is constructed by a rational cubic function with two shape parameters. It also used for side-vertex interpolants. Simple data dependent sufficient condition is deriving on these free parameters to ensure the positivity of scattered data interpolation. This scheme is visually pleasing as well even though it is not local and computationally economical. Chua and Kong [16] discussed the constrained C 1 interpolation scheme to scattered data interpolation using a rational blend of quartic Bezier triangles. Sufficient range restriction conditions are derived to ensure the quartic Bezier triangular patch lies on one side of a given constraints surface. The Bezier ordinates are determined by the data values while the gradients are determined by the data sites. The gradients also are derived if needed so that range restriction condition is satisfied. Ibraheem et al. [17] developed a local C 1 positivity preserving scheme when the points of scattered data obtained from various sources. This scheme using Delaunay triangulation to triangulate the irregular data. Then, they interpolate each boundary and radial curves of the triangle using rational trigonometric cubic function. Orthogonality of trigonometric functions to ensure the smooth surface is generated as compared to polynomial functions. This scheme is worth it in the surface reconstruction of signal processing, CAD/CAM design, and others. However, the interpolating is not smooth. The main objective of this chapter is to interpolate the rainfall scattered data interpolation using quartic rational triangular patches. Rainfall data are obtained from various sources and it be discussed in Sect. 4. This chapter is divided into several sections. Section 2 discusses the proposed method of quartic rational triangular patch. Section 3 discussed the scattered data interpolation by using the proposed method while Sect. 4 discusses the result and discussion.

2 Rational Quartic Triangular Patches Let be u, v, w the barycentric coordinates on triangle T with vertices V 1 , V 2 , V 3 as shown in Fig. 1 such that u + v + w = 1 and u, v, w ≥ 0. Any point V (x, y) inside the triangle (including the vertices) can be expressed as V = uV 1 + vV 2 + wV 3

(1)

Quartic rational is defined as follows. Definition 1 Let α, β, γ ∈ [0, +∞), given control points Px,y,z 3 (x, y, z ∈ N, x + y + z = 3) and a triangular domain D { (u, v, w)|u + v + w = 1, u ≥ 0, v ≥ 0, w ≥ 0} and is defined as

∈ =

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Fig. 1 Triangle

R(u, v, w) =



3 Bx,y,z (u, v, w)Px,y,z , (u, v, w) ∈ D

(2)

i+ j+k=3

3 where Bx,y,z (u, v, w) is basis functions with three shape parameters α, β, γ .

From (2), we simply obtain u2 v2 P3,0,0 + P030 1 + α(1 − u) 1 + β(1 − v) w2 u 2 v[2 + α + 2α(1 − u)] P003 + P210 + 1 + γ (1 − w) 1 + α(1 − u) u 2 w[2 + α + 2α(1 − u)] P201 + 1 + α(1 − u) uv2 [2 + β + 2β(1 − v)] P120 + 1 + β(1 − v) v2 w[2 + β + 2β(1 − v)] P021 + 1 + β(1 − v)   uw2 2 + γ + 2γ (1 − w) P102 + 1 + γ (1 − w)   vw2 2 + γ + 2γ (1 − w) P012 + 6uvw P111 + 1 + γ (1 − w)

R(u, v, w) =

(3)

Figure 2 shows some basis functions of quartic rational triangular patches. Figure 3 shows the control nets for the quartic rational triangular patches, while Fig. 4 shows the basis function of quartic rational triangular patches that lies on a triangular domain.

Rainfall Scattered Data Interpolation …

Fig. 2 Some basis functions of quartic rational triangular patches with α = 1.5, β = 0, γ = 1.0 Fig. 3 Control nets for quartic rational triangular patch

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Fig. 4 Rational quartic triangular basis function

Some properties of rational quartic triangular patches can be described as below [10]: (a) Affine invariance and convex hull properties. Since the basis function rational quartic basis over the triangular domain have the properties of partition of unity and nonnegativity, we can observe that the corresponding patch in (1) has affine invariance and convex hull properties. (b) Interpolating corner points. Direct computation gives that R(1, 0, 0) = P3,0,0 , R(0, 1, 0) = P0,3,0 , R(0, 0, 1) = P0,0,3 . These indicate that the rational quartic triangular patch in (1) interpolates at the corner. (c) Tangent planes at corner points. (d) Boundary property. (e) Shape adjustable property. Without changing its control net, we can adjust the shape by manipulating value α, β, γ .

3 Scattered Data Interpolation The problem statement for scattered data interpolation can be described as follows: Given the functional scattered data (xi , yi , z i ), i = 1, 2, ..., N

Rainfall Scattered Data Interpolation …

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Fig. 5 The directional P1 , P2 , P3

We wish to construct a smooth C 1 surface z = F(x, y) such that z i = F(xi , yi ), i = 1, 2, ..., N This scheme comprising convex combination of three local schemes P1 , P2 and P3 are defined as vw P1 + uw P2 + uv P3 ,u + v + w = 1 vw + uw + uv

(4)

u 2 v2 P1 + v2 w2 P2 + u 2 w2 P3 ,u + v + w = 1 u 2 v2 + v2 w2 + u 2 w2

(5)

P(u, v, w) = and R(u, v, w) =

i where the local scheme Pi , i = 1, 2, 3 is derived by replacing P111 which denote the inner ordinates in the proposed method as show in Fig. 5. Next, we will calculate the inner triangular points (or ordinates) for the local schemes Pi , i = 1, 2, 3. by using Cubic Precision Method [18]. The inner ordinates i such that C 1 condition is satisfied only on are obtained by replacing b111 with b111 the boundary ei of the triangle. Besides that, Goodman and Said [19] scheme is used to calculate the boundary ordinates for each triangle. Figure 6 shows a directional triangle P with three vertices (V1 , V2 , V3 ), three edges (e1 , e2 , e3 ) and one face. Note that, e3 , e2 , e1 are opposite to V3 , V2 , V1 respectively. The vertices V1 , V2 and V3 with corresponding barycentric coordinates (1, 0, 0), (0, 1, 0) and (0, 0, 1) respectively and e1 , e2 and e3 are direction vectors which are (0, −1, 1), (1, 0, 1) and (−1, 1, 0) respectively. The derivation along a triangle edges is derived. Suppose emn is the edge of connecting (xm , ym ) to (xn , yn ) for triangular, then it will define as below [20]

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Fig. 6 Notations on a triangle P

δR δF δF + (ym − yn ) = (xm − xn ) δemn δx δy Let the directional derivatives along e2 and e3 at V1 be     δx δy δR δx δy − Fx (V1 ) + − Fy (V1 ) (V1 ) = δe2 δv δu δv δu = (x1 − x3 )Fx (V1 ) + (x1 − x3 )Fy (V1 )     δR δx δy δx δy − Fx (V1 ) + − Fy (V1 ) (V1 ) = δe3 δv δu δv δu = (x2 − x1 )Fx (V1 ) + (x2 − x1 )Fy (V1 )

(6)

Then, applying directional derivative into (3), yields δR (V1 ) = (2 + α)(P300 − P201 ); δe2 δR (V1 ) = (2 + α)(P210 − P300 ); δe3

(7)

From (6) and (7), we get P201 = P300 −

 1  (x1 − x3 )Fx (V1 ) + (y1 − y3 )Fy (V1 ) (2 + α)

(8)

P210 = P300 +

 1  (x2 − x1 )Fx (V1 ) + (y2 − y1 )Fy (V1 ) + α) (2

(9)

and

Other directional derivatives along e1 , e3 at V2 and e1 , e2 at V3 are

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Fig. 7 Two adjacent rational quartic triangular patches

 1  (x2 − x1 )Fx (V2 ) + (y2 − y1 )Fy (V2 ) − P030 (2 + β)  1  = (x3 − x2 )Fx (V2 ) + (y2 − y1 )Fy (V2 ) + P030 (2 + β)  1  = P003 − (x3 − x2 )Fx (V3 ) + (y3 − y2 )Fy (V3 ) (2 + γ )  1  = P003 + (x1 − x3 )Fx (V3 ) + (y1 − y2 )Fy (V3 ) (2 + γ )

P120 = P021 P012 P102

In order to achieve C 1 continuity along all the edges (Fig. 7), the following equations must be satisfied. 1 + t 2 b012 c201 = r 2 b210 + 2stb021 + 2r sb120 + s 2 b030 + 2r tb111

(10)

1 c210 = r 2 b201 + 2stb012 + 2r tb102 + s 2 b021 + 2r sb111 + t 2 b003

(11)

1 b210 = u 2 c201 + 2vwc012 + 2uwc102 + v2 c021 + 2uvc111 + w2 c003

(12)

1 b201 = u 2 c210 + 2vwc021 + 2uwc120 + v2 c030 + 2uvc111 + w2 c012

(13)

1 in (10) and (11), we need to add these equations together. Thus, we To find b111 obtain

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N. N. C. Draman et al.

  1 c201 + c210 − r 2 (b210 + b201 ) − s 2 (b030 + b021 ) 2r (s + t) + t 2 (b012 + b003 ) − 2st(b021 + b012 ) − r sb120 − 2r tb102 .

1 b111 =

Similarly, with (12) and (13) we obtain the following   1 b201 + b210 − u 2 (c210 + c201 ) − v2 (c030 + c021 ) 2u(v + w) + w2 (c012 + c003 ) − 2vw(c021 + c012 ) − uvc120 − 2uwc102 .

1 = c111

The final scheme can be written as    1 2 3 bi jk Bi,3 j,k + 6uvw c1 b111 + c2 b111 + c3 b111 P(u, v, w) =

(14)

i+ j+k=3 i. j.k=1

with c1 =

vw uw uv , c2 = , c3 = vw + uw + uv vw + uw + uv vw + uw + uv

(15)

In Goodman and Said [19], the other version of convex combination is used such that v2 w2 , v2 w2 + u 2 w2 + u 2 v2 u 2 w2 , c2 = 2 2 v w + u 2 w2 + u 2 v2 u 2 v2 c3 = 2 2 v w + u 2 w2 + u 2 v2 c1 =

(16)

4 Results and Discussions In this section, we applied the proposed method to visualize of rainfall pattern which is the average rainfall in Arthur’s Pass, New Zealand and two cases in Peninsular Malaysia. The data is described below. Visualisation of Arthur’s Pass, New Zealand Arthur’s Pass, New Zealand is a township in the Southern Alps of the South Island of New Zealand where is located in the Selwyn district. The data is taken from Jamil and Piah [21] and is shown in Table 1.

Rainfall Scattered Data Interpolation … Table 1 Average rainfall distribution in Arthur’s Pass, New Zealand

11

Location

Longitude

Latitude

Average rainfall (mm)

A

171.567

−42.950

4.6

B

171.672

−43.470

0.1

C

171.946

−43.529

0.2

D

171.800

−43.800

0.4

E

171.208

−44.035

0.2

Figure 8 shows Delaunay triangulation of Table 1. Figure 9 shows 3D interpolant while Fig. 10 shows surface interpolation of average rainfall distribution in Arthur’s Pass, New Zealand. Table 2 shows the numerical result of minimum error of average rainfall distribution in Arthur’s Pass. Fig. 8 Delaunay triangulation for average rainfall at Arthur’s Pass, New Zealand

Fig. 9 3D interpolation for average rainfall at Arthur’s Pass, New Zealand

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N. N. C. Draman et al.

Fig. 10 Surface interpolant for average rainfall at Arthur’s Pass, New Zealand

Table 2 Minimum value of average rainfall distribution in Arthur’s Pass, New Zealand Method

Shape parameters α

Cubic Ball

Minimum value β

γ

0

0

0

−0.1260

60

50

70

−0.0135

50

50

50

−0.0161

70

70

70

−0.0140

−0.0975

Cubic Bezier Rational quartic

Table 2 shows minimum value of average rainfall distribution in Arthur’Pass, New Zealand. From the table, we can conclude that our proposed method shows that approaching positive value and it is much better than the existing scheme. Average Rainfall in March and May 2007 of 10 Major Rainfall Measuring Stations in Peninsular Malaysia Next, we tested our proposed scheme for the other rainfall application. The data is obtained from Malaysia Meteorological Department. The data is collected at 10 location and represent the average monthly measurement in millimeters. From Table 3, we have chosen two months where are March and May 2007. Figure 11 shows Delaunay Triangulation that contains 11 triangular patches that connected these data sites. Figure 12 shows 3D interpolation of the average amount of rainfall in millimeters while Fig. 13 shows surface interpolation estimated average amount of rainfall (in mm) in March and May 2007 at 10 chosen locations. Tables 4 and 5 shows the numerical result shows the minimum value of the estimated average amount in March and May 2007 respectively. Tables 4 and 5 show that our proposed method give value that can be approaching to the positive value. It is because our proposed method can manipulate the value α, β and γ .

Rainfall Scattered Data Interpolation …

13

Table 3 Estimated average amount of rainfall (in mm) in March and May 2007 at 10 chosen location Name of town

Location Longitude

Estimated average amounts of rainfall (mm) Latitude

March 2007 244.6

May 2007

Jitra

100.4167

6.2667

Kepala Batas

100.4333

5.5167

58.19

139.62

66.12

Kuala Kangsar

109.3333

4.7667

151.17

105.29

Cheras

101.7667

3.0500

283.25

271.29

Genting Highland

101.8000

3.4000

524.35

87.26

Jertih

102.5000

5.7500

122.25

88.80

Nilai

101.8000

2.8167

216.32

229.29

Segamat

102.8167

2.5000

132.57

174.85

Rawang

101.5833

3.3167

398.57

70.81

Kangar

100.2000

6.4333

83.73

95.50

Fig. 11 Delaunay triangulation for the estimated average amount of rainfall (in mm) in March and May 2007 at 10 chosen locations

Average Rainfall (in mm) in March and May 2007 of 25 Major Rainfall Measuring Stations in Peninsular Malaysia In this subsection, we extend the number of locations into 25 major rainfall stations in Peninsular Malaysia. The data is obtained from Malaysia Meteorological Department. Table 6 shows the average rainfall measured in millimeters for 25 major rainfall in March and May 2007. Figure 14 shows Delaunay triangulation for 25 major rainfall where it has 38 triangular patches. Figure 15 shows 3D surface interpolation for March and May 2007. Figure 16 shows surface interpolation for March and May 2007. Tables 7 and 8 shows the minimum value of the estimated average amount of rainfall in March and May 2007 respectively.

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N. N. C. Draman et al.

(b) May 2007

(a) March 2007

Fig. 12 3D interpolation for the estimated average amount of rainfall (in mm) in March and May 2007 at 10 chosen locations

(b) May 2007

(a) March 2007

Fig. 13 Surface interpolation for the estimated average amount of rainfall (in mm) in March and May 2007 at 10 chosen locations Table 4 Minimum value of the estimated average amount of rainfall (in mm) in March 2007 at 10 chosen locations Method

Shape parameters α

Cubic Ball

Minimum value (March) β

γ

0

0

10

10

10

−356.9961

50

50

50

−214.9747

50

50

70

−211.0588

−1405.6694

Cubic Bezier Rational quartic

−2062.6266

0

Rainfall Scattered Data Interpolation …

15

Table 5 The minimum value of the estimated average amount of rainfall (in mm) in May 2007 at 10 chosen locations Method

Shape parameters α

Cubic Ball

Minimum value (May) β

0

0

γ −918.4227

0

−679.1278

Cubic Bezier Rational quartic

−86.9234

30

40

50

50

50

50

−86.8074

10

10

10

−138.0190

Table 6 Average rainfall (in mm) in March and May 2007 of 25 major rainfall measuring stations in Peninsular Malaysia Stations

Location

Amount (in mm)

Longitude

Latitude

March 2007

May 2007

Chuping

100.2667

6.4833

61.0

88.0

Langkawi

99.7333

6.3333

40.6

166.0

Alor Setar

100.4000

6.2000

277.8

67.4

Butterworth

100.2670

5.4667

58.9

143.2

Prai

100.4000

5.3500

208.1

153.4

Bayan Lepas

100.2667

5.3000

125.2

144.4

Ipoh

101.1000

4.5833

364.2

42.6

Cameron Highland

101.3667

4.4667

252.0

223.2

Lubok Merbau

100.9000

4.8000

156.4

98.4

Sitiawan

100.7000

4.2167

44.4

26.8

Subang

101.5500

3.1167

329.2

68.2

Petaling Jaya

101.6500

3.1000

321.0

196.2

KLIA

101.7000

2.7167

186.2

188.8

Malacca

102.2500

2.2667

113.8

183.4

Batu Pahat

102.9833

1.8667

182.0

195.0

Kluang

103.3100

2.0167

92.4

130.2

Senai

103.6667

1.6333

148.6

296.0

Kota Bharu

102.2833

6.1667

115.2

109.2

Kuala Krai

102.2000

5.5333

166.0

238.7

Kuala Terengganu Airport

103.1000

5.3833

121.0

64.8

Kuantan

103.2167

3.7833

79.2

270.4

Batu Embun

102.3500

3.9667

146.2

256.2

Temerloh

102.3833

3.4667

114.2

324.2

Muadzam Shah

103.0833

3.0500

131.6

204.8

Mersing

103.8333

2.4500

183.4

196.2

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N. N. C. Draman et al.

Fig. 14 Delaunay Triangulation average rainfall (in mm) in March and May 2007 of 25 major rainfall measuring stations in Peninsular Malaysia

Fig. 15 3D surface interpolation for March and May 2007

From Tables 7 and 8, we can see that our proposed method can give minimum value that approaches to positive value when α = β = γ = 50 rather than two existing methods.

Rainfall Scattered Data Interpolation …

17

Fig. 16 Surface interpolation for March and May 2007

Table 7 The minimum value of the estimated average amount of rainfall (in mm) in March 2007 at 25 chosen locations Method

Shape parameters

Minimum value (March)

α

β

γ

Cubic Ball

0

0

0

−198.7122

3

3

3

−195.2252

1

2

1

−186.5321

1

5

3

−197.2544

−180.1413

Cubic Bezier Rational quartic

Table 8 The minimum value of the estimated average amount of rainfall (in mm) in May 2007 at 25 chosen locations Method

Shape parameters α

Cubic Ball

0

Minimum value (May) β 0

γ 0

Rational quartic

−191.4534 −148.6903

Cubic Bezier 1

5

3

−90.1262

50

50

30

−13.7212

70

50

60

−16.5467

5 Conclusion In this chapter, rational quartic triangular patches are applied to rainfall scattered data interpolation without considering shape preserving. Our proposed scheme is compared the performance which is minimum value with two existing scheme which is cubic Ball and cubic Bezier. From the result, we conclude that our proposed scheme gives minimum value that approaching to positive. For future works, we can apply

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the proposed scheme for shape preserving of positive scattered data sets and compare with some meshless methods. Acknowledgements This research was fully supported by Universiti Teknologi PTERONAS (UTP) and Ministry of Education, Malaysia through FRGS/1/2018/STG06/UTP/03/1015MA0020 and Universiti Teknologi Petronas (UTP) through a research grant YUTP: 0153AA-H24 (Spline Triangulation for Spatial Interpolation of Geophysical Data). The first author is supported through Graduate Research Assistant (GRA) Scheme.

References 1. Goodman, T.N.T., Said, H.B.: A C1 triangular interpolant suitable for scattered data interpolation. Commun. Appl. Numer. Methods 7(6), 479–485 (1991) 2. Karim, S.A.A., Pang, K.V.: Local control of the curves using rational cubic spline. J. Appl. Math. (2014) 3. Chan, E.S., Ong, B.H.: Range restricted scattered data interpolation using convex combination of cubic Bézier triangles. J. Comput. Appl. Math. 136(1–2), 135–147 (2001) 4. Karim, S.A.A., Pang, K.V., Saaban, A.: Positivity preserving interpolation using rational bicubic spline. J. Appl. Math. (2015) 5. Saaban, A., Majid, A.A., Piah, M., Rahni, A.: Visualization of rainfall data distribution using quintic triangular Bézier patches. Bull. Malays. Math. Sci. Soc. 32(2) (2009) 6. Foley, T.A., Opitz, K.: Hybrid cubic Bézier triangle patches. In Mathematical methods in computer aided geometric design II, pp. 275–286. Academic Press (1992) 7. Ali, F.A.M., Karim, S.A.A., Saaban, A., Hasan, M.K., Ghaffar, A., Nisar, K.S., Baleanu, D.: Construction of cubic timmer triangular patches and its application in scattered data interpolation. Mathematics 8(2), 159 (2020) 8. Franke, R., Nielson, G.M.: Scattered data interpolation of large sets of scattered data. Int. J. Numer. Methods Eng. 15, 1–691 (1991) 9. Shepard, D.: A two-dimensional interpolation function for irregularly-spaced data. In: Proceedings of the 1968 23rd ACM National Conference, pp. 517–524. ACM, Jan 1968 10. Draman, N.N.C., Karim, S.A.A., Hashim, I.: Scattered data interpolation using rational quartic triangular patches with three parameters. IEEE Access (in Press) 11. Zhu, Y., Han, X., Liu, S.: Quartic rational said-ball-like basis with tension shape parameters and its application. J. Appl. Math. (2014) 12. Jamil, S.J., Piah, A.R.M.: Positivity preserving scattered data interpolation using ball triangular patches. MATEMATIKA Malays. J. Ind. Appl. Math. 24, 159–168 (2008) 13. Ibraheem, F., Hussain, M.Z., Bhatti, A.A.: C1 positive surface over positive scattered data sites. PLoS ONE 10(6), e0120658 (2015) 14. Piah, A.R.M., Goodman, T.N., Unsworth, K.: Positivity-preserving scattered data interpolation. In: Mathematics of Surfaces XI, pp. 336–349. Springer, Berlin, Heidelberg (2005) 15. Sulaiman, P.S., Jaafar, A.: Delaunay triangulation of a missing points. J. Adv. Sci. Eng. Res. 7, 58–69 (2017) 16. Awang, N., Rahmat, R. W.: Derivative estimation of triangular patch by using cubic least square method. In: AIP Conference Proceedings, vol. 1870, No. 1, p. 050015. AIP Publishing, Aug 2017 17. Hussain, M.Z., Hussain, M.: C1 positive scattered data interpolation. Comput. Math. Appl. 59(1), 457–467 (2010) 18. Saaban, A., Majid, A.A., Piah, M., Rahni, A.: Visualization of rainfall data distribution using quintic triangular Bézier patches. Bull. Malays. Math. Sci. Soc. 32(2)

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19. Karim, S.A.A., Pang, K.V.: Shape preserving interpolation using rational cubic spline. Res. J. Appl. Sci. Eng. Technol. 8(2), 167–178 (2014) 20. https://www.met.gov.my 21. Chua, H.S., Kong, V.P.: Constrained C1 scattered data interpolation using rational blend. In: AIP Conference Proceedings, vol. 1605, No. 1, pp. 280–285. AIP (2014)

A Hybrid Metaheuristic Algorithm for Truss Structure Domain’s Optimization Problem Kallol Biswas, Pandian Vasant, Jose Antonio Gamez Vintaned, Junzo Watada, Arunava Roy, and Rajalingam Sokkalingam

Abstract For calculating constrained optimization problem various socio/bioinspired algorithms have adopted a penalty function approach to handle linear and nonlinear constraints. In a general sense, the approach is quite easy to understand, but a precise choice of penalty parameter is very much important. It requires a bunch number of primer preliminaries. So as to beat this restriction another self-adaptive penalty function (SAPF) approach will be proposed and incorporated into Particle Swarm Optimization (PSO) algorithm. This approach is referred to as PSO-SAPF. Besides, PSO-SAPF approach will be hybridized with Colliding Bodies Optimization (CBO) referred to as PSO-SAPF-CBO algorithm. The performance of PSOSAPF and PSO-SAPF-CBO algorithm will be distinctly validated by solving discrete and mixed variable problems from truss structure domain and linear and nonlinear domain. The effect of behavior of penalty parameter, penalty function and constrained violation will be analyzed and discussed with the advantages over other algorithms. K. Biswas (B) · P. Vasant · R. Sokkalingam Fundamental and Applied Sciences Department, Universiti Teknologi PETRONAS, Bandar Seri Iskandar, 32610 Seri Iskandar, Perak Darul Ridzuan, Malaysia e-mail: [email protected] P. Vasant e-mail: [email protected] R. Sokkalingam e-mail: [email protected] J. A. G. Vintaned Department of Petroleum Geosciences, Universiti Teknologi PETRONAS, Bandar Seri Iskandar, 32610 Seri Iskandar, Perak Darul Ridzuan, Malaysia e-mail: [email protected] J. Watada · A. Roy Department of Computer and Information Sciences, Universiti Teknologi PETRONAS, Bandar Seri Iskandar, 32610 Seri Iskandar, Perak Darul Ridzuan, Malaysia e-mail: [email protected] A. Roy e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 S. A. Abdul Karim (ed.), Theoretical, Modelling and Numerical Simulations Toward Industry 4.0, Studies in Systems, Decision and Control 319, https://doi.org/10.1007/978-981-15-8987-4_2

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Keywords Particle swarm optimization · Self-adaptive · Penalty function · Linear and nonlinear constraints · CBO

1 Introduction There are some mechanical and truss structure optimization domain problems that have various linear and nonlinear constraints. These problems are complex and complicated to solve. These challenges include discrete and mixed design variables which have contributed to the difficulties in looking for the best set of variables from a small search area. Several artificial intelligence (AI) based optimization techniques such as Particle swarm Optimization (PSO) [1, 2], Firefly Algorithm (FA) [3], Probability Collectives (PC) [4], Colliding Bodies Optimization (CBO) [5], Symbiotic Organism Search (SBO) [6], Mine Blast algorithm (MBA) [7], Harmony search [8], Generalized Hopfield Networks (GHN) [9], etc. were proposed. Furthermore bio-based Genetic Algorithm (GA) [10] and socio based algorithms such as Cohort Intelligence (CI) [11], Ideology Algorithm (IA) [12], Socio evolution and learning Optimization algorithm (SELO) [13], etc. These algorithms included several methods for handling constraints such as probability-based, penalty-based, feasibility based, etc. Using certain meanings, the penalty-based techniques are used to penalize the violated constraints and turn the constrained problem into an unconstrained problem. A large number of preliminary trials are required to establish a suitable penalty parameter. Deb and Agrawal [14] suggested a parameterless approach called a niched penalty function approach. A viable alternative has been preferred in this method based on three parameters, for example choosing the feasible solution rather than infeasible solution based on a smaller number of constraints violation. These three rules then referred to as feasibility-based rule and were used as a constraint handling technique [15]. In order to make fitness bias towards feasibility, the probability distribution based constraints handling technique [16] was suggested. The mathematical structure of this approach, however depends on the problem and must be made more general. Because of straightforward construction and easy implementation, the penalty function solution was commonly used. Many approaches have so far been suggested based on constraint handling technique such as barrier function approach which has demanded the exclusion of the infeasible solution [17], the exact penalty function [18] and dynamic penalty function to penalize the violated constraints by setting the value of penalty parameter and its multiplication factor respectively. There have also been ideas on other approaches such as annealing penalty function [19] which was based on idea of Simulated Annealing (SA) and adaptive penalty function [20] to remove the penalty parameter setting from other penalty function methods. A distinct penalty parameter for different measured fitness functions was set in penalty based segregated GA [21]. To deal with linear and nonlinear constraints, these methods were successfully employed with nature-inspired optimization methodology. These techniques are simple and easy to apply to a wide variety of constrained optimization

A Hybrid Metaheuristic Algorithm for Truss Structure …

23

problems. Yet their output degenerates with the amount of constraints. [17]. An exact penalty approach was adopted by Azad [22] for nonlinear optimization problems having discrete design variables. A number of preliminary trials are necessary to establish an appropriate penalty parameter for each individual problem [18]. This similar approach has been implemented for FA and CL algorithm with static penalty function approach (CI-SPF) [23] for the solution of discrete and mixed variability problems with both linear and nonlinear constraints in engineering design and truss structural domain. It was however observed that the penalty parameter was difficult to pick as number of constraints increased. Kannan [24] suggested an improvement in the Lagrange multiplier approach and Viswanathan used the dynamic penalty function method to solve discrete and mixed variable problems from the field of design engineering. This method multiplied the penalty parameter with an appropriate factor and iteratively penalize the violated constraints. In order to solve non-linear engineering problems with discrete and mixed variables, Shih and Yang [9] have suggested the use of a generalized Hopfield method with an extended penalty approach. The penalty parameter in these methods was initialized using the arbitrary value (0 and 1) and then iteratively modified with an incremental multiplicator factor. There was a limit to the probability that the objective function value may become unstable if the multiplication factor is too large and trapped in the local minima. Curtis and Nocedal [25] implemented flexible penalty function introduced flexible penalty function to handle nonlinear constraints in a way similar to the dynamic penalty function method. In this case, instead of a fixed value influentially guiding convergence, the penalty parameter was arbitrarily chosen from a specified interval. Nanakorn and Meesomklin suggested an adaptive penalty function method using the modified binary scaling method to calculate the fitness value. This fitness value is then measured in three different categories which are minimum fitness, medium fitness value of all feasible solutions and best possible value. Each infeasible value is then penalized by the best infeasible value with a fitness that increase to ϕ times the average fitness value. The parameter ϕ had to be specified based on initial test. The smooth sequential penalty function implemented with the Quasi-Newton method was presented by Broyden and Attia [26] which is then connected to the Jacobian constraints led orthogonal transformation. Parsopoulos and Vrahatis [27] introduced a non-stationary multi-stage penalty approach, followed by Coath and Halgamuge [28] along with the feasibility preservation process to solve nonlinear problems. Michalewicz and M. Schoenauer [19] proposed a self-adaptive penalty function method using evolutionary algorithms. It splits the penalty method into two separate sections, for example, the number of violated constraints and a number of limitation which are violated and penalized individually. Therefore, it was important to choose independent weighting factors that increase the number of working parameters. Therefore several initial trials have been necessary to establish the appropriate penalty parameters for two parts. Nie [29] proposed a new semi penalty approach that addresses the strength of sequential quadratic programming (SQP) method and

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Sequential Penalty Quadratic Programming (S/QP) method in which the constraint on equality and inequality was viewed distinctly. Hasancebi and Azad [30] introduced another alternative penalty function scheme in which the Adaptive dimensional Search (ADS) approach included a relaxation technique. This tactic penalized and regulated the infeasible solution by selecting an infeasible solution to avoid the local minima. The intensity of penalty has been minimized by increasing the reduction factor once the solution is saturated. Using an updated penalty parameter the solution was recalculated and compared after every stagnation escape period (SEP) (i.e., saturation period). The original penalty parameter was used to calculate the solution and continue the process for looking for the best solution in the event that the current alternative is worse than the prior SEP solution. In this situation, the penalty calculation was done continuously, which requires an additional multiplication factor that appears to increase computing cost. Particle Swarm Optimization (PSO) is a metaheuristic optimization technique that makes few assumptions about solving the problem by the ability to search for large spaces and search for better solutions. It was first developed by Kennedy and Eberhart (1995) and Kennedy (1997). PSO technique provides an evolutionary-based search. There are two common penalty functions namely exterior and interior. External penalty function begins with an infeasible solution and convergence from infeasible to feasible region. The roles of internal function begin with a feasible solution and then move to restricted border from feasible area. External penalty functions are preferred over internal penalty function because they do not need to be enforced effectively. Static, fluid, adaptive and death penalty functions are the external penalty functions that have been developed to date. In the proposed work the Self-Adaptive Penalty Function (SAPF) solution will be introduced and integrated into the PSO algorithm in order to overcome the limitations of these approaches. This approach will eliminate the effort to set the penalty parameter and no other supporting parameter is required to do so. This process will make the PSO-SAPF algorithm free from setting of penalty parameter which may further help to generalize the algorithm. Additionally, a PSOSAPF algorithm will be hybridized with CBO in order to overcome the limitation of PSO-SAPF. The proposed PSO-SAPF and PSO-SAPF-CBO will be tested for solving different problems.

2 Methodology 2.1 Self-adaptive Penalty Function (SAPF) At first the proposed SAPF model will be constructed. In general, the constrained optimization problem is expressed as follows: Minimize f(x) = f(x1 , x2 , x3 , . . . , xn )

A Hybrid Metaheuristic Algorithm for Truss Structure …

25

Subject to gi (X ) ≤ 0, i = 1, 2, . . . , n h i (X ) = 0, i = 1, 2, . . . , m ψ lower ≤ X ≤ ψ upper

(2.1)

A static penalty function (SPF) constraint handling approach was widely used. It is expressed as follows PF =  ∗

 n 

gi (X ) +

i=1

m 

 h i (X )

(2.2)

i=1

n  m where  is a penalty parameter and i=1 gi (X ) + i=1 h i (X ) is summation of violated constraint. However, to penalize the violated constraints  value has to be randomly calculated. This practice allows the function value to influence and take longer to calculate the suitable  . These are the major disadvantages of SPF approach. So, to overcome a self-adaptive penalty function (SAPF) is proposed. In SAPF the objective function f(x) is itself utilize as a penalty parameter expresses as follows.   n m   gi (X ) + h i (X ) (2.3) SAPF = f(x) ∗ i=1

i=1

And further forms of the pseudo-objective function is Ø(X) = f(X) + SAPF. An experiment will be done to analyze this.

2.2 Particle Swarm Optimization It is a metaheuristic algorithm. It is also seeking to provide some possible answers. It can search for a solution within a wide search space. Kennedy initially proposed the algorithm [31]. The random iterations are used for this algorithm. Swarm here where the particles are gathered may be created and depends on the neighborhood. Particle information is shared within the community. The interplay between particles is defined by the corresponding matrix. In the matrix row, the same type of swarm is represented, called the informing particles. In themselves, they exchange information. However, the column depicts the informed particles. The information collected by the specific swarm is stored by them. It also contains data on the global position. So when the informant particle communicates with the informed particles, the global position can be achieved. First, PSO measures and optimizes the solution for candidates. This enables an objective function to be optimized. In accordance with

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the mathematical formula, PSO adjusts the position time and again. It then seeks to identify the particle’s position and velocity [32, 33]. Recent investigations have shown that PSO was applied to a very varied application [34–37]. In recent years this approach of optimization has been extended successfully to the prediction of reserve permeability, the prediction of minimal miscibility pressure in the form of carbon injections and parameters for the polypropylene reactor in the cycle, often using hybrid methodology with the aid of other optimizing instruments to solve various problems of petroleum optimization. Apparently, recent research found that the uses of PSO cover numerous different areas of future use, such as communication networks, automation, signal processing, power generation, power transmission and distribution systems, as well as networks, forecasts and projections, telecommunications and software, and investment forecasting. In the pharmaceutical and medical industries, for example. Many uses in genetic engineering include human tremor studies to treat Parkinson’s disease; gene-clustering inference; and the identification of DNA motifs. GA has crossover and mutation operators, but on the other hand PSO has no such operators. In this case, a particle is known as a chromosome-like aspect of GA. Here, an initial value is given for every particle. Each particle is subsequently assigned to two vectors, including velocity and position vector. The particles then travel around the region of search. This attempts to find viable solutions. The next optimal candidate value is also sought. The best preceding position is assigned to memory. After the last test, a general memory is allocated to the best position. Thus, in the quest for the best solution it repeats the approach. It replaces the previous value when a better solution is obtained. This process goes on until the global optimum is found. But it is not guaranteed to find a truly global position [38]. Particles here have been grouped into swarm [39]. The relationship between particles is defined by the adjacency matrix principle. Various PSO forms can be described on the basis of various neighborhood communication topologies. In the PSO algorithm, three important components are cognitive component (ϕ p ), inertia component (ω) social   component, ϕg . Figure 2.1 shows the schematic drawing of position update process of a particle. For a two-dimensional space     Vi,d = ωVi,d + ϕ p r p pi,d − X i,d + ϕg r g gd − X i,d

(2.4)

where Vi,d Velocity of ith particle and for dth dimension r p , r g are random numbers uniformly distributed between 0 and 1, applied to the cognitive and social components, respectively Best particle value found in the current swarm for the dth dimension of ith pi,d particle, up to current iteration X i,d Value of the d th dimension of ith particle Best global value found for the dth dimension by all particles in the current gd and previous swarms up to current iteration, which, at the end of the last iteration, contains the optimal values the dth dimension

A Hybrid Metaheuristic Algorithm for Truss Structure …

27

Fig. 2.1 Schematic drawing of position update process of a particle

X i(updated) = X i( pr ev) + vi

(2.5)

Using Eq. (2.5) the particle updates it’s position. In this case, the inertia element limits the particles within the problem area. Where cognitive components play a role, the particles shift into their best previous position. And with the help of social elements, they will reach their optimal global position.

2.3 PSO with SAPF Here according to the proposal, a PSO-SAPF model will be developed. Consider a bird flock with number of particles C. For every individual candidate c (c = 1, 2, …, C) using the PSO-SAPF approach the pseudo objective function (behavior) (refer to Eq. 2.3) can be expressed as follows:       ∅ Xc = f Xc + S AP F Xc

(2.6)

n  m where S A P F(X c ) = f (X c )∗ i=1 gi (X c ) + i=1 h i (X c ) is penalty function and f (X c ) is the objective function of individual particle. The behavior of the penalty parameter is dependent on the variable sampling space. An independent penalty parameter f (X c ) will be generated by every individual particle C to penalize the violated constraints associated with its behavior. And then, for every learning trial

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of the PSO-SAPF algorithm, upgrades the penalty function. But PSO-SAPF’s disadvantage will be observed. At the end of every learning attempt, the particles update its individual search space. It will be examined. If there is any limitation in that case PSO-SAPF will be hybridized with CBO.

2.4 Colliding Bodies Optimization (CBO) The CBO algorithm was proposed by Kaveh and Mahdavi [40, 41]. It was motivated by colliding bodies (objects) physical behavior. This obeys the law of momentum and energy conservation, in which the momentum of all objects prior to collision is equal to the momentum of all objects after the collision. Two moving bodies having masses and velocities are separated with updated velocities after the collision. This causes an object to move in search space towards a better position. On certain continuous, discrete and mixed variable truss structure and field optimization problems, a metaheuristic CBO algorithm was successfully validated [38]. The CBO was found to be sensitive to the number of objects and to maintain better convergence and a higher level of exploration, more collision was necessary. Moreover, in order to include a memory in it and save the best option for further operations, Kaveh [39] updated the Enhanced CBO. This lets the CBO algorithm increase the pace of investigation and minimize computational costs at faster convergence. In ECBO some parameter was generated arbitrarily between and compared to a uniformly distributed number to hop out of local minima. The adjustment in the location of the colliding bodies was also decided. The hybridization CBO-PSO was proposed to exploit the ability of CBO by incorporating basic features of PSO. The CBO-PSO [38] was successfully validated by solving continuous variable truss structure problems with dynamin constraints incorporated with SPF approach.

2.5 Framework of PSO-SAPF-CBO It is necessary to generalize the problem-solving technique to explore the applicability for diversified real-world applications. PSO has already been validated by solving large groups of problems. However, PSO algorithm required certain preliminary trials to set the parameter to avoid the solution trap into the local minima [23]. In order to overcome this limitation of the PSO algorithm an important characteristic of CBO is incorporated into PSO. The PSO-SAPF-CBO algorithm will employ PSO for global search, SAPF for constraint handling and CBO works as a local search. The natural tendency of PSO candidates is to follow certain candidates probabilistically chosen using a roulette wheel approach to evolve individual behavior. Further, the learning ability of PSO candidates will be refined using CBO. The proposed PSO-SAPF-CBO is mathematically expressed as follows:

A Hybrid Metaheuristic Algorithm for Truss Structure …

29

Step 1: Consider a swarm with C number of particles, every individual particle C(c = 1, 2, …, C) belongs a set of attributes/variables X c = (x1c , x2c , . . . , x Nc ) which makes the behavior of an individual particle f (X c ). The initial solution is randomly generated similar to the other population-based techniques as follows: X c = ψ lower + (ψ upper − ψ lower ) ∗ rand(1, N)

(2.7)

Step 2: The self-adaptive penalty function (SAPF) approach will be incorporated to handle the constrained and obtained pseudo-objective function Ø(X c ) Step 3: The probability of selecting behavior f (X c ) of every associated candidate c(c = 1, 2, 3, …, C) will be evaluated as follows 1

P c = n∅

(X c )

1 c i=1 ∅ (X )

(2.8)

Step 4: Every individual particle c(c = 1, 2, …, C) generates a random number r ∈ [0,1] and using a roulette wheel approach decides to follow the corresponding behaviour and associated attributes. Step 5: On following a suitable candidate, all the candidates will be arranged in descending order. In the context of CBO, the first half is referred to as slow learning candidates and other half is referred to as fast learning candidates. These motivate slow learning candidates to improve their position in the search space. Figure 2.2 has depicted the flow chart for PSO-SAPF-CBO algorithm.

3 Result and Analysis The PSO-SAPF and PSO-SAPF-CBO were used to solve the truss structure problem. Later the solutions were compared with GA. All issues relating to the truss construction were designed to reduce weight overall by following the requirements, total allowed tension both in stress and pressure and maximum permissible movement in both vertical and horizontal directions at each node [42]. In these problems, the number of variables was equal to the number of trusses. For symmetric truss structure problem, the number of variables was equal to number of the truss. The variables were classified in category as described in the respective comparison table for symmetrical truss construction problems, including 25 bar 45 bar and 72 bar. For all truss structure problems, the cross-section area was considered as variable and it was selected from the set of discrete values. Here the truss structure problem is defined as follows: Minimize f = w =

N  i=1

ρi Ai li

(2.9)

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K. Biswas et al. Start

Initialize number of swarm

The probability associated with the behavior being followed by every particle in the swarm will evaluate

No

PSO-SAPF

Evaluate the objective function , Compute the constraints using SAPF approach and obtained a pseudo objective function

No Using roulette wheel approach every particle selects behavior to follow from within the C available choices

CBO

Particles are categorized in slow learning and fast learning based on followed behaviour

New position of each particle C is determined

Satisfy the constraints

Yes Convergence?

Yes Accept the current result as final solution

Stop

Fig. 2.2 Flow chart of PSO-SAPF-CBO

Subject to |σi | ≤ σmax i = 1, 2, 3, . . . , N

(2.10)

  u j  ≤ u max j = 1, 2, 3, . . . , M

(2.11)

Where W Ai ρ L

Function to be optimized (weight) Area of cross section’s set where the area will be calculated for truss structure member I = 1, 2, 3, …, N Weight density of the truss structure material. length of the truss structure member.

A Hybrid Metaheuristic Algorithm for Truss Structure … Table 2.1 Comparison of result for solving 6-bar truss structure

31

Variables

GA [43]

PSO-SAPF

PSO-SAPF-CBO

A1

30

29

29

A2

19.9

19.8

19.9

A3

15.5

15.5

15.5

A4

7.22

7.22

7.22

A5

22

22

22

A6

22

22

22

Truss weight W(lb)

4962.1

4961.9453

4962.0854

Function evaluations

*N/A

2250

1740

*N/A Not Available

σmax Highest stress. u max Highest displacement. Test Example: 1–6 Bar Truss Structure The PSO-SAPF and PSO-SAPF-CBO resolution of six bar structural trusses has achieved nearly similar results to that shown in Table 2.1 for GA [43]. In the 20 experiments using PSO-SAPF and PSO-SAPF-CBO algorithms, the best mean and worst results were about the same, with a standard deviation of zero and average estimation times were 4.58 s and 3.92 s. The average function value reported using GA was 5250 lb. From this it was observed that PSO-SAPF and PSO-SAPF-CBO performed better than GA. The average number of function evaluations for both the proposed technique was 2849 and 2550 respectively. The same problem was attempted using CBO however, unable to get the feasible solution due to premature convergence of the solution.

4 Resources Used See Table 2.2.

5 Conclusion The main advantage of the proposed constrained handling SAPF approach is that it can be directly applicable to a variety of constrained optimization problems without preliminary trials. The PSO-SAPF algorithm was reported to be more efficient which reduced the efforts of trial and error process for setting of suitable penalty parameter. In PSO-SAPF algorithm the penalty parameter was selected based on the available

32 Table 2.2 Detail setting of required system

K. Biswas et al. Name

Setting

Hardware CPU

Intel core™ i5-6400 processor

Frequency

2.70 GHz

RAM

8 GB

Hard Drive

1 TB

Software Operating System

Windows 10

Language

MATLAB R2016A, PYTHON

set of design variables which updated iteratively. The PSO-SAPF-CBO approach also reduced the efforts of trial and error method. And they also give the comparative result. However, only CBO was unable to get the feasible solution.

References 1. Li, L., Huang, Z., Liu, F., Wu, Q.: A heuristic particle swarm optimizer for optimization of pin connected structures. Comput. Struct. 85(7–8), 340–349 (2007) 2. Datta, D., Figueira, J.R.: A real-integer-discrete-coded particle swarm optimization for design problems. Appl. Soft Comput. 11(4), 3625–3633 (2011) 3. Gandomi, A.H., Yang, X.-S., Alavi, A.H.: Mixed variable structural optimization using firefly algorithm. Comput. Struct. 89(23–24), 2325–2336 (2011) 4. Kulkarni, A.J., Tai, K.: Probability collectives: a multi-agent approach for solving combinatorial optimization problems. Appl. Soft Comput. 10(3), 759–771 (2010) 5. Kaveh, A., Mahdavi, V.: Colliding bodies optimization method for optimum design of truss structures with continuous variables. Adv. Eng. Softw. 70, 1–12 (2014) 6. Cheng, M.-Y., Prayogo, D.: Symbiotic organisms search: a new metaheuristic optimization algorithm. Comput. Struct. 139, 98–112 (2014) 7. Sadollah, A., Bahreininejad, A., Eskandar, H., Hamdi, M.: Mine blast algorithm for optimization of truss structures with discrete variables. Comput. Struct. 102, 49–63 (2012) 8. L. Lamberti, C. Pappalettere, An improved harmony-search algorithm for truss structure optimization, in Proceedings of the Twelfth International Conference on Civil, Structural and Environmental Engineering Computing (Civil-Comp Press, Stirlingshire, UK, 2009), Paper 65 9. Shih, C., Yang, Y.: Generalized Hopfield network based structural optimization using sequential unconstrained minimization technique with additional penalty strategy. Adv. Eng. Softw. 33(7– 10), 721–729 (2002) 10. K. Deb, S. Gulati, S. Chakrabarti, Optimal truss-structure design using real-coded genetic algorithms, in Proceedings of the Third Annual Conference Genetic Programming 1998 (1998), pp. 22–25 11. A.J. Kulkarni, I.P. Durugkar, M. Kumar, Cohort intelligence: a self supervised learning behavior, in 2013 IEEE International Conference on Systems, Man, and Cybernetics (IEEE), pp. 1396–1400 12. Huan, T.T., Kulkarni, A.J., Kanesan, J., Huang, C.J., Abraham, A.: Ideology algorithm: a socio-inspired optimization methodology. Neural Comput. Appl. 28(1), 845–876 (2017)

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13. Kumar, M., Kulkarni, A.J., Satapathy, S.C.: Socio evolution & learning optimization algorithm: A socio-inspired optimization methodology. Future Generation Comput. Syst. 81, 252–272 (2018) 14. K. Deb, S. Agrawal, A niched-penalty approach for constraint handling in genetic algorithms, in Artificial Neural Nets and Genetic Algorithms (Springer, 1999), pp. 235–243 15. Deb, K.: An efficient constraint handling method for genetic algorithms. Comput. Methods Appl. Mech. Eng. 186(2–4), 311–338 (2000) 16. Kulkarni, A.J., Shabir, H.: Solving 0–1 knapsack problem using cohort intelligence algorithm. Int. J. Mach. Learn. Cybernet. 7(3), 427–441 (2016) 17. D.G. Luenberger, Y. Ye, Linear and Nonlinear Programming (Springer, 1984) 18. Homaifar, A., Qi, C.X., Lai, S.H.: Constrained optimization via genetic algorithms. Simulation 62(4), 242–253 (1994) 19. Michalewicz, Z., Schoenauer, M.: Evolutionary algorithms for constrained parameter optimization problems. Evol. Comput. 4(1), 1–32 (1996) 20. M. Gen, R. Cheng, A survey of penalty techniques in genetic algorithms, in Proceedings of IEEE International Conference on Evolutionary Computation (IEEE, 1996), pp. 804–809 21. R. Le Riche, C. Knopf-Lenoir, R.T. Haftka, A segregated genetic algorithm for constrained structural optimization, in ICGA (1995), pp. 558–565 22. Azad, S.K., Hasançebi, O.: Upper bound strategy for metaheuristic based design optimization of steel frames. Adv. Eng. Softw. 57, 19–32 (2013) 23. Kale, I.R., Kulkarni, A.J.: Cohort intelligence algorithm for discrete and mixed variable engineering problems. Int. J. Parallel Emerg. Distrib. Syst. 33(6), 627–662 (2018) 24. Kannan, B., Kramer, S.N.: An augmented Lagrange multiplier based method for mixed integer discrete continuous optimization and its applications to mechanical design. J. Mech. Des. 116(2), 405–411 (1994) 25. Curtis, F.E., Nocedal, J.: Flexible penalty functions for nonlinear constrained optimization. IMA J. Num. Analy. 28(4), 749–769 (2008) 26. C.G. Broyden, N.F. Attia, A smooth sequential penalty function method for solving nonlinear programming problems, in System Modelling and Optimization (Springer Berlin Heidelberg, Berlin, Heidelberg, 1984), pp. 237–245 27. Parsopoulos, K., Vrahatis, M.: Initializing the particle swarm optimizer using the nonlinear simplex method. Adv. intell. Syst. Fuzzy Syst. Evol. Comput. 216, 1–6 (2002) 28. G. Coath, S.K. Halgamuge, A comparison of constraint-handling methods for the application of particle swarm optimization to constrained nonlinear optimization problems, in The 2003 Congress on Evolutionary Computation, CEC’03, vol. 4 (IEEE, 2003), pp. 2419–2425 29. Nie, P.-Y.: A new penalty method for nonlinear programming. Comput. Math Appl. 52(6–7), 883–896 (2006) 30. Hasançebi, O., Azad, S.K.: Adaptive dimensional search: a new metaheuristic algorithm for discrete truss sizing optimization. Comput. Struct. 154, 1–16 (2015) 31. A.J. Kulkarni, G. Krishnasamy, A. Abraham, Cohort intelligence for solving travelling salesman problems, in Cohort Intelligence: A Socio-inspired Optimization Method (Springer, 2017), pp. 75–86 32. Shastri, A.S., Kulkarni, A.J.: Multi-cohort intelligence algorithm: an intra-group and intergroup learning behaviour based socio-inspired optimisation methodology. Int. J. Parallel Emerg. Distrib. Syst. 33(6), 675–715 (2018) 33. Krishnasamy, G., Kulkarni, A.J., Paramesran, R.: A hybrid approach for data clustering based on modified cohort intelligence and K-means. Exp. Syst. Appl. 41(13), 6009–6016 (2014) 34. S.M. Gaikwad, R.R. Joshi, A.J. Kulkarni, Cohort intelligence and genetic algorithm along with ahp to recommend an ice cream to a diabetic patient, in International Conference on Swarm, Evolutionary, and Memetic Computing (Springer, 2015), pp. 40–49 35. Patankar, N.S., Kulkarni, A.J.: Variations of cohort intelligence. Soft. Comput. 22(6), 1731– 1747 (2018) 36. Sarmah, D.K., Kulkarni, A.J.: Image steganography capacity improvement using cohort intelligence and modified multi-random start local search methods. Arab. J. Sci. Eng. 43(8), 3927–3950 (2018)

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Electric Field Behavior in Time Domain for Semicircle Antenna in Homogeneous Multi Layered Media Elsayed Dahy, Afza Shafie, and Noorhana Yahya

Abstract Seabed logging is a direct and remote application of control source electromagnetic method to detect the hydrocarbon reservoirs. In this method, electromagnetic waves are used to detect the resistivity contrast between the various sediment layers. In this study, a semicircle antenna is used as the source of electromagnetic waves for detection of the resistive layers under the seabed. Numerical solutions of the electric field wave equations in time domain are derived. Results on the effect of seawater depth, and thickness of hydrocarbon on the electric field have shown that antenna is able to assist in hydrocarbon detection. Keywords Electromagnetics · Seabed logging · Hydrocarbon · Antenna · Resistivity contrast

1 Introduction There are two methods for hydrocarbon exploration in offshore environment: seismic and controlled source electromagnetics (CSEM). In seismic method, acoustic waves are used to generate images of the surface and subsurface; however, the problem in this technique is that, the sensitivity of the waves to variations in fluid saturation is very poor. Hence, unable to discriminate between oil and saline water trapped in the well. In the recent years, controlled source electromagnetic method, also known as seabed logging (SBL) is introduced to detect the hydrocarbon (HC) reservoirs.

E. Dahy Department of Mathematics, Al-Azhar University, Assuit, Egypt e-mail: [email protected] A. Shafie (B) · N. Yahya Fundamental and Applied Sciences Department, Universiti Teknologi PETRONAS, Bandar Seri Iskandar, 32610 Seri Iskandar, Perak Darul Ridzuan, Malaysia e-mail: [email protected] N. Yahya e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 S. A. Abdul Karim (ed.), Theoretical, Modelling and Numerical Simulations Toward Industry 4.0, Studies in Systems, Decision and Control 319, https://doi.org/10.1007/978-981-15-8987-4_3

35

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E. Dahy et al.

The theory behind CSEM surveys is based on the knowledge that the propagation of an electromagnetic (EM) field induced in a conductive subsurface is mainly affected by spatial distribution of resistivity. In marine environments, saltwater-filled sediments typically represent good conductors, whereas hydrocarbon-filled sediments, salt, volcanic rocks and carbonates represent examples of resistive bodies that scatter the EM field. Part of the EM field scattered by subsurface inhomogeneities propagates back to the seafloor where the signal is recorded by receivers equipped with electric and magnetic sensors. CSEM is one of the techniques used to detect the presence of hydrocarbon layers beneath the seabed. This method utilizes man made electric and magnetic field to excite the earth beneath the sea floor [1–4]. In this method, a powerful horizontal electric dipole (HED) used to transmit ultralow frequency (~0.1 to 5 Hz) EM waves is being towed approximately 30 – 40 m above the seabed to detect resistivity contrasts in the subsurface. The relative increment of the lateral magnetic component of electromagnetic signal will be recorded by an array of receivers located at the seafloor. In general, these receivers will record the waves that are from the source or direct waves, the waves that interact with the air or the airwaves and the guided waves, which are waves that are reflected from interaction with hydrocarbon reservoirs as shown in Fig. 1. Direct wave components are signals that are transmitted directly from the antenna to reach the array of receivers. These waves are detected at near offsets [5] and can be neglected by considering offsets, from the source, approximately three times

Fig. 1 Components of waves recorded by the receivers

Electric Field Behavior in Time Domain for Semicircle Antenna …

37

the target depth [6]. The guided waves propagate from the antenna, travel downwards along the sediments, but instead of being reflected upwards, they are guided along the HC reservoir and travel further offset until reflected upwards towards the array of receivers [6]. The guided wave are more dominant and visible at larger offsets [7]. The airwaves are also EM signal component generated by the transmitter that diffuses vertically upwards and propagate as a ‘wave’ through the air at a speed of light with no attenuation before diffusing back down vertically through the water layer to the sea bottom, where it is recorded by the receivers [1]. Typically, hydrocarbon reservoirs has resistivity 1 to 2 orders of magnitude (between 10 and 1000 -m) higher than siltstones with a few ohm meters (1–5 -m) [8–10]. This causes EM fields response at the receiver at large transmitter— receiver separations (offset) to be larger in magnitude where highly resistive layer such as hydrocarbon are present in the subsurface. Therefore, SBL concept relies on the large resistivity contrast between hydrocarbon saturated reservoirs, and the surrounding subsurface layers saturated with aqueous saline fluids. Due to close relation between fluid saturation and electric conductivity, the CSEM method can discriminate between brine and resistive layers in the subsurface [8].Target depth and sea water depth plays an important role in the success of the SBL techniques since these two variables have an effect on the signals recorded by the receivers during the survey. One of the biggest challenges in SBL is the presence of airwaves. The airwave component is predominantly generated by the signal that diffuses vertically upwards from the source to the sea surface. In deep water (>500 m), the up going waves are highly attenuated and hence losing its strength upon reaching the receivers, but in shallow water, the up-going waves or the airwaves, are more dominant than the down going waves. The presence of airwaves affects the data significantly in shallow water ( 0  βi, j > 0, γi, j ≥ 0 A1 = 2αi, j + γi, j Fi, j + αi, j h i Fi,x j ,     A2 = αi, j + γi, j F  i+1, j + βi, j + γi,x j Fi, j , A3 = 2βi, j + γi, j Fi+1, j − βi, j h i Fi+1, j, A4 = βi, j Fi+1, j , q1 (θ ) = (1 − θ )2 αi, j + γi, j θ (1 − θ ) + θ 2 βi, j ; 4   (1 − θ)3−i θ i Bi S x, y j+1 = i=0 q2 (θ )

(17)

with B0 = αi, j+1 Fi, j+1 , αi, j+1 > 0  x βi, j+1 > 0, γi, j+1 ≥ 0 B1 = 2αi, j+1 + γi, j+1 Fi, j+1 + αi, j+1 h i Fi, j+1 ,     AB2 = αi, j+1 + γi, j+1 Fi+1, j+1 + βi, j+1 + γi, j+1 Fi, j+1 , x B3 = 2βi, j+1 + γi, j+1 Fi+1, j+1 − βi, j+1 h i Fi+1, j+1 , B4 = βi, j Fi+1, j+1 , q2 (θ ) = (1 − θ)2 αi, j+1 + γi, j+1 θ (1 − θ ) + θ 2 βi, j+1 4 (1 − ϕ)3−i ϕ i Ci (18) S(xi , y) = i=0 q3 (ϕ) with C0 = αˆ i, j Fi, j , αˆ i, j > 0   y C1 = 2αˆ i, j + γˆi, j Fi, j + αˆi, j hˆ j Fi, j , βˆi, j > 0, γˆi, j ≥ 0   C2 = αˆ i, j + γˆi, j Fi, j+1 + βˆi, j + γˆi, j Fi, j ,  y C3 = 2βˆi, j + γˆi, j Fi, j+1 − βˆi, j hˆ j Fi, j+1 , C4 = βˆi, j Fi, j+1 , q3 (ϕ) = (1 − ϕ)2 αˆ i, j + γˆi, j ϕ(1 − ϕ) + ϕ 2 βˆi, j ; 4 S(xi+1 , y) =

i=0 (1

− ϕ)3−i ϕ i Di q4 (ϕ)

(19)

with D0 = αˆ i+1, j Fi+1, j , αˆ i+1, j > 0   y ˆ D1 = 2αˆ i+1, j + γˆi+1, j Fi+1, j + αˆi+1, j h j Fi+1, j , βˆi+1, j > 0, γˆi+1, j ≥ 0   D2 = αˆ i+1, j + γˆi+1, j Fi+1, j+1 + βˆi+1, j + γˆi+1, j Fi+1, j ,  y D3 = 2βˆi+1, j + γˆi+1, j Fi+1, j+1 − βˆi+1, j hˆ j Fi+1, j+1 , D3 = βˆi, j Fi+1, j+1 , q4 (ϕ) = (1 − ϕ)2 αˆ i+1, j + γˆi+1, j ϕ(1 − ϕ) + ϕ 2 βˆi+1, j

Surface Interpolation Using Partially Blended …

61

Remark 1 When αi, j = αi, j+1 = βi, j = βi, j+1 = 1, αˆ i, j = αˆ i+1, j = βˆi, j = βˆi+1, j = 1 and γi, j = γi, j+1 = γˆi, j = γˆi+1, j = 2, the partially blended rational bi-cubic spline in (15) reduce to standard bi-quartic polynomial spline. Theorem 1 The partially blended rational bi-quartic spline interpolation defined in (15) is C1 everywhere provided that all free parameters are positive. Theorem 2 The proposed rational bi-quartic spline interpolation given in (15) is shape preserving provided that all four curves network defined in (16)–(19) are shape preserving.

4 Surface Interpolation The partially blended rational bi-quartic spline constructed in the previous section can be used to interpolate surface data (organized or unorganized). To do this, an efficient algorithm is presented below: Algorithm   Input: 3D data points xi , y j , Fi, j , i = 0, 1, . . . , n; j = 0, 1, . . . , m. Step 1: For i = 0, 1, . . . , n; j = 0, 1, . . . , m y Calculate the first derivative values Fi,x j and Fi, j by using AMM given in Sect. 2. Step 2: For i = 0, 1, . . . , n − 1; j = 0, 1, . . . , m − 1 Choose suitable positive values for free parameters αi, j , βi, j , αi, j+1 , βi, j+1 , αi, j , αˆ i+1, j , βˆi, j , βˆi+1, j and γi, j , γi, j+1 , γˆi, j , γˆi+1, j . Step 3: Construct the interpolating surface by using (15). Step 4: Calculate root mean square error (RMSE) Output: Interpolating surfaces Steps 2–4 can be repeated for different data sets.

5 Results and Discussion In this subsection, numerical examples for surface data interpolation by using the proposed partially blended rational bi-quartic spline interpolation will be discussed in detail. Two data sets are taken from Abbas et al. [15]. RMSE is used as error measurement to validate the proposed scheme. Example 1 Surface data from the following function are truncated to five decimal places. F1 (x, y) = e−(x

2

+y 2 )/15

(sin(x) + cos(y)) + 0.33,

Table 1 shows the tested data.

0 ≤ x, y ≤ 6

(20)

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Table 1 Surface data from function F1 (x, y) y

x 0

2

4

6

0

1.33000

0.01126

0.10505

0.41710

2

1.79240

0.61930

0.39739

0.45990

4

0.41370

0.02081

0.16294

0.33635

6

0.39537

0.28167

0.30087

0.33560

Figure 2a shows the default bi-quartic polynomial spline when αi, j = βi, j = 1, αˆ i, j = βˆi, j = 1 and γi, j = γˆi, j = 2, for the surface data given in Table 1 . Figures 2b–f shows interpolating surfaces with various values of the shape parameters as listed in Table 2. From Fig. 2e, we notice that, the proposed scheme is approaching bilinear surface since γi, j and γˆi, j are a larger number. Finally Fig. 2g shows the interpolating surface using the scheme of Abbas et al. [15]. Example 2 Consider the function   F2 (x, y) = sin ye−x + 1,

−3 ≤ x, y ≤ 3, x, y = 0

(21)

quoted from Abbas et al. [15]. Positive data from the function are truncated to five decimal places. The data are shown in Table 3. Figure 3 shows the default bi-quartic polynomial spline when αi, j = βi, j = 1, αˆ i, j = βˆi, j = 1 and γi, j = γˆi, j = 2, for the data given in Table 2. Figure 3b–g shows the interpolating surface generated by using the proposed rational bi-quartic spline with various parameters values as shown in Table 4. The produced surface is smooth and continuous everywhere on the given domain. Finally, Fig. 3g shows the interpolating surface using the scheme of Abbas et al. [15]. Root Mean Square Error Comparison In this section, we compare the performance between the proposed partially blended rational bi-quartic spline interpolation with the works of Abbas et al. [15] and Karim et al. [17]. It is made by calculating the Root Mean Square Error (RMSE) on 33 × 33 evaluation points of a uniform rectangular grid. Table 5 summarizes the results of RMSE estimation for the proposed scheme and the schemes of [15, 17]. For Example 2, the proposed scheme is better than [15, 17]. Overall, the proposed partially blended rational bi-cubic spline interpolation produces the interpolating surface as good as Abbas et al. [15] and Karim et al. [17], respectively. A. αi, j = βi, j = 1, αˆ i, j = βˆi, j = 1 and γi, j = γˆi, j = 2. B. αi, j = βi, j = 1, αˆ i, j = βˆi, j = 1 and γi, j = γˆi, j = 1.

Surface Interpolation Using Partially Blended … Fig. 2 Interpolating surfaces for Example 1. Using Abbas et al. [15]

63

64 Fig. 2 (continued)

S. A. A. Karim et al.

Surface Interpolation Using Partially Blended …

65

Fig. 2 (continued)

(g)

Table 2 Parameters value for Fig. 2 Figures

Parameters

2b

αi, j = βi, j = 1, αˆ i, j = βˆi, j = 1 and γi, j = γˆi, j = 5

2c

αi, j = βi, j = 1, αˆ i, j = βˆi, j = 1 and γi, j = γˆi, j = 0.5

2d

αi, j = βi, j = 1, αˆ i, j = βˆi, j = 1 and γi, j = γˆi, j = 1

2e

αi, j = βi, j = 1, αˆ i, j = βˆi, j = 1 and γi, j = γˆi, j = 100

2f

αi, j = βi, j = 1, αˆ i, j = βˆi, j = 0.5 and γi, j = γˆi, j = 0.5

Table 3 Surface data from function F2 (x, y) y

x −3

−2

−1

1

2

3

−3

1.53660

0.37930

0.05553

1.94450

1.62070

0.46344

−2

1.17510

0.19847

0.10615

1.89390

1.80150

0.82489

−1

0.04492

1.74900

0.58922

1.41080

0.25095

1.95510

1

0.10715

0.32885

0.64036

1.35960

1.67110

1.89290

2

0.60506

0.73262

0.86508

1.13490

1.26740

1.39490

3

0.85119

0.90059

0.95023

1.04980

1.09940

1.14880

C. αi, j = βi, j = 1, αˆ i, j = βˆi, j = 1 and γi, j = γˆi, j = 0.5. D. αi, j = βi, j = 1, αˆ i, j = βˆi, j = 0.5 and γi, j = γˆi, j = 0.5.

66 Fig. 3 Interpolating surfaces for Example 2. Surface using [15]

S. A. A. Karim et al.

Surface Interpolation Using Partially Blended … Fig. 3 (continued)

67

68

S. A. A. Karim et al.

Fig. 3 (continued)

(g)

Table 4 Parameters value for Fig. 3 Figures

Parameters

3b

αi, j = βi, j = 1, αˆ i, j = βˆi, j = 1 and γi, j = γˆi, j = 1

3c

αi, j = βi, j = 1, αˆ i, j = βˆi, j = 1 and γi, j = γˆi, j = 5

3d

αi, j = βi, j = 1, αˆ i, j = βˆi, j = 1 and γi, j = γˆi, j = 0.5

3e

αi, j = βi, j = 1, αˆ i, j = βˆi, j = 0.5 and γi, j = γˆi, j = 2

3f

αi, j = βi, j = 1, αˆ i, j = βˆi, j = 1 and γi, j = γˆi, j = 100

Table 5 RMSE estimation Method

Example

Abbas et al. [15] Karim et al. [17] Proposed scheme

1

2

0.0252

0.3900

0.0233

0.3947

A

0.0243

0.2614

B

0.0240

0.2549

C

0.0239

0.2703

D

0.0231

0.2664

The suitable free parameters values are obtained through numerical simulation. However, these values can be obtained by using genetic algorithm (GA) or other neural network (NN) techniques. We also can obtain smaller RMSE value by sampling more tested data. But, this will increase the CPU time (in seconds) to construct the interpolating surface.

Surface Interpolation Using Partially Blended …

69

6 Summary In this chapter, a new partially blended rational bi-quartic spline with twelve free parameters are constructed. These free parameters provide greater flexibility to the user in changing the shape of the surface without the need to change the control points. The final surface is smooth since it has degree of continuity C1 on entire given domain. Numerical results show that, the proposed scheme is on par with some established schemes. Future work will be focusing on the application of the proposed partially blended rational bi-quartic spline interpolation for various types of shape preserving interpolation such as positivity, monotonicity, constrained modelling and convexity. Application in image interpolation also possible by extending the main idea from Zulkifli et al. [24]. Acknowledgements This work is fully supported by Universiti Teknologi PETRONAS (UTP) and Ministry of Education, Malaysia through research grant FRGS/1/2018/STG06/UTP/03/1015MA0020. Special thank you to UTP for providing MATLAB software for computer implementation.

References 1. Brodlie, K.W., Butt, S.: Preserving convexity using piecewise cubic interpolation. Comput. Graph. 15, 15–23 (1991) 2. Brodlie, K.W., Mashwama, P., Butt, S.: Visualization of surface data to preserve positivity and other simple constraints. Comput. Graph. 19(4), 585–594 (1995) 3. Sarfraz, M., Hussain, M.Z., Hussain, M.: Shape-preserving curve interpolation. Int. J. Comput. Math. 89(1), 35–53 (2012) 4. Beliakov, G.: Monotonicity preserving approximation of multivariate scattered data. BIT 45(4), 653–677 (2005) 5. Goodman, T.N.T., Ong, B.H., Unsworth, K.: Constrained interpolation using rational cubic splines. In: Farin, G. (ed.) NURBS for Curve and Surface Design, SIAM, Philadelphia, pp. 59– 74 (1991) 6. Karim, S.A.A., Saaban, A., Skala, V.: Range-restricted surface interpolation using rational bicubic spline functions with 12 parameters. IEEE Access 7, 104992–105007 (2019). https://doi. org/10.1109/ACCESS.2019.2931454 7. Karim, S.A.A., Hasan, M.K., Hashim, I. (2019). Constrained interpolation using rational cubic spline with three parameters. Sains Malaysiana, 48(3), 685–695 (2019) 8. Wu, J., Zhang, X., Peng, L.: Positive Approximation and interpolation using compactly supported radial basis functions. Math. Prob. Eng. Article ID 964528, 10 (2010) 9. Wu, J., Lai, Y., Zhang, X.: Radial basis functions for shape preserving planar interpolating curves. J. Inf. Comput. Sci. 7(7), 1453–1458 (2010) 10. Han, X.: Convexity-preserving piecewise rational quartic interpolation. SIAM J Numer. Anal. 46(2), 920–929 (2008) 11. Zhu, Y., Han, X., Han, J.: Quartic trigonometric Bézier curves and shape preserving interpolation curves. J. Comput. Inf. Syst. 8(2), 905–914 (2012) 12. Zhu, Y., Han, X.: Shape preserving C 2 rational quartic interpolation spline with two parameters. Int. J. Comput. Math. 92(10), 2160–2177 (2014) 13. Han, X.: Shape preserving piecewise rational interpolant with quartic numerator and quadratic denominator. Appl. Math. Comput. 251, 258–274 (2015)

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14. Abbas, M., Majid, A.A., Awang, M.N.H., Ali, J.M.: Shape preserving positive surface data visualization by spline functions. Appl. Math. Sci. 6(6), 291–307 (2012) 15. Abbas, M., Majid, A.A., Ali, J.M.: Shape preserving rational bi-cubic function for positive data. World Appl. Sci. J. 18(11), 1671–1679 (2012) 16. Abbas, M., Majid, A.A., Ali, J.M.: Positivity-preserving rational bi-cubic spline interpolation for 3D positive data. Appl. Math. Comput. 234, 460–476 (2014) 17. Karim, S.A.A., Kong, V.P., Saaban, A.: Positivity preserving interpolation using rational bicubic spline. J. Appl. Math. Article ID 572768, 15. http://dx.doi.org/10.1155/2015/572768 18. Karim, S.A.A., Kong, V.P.: Shape preserving interpolation using rational cubic spline. Res. J. Appl. Sci. Eng. Technol. (RJASET) 8(2), 167–168 (2014) 19. Karim, S.A.A., Kong, V.P.: Convexity-preserving using rational cubic spline interpolation. Res. J. Appl. Sci. Eng. Technol. (RJASET) 8(3), 312–320 (2014) 20. Karim, S.A.A., Kong, V.P.: Monotonicity-preserving using rational cubic spline interpolation. Res. J. Appl. Sci. (RJAS) 9(4), 214–223 (2014) 21. Karim, S.A.A., Kong, V.P.: Shape Preserving interpolation using C 2 rational cubic spline. J. Appl. Math. Article ID 4875358, 12 (2016). http://dx.doi.org/10.1155/2016/4875358 22. Harim, A., Karim, S.A.A, Othman. M., Saaban, A.: High accuracy data interpolation using rational quartic spline with three parameters. Int. J. Sci. Technol. Res. 1219–27432 (2019) 23. Delbourgo, R., Gregory, J.A.: The determination of derivative parameters for a monotonic rational quadratic interpolant. IMA J. Numer. Anal. 5, 397–406 (1985) 24. Zulkifli, N.A., Karim, S.A.A., Shafie, A., Sarfraz, M., Ghaffar, A., and Nisar, K.S.: Image interpolation using a rational bi-cubic Ball. Mathematics 7, 1045 (2019). https://doi.org/10. 3390/math7111045

Cost-Benefit Analysis of Sustainable Solar-Powered Workplace Electric Vehicle Charging Station Kameswara Satya Prakash Oruganti, Chockalingam Aravind Vaithilingam, Gowthamraj Rajendran, and Agileswari Ramasamy

Abstract In the present scenario, carbon emissions from the transportation sector and power sector create alarming situations of a drastic rise in air pollution. Due to the overwhelming response for the Solar photovoltaic system (SPS) and Electric Vehicles (EVs), the context of raise in Solar-powered electric vehicle charging station (SPEVCS) is becoming more favourable. Adopting to EVs has created a paradigm shift for both sectors. Due to the increased growth of the EVs, there are possibilities of global doubling of renewable energy resources. The deployment of SPEVCS for EVs along with information communication technology (ICT) will maximise the technical features and minimise the operation costs. The potential combination of solar energy with rapid charging systems makes the low voltage distribution network free from overloading of distribution transformers, overloading of network feeders. It avoids various impacts caused by the charging of EVs. In this chapter, the design and sizing of SPEVCS for workplace charging system along with performance and cost analysis of the SPECS is carried out. Based on the analysis of the results, energy can be produced at 0.58 MYR/kWh, and 4890 tons of CO2 emissions reduction.

K. S. P. Oruganti · C. A. Vaithilingam (B) · G. Rajendran Faculty of Innovation and Technology, Taylor’s University, 1, Jalan Taylor’s, 47500 Selangor, Malaysia e-mail: [email protected] K. S. P. Oruganti e-mail: [email protected]; [email protected] G. Rajendran e-mail: [email protected] A. Ramasamy Institute of Power Engineering (IPE), Universiti Tenaga Nasional, Kajang, 43000 Selangor, Malaysia e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 S. A. Abdul Karim (ed.), Theoretical, Modelling and Numerical Simulations Toward Industry 4.0, Studies in Systems, Decision and Control 319, https://doi.org/10.1007/978-981-15-8987-4_5

71

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1 Introduction Fossil fuel vehicles have come with many disadvantages such as greenhouse gas emission, and global warming effects. The fossil fuel is non-renewable and nonsustainable and also the demand for fossil fuel is increasing every year [1]. Due to the disadvantages of fossil-fuel vehicle, the different alternatives have come with new technologies and one among them is electric vehicles. The electric vehicles have more attention as a suitable replacement for fossil fuel vehicles [1] which has many advantages such as energy conservation, environmental protection, and efficiency [2]. However, different factors have impacts on the public to shift toward electric vehicles instead of preferring fossil fuel vehicles. In [3], it has been seen that around 18.1% of the people are ready to replace their fossil fuel vehicles to an electric vehicle. However, the design and deployment of the proper charging station for electric vehicles are one of the main problems associated with electric vehicles. The adoption of electric vehicles diminishes due to the unavailability of proper charging stations [4]. So, it is necessary to improve the public acceptance of the electric vehicles by designing an electric vehicle charging stations with enough parking spot [5]. The deployment of electric vehicle charging stations required billions of dollars [6] and needs to provide explicit payback assurances. The required payback assurance can be provided by extended deployment of EVs within the nation, increasing the price of fossil fuel, reducing the electricity price, encouraging to charge at night for free, financial incentives for the EV purchasers, and infrastructure investors [7]. However, integrating EV charging station with renewable power system [8] would be able to help in reducing the burden associated with the payback assurances. Moreover, installing solar PV on the roofs of existing petrol stations can solve the anxiety problem and reduce the initial cost of building new buildings for public charging stations [9]. So, based on the above discussion, installing solar PV on the roofs of the significant number of existing petrol stations associated with charging stations would increase the deployment of charging stations, and increase the penetration of the EVs and solar PVs [10–11]. The charging stations have classified into three types such as level 1 (slow) charging stations, level 2 (medium) charging stations, and DC fast-charging stations [12–15]. The input for the electric vehicle charging stations are from renewable or non-renewable sources depends on the availability. The non-renewable sources like thermal power, nuclear power is supplied by the grid, which has many disadvantages like air pollution and water pollution. So, non-renewable energies are needed to be replaced by renewable energies [16]. There are different literature studies on the technical point of view [17–23] and economical studies [24–31], were conducted on the design and sizing of solar PV based electric vehicle charging stations. However, very few studies [20, 26, 28, 32, 33] were conducted on the coupling of solar PV-Grid-Battery-EV systems. In this chapter, the typical workspace solar photovoltaic electric vehicle charging station is designed using Helioscope (www.helioscope.com) and PVsyst (www.pvs

Cost-Benefit Analysis of Sustainable Solar-Powered …

73

yst.com) for the analysis. The two simulation tools are equal designing tools whereas, the primary difference between the helioscope and PVsyst is helioscope calculates the system behaviour at the module level, and PVsyst calculates the system behaviour at the array level [34]. The following section deals with the methodology, result and discussions and conclusion.

2 Methodology In this section, a complete methodology to design the solar carport is discussed and analysed. PVsyst and Helioscope are used to design and to size and the governing equations involved in the following design is elucidated. The main space utilised for a carport is Taylor’s university open car parking. Based on the available space, the helioscope is used to design the solar arrays required to cover the maximum car park and applied two types of array arrangement to maximize the power production. The two types of module mounting are portrait or landscape mode, depends on the available space the module mounting is adopted. The load acting on the system is purely electric vehicles loads as this carport is designed to supply ample energy for EV charging. The system consists of 874 modules in which 19 modules are connected in series, and 46 strings are connected in parallel which contributes to supply 395.5 MWh/year with an average solar array efficiency of 13.78% including of all losses. The solar array is connected to the five solar inverters of 50 kW each with input nominal Maximum Power Point voltage of 600 V and output maximum AC of 80 A and maximum efficiency of 98.6%. The generated output from the inverter can be connected to point of common coupling (PCC) where from there it can be connected to the grid to feed energy or can connect to the EV chargers like ASEA Brown Boveri (ABB’s) Terra 54 CT [35] as charging equipment. The solar carport is capable of producing 375 MWh/year and on an average of 31.25 MWh/month, which is approximately 1041 kWh/day. Based on the report by InsideEVs [36–37], and battery electric vehicle (BEV) battery size ranging from 17.6 kWh to 100 kWh of “smart EQ fortwo coupe” to “Tesla Model S Long-range” respectively. The energy produced by the system is catered to charge a total of ten numbers of Tesla model S completely. Simplified schematic of the SPEVCS is shown in Fig. 1. The system consists of PV array; Grid-connected inverter and DC fast chargers to charge EVs. The proposed system can carry out two operations which are charging of electric vehicles and feeding the energy to the grid. The proposed layout of the solar carport is shown in Fig. 2 and from the design and sizing in helioscope, the proposed system can be approximate of 280kWp. Layout for the shadow analysis is carried out in helioscope with the typical south-facing view and further the shadow analysis is carried out for both south-western view and south-eastern view as shown in Figs. 3, 4 and 5. The solar path at the designated latitude and longitude with an altitude of 19 m shows the sun’s path for the entire year is shown in Fig. 6, based on the displayed path solar carport is designed to avoid losses due to the shading.

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K. S. P. Oruganti et al. Power electronic Control Source PV Array

Grid Source Inverter (AC)

Chopper-Inverter (DC-DC-AC)

Load Demand

Consumer

Fig. 1 Simplified schematic of SPEVCS

Fig. 2 Layout of the proposed SPEVCS

3 Modelling of the Proposed System 3.1 PV Array Output The modelling of the solar photovoltaic system was carried out by considering the work presented in the article [38] PV array output can be determined from the Eq. (1). Ppv = SI.η.A

(1)

Cost-Benefit Analysis of Sustainable Solar-Powered …

Fig. 3 Layout for the shadow analysis of proposed SPEVCS

Fig. 4 Shading analysis from south-western angle

75

76

Fig. 5 Shading analysis from the south-eastern angle

Fig. 6 Solar paths at Lat 3.1° N, Longitude 101.6° E, altitude 19 m

K. S. P. Oruganti et al.

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77

where SI is the annual solar insolation in kWh/m2 , η is the efficiency of the solar panel module and A is the surface area of the solar array. The effect of temperature can be considered by the Eq. (2) where ηt is the temperature de-rating effect. As we know, solar panels need to function in the appropriate temperature and if the temperature increases, efficiency decreases, which in turn reduces the power output. ηt = 1 − [β.(TC − TS)]

(2)

The system output depends on the temperature and de-rating effect. The governing equation for solar carport output is changed to Eq. (3). Where W is the rated capacity of PV panel in kW, β is the temperature coefficient, Gc is the solar insolation in real-time condition, Gs is the solar insolation in standard test conditions, T c and T s are the temperatures of the PV cell at real-time and standard test conditions.    PPV = W.ηt . G C G S [1 + β(TC − TS )]

(3)

The solar carport is designed for the topographical location of Subang Jaya in the peninsular of Malaysia at Taylor’s University. Average clear sunshine at this location is three to four and a half hours per day which provide yearly irradiation on the collector plane was maximum of 1600 kWh/m2 . Free mounted fixed-tilt flat roof carport is considered in PVsyst for design and analysis with a tilt angle of 3°. Overall the system production for the entire year is given in Table 1 and shaded irradiance on the collector plane for all sections of a carport is given in Table 2. System losses inclusive of module array mismatch, ohmic wiring losses and global inverter losses are given in Table 3. PV array loss behaviour at Irradiance @800 W/m2 , incidence angle at 40°, ambient temperature @20°, wind velocity of 3 m/s is shown in Fig. 7. Sizing of PV array and inverter modules in PVsyst is shown in Fig. 8. The IV characteristics for various irradiation and various temperature and PV characteristics curve is shown in Figs. 9, 10, and 11.

3.2 Inverter The inverters are the main system blocks to convert system-generated DC power to AC power. It carries out majorly two functions; one is to convert DC to AC and other is maximum power point tracking (MPPT). Inversion efficiency can be calculated by simple linear interpolation between the adjacent points is used. If P1 is the nearest power point below the target power, and P2 is the nearest point above the target power and the extrapolated efficiency at the given voltage for power P1 and P2 is given by Eq. (4) and (5) where η is the efficiency at a given voltage level and power point below the target power.  ηv, p1 = ηvmin , p1 + (v − v1 )

ηv2 , p1 − ηv1 , p1 v2 − v1

 (4)

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Table 1 Monthly production of system Balances and main results Horizontal global irradiation (kWh/m2 )

Ambient temperature (°C)

Global incident in collectors’ plane (kWh/m2 )

Effective energy at the output of the array (kWh)

Energy injected into grid (kWh)

January

129.4

27.25

131.6

30,893

30,441

February

131.3

27.73

133

31,035

30,561

March

150.3

28.07

150.9

35,122

34,583

April

140.2

27.57

139.4

32,669

32,181

May

142.5

28.58

140.6

32,858

32,373

June

131

27.81

128.8

30,320

29,881

July

133.3

27.83

131.5

30,811

30,351

August

133.8

27.79

132.6

31,185

30,730

September

131.2

27.2

131.1

30,786

30,326

October

135.5

27.48

136.5

32,020

31,540

November

119.9

26.69

121.7

28,844

28,430

December

118.9

27.17

121.1

28,503

28,089

Year

1597.3

27.6

1598.9

375,046

369,487

Table 2 Shaded irradiance on modules Description

Tilt

Azimuth

CP 1

3.0°

180.0°

62

19.8

1,601.1

CP 2

3.0°

180.0°

62

19.8

1,601.1

CP 3

3.0°

180.0°

64

20.5

1,601.1

CP 4

3.0°

180.0°

64

20.5

1,601.1

CP 5

3.0°

180.0°

44

14.1

1,601.1

CP 6

3.0°

180.0°

61

19.5

1,601.1

CP 7

3.0°

180.0°

61

19.5

1,599.8

CP 8

3.0°

180.0°

232

74.2

1,600.4

CP 9

3.0°

180.0°

232

74.2

1,601.1

882

282.2

1,600.8

Totals, weighted by kWp

Modules

Nameplate (kWp)

 ηv, p2 = ηvmin , p2 + (v − v1 )

ηv2 , p2 − ηv1 , p2 v2 − v1

Shaded irradiance (kWh/m2)

 (5)

Cost-Benefit Analysis of Sustainable Solar-Powered …

79

Table 3 Detailed system losses Detailed system losses Module array mismatch loss (kWh)

Ohmic wiring loss (kWh)

January

314.5

240

30,893

February

316.3

277

31,035

March

358

318

35,122

April

332.7

273.4

32,669

May

334.5

262.4

32,858

June

308.6

227.8

30,320

July

313.7

247.5

30,811

August

317.4

240.2

31,185

September

313.5

251.6

30,786

October

326.1

264.4

32,020

November

293.5

210.9

28,844

December

290

210

28,503

Year

3818.9

3023.3

375,046

Fig. 7 PV array behaviour for each loss effect

Array virtual energy at MPP (kWh)

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K. S. P. Oruganti et al.

Fig. 8 Sizing of PV and inverter modules

Fig. 9 I-V characteristics curves at various incident irradiance for canadian solar Inc. CS6X-320P

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81

Fig. 10 I-V characteristics curves at various temperatures for canadian solar Inc. CS6X-320P

Fig. 11 P–V characteristics curves at 45°C for canadian solar Inc. CS6X-320P

The nearest voltage level above and below the inverter DC voltage, extrapolate the efficiency based on the current DC power and extrapolate the efficiency at the actual DC voltage level using the voltage and efficiency points determined in Eq. (6).  ηv, p = ηv, p1 + ( p − p1 )

ηv, p2 − ηv, p1 p2 − p1

 (6)

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When the system components are connected as per the required array power, all modules and power optimizers are connected in series and parallel to operate at respective same current levels and voltage levels. The output voltage for a series string V load is the sum of the individual voltage contributions (V i ) of each module, less than the total line losses based on the current I i on the string length of L and specific resistivity of the conductor which is shown in Eq. (7). When components are connected in parallel, all are constrained to operate at specific voltage levels. The voltage V i for each component, i can be calculated based on the voltage of the next is given by (L i – L nearest component (V i – 1 ). The distance between the components n i – 1 ) and the total current on the segment is given in Eq. (8) by j=1 I j . Detailed inverter losses are calculated based on the above formulations are given in Table 4. Vload =

n 

  vi − I · 2L p

(7)

i=1

Vi = Vi−1 + 2(L i − L i−1 )ρ

n 

Ij

(8)

j=1

Table 4 Detailed inverter losses

Month

Available energy at inverter output

Operating inverter efficiency

MWh

Global inverter losses

Inverter loss during operation

%

MWh

MWh

January

30.44

98.54

0.452

0.447

February

30.56

98.48

0.473

0.468

March

34.58

98.47

0.538

0.533

April

32.18

98.51

0.487

0.482

May

32.37

98.52

0.485

0.48

June

29.88

98.55

0.439

0.434

July

30.35

98.51

0.46

0.455

August

30.73

98.54

0.455

0.45

September

30.33

98.51

0.459

0.455

October

31.54

98.5

0.48

0.475

November

28.43

98.57

0.413

0.408

December

28.09

98.55

0.414

0.409

369.49

98.52

5.556

5.499

Year

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3.3 Optimizers Power optimizers operate between modules or array of modules which transforms the module output voltage and current allowing a module to be driven at an operating power point which is independent of other modules or array which in-turn operate at or near its peak output. System-level algorithms can also adjust the power transformation for all optimizers as a group to perform at maximum output. Optimizer efficiency is calculated based on the bucking ratio x, which is defined in Eq. (9) where V out is the effective output voltage, and V in is the maximum power point voltage of the attached module. Based on the selected bucking ratio, efficiency is then calculated using a linear extrapolation which is shown in Eq. (10). The transformations for current in response to voltage bucking is given in Eq. (11) and the output power is given in Eq. (12). The summary of the entire solar carport system design is given in Table 5, consists of system orientation parameters like type of the tilt and azimuth angle, nominal rated power for one sub-array and inverters. x= ηx = ηxmin



(9) 

ηxmax − ηxmin + (xmax − xmin ) xmax − xmin Iout =

Pout =

Vout Vin



Iin · η(x) x

 Iin η(x) · (Vin · x) = Iin Vin η(x) x

(10) (11) (12)

Table 5 Summary of system design System orientation parameters Field type

Fixed tilted plane

Plane tilt/azimuth

3°/0°

Compatibility between system definitions Full system, orientation

Tilt/Azimut = 3°/0°

1 Sub-array

PNom = 280 kWp, modules area = 1677 m2

System parameters PV modules

46 strings of 19 modules in series, 874 total

Pnom = 320 Wp

Pnom array = 280 kWp, Area = 1677 m2

Inverters (50.0 kWac)

10 MPPT inputs, Total 250 kWac, PNom Ratio = 1.12

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4 Summary The proposed system consists of 280 kWp solar power plant which able to produce on an average of 3.6 kWh/kWp. The power generated from the SPV is connected to the five numbers of central inverters of 50 kVA each. The converted AC power then fed to the commercially available charging stations. In the proposed SPEVCS, the energy produced per day is 1015 kWh. The energy required to charge a typical electric vehicle which varies from 151 to 240 Wh/km for Tesla 3 and Tesla S 85 respectively [39]. From the energy required per kilometre, the energy supplied from the SPEVCS can provide a minimum of 6675 km drive. By using solar PV, it can assist in reducing the consumption from the grid (TNB) and generate its renewable clean energy. With the solar-powered charging station on the rise, it is feasible for Taylor’s University to apply this new technology so that not only the generated energy from solar PV will be given back to TNB but it can also be used to power up the charging stations which can reduce the well-to-wheel emission and able to generate energy at 0.58 MYR/kWh and reduce 4890 tons CO2 emissions. Acknowledgements This work was supported by Taylor’s University through its TAYLOR’S PhD SCHOLARSHIP Programme through grant TUFR/2017/001/01.

References 1. Mehrjerdi, H., Rakhshani, E.: Vehicle-to-grid technology for cost reduction and uncertainty management integrated with solar power. J. Clean. Prod. 229, 463–469 (2019) 2. Zhang, J., et al.: Design scheme for fast charging station for electric vehicles with distributed photovoltaic power generation. Glob. Energy Interconnect. 2(2), 150–159 (2019) 3. Wang, N., Tang, L., Pan, H.: Analysis of public acceptance of electric vehicles: an empirical study in Shanghai. Technol. Forecast. Soc. Change 126, 284–291 (2018) 4. Bonges, H.A., III., Lusk, A.C.: Addressing electric vehicle (EV) sales and range anxiety through parking layout, policy and regulation. Transp. Res. Part a Policy Pract. 83, 63–73 (2016) 5. Mehrjerdi, H.: Off-grid solar powered charging station for electric and hydrogen vehicles including fuel cell and hydrogen storage. Int. J. Hydrogen Energy 44(23), 11574–11583 (2019) 6. Michael, M., Groll, X., Mosquet, D., Rizoulis, Sticher, G.: The comeback of the electric car. How Real, How Soon, What Must Happen Next. BCG Bost. Consult. Gr. (2009) 7. Kley, F., Lerch, C., Dallinger, D.: New business models for electric cars—a holistic approach. Energy Policy 39(6), 3392–3403 (2011) 8. Silverstein, K.: Solar-powered electric vehicle charging stations are just around the corner. Available https://www.forbes.com/sites/kensilverstein/2020/02/10/solar-powered-electric-veh icle-charging-stations-are-just-around-the-corner/#8ce771c320f6 (2020). Accessed 29 Feb 2020 9. Munnings, C.: Solar -powered electric vehicle charging. Available https://www.csiro.au/en/Res earch/EF/Areas/Grids-and-storage/Intelligent-systems/Solar-charging (2019). Accessed 29 Feb 2020 10. Becker, F.G.: PV for your EV: solar tech powers electric cars through summer. Available https://www.csiro.au/en/News/News-releases/2019/PV-for-your-EV-solar-tech-pow ers-electric-cars-through-summer (2019). Accessed 29 Feb 2020

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11. Wang, L.: Beautiful, solar-powered EV charging stations promise to charge a vehicle in 15 minutes. Available https://inhabitat.com/beautiful-solar-powered-ev-charging-stations-pro mise-to-charge-a-vehicle-in-15-minutes/ (2019). Accessed 29 Feb 2020 12. Yıldız, B., Olcaytu, E., Sen, ¸ A.: The urban recharging infrastructure design problem with stochastic demands and capacitated charging stations. Transp. Res. Part B Methodol. 119, 22–44 (2019) 13. Domínguez-Navarro, J.A., Dufo-López, R., Yusta-Loyo, J.M., Artal-Sevil, J.S., BernalAgustín, J.L.: Design of an electric vehicle fast-charging station with integration of renewable energy and storage systems. Int. J. Electr. Power Energy Syst. 105, 46–58 (2019) 14. Zhang, Y., et al.: Modeling of fast charging station equipped with energy storage. Glob. Energy Interconnect. 1(2), 145–152 (2018) 15. Mike, C.: ABB to build world’s largest nationwide network of EV fast-charging stations in the Netherlands. Available https://new.abb.com/news/detail/13781/abb-to-build-worlds-lar gest-nationwide-network-of-ev-fast-charging-stations-in-the-netherlands (2013). Accessed 29 Feb 2020 16. Mehrjerdi, H., Hemmati, R.: Coordination of vehicle-to-home and renewable capacity resources for energy management in resilience and self-healing building. Renew. Energy 146, 568–579 (2020) 17. Kobayashi, Y., Kiyama, N., Aoshima, H., Kashiyama, M.: A route search method for electric vehicles in consideration of range and locations of charging stations. IEEE Intelligent Vehicles Symposium (IV) 2011, 920–925 (2011) 18. Lee, J., Park, G.L.: Dual battery management for renewable energy integration in EV charging stations. Neurocomputing 148, 181–186 (2015) 19. Locment, F., Sechilariu, M., Forgez, C.: Electric vehicle charging system with PV gridconnected configuration. In: 2010 IEEE Vehicle Power and Propulsion Conference, pp. 1–6 (2010) 20. Lukic, S., Mulhall, P., Emadi, A.: Energy autonomous solar/ battery auto rickshaw. J. Asian Electr. Veh. 6(2), 1135–1143 (2008) 21. Mohamed, A., Salehi, V., Ma, T., Mohammed, O.: Real-time energy management algorithm for plug-in hybrid electric vehicle charging parks involving sustainable energy. IEEE Trans. Sustain. Energy 5(2), 577–586 (2013) 22. Chen, Q. Liu, N. Wang, C., Zhang, J.: Optimal power utilizing strategy for PV-based EV charging stations considering Real-time price. In 2014 IEEE Conference and Expo Transportation Electrification Asia-Pacific (ITEC Asia-Pacific), pp. 1–6 (2014) 23. Satya Prakash Oruganti, K., Aravind Vaithilingam, C. Rajendran, G., R.A.: Design and sizing of mobile solar photovoltaic power plant to support rapid charging for electric vehicles. Energies. 12(18), 2019 24. Bayram, I.S., Michailidis, G., Devetsikiotis, M., Bhattacharya, S., Chakrabortty, A., Granelli, F.: Local energy storage sizing in plug-in hybrid electric vehicle charging stations under blocking probability constraints. In: 2011 IEEE International Conference on Smart Grid Communications (SmartGridComm), pp. 78–83 (2011) 25. Guo, Y., Hu, J., Su, W.: Stochastic optimization for economic operation of plug-in electric vehicle charging stations at a municipal parking deck integrated with on-site renewable energy generation. In: 2014 IEEE Transportation Electrification Conference and Expo (ITEC), pp. 1–6. (2014) 26. Li, J., Yang, B., Xu, Y., Chen, C., Guan, X., Zhang, W.: Scheduling of electric vehicle charging request and power allocation at charging stations with renewable energy. In: Proceedings of the 33rd Chinese Control Conference, pp. 7066–7071 (2014) 27. Miskovski D., Williamson, S.S.: Modeling and simulation of a photovoltaic (PV) based inductive power transfer electric vehicle public charging station. In: 2013 IEEE Transportation Electrification Conference and Expo (ITEC), pp. 1–6. (2013) 28. Chen, Q., Liu, N., Lu, X., Zhang, J.: A heuristic charging strategy for real-time operation of PV-based charging station for electric vehicles. IEEE Innov. Smart Grid Technol.-Asia (ISGT ASIA) 2014, 465–469 (2014)

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29. Tulpule, P., Marano, V., Yurkovich, S., Rizzoni, G.: Energy economic analysis of PV based charging station at workplace parking garage. IEEE EnergyTech 2011, 1–6 (2011) 30. Lee, W., Xiang, L., Schober, R., Wong, V.W.S.: Analysis of the behavior of electric vehicle charging stations with renewable generations. In: 2013 IEEE International Conference on Smart Grid Communications (SmartGridComm), pp. 145–150. (2013) 31. Li, X., Lopes, L.A.C. Williamson, S.S.: On the suitability of plug-in hybrid electric vehicle (PHEV) charging infrastructures based on wind and solar energy: In: 2009 IEEE Power & Energy Society General Meeting, pp. 1–8. (2009) 32. Rasin, Z., Rahman, M.F.: Grid-connected quasi-Z-source PV inverter for electricvehicle charging station. In: 2013 International Conference on Renewable Energy Research and Applications (ICRERA), pp. 627–632. (2013) 33. Zhao, H., Burke, A.: An intelligent solar powered battery buffered EV charging station with solar electricity forecasting and EV charging load projection functions. In: 2014 IEEE Int. Electr. Veh. Conf. IEVC 2014, pp. 4–11 (2014) 34. BEW Engineering: Performance Model Evaluation (2002) 35. ABB: Smarter Mobility Terra 54 multi-standard DC charging station pp. 53–54, (2019) 36. Kane, M.: Compare EVs: guide to range, specs, pricing & more. Eric Loveday. Available https://insideevs.com/reviews/344001/compare-evs/ (2019) Accessed 05 Mar 2020 37. Robinson, J., Brase, G., Griswold, W., Jackson, C., Erickson, L.: Business models for solar powered charging stations to develop infrastructure for electric vehicles. Sustain. 6(10), 7358– 7387 (2014) 38. Karmaker, A.K., Ahmed, M.R., Hossain, M.A., Sikder, M.M.: Feasibility assessment & design of hybrid renewable energy based electric vehicle charging station in Bangladesh. Sustain. Cities Soc. 39(February), 189–202 (2018) 39. Fosdick, R.J.: Electric Vehicle. Automotive Industries AI. Available https://batteryuniversity. com/learn/article/electric_vehicle_ev (1977). Accessed 23 Mar 2020

Positivity-Preserving Interpolation Using Rational Quartic Spline Functions Samsul Ariffin Abdul Karim

and Van Thien Nguyen

Abstract A curve interpolation scheme based on generalized Ball basis functions to visualize scientific data has been developed. The scheme uses piecewise rational quartic Said-Ball functions with linear denominator. The parameter, in the description of the rational interpolant, has been constrained to preserve the shape of the data. We examine the positivity-preserving properties of this rational interpolant to a given data set. The degree of smoothness attained is C 1 . Numerical results show that the proposed scheme works well for all tested data sets. Keywords Visualization · Interpolation · Rational Said-Ball · Positivity-preserving

1 Introduction Positivity is an important geometric property. There are many physical situations where entities only a meaning when their values are positive. For example, in a probability distribution the representation is always positive or when dealing with samples of populations, the data are always positive. Other application areas include monthly rainfall distribution, levels of gas discharge in certain chemical reactions, volume as well as wind energy, etc. [1–3]. This chapter will examine shape preservation of positive data sets. Many authors worked in this area, for examples in [1–9] the authors discussed the positivity S. A. A. Karim (B) Fundamental and Applied Sciences Department and Centre for Systems Engineering (CSE), Institute of Autonomous System, Universiti Teknologi PETRONAS, Bandar Seri Iskandar, 32610 Seri Iskandar, Perak Darul Ridzuan, Malaysia e-mail: [email protected] V. T. Nguyen FPT University, Education zone, Hoa Lac high tech Park, Km 29 Thang Long highway, Thach That ward, Hanoi, Vietnam e-mail: [email protected]

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 S. A. Abdul Karim (ed.), Theoretical, Modelling and Numerical Simulations Toward Industry 4.0, Studies in Systems, Decision and Control 319, https://doi.org/10.1007/978-981-15-8987-4_6

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preserving by using a rational spline and polynomial splinep. Motivated by the work of Wang and Tan [10], Karim [11] and Karim and Piah [12] has introduced monotonicity and convexity preserving scheme using rational quartic generalized Ball with quartic Said-Ball function as numerator and linear denominator. Algorithms presented in [13–19] provide fast computation for this curve. Karim [20] proposed rational cubic Ball function (cubic numerator and cubic denominator) to generate positive interpolating curves and Karim et al. [21] proposed the positivity preserving using GC 1 rational quartic spline. In this chapter, the rational quartic generalized Ball (Said-Ball) with linear denominator from Karim [10] will be used to visualize positive data. The method in this chapter has several advantageous features. The curve which preserves the positivity of data has an explicit representation and is C 1 continuous. No further restriction is needed to be exerted onto the data and the method is local. Numerical examples show that the curves obtained are very smooth with inherent feature (namely positivity) of given data. The remainder of the chapter is organized as follows; Sect. 2 reviews the rational generalized Ball interpolant (quartic Said-Ball as numerator and linear denominator). The derivative parameter estimation will be discussed in Sect. 3. The positivity problem is discussed in Sect. 4 for the generation positivity-preserving. The numerical results will be presented in Sect. 5 with comparison to Karim [20] method. Finally, Sect. 6 concludes the chapter.

2 Rational Quartic Said-Ball Interpolant In this section, the rational quartic Said-Ball interpolant with linear denominator will be discussed. More detail can be found Karim [11]. Suppose {(xi , f i ), i = 1, ..., n} is a given set of data points, where x1 < x2 < fi ) i) and a local variable, θ = (x−x , hence ... < xn . Let h i = xi+1 − xi , i = ( fi+1h− hi i 0 ≤ θ ≤ 1.   For x ∈ xi , xi+1 , i = 1, 2, . . . , n − 1, s(x) = s(xi + h i θ ) ≡ Si (θ ) =

Pi (θ ) , Q i (θ )

where Pi (θ ) = Vi0 (1 − θ )4 + Vi1 (1 − θ )3 θ + Vi2 (1 − θ )2 θ 2 + Vi3 θ 3 (1 − θ ) + Vi4 θ 4 Q i (θ ) = αi (1 − θ) + βi θ

(1)

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89

with Vi0 = αi f i , Vi1 = (2αi + βi ) f i + αi h i di , Vi2 = 3αi f i+1 + 3βi f i , Vi3 = (αi + 2βi ) f i+1 − βi h i di+1 , and Vi4 = βi f i+1 . Meanwhile αi , βi > 0 are shape parameters, and di denotes a given first derivative value at knot xi . The rational quartic interpolant (1) satisfies the following properties: s(xi ) = f i , s(xi+1 ) = f i+1 , (1)

s (xi ) = di , s (1) (xi+1 ) = di+1 . thus, s(x) is C 1 in [x1 , xn ]. Note: Parameters αi , βi are used for s(x) to be symmetric. By letting ei = can then be expressed in term of a single independent parameter, ei > 0:

(2)

αi βi

, (1)

P1i (θ ) = Vi0 (1 − θ )4 + V1∗ (1 − θ )3 θ + V2∗ (1 − θ )2 θ 2 + V3∗ θ 3 (1 − θ ) + Vi4 θ 4 Q 1i (θ ) = ei (1 − θ ) + θ. with V1∗ = (2ei + 1) f i + ei h i di , V2∗ = 3ei f i+1 + 3 f i , V3∗ = (ei + 2) f i+1 − h i di+1 . We can rewrite Eq. (1) as Eq. (3), given below: s(x) = s(xi + h i θ ) ≡ Si (θ ) =

P1i (θ ) . Q 1i (θ )

(3)

This form of the rational interpolant is used throughout the remainder of this chapter. Obviously when ei = 1 (or when αi = βi ) the rational quartic interpolant in (1) will reduce to the non-rational quartic Said-Ball polynomial interpolation [17].

3 Determination of Derivatives In most applications, the derivative values are not given, and it must be determined either from the data or through some other mean. In this chapter, the derivative values will be computed from the given data in such a way with the C 1 smoothness of the interpolant (3). There exist various methods to determine the derivative values such as arithmetic mean method, mean geometric method, mean harmonic method, Butland’s method etc. [22, 23]. In this chapter the arithmetic mean method will be utilized.

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The arithmetic mean method (AMM) is based from the three-point difference approximation and on arithmetic manipulation. The method is given as follow: At the end points x1 and xn .  d1 = 1 + (1 − 2 )

 h1 , h1 + h2

(4)

and  dn = n−1 + (n−1 − n−2 )

 h n−1 . h n−1 + h n−2

(5)

At interior points, xi , i = 2, ..., n − 1, the values of di are given as  di =

0,

h i−1 i +h i i−1 , h i−1 +h i

if i−1 = 0 or i = 0 otherwise i = 2, 3, . . . , n − 1.

(6)

4 Positivity-Preserving Rational Said-Ball Interpolant The rational quartic Said-Ball interpolant with linear denominator described in Sect. 2, has some deficiencies as far as the shape preserving positivity issue is concerned. For example, the rational quartic Said-Ball of Sect. 2 does not preserve the shape of the positive data given in Tables 1 and 2, respectively. Figures 1 and 2 show clearly that, the proposed rational quartic interpolant fail to preserve the positivity everywhere along the interval [3,7] and [3,4] for data given in Tables 1 and 2, respectively. The ordinary cubic spline interpolation also does not guarantee to preserve the positivity of the data sets (Figs. 3 and 4). Along the whole interval, both rational quartic Said-Ball interpolation and cubic splines gives some negatives Table 1 A positive data from [9] i

1

2

3

4

5

6

7

xi

2

3

7

8

9

13

14

fi

10

2

3

7

2

3

10

Table 2 A positive data from [21] i

1

2

3

4

5

6

xi

1

2

3

4

5

6

fi

0.058

1.177

0.070

0.923

3.270

0.428

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Fig. 1 Quartic Said-Ball interpolation for data in Table 1 with ei = 1.

Fig. 2 Quartic Said-Ball interpolation for data in Table 2 with ei = 1.

values. Now, the positivity-preserving property of rational interpolant to a given positive (or correspondingly negative) data set will be examined in detail. It needs to impose conditions on a rational quartic defined in (3) to preserve positivity of a data set. Assume a strictly positive data set is given

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Fig. 3 Cubic Spline interpolation for data in Table 1

Fig. 4 Cubic Spline interpolation for data in Table 2

{(x1 , f 1 ), (x2 , f 2 ), ..., (xn , f n )} where x1 < x2 < ... < xn ,

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and f i > 0: i = 1, 2, . . . , n.

(7)

Since the shape parameter ei is positive, the interpolant (3) will be positive if the quartic polynomial (numerator) is positive since the denominator, Q 1i (θ ) in (3) is always positive. Determination of suitable value of ei is significant to make sure the rational interpolant (3) is positive on [x1 , xn ]. Now by extending the results from Hess and Schmidt [6], the sufficient condition for the quartic Said-Ball polynomial to be positive will be derived. Proposition 1 (Positivity of Quartic Said-Ball polynomial) For the strict inequality positive data in (7),P1i (θ ) > 0 if and only if:  P1i (0), P1i (1) ∈ R1

(8)

4 f i+1 −4ei f i . R1 = (a, b): a > ,b < hi hi

(9)



where

Now, by differentiate P1i (θ ) w.r.t. θ , it can easily be show that P1i (0) =

−3ei f i + (2ei + 1) f i + ei h i di hi

P1i (1) =

3 f i+1 − (ei + 2) f i+1 − h i di+1 hi

and

  Now from Proposition 1, it can be deduced that P1i (0), P1i (1) ∈ R1 if and only if P1i (0) =

−4ei f i −3ei f i + (2ei + 1) f i + ei h i di > hi hi

(10)

4 f i+1 3 f i+1 − (ei + 2) f i+1 + h i di+1 < hi hi

(11)

and P1i (1) =

The inequality (10) leads to the following relation:

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ei > −

fi 3 f i + h i di

(12)

The inequality (11) leads to the following restriction on the parameter ei : ei >

h i di+1 −3 f i+1

(13)

Equations (12) and (13) can be summarized as Proposition 2. Proposition 2 For a strictly positive data given in (7), the rationalquartic interpolant (defined over the interval [x1 , xn ]is positive in each subinterval xi , xi+1 , i = 1, 2, . . . , n − 1,if the following sufficient conditions are satisfied: ei = li + Max 0, −

fi h i di+1 , − 3 , li > 0. 3 f i + h i di f i+1

(14)

An algorithm to generate C 1 positivity-preserving curves using Proposition 2 is given below. Algorithm 1 Positivity-Preserving Interpolation n 1. Input the number of data points,n and data points {xi , f i }i=0 where all f i > 0. 2. For i = 1, 2, . . . , n, estimate di using arithmetic mean method (AMM) given in Sect. 3. 3. For i = 1, 2, . . . , n − 1 Construct the piecewise positive interpolating curves Si (x) using (3) by.

• Calculate h i and i • Calculate the shape parameter ei using (14) with suitable choice of li > 0. 4. Plot positive interpolating curve using (3).

5 Results and Discussion In order to illustrate the shape preserving interpolation by using rational quartic Said-Ball interpolation (quartic numerator and linear denominator), two sets of data were taken from Sarfraz et al. [9] and Karim et al. [21]. Both are positive data sets. The first derivatives are estimated by using AMM method. Mathematica Version 11 installed on Intel® Core™i5-8250U@ CPU 1.8 GHz is used to produce all graphical and numerical results. Figures 5 and 6 show the resultant shape preserving interpolation using the proposed rational quartic Said-Ball with linear denominator. From Fig. 6, we cannot see clearly whether on the interval [9, 13] the rational quartic Said-Ball spline are positive or not. By zooming on the interval [9, 13], we have noticed that from Fig. 7

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Fig. 5 Shape preserving intrepolation using rational quartic Said-Ball for data in Table 1

Fig. 6 Shape preserving interpolation using rational quartic Said-Ball for data in Table 2

that, the interpolating curves are positive on [9, 13], indeed it is positive everywhere on the whole interval [2, 14]. Meanwhile, Figs. 8 and 9 show the shape preserving interpolation using rational cubic Ball with cubic numerator and cubic denominator proposed by Karim [20]. From Figs. 5, 6, 8 and 9, it is clear that the proposed method works well and the results also are comparable with Karim [20]. One of the advantages of the proposed method is in terms of computation where rational quartic Said-Ball required less

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Fig. 7 The magnificent of shape preserving interpolation in Fig. 5 on interval [9, 13]

Fig. 8 Shape preserving intrepolation using rational cubic Ball from Karim [20] for data in Table 1

compared to the established method with cubic numerator and cubic denominator in Sarfraz et al. [2], Sarfraz [3] and rational cubic Ball with cubic denominator in Karim [20]. Thus, the proposed scheme can be used to interpolate thousands positive data sets i.e. big data sets.

6 Conclusions In this chapter the problem of positivity-preserving interpolation using rational quartic Said-Ball (with quartic numerator and linear denominator) with shape parameters ei > 0 are discussed. The sufficient condition in Proposition 2 will ensure that

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Fig. 9 Shape preserving intrepolation using rational cubic Ball from Karim [20] for data in Table 2

the rational quartic Said-Ball interpolant will positive on entire given interval i.e. everywhere. Furthermore, the proposed scheme is local, simple to use, requires few computation steps and the output is comparable to some existing works. An extension from this work can be done in other shape preserving properties such as monotonicity and convexity interpolation. Acknowledgements The authors would like to acknowledge Universiti Teknologi PETRONAS (UTP) and Ministry of Education (MOE), Malaysia for the financial support received in the form of a research grant: FRGS/1/2018/STG06/UTP/03/1/015MA0-020.

References 1. Hussain, M.Z., Sarfraz, M., Hussain, M.: Scientific data visualization with shape preserving rational cubic interpolation. Euro. J. Pure Appl. Math. 3(2), 194–212 (2010) 2. Sarfraz, M., Butt, S., Hussain, M.Z.: Visualization of shaped data by a rational cubic spline interpolation. Comput. Graph. 25, 833–845 (2001) 3. Sarfraz, M.: Visualization of positive and convex data by a rational cubic spline interpolation. Inf. Sci. 146(1–4), 239–254 (2002) 4. Hussain, M.Z., Sarfraz, M.: Positivity-Preserving interpolation of positive data by rational cubics. J. Comput. Appl. Math. 218, 446–458 (2008) 5. Butt, S., Brodlie. K.W.: Preserving positivity using piecewise cubic interpolation. Comput. Graph. 17(1), 55–64 (1993) 6. Hess, W., Schmidt, J.W.: Positive quartic, monotone quintic C 2 -spline interpolation in one and two dimensions. J. Comput. Appl. Math. 55(1), 51–67 (1994) 7. Dougherty, R.L., Edelman, A., Hyman, J.M.: Nonnegativity-, Monotonicity-, or convexitypreserving cubic and quintic hermite interpolation. Math. Comput. 52(186), 471–494 (1989)

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8. Sarfraz, M., Hussain, M.Z., Nisar, A.: Positive data modeling using spline function. Appl. Math. Comput. 216, 2036–2049 (2010) 9. Sarfraz, M., Hussain, M.Z., Chaudary, F.S.: Shape preserving cubic spline for data visualization. Comput. Graph. CAD/CAM 01, 185–193 (2005) 10. Wang, Q., Tan, J.: Rational quartic involving shape parameters. J. Inf. Comput. Sci. 1(1), 127–130 (2004) 11. Karim, S.A.A.: Rational generalized ball functions for monotonic and convex interpolating curves. In Malaysia master sciences theses, Universiti Sains Malaysia (2008) 12. Karim, S.A.A., Piah, A.R.M.: Rational generalized ball functions for convex interpolating curves. J. Qual. Measure. Anal. JQMA 5(1), 65–74 (2009) 13. Phien, H.N., Dejdumrong, N.: Efficient algorithms for Bézier curves. Comput. Aided Geometr. Des. 17, 247–250 (2000) 14. Delgado, J., Pena, J.M.: A shape preserving representation with and evaluation algorithm of linear complexity. Comput. Aided Geometr. Des. 20(1), 1–10 (2003) 15. Dejdumrong, N.: A shape preserving verification technique for parametric curves. In: Computer Graphics, Imaging and Visualization, Techniques and Applications, pp. 163–168. CGIV 2007 Bangkok, Thailand (2007) 16. Hu, S.M., Wang, G.Z., Jin, T.G.: Properties of two generalized ball curves. Comput. Aided Des. 28, 125–133 (1996) 17. Said, H.B.: Generalized Ball Curve at its recursive algorithm. ACM Trans. Graph. 8, 360–371 (1989) 18. Wang, G.J.: Ball curve of high degree and its geometric properties. Appl. Math. J. Chin. Univ. 2, 126–140 (1987) 19. Goodman, T.N.T., Said, H.B.: Shape preserving properties of the generalised Ball basis. Comput. Aided Geometr. Des. 8, 115–121 (1991) 20. Karim, S.A.A.: Rational cubic ball functions for positive interpolating curves. Far East J. Math. Sci. FJMS 82(2), 193–207 (2013) 21. Karim, S.A.A., Kong, V.P., Hashim, I.: Positivity preserving using rational quartic spline. AIP Conf. Proc. 1522, 518–525 (2013) 22. Delbourgo, R., Gregory, J.A.: The determination of derivative parameters for a monotonic rational quadratic interpolant. IMA J. Numer. Anal. 5, 397–406 (1985a) 23. Delbourgo, R., Gregory, J.A.: Shape preserving piecewise rational interpolation. SIAM J. Sci. And Statist. Comput. 6, 967–976 (1985b)

Decision Support Method for Agricultural Irrigation Scenarios Performance Using WEAP Model Saiful Azmi Husain and Nor Hamizah Mohd Rhyme

Abstract The Water Evaluation and Planning (WEAP) model was invented by the Stockholm Environment Institute (SEI) to assist planning and management issues related with the development of water resources. The WEAP model can be applied to agricultural site and other sectors; and it can also tackle a large scope of problems encompassing water demand analyses, water saving, optimizing available water distribution and cost-profit analyses [1]. In this study, the WEAP model is introduced to look at the performance of the effect of different agricultural irrigation scenarios on special type of rice variety, i.e. MRQ76, planted in Wasan padi field, situated at Brunei-Muara District, Brunei Darussalam. Specifically, we apply the WEAPMABIA model (based on soil-water balance approach) to perform evaluation of irrigation scheduling for three different rice growing seasons. We found that the WEAP model is a good model for providing the best irrigation scheduling strategy for optimum rice yield and efficient water management.

1 Introduction Irrigation scheduling is the process used by irrigation system managers to determine the correct frequency and duration of watering. It is convenient for irrigation system managers to know the amount of water consumed by plants that is required for irrigation scheduling in order to produce optimal yield. Studies has revealed that irrigation scheduling using water balance approach can save 15–35% of the water usually pumped without reducing production [2]. Also, recent study on profit efficiency in rice yields in Brunei Darussalam has reported that those farmers who have S. A. Husain (B) · N. H. M. Rhyme Faculty of Science, Universiti Brunei Darussalam, Jalan Tungku Link, Gadong BE1410, Brunei Darussalam e-mail: [email protected] N. H. M. Rhyme e-mail: [email protected]

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 S. A. Abdul Karim (ed.), Theoretical, Modelling and Numerical Simulations Toward Industry 4.0, Studies in Systems, Decision and Control 319, https://doi.org/10.1007/978-981-15-8987-4_7

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used proper irrigation contributed as one of the major factors that resulted in better rice yields [3]. But irrigation scheduling is usually difficult to be evaluated manually, without the support of computer programs, since it is a time-consuming and complex process [4]. In this study, the WEAP model is used as it can be applied to agricultural site and other sectors; and can also tackle a large scope of problems encompassing water demand analyses, water saving, optimizing available water distribution and cost-profit analyses [1]. The main objectives of the present studies are: (a) To perform evaluation of irrigation scheduling for 3 different rice growing seasons using the WEAP-MABIA model; (b) To provide the best irrigation scheduling strategy for optimum rice yield and efficient water management, during the three different rice-growing seasons. This chapter is organized as follows. In Sect. 7.1, we give some basic introduction of the importance of irrigation scheduling for efficient water management and optimum rice yields. Section 7.2 presents the literature review of the related work of using WEAP model as a decision support method to find for best irrigation strategy. Section 7.3 discuss the methodology to look at different irrigation schemes using WEAP model for special type of rice variety, i.e. MRQ76 planted in Wasan padi field, situated in Brunei-Muara district, Brunei Darussalam. Section 7.4 is dedicated for Results and Discussion. Finally, Conclusion and Future work is presented in Sect. 7.5.

2 Literature Review of Related Work of Using WEAP Model There exist many researches regarding the use of WEAP model as a decision support method to find the best irrigation scheduling strategy for optimum crop yield and efficient water management. For instance, a case study on the consequences of climate change on crop water usage was undertaken by Karamouz et al. [5], in which WEAP model was applied to project future irrigation demands and also to examine various management scenarios, such as expansion of future agricultural areas, the impacts of climate change on agricultural water demand and the advancement of irrigation efficiency. Bhatti and Patel [6] found out that by adopting WEAP model, deficit irrigation strategy was favourable in irrigation scheduling for cotton crop compared at different irrigation approaches. Bhatti [7] also assessed varying irrigation programmes by evaluating irrigation demands in real time circumstances, and found out that the best irrigation strategy was to utilize less amount of water for certain crops by implementing WEAP model.

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3 Methodology for Agricultural Irrigation Development Using WEAP Model The area of study is Wasan in Brunei-Muara District, Brunei Darussalam which lies between 4°47 N, 114°48 E. The rice-based cropping system was normally taken to be three times per year and in this study, the data was provided by the Department of Agriculture and Agrifood are only for growth stages of MRQ76 rice and length of days during its growing stages until maturity from 2017 until 2018. There are three seasons: season 1 from January 2017–April 2017, season 2 from June 2017– Sept 2017 and lastly, season 3 from Nov 2017–February 2018. Wasan has a low land area which is suitable for rice plantation and the soil type of this area is heavy cracking clay [8]. The Ministry of Industry and Primary Resources, Brunei Darussalam has informed that a new variety of locally produced rice called MRQ76 is now sold at local stores and supermarkets. The MRQ76 has been tested under a joint research project conducted by the Department of Agriculture and Agrifood and the Malaysian Agricultural Research and Development Institute. The MRQ76 was introduced in Malaysia in 2012 as a great quality rice variety that was chosen based on the fragrance and texture of the rice. According on the result of the taste evaluation by the department, they discovered that the quality of MRQ76 variety is as good as that of local rice namely “Laila” rice along with the fragrant rice imported from Thailand [9]. Figure 7.1 shows the comparison between MRQ76 rice and imported rice from “Khao Dawk Mali 105” (KDML105) variety. Observational meteorological data of sunshine hours, maximum and minimum temperature, and relative humidity, was sourced from Soil Science and Plant Nutrient Unit, Brunei Agriculture Research Centre. As for solar radiation, the data was taken from the NASA website for the Prediction of Worldwide Energy Resource (POWER) and wind speed data is obtained from Brunei Darussalam Meteorological Department (BDMD).

Fig. 7.1 Local grown rice variety, MRQ76 (on the left) [10] and KDML 105 rice sown in Thailand (on the right) [11]

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4 Results and Discussion The objectives of the study is to perform evaluation of irrigation scheduling for 3 different rice growing seasons (January 2017–February 2018) using the WEAPMABIA model and to provide the best irrigation scheduling strategy for optimum rice yield and efficient water management, during the three different rice-growing seasons. This will help the irrigation system managers to make decision taking into account various scenarios, in order to boost yield without causing stress moisture and also improve water use efficiency. Following WEAP-MABIA method, Evapotranspiration (ET) was also estimated for three different growing seasons to meet the irrigation requirement with demand. Daily weather parameters were used to calculate reference evapotranspiration using FAO-56 Penman-Monteith equation employed in MABIA method featured in WEAP, assimilating dual crop coefficient (Kcb ) techniques along with water balance model in the root zone to develop irrigation schemes under different condition [12]. The soil under study is saturated due to the necessity of paddy rice to be in flooding conditions, hence the initial depletion is considered as zero for the base year 2016 for calculating soil water balance. The potential yield of MRQ76 is 4230 kg/ha, according to the data provided by Department of Agriculture and Agrifood, Brunei Darussalam for the year 2013 [13]. The simulated irrigation scenarios using WEAP-MABIA model employed for MRQ76 rice in Wasan for three different seasons (season 1, 2 and 3) during 2017–2018 are shown on Table 7.2. IR 1 depicts deficit irrigation to encourage water savings, the irrigation amount is applied at 150% RAW. In scenario IR 2 represents full irrigation to avoid soil moisture stress, at varying amount of irrigation applied at 100% of RAW. Similarly, both scenarios IR 1 and IR 2 are model-based approach whereby varying irrigation depth is applied at 100% moisture depletion, administered at specific day during the Table 7.1 List of some literature review and its advantages of using WEAP model Literature review of using WEAP model Advantages of WEAP model (1) Karamouz et al. [5]

Able to project future irrigation demands and examine various management scenarios, such as expansion of future agricultural areas, the impacts of climate change on agricultural water demand and the advancement of irrigation efficiency.

(2) Bhatti and Patel [6]

Able to predict the best irrigation strategy in irrigation scheduling for cotton crop compared at different irrigation approaches. In this case, deficit irrigation (DI) was favourable for the study.

(3) Bhatti [7]

Able to assess varying irrigation programmes by evaluating irrigation demands in real time circumstances, and found out that the best irrigation strategy was to utilize less amount of water for certain crops.

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103

growing periods according to their soil moisture depletion simulated by the water balance model (as in Table 7.3). However, the difference in IR 1 from IR 2 lies in their trigger method. The irrigation under scenario IR 1 is triggered when the readily available water (RAW) reaches 150%, allowing slight water stress. Whereas in scenario IR 2, the irrigation is applied when RAW reaches 100%, allowing no stress. The development of IR 3 and IR 4 scenarios were based on conventional irrigation where the suggested daily crop water requirement for paddy rice in Wasan area was estimated as 10 mm [14]. Hence, irrigation depth for IR 3 should be 70 mm on a weekly basis and 100 mm would be applied at an interval of 20 days for scenario IR 4. To provide best irrigation strategy for different scenarios, the built-in daily soil moisture balance in WEAP is also quantified, for MRQ76 rice crop as well as the means of ETc actual, rainfall, irrigation, surface runoff, and flow to groundwater, yield, water use efficiency (WUE), and Irrigation water use efficiency (IWUE) were calculated out using WEAP for period from January 2017 until February 2018 for all four irrigation scenarios. The soil water balance in the root zone is provided in Table 7.2 Summary of irrigation scenario for rice in Wasan Irrigation scenario (IR)

Irrigation scheduling (Days After Sowing)

Irrigation amount (mm)

Description

IR 1

1, 7, 9, 12, 13, 14, 16, 18, 29, 41, 46, 77 and 104

Varying

Deficit Irrigation that is decided by model: 150% of Readily Available Water (RAW), depth applied at 100% Depletion. Stress is allowed

IR 2

1, 7, 9, 11, 13, 15, 17, 18, 28, 40, 44, 47, 75 and 102

Varying

Full Irrigation: No stress allowed. Soil moisture is allowed to deplete when 100% RAW for trigger with irrigation depth applied at 100% Depletion

IR 3

1–105

70 mm

Weekly Irrigation: Every 7 days, 70 mm of irrigation water is applied. Last Irrigation is halted 15 days before harvesting

IR 4

1–105

100 mm

Deficit irrigation where 100 mm of fixed irrigation depth is applied at fixed interval of 20 days. Irrigation is discontinued 15 days prior to harvest time

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Table 7.3 and Eq. 7.1 shows how the soil moisture depletion in Table 7.3 is calculated. Dr,i − Dr,i−1 = −Ri + S Ri − Ii − C Ri + E Tc,i + D Pi Dr,i − Dr,i−1 Ri S Ri Ii CRi ETc,i DPi

(7.1)

represents Change in soil-moisture depletion in the root zone represents Rainfall depth represents Surface runoff represents Irrigation depth represents Capillary rise represents Computed crop evapotranspiration represents Deep percolation due to gravity.

For instance, calculating the average of soil moisture depletion for IR 1 according to Eq. 7.1: 259 = −1483 + 294 − 184.33 − 0 + 542 + 1091, where the value of CRi is assumed to be zero. WUE is calculated by dividing crop yield produced per hectares (kg/ha) by ETc (mm), and IWUE is measured by dividing crop yield by irrigation water applied. Both average WUE and IWUE get larger as the irrigation water application increases due to effective use of soil water storage. In terms of average yield, WUE and IWUE, IR 1 and IR 2 are amongst the irrigation scenarios to achieve the highest average value. In irrigation scenario, IR 2 has the average values of ETc (554 mm), WUE (7.64 kg/ha/mm), and IWUE (22.39 kg/ha/mm) with the highest average yield of 4223 kg/ha. The second highest average yield of 4200 kg/ha was achieved by IR 1, but with higher values of WUE (7.77 kg/ha/mm) and IWUE (23.77 kg/ha/mm). The remaining scenarios IR 3 and IR 4 do not manage to produce higher yield. Furthermore their average seasonal irrigation water depths were 1530 mm (IR 3) and 781 mm (IR 4), which were much higher than those of IR 1 and IR 2, decreasing their IWUE. This shows that lower water consumption is sufficient enough for crop growth without experiencing yield reduction.

5 Conclusion and Future Work To conclude which irrigation strategy is the best, Table 7.4 shows the irrigation scenarios according to descending mean MRQ76 yields as shown in Table 7.4. Since IR 2 produced the highest yield with high WUE and IWUE (from Table 7.3), it should be chosen as the optimal strategy among the three. The remaining scenarios IR 3 and IR 4 do not manage to produce higher yield. Furthermore their average seasonal irrigation water depths which were much higher than those of IR 1 and IR 2, decreasing their IWUE. This shows that lower water consumption is sufficient enough for crop growth without experiencing yield reduction. Future work is to gather the relevant and complete observational irrigation data from the Department

IR4

IR3

533

557

498

529

Season 3

Average

553

Average

Season 2

520

Season 1

574

Season 3

554

Average

Season 2

521

Season 3

564

575

Season 1

565

542

Average

Season 2

509

Season 3

Season 1

570

Season 2

IR2

547

Season 1

IR1

ETc actual (mm)

Season

Irrigation

781

757

800

785

1530

1573

1680

1337

190

172

184

215

184.33

156

156

241

Irrigation (mm)

1483

1879

1357

1214

1483

1879

1357

1214

1483

1879

1357

1214

1483

1879

1357

1214

Rainfall (mm)

268

213

307

284

254

194

286

281

238

176

266

271

259

203

293

281

Decrease in soil moisture (mm)

Table 7.3 Water balance, yields and efficiencies for rice in Wasan

448

393

499

452

617

580

694

577

270

218

294

297

294

235

316

331

Runoff (mm)

1555

1958

1408

1298

2097

2546

2055

1691

1088

1488

938

838

1091

1494

920

858

Flow to groundwater (mm)

3582

3780

3636

3331

4093

4145

4167

3966

4223

4230

4230

4209

4200

4213

4203

4183

Yield (kg/ha)

6.79

7.59

6.53

6.25

7.42

7.97

7.26

7.03

7.64

8.12

7.36

7.44

7.77

8.28

7.37

7.65

WUE Ec (Kg/ha/mm)

4.59

4.99

4.55

4.24

2.69

2.64

2.48

2.97

22.39

24.59

22.99

19.58

23.77

27.01

26.94

17.36

IWUE (Kg/ha/mm)

Decision Support Method for Agricultural Irrigation Scenarios … 105

106 Table 7.4 Average yields of MRQ76 rice for various irrigation scenarios in descending order

S. A. Husain and N. H. M. Rhyme Irrigation scenarios

IR 2

IR 1

IR 3

Mean yield

4223.00

4200.00

4092.67

of Agriculture and Agrifood and compared with the simulated irrigation scenarios results from WEAP model. Despite limited data, our simulated result shows that the mean yield, i.e. 4223 kg/ha is nearly similar to the observational yield of MRQ76, i.e. 4230 kg/ha for year 2013, indicating WEAP model is a suitable model, not only to provide best irrigation strategy, but also a decision support method to predict crop yields and provide efficient information of water usage for irrigation without affecting the crop yield.

References 1. Yates, D., Sieber, J., Purkey, D., Huber-Lee, A.: WEAP21—a demand-, priority-, and preference-driven water planning model. Water Int. 30(4), 487–500 (2005) 2. R. Evans, R.E. Sneed, D.K. Cassel, Irrigation scheduling to improve water-use and energy-use efficiency. in North Carolina Cooperative Extension Service (AG, 1996), pp. 452–454 3. F. Galawat, M. Yabe, R. Economics, F. City, Profit efficiency in rice production in Brunei Darussalam 18(1), 100–112 (2012) 4. C. Brouwer et al., Irrigation Water Management: Irrigation Scheduling (Training Manual no 4) (1989), http://www.fao.org/3/T7202E/t7202e00.htm#Contents 5. M. Karamouz, Climate change impacts on crop water requirements: a case study, in 1st IWA Malaysia Young Water Professionals Conference, International Water Association, Johor Bahru, Malaysia (2010), pp. 1–11 6. Bhatti, G.H., Patel, H.M.: Irrigation scheduling strategies for cotton crop in semi-arid climate using WEAP model. J. Indian Water Res. Soc. 35(1), 7–15 (2015) 7. G.H. Bhatti, Developing Irrigation Management Strategies for Agro-Climatic Region I and II of Sardar Sarovar Project Command Area (Doctoral dissertation) (2016), Retrieved from http:// hdl.handle.net/10603/146765 8. Soil Fertility Evaluation/Advisory Service in Negara Brunei Darussalam Report: Soil Management in the Agricultural Development Areas, 1-233, https://publications.csiro.au›rpr›dow nload 9. Online News (2015), https://www.asiaone.com/food/bruneis-locally-grown-rice-now-superm arkets 10. Department of Agriculture and Agridfood: Business Opportunities in Agriculture and Agrifood. Ministry of Primary Resources and Tourism, Brunei Darussalam (2019) 11. Rice Department, Khao Dawk Mali 105. Bureau of Rice Research and Development, Thailand (2010), http://www.ricethailand.go.th/library/document/E-book/brrd5301007c.pdf 12. R.K. Chokshi, G.H. Bhatti, H.M. Patel, Estimation of evapotranspiration in sardar sarovar command area using WEAP, in 2012 International SWAT Conference (Indian Institute of Technology, Delhi, 2012) 13. Department of Agriculture and Agrifood (National Report), (Ministry of Primary Resources and Tourism, Bandar Seri Begawan, 2018) 14. Country Report for APEC Conference: New Irrigation Scheme to Increase Agriculture Production in Brunei Darussalam (Bangkok, Thailand, 2012)

Bayesian Variable Selection for Linear Models Using I-Priors Haziq Jamil and Wicher Bergsma

Abstract The Bayesian approach to modelling differs from the frequentist approach primarily in the supplementation of additional information about the parameters to the data. If we specify a “good” prior, in the sense that the prior nudges the likelihood in the right direction, then the estimates will also be good. This is what we aim to do in the case of variable selection problems, whereby the Bayesian method reduces the selection problem to one of estimation from a true search of the variable space for the model which optimises a certain criterion. We contribute to the vastly available literature of variable selection methods by using I-priors [5]— a class of Gaussian distributions which has the distinguishing property of having covariance proportional to the Fisher information (of the model parameters). The original motivation behind the I-prior methodology was to develop a novel unifying approach to various regression models. In this work, we detail the I-prior model used, and showcase some simulation results and several real-world applications in which the I-prior performs favourably compared to other prior distributions and/or variable selection techniques in terms of model size, R2 , predictive ability, and so on. Keywords Bayesian · Variable selection · MCMC · Collinearity · Linear regression In statistical modelling, there is often a genuine interest to learn the most reasonable, parsimonious, and interpretable model that fits the data. We turn our attention to the problem of variable selection in the context of ordinary linear regressions. Model selection is indeed a vastly covered topic, so our focus is on the Bayesian approach to model selection, emphasising the selection of variables through inferences on model probabilities. The appeal of Bayesian methods are that it reduces the selection problem to one of estimation, rather than a true search of the variable space for the H. Jamil (B) · W. Bergsma Department of Statistics, London School of Economics and Political Science, Houghton Street, London WC2A 2AE, United Kingdom e-mail: [email protected] W. Bergsma e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 S. A. Abdul Karim (ed.), Theoretical, Modelling and Numerical Simulations Toward Industry 4.0, Studies in Systems, Decision and Control 319, https://doi.org/10.1007/978-981-15-8987-4_8

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model that optimises a certain criterion. The I-prior [4] further enhances this appeal, as we discover that with it, existing Bayesian variable selection models performance improves, especially in problems where multicollinearity is present in the data set. This is shown through a simulation study and also real data examples.

1 Introduction Consider the normal linear regression model, where response variables y1 , . . . , yn relate to several independent variables linearly through the following equation: yi =

p 

xik βk + i

k=1

(1)

iid

i ∼ N(0, σ 2 ). The topic of interest for this chapter is model selection for linear regression models. That is, from a set of p covariates or predictors {X1 , . . . , Xp }, the task is to determine the best choice of subset(s) of variables that should be included in a regression model used to explain the variation in the response variable. As such, the term variable selection is synonymous to model selection for linear regression models. Fundamental to this notion of variable selection is an inherent belief in sparseness of the true data generative process surrounding the response variable, i.e. not all of the variables need be used to predict the response. Model selection is indeed a huge topic to cover fully. We broadly classify variable selection into three categories: (1) (pairwise) model comparison using some criterion; (2) shrinkage to induce sparsity; and (3) Bayesian model selection. We understand that different categorisations and hence categories of model selection exist in the literature, but our focus is on the discussion of the three types as mentioned. Model selection criteria, both from a frequentist and Bayesian standpoint, can either be of a predictive nature (e.g. R2 , mean squared error of prediction (MSEP), Cp [30], k-fold cross-validation MSEP, etc.), or be based on likelihood (e.g. likelihood ratios, Bayes factors, Akaike information criterion (AIC [1]), Bayesian information criterion (BIC [42], etc.). Selecting a model based on either of these criteria requires comparison of all 2p criteria, which is not feasible for large p. Typically, these criteria are used in conjunction with stepwise procedures such as forward-selection or backward-deletion to restrict attention to a smaller number of potential subsets [15, 32]. On the other hand, regularised least squares regression (ridge regression [17, 44], or a convex combination of the two via elastic nets [45], etc.) provide additional information to the regression model in order to provide a sparse solution to linear system of equations in β. These methods are proven to be popular as they are fast and perform exceptionally well in many situations, even in cases where p > n. Additionally, the Lasso produces solutions for β which are exactly zero. However, the

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109

Lasso in general produces estimates which are biased towards zero, are inconsistent, and have no valid standard errors [14, 25]. Further criticisms of the Lasso include its inability to select more than n predictors in a p > n situation, and poor performance when multicollinearity exists among the covariates. From a Bayesian perspective, regularisation is akin to placing priors on the βk ’s to shrink the effects of the βk ’s: the ridge regression has a Bayesian interpretation of placing normal priors on the regression coefficients, while for the Lasso, a Laplace or double exponential prior [37]. The term adaptive shrinkage has been used for the method in which hyperpriors are placed on the scale parameter of the prior for the βk ’s. The idea is to adaptively shape the prior depending on the importance of the variable in the regression model. Bayesian shrinkage includes the task of specifying tuning parameters. This could potentially affect chain mixing in a Markov chain Monte Carlo method (MCMC) procedure, the estimation method that is commonly used. True Bayesian model selection is probabilistic in nature: a priori, one assigns probabilities over the set of models, and then after observing the data, posterior model probabilities (PMPs) are used to discern which of the models was likeliest to have been behind the data generative process of the observed responses. Of course, with large p, calculation of all 2p posterior model probabilities to ascertain which is highest will be a challenge, if not impossible. But, as with most Bayesian applications, MCMC can be applied as a practical means of overcoming this intractability. This stochastic approach to variable selection was pioneered by George and Mc-Culloch [15], and studied by others such as Kuo and Mallick [24], Dellaportas et al. [12], and Ntzoufras [34]. Unlike shrinkage methods, Bayesian model selection is able to quantify the amount of times a variable “enters the model” (inclusion probabilities), and thereby measuring its worth as a predictor. Note that, in addition to model probabilities and inclusion probabilities, estimates of regression coefficients are obtained simultaneously in Bayesian variable selection. When several competing models have high posterior probabilities, regression coefficients from each model, or indeed any quantity of interest, may be combined and weighted by their posterior model probabilities—a technique known as Bayesian model averaging [18, 29]. Averaging over a set of models takes into account the uncertainty surrounding model selection, which other standard statistical procedures ignore upon selection of a single model from which to do inference. It is known to be the case that predictive accuracy of the model-averaged quantity is improved, as measured by a logarithmic scoring rule [39]. Bayesian model selection is not without criticism, however. For complex models with many predictors or samples, MCMC is slow and may mix poorly [35]. Often, there are a lot of tuning parameters that need to be set correctly for the problem at hand. Also, the choice of priors for model parameters affects consistency of Bayesian model selection procedures. Specifically, improper priors cannot be used to calculate posterior model probabilities [9]—otherwise, one risks running into Lindley’s paradox1 [28]. 1 Briefly,

in testing a point null hypothesis of the mean of a normally distributed parameter, the null hypothesis is increasingly accepted as the prior variance of the parameter approaches infinity,

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The plan for this chapter is to describe a fully Bayesian model for variable selection using I-priors, a novel objective prior for the regression function proposed by Bergsma [4]. The approach that we take is a stochastic search of the model space due to Kuo and Mallick [24], realised through a simple Gibbs sampling procedure. The main motivation behind using I-priors in Bayesian variable selection is its suitability in accommodating to datasets with strong multicollinearity and being able to run with little to no prior information about the parameters. A simulation study is conducted and several real-world examples presented to demonstrate this fact.

2 Brief Introduction to I-Priors For n pairs of data {(y1 , x1 ), . . . , (yn , xn )}, a general regression model is considered: yi = f (xi ) + i

(2)

iid

i ∼ N(0, σ 2 ).

where yi ∈ R, xi ∈ Rp and f belongs to some reproducing kernel Hilbert space (RKHS) F with reproducing kernel h : Rp × Rp → R. We are interested in estimating the regression function f . For this purpose, we define the I-prior as the distribution which maximises the entropy of all distributions of f subject to suitable constraints. For the present model, the I-prior on f is Gaussian with covariance kernel proportional to the Fisher information for f , and mean chosen a priori (e.g., 0). Practically, F may be any arbitrarily appropriate RKHS for which the regression function f lies in, examples of which include the space of polynomial functions, smooth exponential functions, or even the RKHS containing Brownian motion paths (see Chap. 2 of Jamil [20]). The emphasis of the current work is on the space of linear functions for linear regression.. For the present model, it is Gaussian with covariance kernel proportional to the Fisher information, and mean chosen a priori (e.g., 0). Note that an RKHS F is spanned by the functions h(·, x) [6]. Therefore, f belonging to a linear RKHS F with kernel h(x, x ) = x x , f can be represented by f (xi ) =

n  j=1

h(xi , xj )wj =

n 

xi xj wj

(3)

j=1

for some real-valued constants w1 , . . . , wn . We see that x xi = therefore

p

k=1 xik xjk ,

and

regardless of evidence for or against the null. The paradox is also termed Jeffreys-Lindley paradox [40].

Bayesian Variable Selection for Linear Models Using I-Priors

f (xi ) =

n 

xi xj wj

j=1

=

111

 p n   j=1

 xik xjk wj

k=1

⎛ ⎞ p n   ⎝ = xjk wj ⎠xik k=1



j=1

 βk

which is identical to (1), the normal linear regression model described at the beginning of this chapter. See Chap. 4 of Jamil [20] for more details. It is well known that the Fisher information for the regression coefficients of model (1) is I(β) ∝ X  X where X is an n × p matrix whose rows are xi . Therefore, an I-prior for β = (β1 , . . . , βp ) is β ∼ Np (0, κσ 2 X  X )

(4)

where κ is some scale parameter that needs to be estimated. Having the σ 2 appear in the covariance of β proves to be an advantage when estimating the variable selection model fully Bayesian under a Gibbs sampling approach (see Sect. 5). The I-prior has the intuitively appealing property that the more information is available about a linear functional of the regression function, the larger its prior variance, and, broadly speaking, the less influential the prior is on the posterior. This is in contrast to the g-prior [46], which has covariance proportional to the inverse Fisher information. Analysis of some real data sets and a small-scale simulation study show competitive performance of the I-prior methodology [4, 20].

3 Preliminary: Model Probabilities, Model Evidence and Bayes Factors The paradigm of model selection is as follows. From a finite set of models M = {M1 , . . . , MK }, pairs of data {(y1 , x1 ), . . . , (yn , xn )}, yi ∈ R and xi ∈ X ≡ Rp , had been generated according to the generative process dictated by one of the models Mk ∈ M and its respective parameters k . Having observed only this data set, the goal is to infer which of the models had generated the data, and consequently obtain estimates for the parameters. It is perhaps most natural to ponder which of the models is most likely to be the “true” one given the data presented, and thus, this natural

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way of thinking leads one to the concept of model probabilities. From a Bayesian perspective in particular, posterior model probabilities allow us to quantify the certainty to which any model is behind the data generative process, after taking into account relevant evidence (observation of the data) and prior beliefs about model and parameter uncertainty. Let p(M1 ), . . . , p(MK ) be prior probabilities assigned to the model space M, and p(k |Mk ) be the prior on the parameters of model Mk . For any model Mk ∈ M, the posterior model probability for model m is p(y|Mk )p(Mk ) p(Mk |y) = K k=1 p(y|Mk )p(Mk )

(5)

where  p(y|Mk ) =

p(y|Mk , k )p(k |Mk ) dk

(6)

is known as the marginal likelihood, or evidence, for model Mk . As a remark, the prior distributions for the parameters do not necessarily need to depend on the model, so we might have that p(k |Mk ) = p(k ). A natural strategy for model selection is to select the model such that p(Mk |y) is largest (the highest probability model, HPM), but several models rather than just a single one may be reported to convey model uncertainty [11]. Note, that models may be pairwise compared based on these posterior model probabilities, for which the posterior odds Bayes factor

prior odds

  p(y|Mk ) p(Mk ) p(Mk |y) = × p(M0 |y) p(y|M0 ) p(M0 )

(7)

provide a point summary for comparing model Mk against model M0 . The first term on the right-hand side is the Bayes factor for comparing any model Mk ∈ M to another model M0 ∈ M, and is denoted by BF(Mk , M0 ). Thus, model selection based on posterior model probabilities can be formalised as the Bayesian alternative to classical hypothesis testing using Bayes factors [23]. The issue that is faced with Bayesian model selection is that all posterior model probabilities must be calculated in order for a full comparison to be made. When the model space is very large, this can prove to be an insurmountable task. In the case of linear regression, where each of the p variables may be selected or not, the size of the model space is 2p . Even for moderate sized p this can already be a challenge computationally. In the coming sections, we shall see that this problem is alleviated by the use of MCMC methods to evaluate posterior model probabilities.

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4 The Bayesian Variable Selection Model We shall loosely refer to a model as a subset of variables selected from the full set of variables {X1 , . . . , Xp }. It would be useful to be able to index each of these 2p possible models somehow, and we achieve this by the use of indicator variables γ = (γ1 , . . . , γp ) ∈ {0, 1}p . Let γj = 1 if the variable Xj is selected, and γj = 0 otherwise, for j = 1, . . . , p. As an example, the full model, where all the variables are included in the model, is denoted by γ = (1, . . . , 1), while the intercept only model is denoted by γ = (0, . . . , 0). Note that we do not consider the intercept to be selectable. Following Kuo and Mallick [24], the linear model in Eq. 1 is expanded to include the indicator variables to form yi = α +

p 

xik γk βk + i

k=1

(8)

iid

i ∼ Nn (0, σ 2 ). Hence, in addition to the usual model parameters (β, σ, α), we are also interested in conducting model inferences through the posterior distribution of the γ ’s. The priors for the parameters are described below: • Model indicators γj . An independent Bernoulli prior is specified for the model indicators p  γ πj j (1 − πj )1−γj . (9) p(γ ) = j=1

We may choose to set all πj = 0.5 a priori to reflect equally likely probabilities that any model may be chosen. Alternatively, we might have some subjective beliefs about which predictor is more likely or unlikely to be included in the model. We may also choose to include πj in the estimation procedure by assigning a hyperprior on πj such as the Beta(1, 1) (uniform distribution), Beta(1/2, 1/2) (Jeffreys prior), or any other suitable hyperprior. In any case, in this chapter we consider the simplest case of setting all πj = 0.5. • Regression coefficients β. The Kuo and Mallick [24] model is often known as the independent sampler due to the independence of model parameters and the indicator variables, i.e., p(β, γ ) = p(β)p(γ ). As such, prior choices for the regression coefficients can be any of the usual priors on β, including (but not limited to) – the independent prior β ∼ Np (0, c2 Ip ) for some choice of c (e.g. c = 10); – the g-prior β|σ, g ∼ Np (0, gσ 2 (X X)−1 ) for some g either chosen a priori or estimated (Bayes or empirical Bayes); or – the I-prior β|σ, κ ∼ Np (0, κσ 2 X X), which is the focus of this chapter. • Intercept α. A normal prior α ∼ N(0, σ 2 A). • Scale σ . An inverse gamma prior σ ∼ −1 (c, d ).

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Priors for the intercept and scale parameters are chosen so as to maintain conjugacy to the normal regression model. Choices for the prior hyperparameters depend on the user’s prior beliefs, but it is reasonable to set vague and uninformative hyperparameters to let the data speak as much as it can, especially in the absence of prior information. With this in mind, we may choose large values of A (e.g. 100) and small values of the shape and scale parameters for the inverse gamma (e.g. 0.001). Note that as c, d → 0 in the inverse gamma distribution we get the Jeffreys prior2 for scale parameters. Remark 8.1 The BVS model Eq. (8) together with the choice of Bernoulli priors on γ and a normal prior Np (0, Vβ ) for β can be seen a spike-and-slab prior prior for linear regression models, a mixture of a point mass at zero and a normal density [16, 33]. Write θ = (γ1 β1 , . . . , γp βp ) , which is interpreted as the “model-specific” regression coefficients. Then, the prior on θ is equivalently written  θ |γ ∼

Np (0, Vβ ) w.p. p(γ ) 0 w.p. 1 − p(γ ).

A subtle fact of these spike-and-slab priors is that the posterior distribution for θ will also be a combination of a point mass and a normal density (with appropriate posterior parameters). Looking at it from this perspective, regression coefficients are assigned zero values with positive probability, and it is this fact that allows covariates to be dropped from the model. As pointed out by Kuo and Mallick [24], the form of the variable selection model allows the selection of important variables, while simultaneously shrinking the coefficients via prior information.

5 Gibbs Sampling for the I-Prior BVS Model The Bayesian variable selection model can be estimated using Gibbs sampling, as demonstrated originally by Kuo and Mallick [24]. In this section, we describe the Gibbs sampling procedure to obtain posterior samples of the parameters. For the I-prior specifically, the joint density of the responses and the priors is p(y, γ , β, α, σ 2 , κ) = p(y|γ , β, α, σ 2 )p(β|σ 2 , κ)p(α|σ 2 )p(γ )p(σ 2 )p(κ), where the distribution of the model p(y|γ , β, α, σ 2 ) and of the priors have been described in the previous section (except for κ, which we now assign an inverse gamma distribution). Let us denote  = {α, β, γ , σ 2 , κ} to be the full set of parameters that we wish to obtain posterior samples for. Starting with suitable initial values (0) , we then proceed to obtain samples (1) , . . . , (T ) by sampling each parameter from the conditional posterior density of that parameter given the rest of the param2 The

Jeffreys prior for a parameter θ is defined as p(θ) ∝ |I (θ)|1/2 [21].

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eters. A suggested set of initial values are the maximum likelihood (ML) estimates of  or the posterior mean estimate of  under the full model γ = (1, . . . , 1) after an initial MCMC run. The Gibbs conditional densities are straightforward to obtain on account of model conjugacy (details of the derivation are given in [20]). We start with β. The conditional ˜ − α1n ) and density of β given α, γ , σ 2 , κ is multivariate normal with mean B(y ˜ where B˜ = Xγ Xγ + (κX X)−1 , and Xγ = (γ1 X1 · · · γp Xp ). covariance matrix σ 2 B, Interestingly, when Xj is dropped from the model (γj = 0), the posterior mean and variance for jth component of β is entirely informed by the prior [24]. The datadriven I-prior incorporates information from the covariates into the prior, which then informs the posterior. manner, the for the intercept α  conditional density  nIn a similar  2˜ −1 ˜ ˜ A , where A = n + A (y − x θ )/ A, σ and A is the prior is found to be N i i i=1 variance for α. The (conditional) posterior samples of γ = (γ1 , . . . , γp ) are obtained componentwise, and each conditional probability mass function for γj is Bernoulli with success probability π˜ j = uj /(uj + vj ), where   1 [1] 2 uj = πj exp − 2 y − α1n − Xθ j 2σ and   1 [0] 2 vj = (1 − πj ) exp − 2 y − α1n − Xθ j . 2σ Here, we have used the notation θ [1] j to indicate the vector θ with the j’th component replaced by β j , and θ [0] to indicate the vector θ with the j’th component replaced by j 0. Values of 1 for γ are more likely to be sampled when the ratio uj /vj is greater than the prior odds πj /(1 − πj ). Specifically when the prior probabilities πj are all set to be 0.5, then γj will be more likely to be sampled as ‘1’ if uj > vj , i.e. if the residual sum of squares (RSS) y − α1n − Xθ 2 is smaller when the jth component of θ is non-zero, compared to the RSS when the jth component of θ is zero. We can in fact draw parallels to a Bayesian hypothesis test, with the null hypothesis being H0 : βj = 0 and the alternative being H1 : βj = 0, conditional on knowing all 2 other values of the parameters. Under Hk , y| ∼ Nn (α1n + Xθ [k] j , σ In ), k = 0, 1. The conditional Bayes factor comparing the model in the alternative hypothesis M1 to the model in the null hypothesis M0 is therefore BF(M1 , M0 ) =

π˜ j uj /πj = vj /(1 − πj ) 1 − π˜ j



πj . 1 − πj

Thus, it can be seen that the decision to include or exclude the jth variable from the model relates a hypothesis test using the Bayes factor rule, and this decision is embedded in the conditional posterior probabilities π˜ j . The Gibbs sampling procedure

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does something that can be described as “an automated stochastic F-test for subset selection” [24]. Both scale parameters σ 2 and κ follow the conditional inverse gamma distributions   σ 2 |α, β, γ , κ ∼ −1 n/2 + cσ + 1, y − α1n − Xθ 2 /2 + dσ and   κ|α, β, γ , σ ∼ p/2 + cκ + 1, β  (X X)−1 β/σ 2 + dκ . 2

−1

Note that the inverse gamma distribution that we specify here is defined by its shape (s) and scale parameter (σ ), and has the density function p(x) = (s)−1 σ s x−(s+1) e−σ/x . Here, {cσ , dσ } and {cκ , dκ } are the shape and scale hyperparameters of the inverse gamma priors on σ 2 and κ respectively.

6 Posterior Inferences Having obtained posterior samples (t) = {α (t) , β (t) , γ (t) , σ 2(t) , κ (t) }, there are two quantities of interest in relation to model inferences. The first is an estimate of posterior model probabilities, given by  P(γ = γ  |y) =

T 1  (t) [γ = γ  ], T i=1

(10)

where [·] is the Iverson bracket. This gives an estimate of the probability of a model coded by γ  appearing in the posterior state space of models. The second is a quantification of the posterior inclusion for each of the p variables X1 , . . . , Xp , known as posterior inclusion probabilities (PIPs) for a variable being selected in any model. This is given by  P(γj = 1|y) =

T 1  (t) [γ = 1], T i=1 j

j = 1, . . . , p.

(11)

Posterior inclusion probabilities are the marginals of the posterior model probabilities across each variable (See Table 1 for an illustration). Note, that the regression coefficient of interest is not β, but rather the “model averaged” regression coefficients θ = (γ1 β1 , . . . , γp βp ) [29]. Posterior variances for θ will typically be larger than variances for β, because posterior estimates surrounding θ will have incorporated model uncertainty, but β on the other hand, will not. Thus, any inferential procedure surrounding the regression coefficients avoids the risk of over-confidence. Note that, since θ will contain values of exactly zero when predictors are dropped out of the model, the posterior density for θ is a mixture of a point

Bayesian Variable Selection for Linear Models Using I-Priors Table 1 Illustration of samples of γ from the Gibbs sampler for p = 3 (t) (t) t γ1 γ2

117

(t)

γ3

1 2 3 .. .

1 1 1 .. .

0 0 1 .. .

1 0 0 .. .

T

1

0

1

As an example, to estimate the posterior model probability of {X1 ,X3 }, we count the occurrences of the combination γ (t) = (1, 0, 1) in the sample and divide by T . To estimate posterior inclusion probabilities for any of the three variables, we take the sample mean of the binary variates columnwise

mass at zero and a normal density. In any case, the likelihood only provides sufficient information to identify the product of β and γ , but not each of them separately [24]. Remark 8.2 The intention of computing model-averaged regression coefficients θ is solely for the inclusion of model uncertainty. There is a strong agreement in the Bayesian variable selection literature that that such coefficients are practically meaningless when it comes to explanatory inferences. Banner and Higgs [2] writes that “regression coefficients... may not hold equivalent interpretations across all of the models in which they appear”, and one reason for this might be “interpretation of partial regression coefficients can depend on other variables that have been included in the model”. The use of model-averaged effect sizes may thus result in misleading inferences [8]. Finally, any quantity of interest can be incorporated as part of the Gibbs sampling procedure. That is, at each Gibbs iteration t = 1, . . . , T , calculate (t) as a function of the parameter values at iteration t. This can be done during the Gibbs sampling process, or even after the fact as part of a post-processing procedure. Any inference on the posterior of will then have incorporated the model uncertainty from a model averaging standpoint, as discussed earlier. As an example, suppose we are interested in the predicted value at a new covariate value xnew ∈ Rp . For each Gibbs sample, calculate (t)  = α (t) + xnew (γ1 β1 , . . . , γp βp ), ynew

and obtain a point estimate yˆ new using the posterior mean or mode. The uncertainty for this estimate is contained in the standard deviation calculated from the sample (1) (T ) , . . . , ynew , from which a 95% credibility interval for this estimate can be obtained ynew from the empirical upper and lower 0.025 cut off points.

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7 Two Stage Procedure The variable selection procedure can be improved by a “preselection” of variables to trim off unimportant variables which reduces the size of the model space being explored. Without appealing to other external preselection methods, there is actually information that we could use from Bayesian variable selection models in the form of posterior inclusion probabilities. The procedure would work as follows: 1. Run the Bayesian variable selection model and obtain posterior inclusion probabilities for each variable. 2. Discard variables with inclusion probabilities less than a certain threshold, τ . 3. Re-run the Bayesian variable selection model on the set of reduced variables. A natural choice for τ would be 0.5, and therefore a two-stage approach to Bayesian variable selection can then be motivated as selecting the subset of variables which constitutes what is known as the median probability model. The median probability model is obtained by selecting all variables with a posterior inclusion probability of greater than or equal to a half. Barbieri and Berger [3] show that the median probability model has the property of being optimally predictive (minimises squared error loss for predictions) under certain strict conditions. The notion of a two-stage approach is not new, as many variable selection methods in the literature generally employ a preselection method of some kind before running their selection process proper. This can be based on subjective preconceptions about which variables to retain, substantive theory, or even an objective preselection criterion. Two-stage procedures for Bayesian variable selection models have been used in works by Fouskakis and Draper [13] and Ntzoufras [34]. In the simulation studies conducted and observations from real-data examples, this two-stage approach does seem to provide a benefit. The complexity of estimating all model probabilities grows exponentially with p, therefore reducing this benefits the model selection procedure because the search of the model space is less cluttered. Of course, this idea works if the “correct” variables are deleted when proceeding to the second stage. We posit that the p posterior inclusion probabilities are easier to estimate than the 2p posterior model probabilities from the same amount of information coming from the MCMC samples. As a result, information summarised through the posterior inclusion probabilities are more precise than the posterior model probabilities.

8 Simulation Study In this section, we conduct a simulation study to compare the performance of different priors in the Bayesian variable selection framework described above. The priors on β that are compared are those mentioned in Sect. 4, i.e. the I-prior, the independent prior with large prior variance (flat/uninformative prior), and the g-prior

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with g = n (unit information prior [34]). We also make a comparison the variable selection performance of the Lasso, which, from a Bayesian perspective, is similar to setting a double-exponential or Laplace priors on the regression coefficients [37]. For clarity, the Lasso model employed in the simulations is of a frequentist regularisation framework as per Tibshirani [44], and is neither a Bayesian variable selection model as described earlier, nor a fully Bayes implementation as per Park and Casella [37]. We felt it interesting to compare the Lasso as it is widely used for variable selection of linear models. The experiment is to select from a total of p = 100 variables of an artificial dataset of sample size n = 150, which has pairwise correlations induced between the variables. This was inspired by the studies done by George and Mc-Culloch [15] and Kuo and Mallick [24] in their respective papers, albeit on a larger scale (in theirs, p = 30). Five different scenarios were looked at. For each scenario, only s out of 100 variables were selected to form the “true” model and generate the responses according to the linear model y ∼ N100 (Xβ, σ 2 I150 ). The signal-to-noise ratio (SNR) as a percentage is defined as s%, and the five scenarios are made up of varying SNR from high to low: 90, 75, 50, 25, and 10%. Variables that were included in the model had true β coefficients equal to one. That is, β true = (1s , 0100−s ) , where 1s is a row-vector of s ones, and 0100−s is a row-vector of 100 − s zeroes. The data generation process is summarised as follows: iid

• Draw Z1 , . . . , Z100 ∼ N150 (0, I150 ). • Draw U ∼ N150 (0, I150 ). • Set X = (Z1 + U, . . . , Z100 + U). This induces pairwise correlations of about 1/2 between the columns of X.3 • Draw y ∼ N150 (Xβ true , σ 2 I150 ), with σ = 2. In each scenario, we are interested in obtaining the highest probability model and counting the number of false choices made in this model after a two-stage procedure of variable selection. False choices can either be selecting variables wrongly (false inclusion) or failing to select a variable (false exclusion). Each scenario was repeated a total of 100 times to account for variability in the data generation process, and the results averaged. A summary of the results is presented in Table 2. The overall results are also plotted in the form a frequency polygon (see Fig. 1). The simulation results seem to indicate that the I-prior performs consistently well across all five scenarios, making no more than five false choices out of 100 (i.e. a 95% correct selection rate) in at least 82% of the time in the worst scenario. We do not observe much difference between the g-prior and the independent prior, and while they behave poorly in high SNR scenarios, these two priors seem to perform extremely well in low SNR scenarios. A high propensity to drop variables in these scenarios is a likely explanation, which does not necessarily indicate good performance—they perform well by contentiously omitting of a large number of unnecessary variables, especially in a two-stage procedure. Finally, the Lasso is well 3 For any row of X, Cov[X , X ] = Cov[Z + U, Z + U ] = Var [U ] = 1, and Var[X ] = Var[Z + j j j j k k U ] = 2. Thus, Corr[Xj , Xk ] = Cov[Xj , Xk ]/(Var[Xj ]Var[Xk ])1/2 = 1/2.

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Table 2 Simulation results (proportion of false choices) for the Bayesian variable selection experiment using the I-prior, an independent prior, the g-prior and the Lasso across varying SNR False choices Signal-to-noise ratio (%) 90 75 50 25 10 I-prior 0–2 3–5 >5 Ind. prior 0–2 3–5 >5 g-prior 0–2 3–5 >5 Lasso 0–2 3–5 >5

0.93 (0.03) 0.07 (0.03) 0.00 (0.00)

0.92 (0.03) 0.07 (0.03) 0.01 (0.01)

0.90 (0.03) 0.10 (0.03) 0.00 (0.00)

0.79 (0.04) 0.20 (0.04) 0.01 (0.01)

0.55 (0.05) 0.27 (0.04) 0.18 (0.04)

0.00 (0.00) 0.00 (0.00) 1.00 (0.00)

0.00 (0.00) 0.00 (0.00) 1.00 (0.00)

0.00 (0.00) 0.00 (0.00) 1.00 (0.00)

0.44 (0.05) 0.30 (0.05) 0.26 (0.04)

1.00 (0.00) 0.00 (0.00) 0.00 (0.00)

0.00 (0.00) 0.00 (0.00) 1.00 (0.00)

0.00 (0.00) 0.00 (0.00) 1.00 (0.00)

0.00 (0.00) 0.00 (0.00) 1.00 (0.00)

0.78 (0.04) 0.14 (0.03) 0.08 (0.03)

0.86 (0.03) 0.13 (0.03) 0.01 (0.01)

0.03 (0.02) 0.19 (0.04) 0.78 (0.04)

0.00 (0.00) 0.02 (0.01) 0.98 (0.01)

0.00 (0.00) 0.00 (0.00) 1.00 (0.00)

0.00 (0.00) 0.00 (0.00) 1.00 (0.00)

0.00 (0.00) 0.00 (0.00) 1.00 (0.00)

Standard errors are given in parentheses

known to yield poor selection performance under multicollinearity, so the results are expected. The Lasso procedure was not subject to a two-stage approach because the Lasso does not provide information regarding posterior inclusion probabilities for individual variables. We also inspect the sensitivity of the hyperprior choice of πj for the indicator variables on the number of false choices made. Figure 2 plots the mean number of false choices made in each of the five SNR scenarios with varying hyperprior setting for πj . From the plot, it is seen that for high SNR scenarios, setting πj too low increases the number of false exclusions. Conversely, for low SNR scenarios, setting πj too high increases the number of false inclusions. This makes sense: when the true model size is small, then setting πj too high encourages variables to be retained in the model, and vice versa. While the optimal πj corresponds directly to the true SNR (e.g. SNR = 10% performs best under πj = 0.10), Fig. 2 makes a case for πj = 0.5 to be a “safe choice” in the face of prior ignorance on model size.

9 Examples Now, we apply our I-prior Bayesian variable selection model to three real-world data sets that have all been previously analysed in the variable selection literature. All examples were analysed in R using our ipriorBVS package [19] which contains a

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I−prior 0.8

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Fig. 1 Frequency polygons for the number of false choices for each of the four priors. The I-prior performs robustly well across the five scenarios tested, mostly yielding five or fewer false inclusions or exclusions. Spurious exclusions led to the independent and g-prior simultaneously performing well in low SNR and badly in high SNR scenarios. The Lasso is known to be unreliable in the presence of collinearity

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90% 75%

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Fig. 2 Average number of false choices (false inclusions or false exclusions) for the five different scenarios (SNR varied between 90, 75, 50, 25 and 10%) with different hyperprior settings for γj ∼ Bern(πj )

wrapper to JAGS [38]. Reproducible code is available at http://myphdcode.haziqj. ml. In all analyses, a two-stage procedure was conducted for the I-prior model, where each stage consists of obtaining 10,000 MCMC samples.

9.1 Aerobic Data Set This dataset appeared in the SAS/STAT ® User’s Guide [41] and was also analysed by Kuo and Mallick [24]. It involves understanding the factors which affect aerobic fitness, which is measured by the ability to consume oxygen. A sample of n = 30 male participants’ had their physical fitness measured by means of simple exercise tests. The response variable contains measurement of oxygen uptake rate in mL/kg body weight per minute. The six covariates were the participants’ age (X1 ), weight (X2 ), time taken to run one mile (X3 ), resting heart rate (X4 ), heart rate while running (X5 ), and maximum heart rate during the exercise (X6 ). This dataset, although small in size, is interesting to analyse because of the correlations between the variables, mainly due to the measurements being taken during the same exercise test. The sample correlations of interest are shown in Fig. 3. Notice that Table 3 reports only on four out of 26 = 64 possible models, but the sum of the posterior model probabilities add to one. Naturally, models which are deemed important by virtue of data evidence are sampled more often, and in fact, models which are unpromising may not even get sampled. So, MCMC methods does not need to list out all possible models because models which are never visited in the posterior state space are assigned a probability of zero. The highest posterior model was found to be the model with the variables X1 , X3 and X5 (PMP = 0.564). In Fig. 4, we can see that the point mass at zero overwhelms the rest of the values in the density plots for X2 , X4 and X6 , and hence these variables were dropped.

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Fig. 3 The sample correlations of interest in the aerobic fitness dataset. These show variables with correlations greater than 0.4 in magnitude Table 3 Results for variable selection of the Aerobic data set PIP Coef. (SD) Model 1 Model 2 X1 X2 X3 X4 X5 X6

0.685 0.205 1.000 0.168 0.663 0.275

−0.169 (0.14) −0.017 (0.05) −0.745 (0.12) −0.013 (0.05) −0.163 (0.15) 0.003 (0.10) PMP BF

✓ ✓

Model 3

✓

✓

✓ 0.564 1.000

Model 4

✓ ✓ ✓ 0.235 0.418

0.105 0.187

0.096 0.170

Note that the Bayes factors reported are the Bayes factors comparing any of the models to Model 1 (base model)

9.2 Mortality and Air Pollution Data The next real world application comes from a paper by McDonald and Schwing [31]. In it, the effects of air pollution on mortality in a US metropolitan area (n = 60 and p = 15) were studied. The response variable is the total age adjusted mortality rate, and the main pollution effects of interest were that of hydrocarbons (HC), oxides of nitrogen (NOx ) and sulphur dioxide (SO2 ). Several other environmental and socioeconomic considerations were taken into account, otherwise the model may include unexplained variation caused by factors other than pollution. For example, a metropolitan area with a high proportion of the elderly should expect to have a higher mortality rate than one with a low proportion. All of the variables can be considered as continuous and real; Table 4 provides a description of the variables. This dataset also contains several highly correlated variables which impedes a meaningful regression analysis. When the full model is fitted using ordinary least squares, none of the pollutant effects were found to be significant. Clearly, a variable selection method was required. McDonald and Schwing [31] used a ridge regression technique to determine which variables to select and eliminate “unstable” coefficients found from a trace analysis. In addition, the authors also looked at a variable

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X5

X6

1.00 0.75 0.50 0.25 0.00

Density

1.00 0.75 0.50 0.25 0.00 1.00 0.75 0.50 0.25 0.00 −1.0

−0.5

0.0

0.5

−1.0

Coefficient

−0.5

0.0

0.5

Fig. 4 Posterior density plots of the regression coefficients θ for the aerobic data set. The spike at zero observed in the density plots for X2 , X4 and X6 is indicative of these variable being dropped often in the posterior samples

elimination method based on total squared error via Mallow’s Cp . The results are summarised in Table 5. In this case, the I-prior BVS model concurred with the overall finding of McDonald and Schwing [31], in that SO2 was found to be a significant contributing factor towards mortality rates, while the rest of the pollutants were not. The I-prior BVS model also obtained a model with the largest R2 and the smallest size. We note that the effect size for SO2 is slightly larger under an I-prior, but generally, the rest of the I-prior coefficients are similar in magnitude and sign to the coefficients of the other two models.

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Table 4 Description of the air pollution data set Variable Description Mortality Precipitation Relative humidity January temperature July temperature Population density Household size Education Sound housing units Age >65 years Non-white White collar Income 65 years −0.213 (0.20) Non-white 0.640 (0.19) White collar −0.014 (0.12) Income n cases. Typically, when there is insufficient information in the data to inform the estimation, then additional information is sought from the priors. In our case, the I-prior covariance involves the inverse of a low rank matrix which is not invertible. A p-variate normal distribution with a singular covariance matrix will only have a probability distribution defined on a low dimensional subspace. The issue may however be computational—it might be worth exploring the generalised inverse, or study ways in which to avoid the inverse computation in the Gibbs sampler. As a matter of fact, we note that the posterior precision for β can be written as   ˜ −1 = Xγ Xγ + (κX X)−1 −1 B  −1 = Xγ Xγ (Xγ Xγ )2 + κIp which avoids the need for inverting the low-rank matrix Xγ Xγ . 2. Improvement in computational time. Although the model itself is not computationally intensive to run (roughly O(np2 ) in time per Gibbs iteration), the main bottleneck is the reliance on a stochastic sampling algorithm. As in the previous chapter, variational inference is a promising area to look into, especially given that the Gibbs conditional distributions were straightforward to obtain, and these might be similar to a mean-field variational distribution. If this is successful, then it is expected to reduce computational time and avoid convergence issues that comes with traditional MCMCs. Variational inference with spike-and-slab priors on regression coefficients was studied by Ormerod et al. [36]. 3. Extension to generalised linear models. Kuo and Mallick [24] in their paper already provided a sketch of how the variable selection model would work.

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In a similar fashion, we can extend the I-prior variable selection to categorical responses when the continuous latent propensities are modelled using linear functions. Such an approach has been implemented in gene selection studies, for which the variables are gene expressions and the responses are presence of a particular disease [26]. Finally, it should be mentioned that more complex variable selection models can be coded with the γ indicators. For instance, in selecting squared or interaction terms, we can insist on having the model select the main term if the squared or interaction term is selected, by specifying yi = α + max(γ1 , γ3 )β1 x1i + max(γ2 , γ3 )β2 x2i + γ3 β3 x1i x2i . Or perhaps, we could use a single γ indicator for the dummy variables which make up a single categorical covariate, which we would then infer on the selection of the single covariate rather than each individual category of the covariate.

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14. Friedman, J.H., Hastie, T., Tibshirani, R.: The Elements of Statistical Learning: Data Mining, Inference, and Prediction, 2nd edn. Springer, New York (2001). ISBN 978-0-387-84857-0. https://doi.org/10.1007/978-0-387-84858-7 15. George, E.I., McCulloch, R.E.: Variable selection via Gibbs sampling. J. Am. Stat. Assoc. 88(423), 881–889 (1993). https://doi.org/10.2307/2290777 16. Geweke, J.: Variable selection and model comparison in regression. In: Bernardo, J.M., Berger, J.O., Philip Dawid, A., Smith, A.F.M. (eds.) Bayesian Statistics 5. Proceedings of the Fifth Valencia International Meeting. Oxford University Press (1996). ISBN: 978-0-19-852356-7 17. Hoerl, A.E., Kennard, R.W.: Ridge regression: biased estimation for nonorthogonal problems. Technometrics 12(1), 55–67 (1970). https://doi.org/10.2307/1267351 18. Hoeting, J.A., Madigan, D., Raftery, A.E., Volinsky, C.T.: Bayesian model averaging: a tutorial. Stat. Sci. 14(4), 382–401 (1999). https://doi.org/10.1214/ss/1009212519 19. Jamil, H. (2018). ipriorBVS: Bayesian Variable Selection Using I-priors. R package version 0.1.1. https://github.com/haziqj/ipriorBVS 20. Jamil, H.: Regression modelling using priors depending on Fisher information covariance kernels (I-priors). Ph.D. thesis, London School of Economics and Political Science (2018) 21. Jeffreys, H.: An invariant form for the prior probability in estimation problems. Proc. Roy. Soc. A 186(1007), 453–461 (1946). https://doi.org/10.1098/rspa.1946.0056 22. Kahle, D., Wickham, H.: ggmap: spatial visualization with ggplot2. R J. 5(1), 144–161 (2013) 23. Kass, R.E., Raftery, A.E.: Bayes factors. J. Am. Stat. Assoc. 90(430), 773–795 (1995). https:// doi.org/10.2307/2291091 24. Kuo, L., Mallick, B.: Variable selection for regression models. Sankhy Indian J. Stat. Ser. B 601, 65–81 25. Kyung, M., Gill, J., Ghosh, M., Casella, G.: Penalized regression, standard errors, and Bayesian lassos. Bayesian Anal. 5(2), 369–411 (2010). https://doi.org/10.1214/10-BA607 26. Lee, K.E., Sha, N., Dougherty, E.R., Vannucci, M., Mallick, B.: Gene selection: a Bayesian variable selection approach. Bioinformatics 19(1), 90–97 (2003). https://doi.org/10.1093/ bioinformatics/19.1.90 27. Leisch, F., Dimitriadou, E.: mlbench: Machine Learning Benchmark Problems. R package version 2.1-1 (2010) 28. Lindley, D.V.: A statistical paradox. Biometrika 44(1–2), 187–192 (1957). https://doi.org/10. 1093/biomet/44.1-2.187 29. Madigan, D., Raftery, A.E.: Model selection and accounting for model uncertainty in graphical models using Occam’s window. J. Am. Stat. Assoc. 89(428), 1535–1546 (1994). https://doi. org/10.2307/2291017 30. Mallows, C.L.: Some comments on CP. Technometrics 15(4), 661–675 (1973). https://doi.org/ 10.2307/1267380 31. McDonald, G.C., Schwing, R.C.: Instabilities of regression estimates relating air pollution to mortality. Technometrics 15(3), 463–481 (1973). https://doi.org/10.2307/1266852 32. Miller, A.: Subset selection in regression. Chapman & Hall/CRC (2002). ISBN: 978-1-58488171-1 33. Mitchell, T.J., Beauchamp, J.J.: Bayesian variable selection in linear regression. J. Am. Stat. Assoc. 83(404), 1023–1032 (1988). https://doi.org/10.2307/2290129 34. Ntzoufras, I.: Bayesian modeling using WinBUGS. Wiley (2011). ISBN 978-0-470-14114-4. https://doi.org/10.1002/9780470434567 35. O’Hara, R.B., Sillanpää, M.J.: A review of Bayesian variable selection methods: what, how and which. Bayesian Anal. 4(1), 85–117 (2009). https://doi.org/10.1214/09-BA403 36. Ormerod, J.T., You, C., Mäller, S.: A variational Bayes approach to variable selection. Electron. J. Stat. 11(2), 3549–3594 (2017). https://doi.org/10.1214/17-EJS1332 37. Park, T., Casella, G.: The Bayesian Lasso. J. Am. Stat. Assoc. 103(482), 681–686 (2008). https://doi.org/10.1198/016214508000000337 38. Plummer, M.: JAGS: a program for analysis of Bayesian graphical models using Gibbs sampling. In: Hornik, K., Leisch, F., Zeileis, A. (eds.) Proceedings of the Third International Workshop on Distributed Statistical Computing (DSC 2003), Vienna, Austria (2003)

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On the Space of m-Subharmonic Functions Samsul Ariffin Abdul Karim

and Van Thien Nguyen

Abstract We consider radially symmetric m-subharmonic functions on the unit ball. We study their convexity and their relation with a solution of the complex Hessian equations. Furthermore, we consider the ordered vector space of δ radially symmetric m-subharmonic functions which is a Riesz space. Moreover, we shall define a space of m-subharmonic functions along with the Mabuchi metric. We also introduce a geodesic between two points in this space and give an equivalent condition when a curve on the space is geodesics. Keywords m-Subharmonic functions · Complex Hessian equations · Dirichlet problems · Radial functions

1 Introduction The real Hessian equation was studied for the first time in [12] by CaffarelliNirenberg-Spruck. They showed that the Dirichlet problem can be solved for smooth, strictly positive right hand side assumption, and certain convexity on the boundary of the domain and smooth boundary data. Then theory of real Hessian equation was developed by many authors in the near three decades. We recommend the reader a nice survey of Wang [40] and the references in it. The complex Hessian equation was investigated by Li [24] in 2004. After that Błocki extended the definition of the complex Hessian operator to non-smooth m-subharmonic functions. Later, many authors developed the theory of the comS. A. A. Karim (B) Fundamental and Applied Sciences Department and Centre for Systems Engineering (CSE), Institute of Autonomous System, Universiti Teknologi PETRONAS, Bandar Seri Iskandar, 32610 Seri Iskandar, Perak Darul Ridzuan, Malaysia e-mail: [email protected] V. T. Nguyen Department of Mathematics, FPT University, Education Zone, Hoa Lac High Tech Park, Thach That Ward, Hanoi, Vietnam e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 S. A. Abdul Karim (ed.), Theoretical, Modelling and Numerical Simulations Toward Industry 4.0, Studies in Systems, Decision and Control 319, https://doi.org/10.1007/978-981-15-8987-4_9

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plex Hessian operator: Dinew-Kołodziej [19, 20], Lu [25–27], Nguyen [29–31], Sadullaev-Abdullaev [37], Åhag-Czy˙z-Hed [6, 7]. Let SHmR (B) denote the family of m-subharmonic functions defined on the unit ball B that are non-positive, radially symmetric, and have zero value on ∂B. In Sect. 3, we show that the assertion u ∈ SHmR (B) is equivalent to the condition u˜ (r) = u(|z|) is an 2n increasing function that is convex with respect to r 2− m , where r = |z| and 1 ≤ m < n (see Theorem 3.8). In Sect. 4 we prove the following theorem. Theorem 1.1 Let F : [0, 1) → [0, ∞) be a non-decreasing function that is leftcontinuous and such that 1  m F(t)dt < ∞. 1 2

Let



1

u˜ (r) =

 F(t)dt.

2n m

−t 1− m

r

Then the function u defined on B by u(z) = u˜ (|z|) is in SHmR (B) and Hm (u) is the unique invariant measure that satisfies 1 Hm (u)(Br ) = F(r). 22n−m π n These are taken from [33]. We shall introduce the space of δ m-subharmonic functions in Sect. 5. We consider the vector space δEp,m = Ep,m − Ep,m generated by the cone Ep,m . By some straightforward calculations it can be checked that δEp,m is a vector space under point-wise addition and usual scalar multiplication. We also show that δEp,m along with a certain quasi-norm is a quasi-Banach space, and for p = 1 a Banach space (see Theorem 5.17). We also prove that there exists a decomposition of each element in δEp,m with control of the quasi-norm (see Theorem 5.18). This section is a part of [32]. In Sect. 6, we investigate the space of m-subharmonic functions. We introduce the Mabuchi space of strictly m-subharmonic functions in an m-hyperconvex domain. We also study the metric properties of this space using Mabuchi geodesics. Some theoretical results are given in this study. In Sect. 7, we show the connection between geodesics and homogeneous Hessian equations. The main result is important in the geometry of m-subharmonic functions.

2 Preliminaries Let  be an open set in Cn and let m be a natural number 1 ≤ m ≤ n. ¯ d c = i(∂¯ − ∂), and let β = dd c |z|2 be the canonical As usual let d = ∂ + ∂, n Kähler form in C . Denote by SHm () the set of all m-subharmonic functions in , and SHm− () for the set of all nonpositive m-subharmonic functions in . For

On the Space of m-Subharmonic Functions

135

u1 , . . . , um ∈ SHm () ∩ L∞ loc (), the operator Hm (u1 , · · · , um ) : = dd c u1 ∧ · · · ∧ dd c um ∧ β n−m = dd c (u1 dd c u2 ∧ · · · ∧ dd c um ∧ β n−m ) is a nonnegative Radon measure. In particular, when u = u1 = · · · = um , the measure Hm (u) := (dd c u)m ∧ β n−m is well defined for u ∈ SHm () ∩ L∞ loc () (see [11, 19], etc.). We list some elementary facts about m-subharmonic functions (see [25, 30]). Proposition 2.1 Let  ⊂ Cn be a bounded domain. Then we have (1) PSH () = SHn () ⊂ SHn−1 () ⊂ · · · ⊂ SH1 () = SH (). (2) If u, v ∈ SHm () then λu + μv ∈ SHm (), ∀λ, μ ≥ 0. (3) If u ∈ SHm () then its standard regularization u  χ is also m-subharmonic in  := {x ∈  : d (x, ∂) > }. (4) If u ∈ SHm () and γ : R → R is convex, nondecreasing function then γ ◦ u ∈ SHm (). (5) If u, v ∈ SHm () then max{u, v} ∈ SHm (). (6) Let {uα } ⊂ SHm () be a locally uniformly bounded sequence from above and u = sup uα . Then the upper semi-continuous regularization u is m-subharmonic function and equal to u almost everywhere. In [25, 28], Lu extended the results from [4, 14, 15] to some classes of m-subharmonic functions. Assume that  is an m-hyperconvex domain. With this assumption, the following class 

∞

E0,m () = {u ∈ SHm () ∩ L () : lim u(z) = 0,

Hm (u) < +∞},

z→∂



is nonempty. A function u ∈ SHm () belongs to Em () if for each z0 ∈ , there exists an open neighborhood U⊂  of z0 and a decreasing sequence {uj } ⊂ E0,m () such that uj ↓ u in U and supj  Hm (uj ) < +∞. Denote by Fm () the class of functions u ∈ SHm () such that  there exists a sequence {uj } ⊂ E0,m () decreasing to u in  and satisfying supj  Hm (uj ) < +∞. We note that each of the above classes is a convex cone. The complex Hessian operator is well-defined for the class Em () and Em () is the maximal domain of definition for the complex Hessian operator. For r > 0, let B(z0 , r) = {z ∈ Cn : |z − z0 | < r}, and to simplify the notation set B = B(0, 1) and Br = B(0, r). A function u : B → [−∞, ∞) is said to be radially symmetric if u(z) = u(|z|), ∀z ∈ B. For each radially symmetric function u : B → [−∞; ∞), we define the function u˜ : [0, 1) → [−∞, ∞) by u˜ (r) = u(|z|), where r = |z|.

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Definition 2.2 We say that a function f : I → [−∞, ∞) is convex with respect to h(r) (h is an invertible function) if the function g : a → f (h−1 (a)) is convex in h(I ), where I is an open set of R. Lemma 2.3 If u is radial symmetric m-subharmonic function in B, then (i) (ii) (iii) (iv)

u˜ in an increasing function. limt→1− u˜ (t) exists. {z ∈ B : u(z) = −∞} ⊂ {0}. u is continuous.

Proof Because each m-subharmonic function is subharmonic, the proof follows from [5, Lemma 5.2].  From the properties (ii) and (iv) of Lemma 2.3 without loss of generality, we can assume that (1) lim u(z) = 0 ∀ξ ∈ ∂B. z∈B,z→ξ

Definition 2.4 Denote by SHmR (B) the set of all nonpositive radially symmetric msubharmonic functions in B that satisfy (2). We denote by SH R (B) and PSH R (B) the cases when m = 1 and m = n respectively.

3 Convexity of Radial Symmetric m-Subharmonic Functions For r > 0, let B(z0 , r) = {z ∈ Cn : |z − z0 | < r}, and to simplify the notation set B = B(0, 1) and Br = B(0, r). A function u : B → [−∞, ∞) is said to be radially symmetric if u(z) = u(|z|, 0, . . . , 0), ∀z ∈ B. For each radially symmetric function u : B → [−∞; ∞), we define the function u˜ : [0, 1) → [−∞, ∞) by u˜ (r) = u(|z|, 0, . . . , 0), where r = |z|.

Definition 3.1 We say that a function f : I → [−∞, ∞) is convex with respect to h(r) (h is an invertible function) if the function g : a → f (h−1 (a)) is convex in h(I ), where I is an open set of R. Lemma 3.2 [33] If u is a radially symmetric m-subharmonic function in B, then (i) (ii) (iii) (iv)

u˜ in an increasing function. limt→1− u˜ (t) exists. {z ∈ B : u(z) = −∞} ⊂ {0}. u is continuous.

On the Space of m-Subharmonic Functions

137

Proof Since each m-subharmonic function is subharmonic, the proof follows from [5, Lemma 5.2].  From the properties (ii) and (iv) of Lemma 2.3 without loss of generality, we can assume that lim u(z) = 0 ∀ξ ∈ ∂B. (2) z∈B,z→ξ

Definition 3.3 Denote by SHmR (B) the set of all non-positive radially symmetric msubharmonic functions in B that satisfy (2). We denote by SH R (B) and PSH R (B) the cases when m = 1 and m = n respectively. In [5], the authors showed that u ∈ PSH R (B) is equivalent to the condition u˜ is an increasing function that is convex with respect to ln(r). In this section, we will prove the corresponding fact for m-subharmonic functions, 1 ≤ m < n. For u ∈ SHmR (B) ∩ C 2 (B), we can compute (see [18, 38])  ∂ 2u 1   u˜  (r) δjk , (z) = 3 r u˜ (r) − u˜  (r) zj z¯k + ∂zj ∂ z¯k 4r 2r where r = |z|, 1 ≤ j, k ≤ n and δjk is the Kronecker delta. Thus all n eigenvalues of the Hessian matrix of u can be computed as follows: 

λ1 = · · · = λn−1

u˜  (r) r u˜ (r) + u˜  (r) , λn = . = 2r 4r

Hence, for r = |z| = 0, we have 

σm (u)(z) =

1≤i1 1 we have gm (a) = u˜ (a

m 2m−2n

1 )=



2n m

−t 1− m

F(t)dt

m a 2m−2n

m = 2n − 2m

1  m

m

F(y 2m−2n )dy.

a

Thus we can compute 

gm (a) =

  m1 −1 m 2n−m m m 2m−2n 2m−2n ) F(a a F  (a 2m−2n ), 2 4(m − n) 2n

which implies gm is convex in (1, +∞). Thus u˜ is convex with respect to r 2− m , and therefore u(z) = u˜ (|z|) ∈ SHmR (B). Now we compute the Lelong number of u at 0. We have   dd c u ∧ (dd c ω0 )m−1 ∧ β n−m = d c u ∧ (dd c ω0 )m−1 ∧ β n−m Br

= =

n m n m

∂Br

m−1 2n(m−1)  −1 r− m d c u ∧ β n−1 ∂Br

 m−1 2n(m−1) −1 r − m 4n (n − 1)! u Br

⎞ ⎛  m−1 2n(m−1) 1 ∂ = −1 r − m 4n (n − 1)!r 2n−1 ⎝ 2n−1 u⎠ m ∂r r n

∂Br

⎞ ⎛ 1  m−1 2n n ∂ 2n  −1 = r m −1 22n+1 π n ⎝ −t 1− m m F(t)dt ⎠ m ∂r =

n m

m−1  −1 22n+1 π n m F(r).

r

This formula shows that if F(0) = 0 then νu (0) = 0. By Lemmas 4.2 and 4.3 if F(0) = 0 then Hm (u)({0}) = 0.

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For F(0) > 0, we have 1 u(z) =



2n m

−t 1− m

F(t)dt

|z|

1 =

−t

1− 2n m

 m

1 F(0)dt +

|z|

 2n  −t 1− m ( m F(t) − m F(0))dt

|z|

 m = ω0 (z) m F(0) + u0 (z), 2(n − m) and by the argument above Hm (u0 )({0}) = 0. Thus we proved that, Hm (u)({0}) = 22n−m π n F(0). For z = 0, we have Hm (u) = 22n−m−1 (n − 1)!

F  (|z|) d Vn . |z|2n−1

Thus, 1 1 Hm (u)(Br ) = 2n−m n Hm (u)({0}) 22n−m π n 2 π  F  (|z|) 1 22n−m−1 (n − 1)! 2n−1 d Vn + 2n−m n 2 π |z| Br \{0}

r = F(0) +

F  (t)dt = F(r).

0

For an arbitrary F, let {Fj } be a sequence of smooth, non-decreasing functions and Fj ↓ F. By the first step, we have uj (z) = u˜j (|z|) ∈ SHmR (B),

1 22n−m π n

Hm (uj )(Br ) = Fj (r).

and by monotone convergence theorem {uj } increases to u ∈ SHmR (B), such that 1 u˜ (r) = r



2n m

−t 1− m

F(t)dt.

On the Space of m-Subharmonic Functions

145

Let G(r) =

1 22n−m π n

Hm (u)(Br ).

As in the proof of Lemma 3.10, G(t) = F(t) almost everywhere. Since both functions are left-continuous, we have F ≡ G. Definition 4.4 Let MRm denote the set of non-negative Radon measures μ defined on B such that there exists a function u ∈ SHmR (B) with Hm (u) = μ. We can see that μ ∈ MRm is a unitary invariant measure. The following corollary is a consequence of Theorem 1.1. Corollary 4.5 [33] Let μ be a unitary invariant measure defined on B, and let F(t) = 2m−2n π −n μ(Bt ). Then μ ∈ MRm if and only if

1 √ m 1/2

F(t)dt is bounded.

5 The Space of Delta m-Subharmonic Function The δ-plurisubharmonic functions were studied by Cegrell (see [13]) and Kiselman (see [23]). Cegrell and Wiklund in [16] investigated the vector space δF = F − F equipped with a suitable norm. They proved that it is a non-separable Banach space and provided the characterization of its dual space. Hai and Hiep in [21] introduced a metric which defines a locally convex topology on the space δE of δ-plurisubharmonic functions from the Cegrell class E (see [15] for the definition of this class). They proved that with this topology δE is non-separable and non-reflexive Fréchet space. In [2], Åhag and Czy˙z proved that the vector space δEp with the vector ordering induced by the cone Ep is σ - Dedekind complete, and with suitable quasi-norm this space is a non-separable quasi-Banach space. They also characterized its topological dual. Recently in [3], Åhag, Cegrell and Czy˙z generalized these results for a cone K of negative plurisubharmonic functions with E0 ⊂ K ⊂ E. In this chapter, we want to extend the results in [2] to the case of m-subharmonic functions. We consider the vector space δEp,m = Ep,m − Ep,m generated by the cone Ep,m . By some straightforward calculations it can be checked that δEp,m is a vector space under point-wise addition and usual scalar multiplication. A way of dealing with the situation −∞ − (−∞) in this vector space is to implement the convention −∞ − (−∞) = −∞. We shall consider δEp,m with two vector orders; the order induced by the positive cone , and the classical point-wise ordering ≥. The order relations  and ≥ on δEp,m are related as follows: if u  v, then u ≤ v. There are functions u, v in δEp,m with u ≥ v, such that u and v are not comparable with respect to  (see Example 1).

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In Sect. 5.2, for u ∈ δEp,m we define

||u||p,m

⎧⎛ ⎫ 1 ⎞ m+p ⎪ ⎪ ⎨  ⎬ ⎝ [−(u1 + u2 )]p Hm (u1 + u2 )⎠ . = inf u=u1 −u2 ⎪ ⎪ ⎭ u1 ,u2 ∈Ep,m ⎩

(14)



Our aim is show that (δEp,m , || · ||p,m ) is a quasi-Banach space, and for p = 1 a Banach space (see Theorem 5.17). We also prove that there exists a decomposition of each element in δEp,m with control of the quasi-norm (see Theorem 5.18).

5.1 Riesz Spaces Let us start with giving some background on ordered vector spaces. For further information on ordered vector spaces and duality we refer the readers to [10]. Definition 5.1 A binary relation  on a set X is said to be an order relation on X if it satisfies the following three properties: (1) reflexivity: x  x, ∀x ∈ X , (2) antisymmetry: x  y and y  x imply x = y, (3) transitivity: x  y and y  z imply x  z. Definition 5.2 A non-empty subset K of a vector space X is a cone if it satisfies: (1) K + K ⊆ K, (2) rK ⊆ K, ∀r ≥ 0, and (3) K ∩ {−K} = {0}, here 0 is the additive identity element of X . Definition 5.3 An order relation X on a vector space X is said to be a vector ordering if X is compatible with the algebraic structure of X , i.e. (i) If x X y, then x + z X y + z, ∀z ∈ X ; (ii) If x X y, then rx X ry, ∀r ≥ 0. An order vector space (X , X ) is a vector space X with a vector ordering X . We denote X + = {x ∈ X : x X 0} for the positive cone of X . Let K be any cone in X , then it generates a vector ordering K on X defined by letting x K y whenever x − y ∈ K. To simplify the notation we shall use  instead of K . Definition 5.4 An ordered vector space (X , ) is a Riesz space (or a vector lattice space) if every pair of vectors x, y of X have a supremum x ∨ y and an infimum x ∧ y in X .   Remark 5.5 Since x ∧ y = − (−x) ∨ (−y) , to show that an ordered vector space is a Riesz space it is enough to prove that any two vectors have a supremum.

On the Space of m-Subharmonic Functions

147

Definition 5.6 An ordered vector space (X , ) is Dedekind σ -complete if every increasing sequence bounded from above has a supremum. Let δEp,m = Ep,m − Ep,m . We make a convention that −∞ − (−∞) = −∞ to handle the case −∞ − (−∞). Then δEp,m is a vector space over R equipped with pointwise addition of functions, and real scalar multiplication. We consider δEp,m with the vector ordering induced by the positive cone, i.e. for u, v ∈ δEp,m , we write u  v if u − v ∈ Ep,m . Note that u  0, for all u ∈ Ep,m although u(x) ≤ 0, for all x ∈ . One of the major advantages of this construction is that (δEp,m )+ = Ep,m . The usual point-wise vector ordering ≥ is defined as u ≥ v if and only if u(x) ≥ v(x), for all x ∈ . These two vector orderings on δEp,m are related as follows : if u  v then v ≥ u. But we do not have the inverse implication. Example 1 (see also [2, Example 3.1]) shows there are functions u, v in δEp,m with u ≥ v, but u, v are not comparable with respect to . In particular, δEp,m is not totally ordered vector space. Along with Mp,m , we are interested in the following set of measures Mp,m = {μ : μ = Hm (u) for some u ∈ Ep,m }. Then Mp,m is a cone. The ordered vector spaces (δMp,m , ) is defined similarly, i.e. μ, ν ∈ δMp,m , μ  ν if μ − ν ∈ Mp,m . Remark 5.7 [25, Theorem 0.0.1] implies that Mp,m is a cone, and if μ ∈ Mp,m and ν is any positive Radon measure such that μ ≥ ν then ν ∈ Mp,m . The usual ordering ≥ on δMp,m is defined as follows : if μ, ν ∈ δMp,m , then we say that μ ≥ ν if μ(A) ≥ ν(A) for every measurable subset A ⊆ . Theorem 5.8 [32] We have (a) The classical order and the order induced by the cone Mp,m coincide. (b) (δEp,m , ≥) and (δMp,m , ≥) are Riesz spaces. (c) (δEp,m , ) is Dedekind σ -complete. Proof In this proof we shall use an idea from [2]. (a) Let μ, ν ∈ Mp,m . If μ  ν, then μ − ν ∈ Mp,m , so μ ≥ ν. Now suppose that μ ≥ ν, then μ ≥ μ − ν ≥ 0, Remark 5.7 implies μ − ν ∈ Mp,m , so μ  ν. (b) Let u, v ∈ (δEp,m , ≥), we have u = u1 − u2 , v = v1 − v2 for some uj , vj ∈ Ep,m , j = 1, 2. Then u ∨≥ v = max(u, v) = max(u1 − u2 , v1 − v2 ) = max(u1 + v2 , u2 + v1 ) − (u2 + v2 ).

We get that u ∨≥ v ∈ δEp,m . Similarly, let μ, ν ∈ (δMp,m , ≥), then there exist μ1 , μ2 , ν1 , ν2 ∈ Mp,m such that μ = μ1 − μ2 , ν = ν1 − ν2 . We have μ ∨≥ ν = sup(μ1 − μ2 , ν1 − ν2 ) = sup(μ1 + ν2 , μ2 + ν1 ) − (μ2 + ν2 ),

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where sup(α, β)(A) := supB⊂A {α(B) + β(A\B)}, for positive measures α, β. We can see that sup(α, β) is the smallest measure majorant of α and β. Remark 5.7 implies that μ ∨≥ ν ∈ δMp,m . (c) Assume that {uj } is an increasing sequence in (δEp,m , ) which is bounded from above by φ, i.e. φ  uj for all j ∈ N. By the definition, for each j ∈ N, we have uj+1 − uj , φ − uj ∈ Ep,m . For k ≥ 2 k−1 

(uj+1 − uj ) ≥ (φ − uk ) +

k−1  (uj+1 − uj ) = φ − u1 ∈ Ep,m .

j=1

j=1

+∞ +∞ Letting k → +∞, we get j=1 (uj+1 − uj ) ≥ φ − u1 . The function γ = j=1 (uj+1 − uj ) is the limit of a decreasing sequence of m-subharmonic functions, so it is a negative m-subharmonic function and γ ≥ φ − u1 ∈ Ep,m . Hence, γ ∈ Ep,m . We put u = u1 + γ ∈ δEp,m . Now we shall prove that u = supj {uj }. In a similar way +∞ as above we have j=k (uj+1 − uj ) ∈ Ep,m for all k ≥ 2, so

u − uk = γ + u1 −

k−1 ∞   (uj+1 − uj ) − u1 = (uj+1 − uj ) ∈ Ep,m , ∀k ≥ 2. j=1

j=k

Thus u  uk for all k. Now suppose that v ∈ δEp,m is any upper bound of {uj }, it means that v  uj , or v − uj ∈ Ep,m , for all j ∈ N. For all k we have (v − uk+1 ) − (v − uk ) = uk − uk+1 ≥ 0, which means that {v − uk } is an increasing sequence of m-subharmonic functions with respect to the usual point-wise order ≥. Furthermore, the following limit exists: α = lim (v − uk ) = (v − u1 ) − k→+∞

+∞ 

(uj+1 − uj ) = (v − u1 ) − γ ,

j=1

therefore it follows that α ∗ = (v − u1 ) − γ ≥ (v − u1 ), where α ∗ denotes the upper semi-continuous regularization of α. It follows that α ∗ ∈ Ep,m . Thus, v − u = α ∗ , i.e. v  u, which completes part (c).  Remark 5.9 Example 3.3 in [3] shows that (δE0,n (B), ) is not a Riesz space. Example 1 Let ρ ∈ E0,m be a m-subharmonic function defining , and let z0 ∈ . Take a, b such that inf ρ < a < b < ρ(z0 ) < 0. Then two functions u = max(ρ, a) 

and v = max(ρ, b) are in E0,m (), and v ≥ u. Since u(z0 ) = v(z0 ) and the subharmonicity, we get that the function u − v is not subharmonic. Thus, u and v are not comparable with respect to the order .

On the Space of m-Subharmonic Functions

149

5.2 Normality We want to show that the formula in (14) defining a quasi-norm on the space δEp,m for p = 1, and a norm for p = 1. First, we prove a Hölder type inequality for functions in the class Ep,m . For m = n, p ≥ 1, Theorem 5.10 was proved in [34] and for m = n, 0 < p < 1 in [4]. The case p ≥ 1 was proved in [25, Lemma 1.7.8]. By using the idea of [4, Lemma 2.1] we will prove it for the case 0 < p < 1. Theorem 5.10 Let u0 , u1 , . . . , um ∈ Ep,m . Then there exists a constant D(p, m) depending only on p and m such that  (−u0 )p dd c u1 ∧ · · · ∧ dd c um ∧ β n−m  p

1

1

≤ D(p, m)ep,m (u0 ) p+m ep,m (u1 ) p+m · · · ep,m (um ) p+m , where

⎧ − α(p,m) ⎪ ⎨p 1−p , if 0 < p < 1, D(p, m) = 1, if p = 1, ⎪ ⎩ pα(p,m) p−1 , if p > 1, p

and α(p, m) = (p + 2)



p+1 p

m−1

− (p + 1).

Proof By using standard approximation, without loss of generality, we can assume that u0 , u1 , . . . , um ∈ E0,m . If 0 < p < 1, then −(−u0 )p ∈ E0,m . Now we let w = −(−u1 )p ∈ E0,m and T = dd c u2 ∧ · · · ∧ dd c um ∧ β n−m . We have 

 (−u0 ) dd u1 ∧ T = − p



1 =− p

1

(−u0 )p dd c (−w) p ∧ T

c



 (−u0 )p (−w)

1 p −1

dd c (−w) ∧ T

(15)



 1 1−p − (−u0 )p (−w) p −2 d (−w) ∧ d c (−w) ∧ T 2 p    1 1 1 −1 p c p ≤ (−u0 ) (−w) dd w ∧ T = (−u0 )p (−u1 )1−p dd c w ∧ T . p p 



Applying Hölder inequality and the integration by parts in E0,m we have

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⎤p ⎡ ⎤1−p  1 (−u0 )p dd c u1 ∧ T ≤ ⎣ (−u0 )dd c w ∧ T ⎦ ⎣ (−u1 )dd c w ∧ T ⎦ p ⎡











⎤p ⎡

⎤1−p  1⎣ = (−w)dd c u0 ∧ T ⎦ ⎣ (−w)dd c u1 ∧ T ⎦ (16) p ⎡







⎤p ⎡

⎡ =

⎤1−p





1⎣ (−u1 )p dd c u0 ∧ T ⎦ ⎣ (−u1 )p dd c u1 ∧ T ⎦ p 

.



From (15) and (16) we get  

≤

⎤p ⎡ ⎤1−p ⎡   1 (−u0 )p dd c u1 ∧ T ≤ ⎣ (−u1 )p dd c u0 ∧ T ⎦ ⎣ (−u1 )p dd c u1 ∧ T ⎦ p 

⎡ 1 p1+p





⎤p 2 ⎡



⎤p(1−p)

⎣ (−u0 )p dd c u1 ∧ T ⎦ ⎣ (−u0 )p dd c u0 ∧ T ⎦ 



⎡

⎤1−p

 × ⎣ (−u1 )p dd c u1 ∧ T ⎦

.



This implies that ⎛

 (−u0 )p dd c u1 ∧ T ≤ p 

1 − 1−p

p ⎞ 1+p



⎝ (−u0 )p dd c u0 ∧ T ⎠

(17)



1 ⎞ 1+p ⎛  × ⎝ (−u1 )p dd c u1 ∧ T ⎠ .



The function F : (E0,m )m+1 → R+ defined by  F(u0 , u1 , . . . , um ) =

(−u0 )p dd c u1 ∧ · · · ∧ dd c um ∧ β n−m 

is symmetric in the last m variables. By inequality (17) this function satisfies p

F(u0 , u1 , . . . , um ) ≤ p− 1−p F(u0 , u0 , u2 , . . . , um ) 1+p F(u1 , u1 , u2 , . . . , um ) 1+p . 1

1

On the Space of m-Subharmonic Functions

151

The rest of the proof goes verbatim as the proof of [34, Theorem 4.1]. Thus we get the desired result.  We want to show that the formula in (14) defining a quasi-norm on the space δEp,m for p = 1, and a norm for p = 1. Lemma 5.11 [32] For u, v ∈ Ep,m , we have   1 1 1 ep,m (u + v) p+m ≤ C(p, m) ep,m (u) p+m + ep,m (v) p+m ,

(18)

where C(p, m) > 1 is a constant depending only on m and p = 1, and C(1, m) = 1. Proof By Theorem 5.10 we have  ep,m (u + v) = =

m (−u − v)p dd c (u + v) ∧ β n−m

 m   k=0

 m (−u − v)p (dd c u)k ∧ (dd c v)m−k ∧ β n−m k 

≤ D(p, m)

m   m k=0

k

p

k

m−k

ep,m (u + v) p+m ep,m (u) p+m ep,m (v) p+m

 m p 1 1 = D(p, m)ep,m (u + v) p+m ep,m (u) p+m + ep,m (v) p+m . Hence ep,m (u + v) ≤ D(p, m)

p+m m

 m+p 1 1 ep,m (u) p+m + ep,m (v) p+m . 1

Thus we get inequality (18) with C(p, m) = D(p, m) m .



Remark 5.12 In general, if u1 , . . . , uk ∈ Ep,m , then 1

ep,m (u1 + · · · + uk ) p+m ≤

k−2 

1

C(p, m)j ep,m (uj ) p+m + C(p, m)k−1 (ep,m (uk−1 )

j=1 1

+ ep,m (uk )) p+m ≤

k 

1

C(p, m)j ep,m (uj ) p+m .

j=1

Lemma 5.13 [32] Let u, v ∈ Ep,m be such that v ≤ u. Then ep,m (u) ≤ D(p, m)

p+m p

ep,m (v),

where D(p, m) is the constant defined in Theorem 5.10. In addition if p ≤ 1, then ep,m (u) ≤ ep,m (v).

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Proof By Theorem 5.10 we have  ep,m (u) =

 (−u)p (dd c u)m ∧ β n−m ≤



(−v)p (dd c u)m ∧ β n−m 

p p+m

ep,m (u)

m p+m

,

ep,m (u) ≤ D(p, m)

p+m p

ep,m (v).

≤ D(p, m)ep,m (v) which implies that

If p ≤ 1, then by [28, Theorem 3.23] there exist decreasing sequences {uj }, {vj } ∈ E0,m such that uj ≥ vj and uj → u, vj → v, ep,m (uj ) → ep,m (u) and ep,m (vj ) → ep,m (v) as j → +∞. The function −(−uj )p ∈ SHm () ∩ L∞ () and limx→ξ [−(−uj )(x)] = 0 for all ξ ∈ ∂ by Proposition 2.1. Integrating by parts, we have  ep,m (uj ) =

 (−uj )p (dd c uj )m ∧ β n−m ≤



(−uj )p (dd c vj )m ∧ β n−m ≤ ep,m (vj ). 

By letting j → +∞ we get ep,m (u) ≤ ep,m (v).



For u ∈ δEp,m , the formula in (14) can be rewritten as follows 1

||u||p,m = inf{ep,m (u1 + u2 ) p+m : u = u1 − u2 , u1 , u2 ∈ Ep,m }.

(19)

1

Lemma 5.14 [32] If u ∈ Ep,m then ||u||p,m = ep,m (u) p+m . 1

Proof Since u = u − 0, then ||u||p,m ≤ ep,m (u) p+m . Let u1 , u2 ∈ Ep,m be such that u = u1 − u2 , then u ≥ u1 − u2 + 2u2 . We have  

m ep,m (u) = (−u)p (dd c u)m ∧ β n−m ≤ (−u)p dd c (u + 2u2 ) ∧ β n−m 

 ≤



m (−u1 − u2 ) dd c (u1 + u2 ) ∧ β n−m = ep,m (u1 + u2 ). p



Hence

1

1

ep,m (u1 + u2 ) p+m ≥ ep,m (u) p+m .

On the Space of m-Subharmonic Functions

153

Taking infimum over u1 , u2 ∈ Ep,m , u1 − u2 = u, we get 1

||u||p,m ≥ ep,m (u) p+m .  Now we recall the definition of a quasi-Banach space. Definition 5.15 A function || · ||: X → [0, +∞) is called a quasi-norm on a vector space X if it satisfies the following properties: (i) ||x|| = 0 if and only if x = 0; (ii) ||rx|| = |r|||x||, for all x ∈ X , r ∈ R; (iii) there exists a constant C ≥ 1 such that ||x + y|| ≤ C(||x|| + ||y||), ∀x, y ∈ X . Aoki [8] and Rolewicz [36] characterized quasi-norm as follows: Theorem 5.16 [8, 36] Let || · || be a quasi-norm on X . Then there exist 0 < q ≤ 1 and an equivalent quasi-norm ||| · ||| on X that satisfies for every x, y ∈ X |||x + y|||q ≤ |||x|||q + |||y|||q . Hence for given quasi-norm || · || on X , we can define the metric d (x, y) = |||x − y|||q on X . The vector space X is called a quasi-Banach space if it is complete with respect to the metric induced by the quasi-norm || · ||. Theorem 5.17 [32] (δEp,m , || · ||p,m ) is a quasi-Banach space for p = 1 and (δE1,m , || · ||1,m ) is a Banach space. Proof (i) If u = 0 ∈ Ep,m , then Lemma 5.14 implies ||u||p,m = 0. Assume that u ∈ δEp,m with ||u||p,m = 0. Let > 0. Then by the definition of ||u||p,m , there exist functions u1 , u2 ∈ Ep,m such that u = u1 − u2 and ep,m (u1 + u2 ) < . Since u1 + u2 ∈ Ep,m , then by [28, Theorem 3.23], there exists a decreasing sequence {vj } ⊂ E0,m , vj ↓ (u1 + u2 ) and supj ep,m (vj ) < . Let φ ∈ Ep,m be such that Hm (φ) = d Vn (by [28, Theorem 5.4]). It follows from Theorem 5.10 that   p m p ||vj ||Lp = (−vj )p d Vn = (−vj )p Hm (φ) ≤ D(p, m)ep,m (vj ) p+m ep,m (φ) p+m 

≤ C

p p+m



,

where C is a constant that does not depend on j. Hence p

p

p

||u||Lp ≤ ||u1 + u2 ||Lp ≤ C p+m .

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Letting → 0+ yields ||u||Lp = 0, thus u = 0 almost everywhere. It means that u1 = u2 almost everywhere in . Moreover u1 and u2 are subharmonic functions on , so we have u1 = u2 in , i.e. u = 0 in . (ii) Let u ∈ δEp,m . For t ∈ R, t > 0, we have 1

||tu||p,m = inf{ep,m (u1 + u2 ) p+m : tu = u1 − u2 , u1 , u2 ∈ Ep,m } 1

= inf{ep,m (tv1 + tv2 ) p+m : u = v1 − v2 , v1 , v2 ∈ Ep,m } = t||u||p,m . The case t < 0 is similar, and the case t = 0 is clear. (iii) Let u, v ∈ δEp,m and > 0. Then there exist functions u1 , u2 , v1 , v2 ∈ Ep,m such that u = u1 − u2 , v = v1 − v2 and 1

1

ep,m (u1 + u2 ) p+m ≤ ||u||p,m + , ep,m (v1 + v2 ) p+m ≤ ||v||p,m + . By Lemma 5.11, 1

||u + v||p,m ≤ ep,m (u1 + u2 + v1 + v2 ) p+m   1 1 ≤ C ep,m (u1 + u2 ) p+m + ep,m (v1 + v2 ) p+m ≤ C(||u||p,m + ||v||p,m ) + 2C , where C = C(p, m) is given in Lemma 5.11. Let → 0+ , we have ||u + v||p,m ≤ C(||u||p,m + ||v||p,m ). If p = 1 then C = C(1, m) = 1. This implies that || · ||1,m is a norm. (iv) Now we shall prove that the space (δEp,m , || · ||p,m ) is complete. Assume that {uj } is a Cauchy sequence in (δEp,m , || · ||p,m ). For each integer i, there is an integer ji such that (20) ||uji+1 − uji ||p,m ≤ (2C)−i . We can choose the ji to form an increasing sequence. Moreover, for each i, there exist vi , wi ∈ Ep,m such that 1

uji+1 − uji = vi − wi and ep,m (vi + wi ) p+m ≤ ||uji+1 − uji ||p,m + (2C)−i .

(21)

Note that ujk+1 = uj1 +

k k k k       uji+1 − uji = uj1 + (vi − wi ) = uj1 + vi − wi . i=1

i=1

By combining Remark 5.12, (21) and (20) we get that

i=1

i=1

(22)

On the Space of m-Subharmonic Functions

155

⎧ ⎨

1 1 ⎫ 1 & p+m % k & p+m % k & p+m % k ⎬    max ep,m vi , ep,m wi (vi + wi ) ≤ ep,m ⎩ ⎭

i=1

≤

k 

i=1

1

C i ep,m (vi + wi ) p+m ≤

k 

i=1

≤

k 

i=1

 C i (2C)−i + ||uji+1 − uji ||p,m

i=1



C (2C) i

−i

+ (2C)

−i



≤2

i=1

∞ 

2−i = 1.

i=1

The sequences { ki=1 vi }k and { ki=1 wi }k are decreasing sequences in Ep,m with bounded m-pluricomplex p-energy. Thus there exist functions ϕ, ψ ∈ Ep,m such that k k i=1 vi → ϕ, i=1 wi → ψ in (δEp,m , || · ||p,m ). By (22), we have ujk → uj1 + ϕ − ψ: = u ∈ δEp,m . Since {uj } is Cauchy sequence, it follows that uj → u.



The following theorem says that there exists a decomposition of each element in δEp,m with explicit control of quasi-norm. Theorem 5.18 [32] For each u ∈ δEp,m , there exist unique functions u+ , u− ∈ Ep,m such that u = u+ − u− , and 1

||u||p,m ≤ ||u+ + u− ||p,m ≤ D(p, m) p ||u||p,m . Furthermore, if p ≤ 1, then ||u||p,m = ||u+ + u− ||p,m . Proof Let u = u1 − u2 ∈ δEp,m , define u+ = sup{α ∈ Ep,m : there exists β ∈ Ep,m such that u2 + α = u1 + β}, and u− = sup{β ∈ Ep,m : there exists α ∈ Ep,m such that u2 + α = u1 + β}. Then (u+ )∗ , (u− )∗ ∈ Ep,m . By Choquet’s lemma, there exist sequences {αj }, {βj } ∈ E0,m such that (supj αj )∗ = (u+ )∗ and (supj βj )∗ = (u− )∗ . Furthermore, we can assume u2 + αj = u1 + βj . By passing to limits we obtain u2 + u+ = u1 + u− . Since u+ = (u+ )∗ and u− = (u− )∗ almost everywhere, we obtain u2 + (u+ )∗ = u1 + (u− )∗ . Hence u+ = (u+ )∗ and u− = (u− )∗ .

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If α, β ∈ Ep,m are such that u = α − β, then α ≤ u+ and β ≤ u− , so α + β ≤ u+ + u− . By Lemma 5.14 and Lemma 5.13 1

1

||u||p,m ≤ ep,m (u+ + u− ) p+m = ||u+ + u− ||p,m ≤ D(p, m) p ep,m (α + β). Taking infimum over all decomposition u = α − β, we get 1

||u||p,m ≤ ||u+ + u− ||p,m ≤ D(p, m) p ||u||p,m . If p ≤ 1, then by Lemma 5.13 ||u||p,m = ||u+ + u− ||p,m .



Remark 5.19 In general, let u = u1 − u2 be in δSHm− (), where  is a bounded domain in Cn . Then u+ = sup{α ∈ SHm− (): there exists β ∈ SHm− () such that u2 + α = u1 + β}, and u− = sup{β ∈ SHm− (): there exists α ∈ SHm− () such that u2 + α = u1 + β}. By reasoning as above, we can show that u+ , u− ∈ SHm− () and u = u+ − u− . For μ ∈ δMp,m , we define ( ' m m + ||uμ2 ||p,m : μ = μ1 − μ2 ; μ1 , μ2 ∈ Mp,m , |μ|p,m = inf ||uμ1 ||p,m where uμj ∈ Ep,m , j = 1, 2 are the uniquely determined solutions to Hm (uμj ) = μj , as in [25, Theorem 1.7.18]. Lemma 5.20 [32] Let μ = μ+ − μ− be the Jordan decomposition of μ, where μ+ =

1 1 (|μ| + μ) and μ− = (|μ| − μ). 2 2

Then m m |μ|p,m = ||uμ+ ||p,m + ||uμ− ||p,m .

Proof Suppose μ = μ1 − μ2 is any representation of μ ∈ δMp,m , then we see that μ+ ≤ μ1 and μ− ≤ μ2 . This implies that μ+ , μ− ∈ Mp,m and Hm (uμ+ ) ≤ Hm (uμ1 ). Hence, we have uμ+ ≥ uμ1 . Now ⎛



m ⎞ p+m

m = ⎝ (−uμ+ )p Hm (uμ+ )⎠ ||uμ+ ||p,m 

=

m ||uμ1 ||p,m .

m ⎛ ⎞ p+m  ≤ ⎝ (−uμ1 )p Hm (uμ1 )⎠



On the Space of m-Subharmonic Functions

157

m m Similarly, we also have ||uμ− ||p,m ≤ ||uμ2 ||p,m . Thus m m + ||uμ− ||p,m . |μ|p,m = ||uμ+ ||p,m

 Theorem 5.21 [32] (δMp,m , | · |p,m ) is a quasi-Banach space for p = 1, and it is a Banach space if p = 1. Proof (i) Suppose that μ ∈ δMp,m and |μ|p,m = 0. From Lemma 5.20 ||uμ+ ||p,m = ||uμ− ||p,m = 0. By the part (i) in Theorem 5.17, we have uμ+ = uμ− = 0. Thus μ+ = μ− = 0, so μ = 0. (ii) For t ≥ 0, we have (tμ)+ = tμ+ , (tμ)− = tμ− , 1

1

utμ+ = t m uμ+ and utμ− = t m uμ− . Hence 1

1

m m m m + ||u(tμ)− ||p,m = ||t m ||p,m + ||t m uμ− ||p,m = t|μ|p,m . |tμ|p,m = ||u(tμ)+ ||p,m

Similarly, if t < 0 then |tμ|p,m = (−t)|μ|p,m . (iii) Let μ, ν ∈ δMp,m , we have μ + ν = μ+ − μ− + ν + − ν − = (μ+ + ν + ) − (μ− + ν − ). Thus (μ + ν)+ ≤ μ+ + ν + and (μ + ν)− ≤ μ− + ν − . There exist functions u(μ+ν)+ , u(μ+ν)− ∈ Ep,m such that Hm (u(μ+ν)+ ) = (μ + ν)+ and Hm (u(μ+ν)− ) = (μ + ν)− . Applying Theorem 5.10, we obtain   ep,m u(μ+ν)+ =  ≤ 

 

 p   −u(μ+ν)+ Hm u(μ+ν)+ =

 p −u(μ+ν)+ (μ+ + ν + ) =

 



 p −u(μ+ν)+ (μ + ν)+



 p   −u(μ+ν)+ Hm (uμ+ ) + Hm (uν + )

   p  m m ≤ D(p, m)ep,m u(μ+ν)+ p+m ep,m (uμ+ ) p+m + ep,m (uν + ) p+m .

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    m m m ep,m u(μ+ν)+ p+m ≤ D(p, m) ep,m (uμ+ ) p+m + ep,m (uν + ) p+m .

Thus

Similarly,     m m m ep,m u(μ+ν)− p+m ≤ D(p, m) ep,m (uμ− ) p+m + ep,m (uν − ) p+m . We have   m   m m m + ||u(μ+ν)− ||p,m = ep,m u(μ+ν)+ p+m + ep,m u(μ+ν)− p+m |μ + ν|p,m = ||u(μ+ν)+ ||p,m   m m m m ≤ D(p, m) ep,m (uμ+ ) p+m + ep,m (uμ− ) p+m + ep,m (uν + ) p+m + ep,m (uν − ) p+m   m m m m = D(p, m) ||uμ+ ||p,m + ||uμ− ||p,m + ||uν + ||p,m + ||uν − ||p,m   = D(p, m) |μ|p,m + |ν|p,m , where D(p, m) is the constant given in Theorem 5.10. As D(1, m) = 1, | · |1,m is a norm. (iv) Now we shall prove that the space (δMp,m , | · |p,m ) is complete. Assume that {μj } is a Cauchy sequence in (δMp,m , | · |p,m ). For each integer i, there is an integer ji such that ) )μj

i+1

)) )) ))m ))m ) mi )) )) )) )) − μji )p,m = ))u(μji+1 −μji )+ )) + ))u(μji+1 −μji )− )) ≤ (2C)− p+m , p,m

p,m

where C = C(p, m) is constant given in Lemma 5.11. We can choose {ji } to be an increasing sequence. In particular, we have )) )) )) )) ))u(μji+1 −μji ) ))

p,m

≤ (2C)− p+m . i

(23)

Define μ = μj1 +

∞  

 μji+1 − μji ,

i=1

then μ+ ≤ μj+1 +

∞   + μji+1 − μji . i=1

(24)

On the Space of m-Subharmonic Functions

159

Now we have for any k ep,m

% k  i=1

& u(μj −μj ) i+1 i

+

≤

k  i=1

=

k  i=1

≤

k 

  C i ep,m u(μj −μj )+ (by Remark 5.12) i+1 i )) )) C i ))u(μj

−μji ) i+1

+

))p+m )) (by Lemma 5.14) )) p,m

C i (2C)−i ≤ 1 (by (5.10)).

i=1

'

( u(μj −μj )+ is a decreasing sequence in Ep,m with bounded m-pluricomi+1 i plex p-energy. Then there is a function u+ ∈ Ep,m such that ki=1 u(μj −μj )+ → u+ . i+1 i From this and (24) we obtain Thus

k i=1

  μ+ ≤ Hm uμj1 + u+ . Hence μ+ ∈ Mp,m . In a similar way one can prove that μ− ∈ Mp,m . Hence μji →  μ = μ+ − μ− in (δMp,m , | · |p,m ). This ends the proof. Corollary 5.22 [32] The cones Ep,m and Mp,m are closed in (δEp,m , || · ||p,m ) and (δMp,m , | · |p,m ) respectively. Theorem 5.23 [32] Let p > 0. Then the interior of Ep,m in (δEp,m , || · ||p,m ) is empty. The corresponding statement for (δMp,m , | · |p,m ) is also valid. Proof (i) We see that 0 is not an interior point of Ep,m . Assume that 0 = u ∈ Ep,m is an interior point of Ep,m in (δEp,m , || · ||p,m ). Then there exists > 0 such that if ||u − v||p,m < , then v ∈ Ep,m . We can find a subset B in  such that Hm (u)(B) > 0 and  1 1  2 m B (−u)p Hm (u) p+m < . Let w ∈ Ep,m be such that Hm (w) = 2χB Hm (u). Then / Ep,m . we have Hm (w)(B) > Hm (u)(B) which implies that the function v = u − w ∈ Now we have 1 Hm (w) ≤ 2Hm (u) = Hm (2 m u). 1

We obtain that 2 m u ≤ w. Hence ⎛ ||u − v||p,m = ||w||p,m = ep,m (w) ⎛ = ⎝2

 B

1 ⎞ p+m

(−w)p Hm (u)⎠

1 p+m

⎛ ≤ ⎝2

= ⎝ (−w)p Hm (w)⎠ 



1 ⎞ p+m



1 ⎞ p+m

(−2 u)p Hm (u)⎠ 1 m

B

This contradicts our assumption that u is an interior point of Ep,m .

< .

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(ii) We argue as above. The point 0 ∈ Mp,m is not an interior point of (δMp,m , | · |p,m ). Assume that 0 = μ ∈ Mp,m is an interior point of Mp,m in (Mp,m , | · |p,m ). Then there exists > 0 such that if |μ − ν|p,m < , then ν ∈ Mp,m . Let uμ ∈ Ep,m be such that Hm (uμ ) = d μ. As before, we can find B ⊂  such that μ(B) > 0 and  m  2 B (−uμ )p d μ p+m < . The measure ν = χ\B μ − χB μ is not an element of Mp,m since ν(B) < 0. [25, Theorem 1.7.18] implies that uμ ≤ uχB μ , where uχB μ ∈ Ep,m is such that Hm (uχB μ ) = χB μ. Hence,

m |μ − ν|p,m = 2|χB μ|p,m = 2||uχB μ ||p,m

⎛



m ⎞ p+m

 ≤ 2 ⎝ (−uμ )p d μ⎠

m ⎛ ⎞ p+m  = 2 ⎝ (−uχB μ )p Hm (uχB μ )⎠

< .

B



6 Space of Strictly m-Subharmonic Functions In this section, we shall define the metric in the space of strictly m-subharmonic functions on the m-hyperconvex domain. We introduction the readers to nice results [1, 17] for plurisubharmonic functions and Kähler manifolds settings. Let  be an m-hyperconvex domain in . We are going to define the Mabuchi space of m-subharmonic functions on . Definition 6.1 The Mabuchi space of m-subharmonic functions in  is M : = {ϕ ∈ C ∞ (, R) ∩ SHm,+ () : ϕ = 0 on ∂},

(25)

where SHm,+ () is the set of all strictly m-subharmonic functions defined on . And the tangent space of M at a point ϕ, denoted by Tϕ M as followed Tϕ M = {γ  (0) : γ : [− , ] → M, γ (0) = ϕ}.

(26)

Proposition 6.2 The tangent space of M at ϕ can be identified with Tϕ M ≡ {u ∈ C ∞ (, R) : u = 0 on ∂}. Proof Let v ∈ C ∞ (, R), v = 0 on ∂. Put γ (s) : = ϕ + sv. Then we have γ (0) = ϕ and γ  (0) = v. Moreover, since ϕ is strictly m-subharmonic, we can choose s is close enough to 0 that γ (s) ∈ M, which implies that v ∈ Tϕ M.

On the Space of m-Subharmonic Functions

161

Conversely, let γ : [− , ] → M. We have that γ (s) ∂ = 0 for every s ∈ [− , ]. This implies that γ  (0) ∂ = 0. Therefore, v = γ  (0) ∈ {u ∈ C ∞ (, R) : u = 0 on ∂}.  Definition 6.3 We define the Mabuchi metric on Tϕ M for each ϕ ∈ M as follows  v1 , v2 ϕ : =



v1 v2 Hm (ϕ),

where v1 , v2 ∈ Tϕ M. Definition 6.4 The energy functional on M is defined by 1 E(ϕt ) = 2

1  0



(ϕ˙t )2 Hm (ϕt ),

where ϕt is a path in M joining ϕ0 and ϕ1 . Definition 6.5 Geodesics between two points ϕ0 , ϕ1 in M are defined as the extremals of the energy functional of a path joining ϕ0 to ϕ1 . Theorem 6.6 The geodesics equation is obtained by computing the Euler-Lagrange equation of the functional E as following ϕ¨t Hm (ϕt ) = md ϕ˙t ∧ d c ϕ˙ t ∧ (dd c ϕt )m−1 ∧ β n−m . Proof Let (φs,t ) be a variation of ϕt , it means that φ0,t = ϕt , φs,0 = ϕ0 , φs,1 = ϕ1 and φs,t = 0 on ∂. We put ψt :=

∂φ ∂s

s=0 . We can see that ψ0 ≡ ψ1 ≡ 0 and ψt = 0 on ∂. Thus, φs,t = ϕt + sψt + o(s),

∂φs,t = ϕ˙ t + sψ˙ t + o(s), ∂t

and Hm (φs,t ) = [dd c (ϕt + sψt )]m ∧ β n−m + o(s) =Hm (ϕt ) + smdd c ψt ∧ (dd c ϕt )m−1 ∧ β n−m + o(s).

(27)

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S. A. A. Karim and V. T. Nguyen

We can compute 1 E(φs,t ) = 2

1   

0

=

1 2

1  0



2

∂φs,t ∂t

Hm (φs,t )

 (ϕ˙ t + sψ˙ t )2 Hm (ϕt ) + smdd c ψt ∧ (dd c ϕt )m−1 ∧ β n−m dt + o(s) 1 

=E(ϕt ) + s 0

sm ϕ˙t ψ˙ t Hm (ϕt )dt + 2 

(28)

1  0



(ϕ˙ t )2 dd c ψt

∧ (dd c ϕt )m−1 ∧ β n−m dt + o(s). We have known that ψ0 ≡ ψ1 ≡ 0. Using the integration by parts, we obtain that 1  0



ϕ˙t ψ˙ t Hm (ϕt )dt = −

1  0



 ψt d ϕ˙t ∧ d c ϕ˙t + ϕ˙t dd c ϕ˙t ∧ (dd c ϕt )m−1 ∧ β n−m dt. (29)

Moreover, by Stokes formula, 1  

0

(ϕ˙t )2 dd c ψt ∧ (dd c ϕt )m−1 ∧ β n−m dt

1  =2 0



ψt (d ϕ˙t ∧ d c ϕ˙t + ϕ˙t dd c ϕ˙ t ) ∧ (dd c ϕt )m−1 ∧ β n−m dt.

Combining (28), (29) and (30), we get that 1  E(φs,t ) = E(ϕt ) + s ∧ (dd c ϕt )

0  m−1



ψt − ϕ¨t (dd c ϕt )m + md ϕ˙t ∧ d c ϕ˙t

∧ β n−m dt + o(s).

Hence, E(φs,t ) − E(ϕt ) s→0 s 1 

 ψt −ϕ¨t (dd c ϕt )m + md ϕ˙t ∧ d c ϕ˙t ∧ (dd c ϕt )m−1 ∧ β n−m dt. = lim

0



(30)

On the Space of m-Subharmonic Functions

163

Therefore, (ϕt ) is a critical point of E if and only if ϕ¨t Hm (ϕt ) = md ϕ˙t ∧ d c ϕ˙ t ∧ (dd c ϕt )m−1 ∧ β n−m . 

7 Geodesics and Homogeneous Hessian Equations In this section, we shall give a bridge connecting geodesics and homogeneous Hessian equations. For each point (ϕt ), t ∈ [0, 1] in M, we set (z, ζ ) = ϕt (z), z ∈  and ζ = et+is ∈ T = {ξ ∈ C : 1 < |ξ | < e}. We shall prove that c Theorem 7.1 A curve (ϕt )0≤t≤1 is a geodesic if and only if [ddz,ζ (z, ζ )]m+1 ∧ n−m c 2 (z, ζ ) = 0, where β(z, ζ ) = dd |(z, ζ )| . β

Proof We see that c dz,ζ (z, ζ ) = dz  + dζ , dz,ζ  = dzc  + dζc .

Now, computing by hands, c  = (dz + dζ )(dzc  + dζc ) = dz dzc  + dz dζc  + dζ dzc  + dζ dζc  ddz,ζ

= dz dzc  + R(z, ζ ), where R(z, ζ ) = dz d c ζ  + dζ dzc  + dζ d c ζ . Thus, c )m+1 ∧ β n−m (z, ζ ) = [dz dzc  + R(z, ζ )]m+1 ∧ β n−m (z, ζ ) (ddz,ζ m+1   m+1 (dz dzc )k ∧ Rm+1−k ∧ β n−m (z, ζ ) = k k=0

=(dz dzc )m+1 ∧ β n−m (z, ζ ) + (m + 1)(dz dzc )m ∧ R ∧ β n−m (z, ζ ) m(m + 1) + (dz dzc )m−1 ∧ R2 ∧ β n−m (z, ζ ). 2 We have used the fact that R3 = R ∧ R ∧ R = 0.

(31)

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S. A. A. Karim and V. T. Nguyen

We shall using the techniques in [1] to show that the right hand side of (31) is zero. The first term in the right hand side of (31) equal to 0 by the bi-degree computation. Otherwise, we have ∂ ∂ dζ + = ϕ˙t (z)(d ζ + d ζ¯ ), dζ  = ∂ζ  + ∂¯ζ  = ∂ζ ∂ ζ¯

i i ∂ ∂ d ζ = ϕ˙t (z)(d ζ − d ζ¯ ), dζc  = d ζ¯ − 2 ∂ ζ¯ ∂ζ 2 c ¯ dζ dζ  = iϕ¨t (z)d ζ ∧ d ζ . These imply that i i R = R(z, ζ ) = iϕ¨t d ζ ∧ d ζ¯ + dz ϕ˙t ∧ d ζ¯ − dz ϕ˙t ∧ d ζ + dzc ϕ˙t ∧ d ζ + dzc ϕ˙t ∧ d ζ¯ , 2 2 R2 = 2idz ϕ˙t ∧ dzc ϕ˙t ∧ d ζ ∧ d ζ¯ .

Hence the second term in the right hand side of (31) is computed as (dz dzc )m ∧ R ∧ β n−m (z, ζ )

m

i = dz dzc ϕt (z) ∧ iϕt ¨(z)d ζ ∧ d ζ¯ + dz ϕ˙t ∧ d ζ¯ 2  i − dz ϕ˙t ∧ d ζ + dzc ϕ˙t ∧ d ζ + dzc ϕ˙t ∧ d ζ¯ ∧ β n−m (z, ζ ) 2 =iϕt ¨(z)(dz dzc ϕt (z))m ∧ d ζ ∧ d ζ¯ ∧ β n−m (z).

(32)

For the third term, we compute ˙ t ) ∧ dzc ϕ˙t ∧ d ζ ∧ d ζ¯ (dz dzc )m−1 ∧ R2 ∧ β n−m (z, ζ ) = (dz dzc ϕt (z))m−1 ∧ 2idz (ϕ (33) c m−1 c n−m = − 2i(dz dz ϕt (z)) ∧ dz ϕ˙t ∧ dz ϕ˙t ∧ d ζ ∧ d ζ¯ ∧ β (z). From (32) and (33), expression (31) is rewritten by

c )m+1 ∧ β n−m (z, ζ ) = (m + 1)i ϕ¨t (dz dzc ϕt )m ∧ β n−m (z) (ddz,ζ  − m(dz dzc ϕt )m−1 ∧ dz ϕ˙t ∧ dzc ϕ˙t ∧ β n−m (z) ∧ d ζ ∧ d ζ¯ . By Theorem 6.6, we obtain c )m+1 ∧ β n−m (z, ζ ) = 0 (ddz,ζ

if and only if the curve (ϕt )0≤t≤1 is a geodesic. These concludes the prove of Theorem 7.1. 

On the Space of m-Subharmonic Functions

165

8 Conclusion In this study we have established a connection between geodesics and homogeneous Hessian equations. This connection is important because the homogeneous Hessian equations has been investigated and it is easy to check when an m-subharmonic function is maximal, while geodesics notation in the space of m-subharmonic functions belongs to geometric category. This connection helps us more understand about m-subharmonic functions as well as their geometry.

References 1. Abja, S.: Geometry and topology of the space of plurisubharmonic functions. J. Geom. Anal. 29(1), 510–541 (2019) 2. Åhag, P., Czy˙z, R.: Modulability and duality of certain cones in pluripotential theory. J. Math. Anal. Appl. 361(2), 302–321 (2010) 3. Åhag, P., Cergell, U., Czy˙z, R.: Vector spaces of delta-plurisubharmonic functions and extensions of the complex Monge-Ampère operator. J. Math. Anal. Appl. 422(2), 960–980 (2015) 4. Åhag, P., Czy˙z, R., Hiep, P.H.: Concerning the energy class Ep for 0