EM E P O IN TS O F T H E V A N D ER M O N D E D ET ER M IN A N T A N D P H EN O M EN O LO G IC A L M O D EL LIN G W IT H P O W ER E X P O N EN TIA L F U N C TIO N S 2 019 ISBN 978-91-7485-431-2 ISSN 1651-4238
Address: P.O. Box 883, SE-721 23 Västerås. Sweden Address: P.O. Box 325, SE-631 05 Eskilstuna. Sweden E-mail: info@mdh.se Web: www.mdh.se
functions
Mälardalen University Press Dissertations No. 293
EXTREME POINTS OF THE VANDERMONDE
DETERMINANT AND PHENOMENOLOGICAL
MODELLING WITH POWER EXPONENTIAL FUNCTIONS
Karl Lundengård 2019
School of Education, Culture and Communication
Mälardalen University Press Dissertations No. 293
EXTREME POINTS OF THE VANDERMONDE
DETERMINANT AND PHENOMENOLOGICAL
MODELLING WITH POWER EXPONENTIAL FUNCTIONS
Karl Lundengård 2019
Copyright © Karl Lundengård, 2019 ISBN 978-91-7485-431-2
ISSN 1651-4238
Printed by E-Print AB, Stockholm, Sweden
Copyright © Karl Lundengård, 2019 ISBN 978-91-7485-431-2
ISSN 1651-4238
Mälardalen University Press Dissertations No. 293
EXTREME POINTS OF THE VANDERMONDE DETERMINANT AND PHENOMENOLOGICAL MODELLING WITH POWER EXPONENTIAL FUNCTIONS
Karl Lundengård
Akademisk avhandling
som för avläggande av filosofie doktorsexamen i matematik/tillämpad matematik vid Akademin för utbildning, kultur och kommunikation kommer att offentligen försvaras torsdagen den 26 september 2019, 13.15 i Delta, Mälardalens högskola, Västerås.
Fakultetsopponent: Professor Palle Jorgensen, University of Iowa
Akademin för utbildning, kultur och kommunikation
Mälardalen University Press Dissertations No. 293
EXTREME POINTS OF THE VANDERMONDE DETERMINANT AND PHENOMENOLOGICAL MODELLING WITH POWER EXPONENTIAL FUNCTIONS
Karl Lundengård
Akademisk avhandling
som för avläggande av filosofie doktorsexamen i matematik/tillämpad matematik vid Akademin för utbildning, kultur och kommunikation kommer att offentligen försvaras torsdagen den 26 september 2019, 13.15 i Delta, Mälardalens högskola, Västerås.
Fakultetsopponent: Professor Palle Jorgensen, University of Iowa
Abstract
This thesis discusses two topics, finding the extreme points of the Vandermonde determinant on various surfaces and phenomenological modelling using power-exponential functions. The relation between these two problems is that they are both related to methods for curve-fitting. Two applications of the mathematical models and methods are also discussed, modelling of electrostatic discharge currents for use in electromagnetic compatibility and modelling of mortality rates for humans. Both the construction and evaluation of models is discussed.
In the first chapter the basic theory for later chapters is introduced. First the Vandermonde matrix, a matrix whose rows (or columns) consists of monomials of sequential powers, its history and some of its properties are discussed. Next, some considerations and typical methods for a common class of curve fitting problems are presented, as well as how to analyse and evaluate the resulting fit. In preparation for the later parts of the thesis the topics of electromagnetic compatibility and mortality rate modelling are briefly introduced.
The second chapter discusses some techniques for finding the extreme points for the determinant of the Vandermonde matrix on various surfaces including spheres, ellipsoids and cylinders. The discussion focuses on low dimensions, but some results are given for arbitrary (finite) dimensions.
In the third chapter a particular model called the p-peaked Analytically Extended Function (AEF) is introduced and fitted to data taken either from a standard for electromagnetic compatibility or experimental measurements. The discussion here is entirely focused on currents originating from lightning or electrostatic discharges.
The fourth chapter consists of a comparison of several different methods for modelling mortality rates, including a model constructed in a similar way to the AEF found in the third chapter. The models are compared with respect to how well they can be fitted to estimated mortality rate for several countries and several years and the results when using the fitted models for mortality rate forecasting is also compared.
ISBN 978-91-7485-431-2 ISSN 1651-4238
Abstract
This thesis discusses two topics, finding the extreme points of the Vandermonde determinant on various surfaces and phenomenological modelling using power-exponential functions. The relation between these two problems is that they are both related to methods for curve-fitting. Two applications of the mathematical models and methods are also discussed, modelling of electrostatic discharge currents for use in electromagnetic compatibility and modelling of mortality rates for humans. Both the construction and evaluation of models is discussed.
In the first chapter the basic theory for later chapters is introduced. First the Vandermonde matrix, a matrix whose rows (or columns) consists of monomials of sequential powers, its history and some of its properties are discussed. Next, some considerations and typical methods for a common class of curve fitting problems are presented, as well as how to analyse and evaluate the resulting fit. In preparation for the later parts of the thesis the topics of electromagnetic compatibility and mortality rate modelling are briefly introduced.
The second chapter discusses some techniques for finding the extreme points for the determinant of the Vandermonde matrix on various surfaces including spheres, ellipsoids and cylinders. The discussion focuses on low dimensions, but some results are given for arbitrary (finite) dimensions.
In the third chapter a particular model called the p-peaked Analytically Extended Function (AEF) is introduced and fitted to data taken either from a standard for electromagnetic compatibility or experimental measurements. The discussion here is entirely focused on currents originating from lightning or electrostatic discharges.
The fourth chapter consists of a comparison of several different methods for modelling mortality rates, including a model constructed in a similar way to the AEF found in the third chapter. The models are compared with respect to how well they can be fitted to estimated mortality rate for several countries and several years and the results when using the fitted models for mortality rate forecasting is also compared.
ISBN 978-91-7485-431-2 ISSN 1651-4238
Acknowledgements
Many thanks to all my coauthors and supervisors. My main supervisor, Pro-fessor Sergei Silvestrov, introduced me to the Vandermonde matrix and fre-quently suggested new problems and research directions throughout my time as a doctoral student. I have learned many lessons about mathematics and academia from him and my co-supervisor Professor Anatoliy Malyarenko. My other co-supervisor Dr. Milica Ranˇci´c played a crucial role and she is a role model with regards to conscientiousness, work ethic, communication and patience. I have learned invaluable lessons about interdisciplinary re-search, communication and time and resource management from her. I also want to thank Dr. Vesna Javor for her regular input that improved the research on electromagnetic compatibility considerably.
Cooperating with other doctoral students was very valuable. Jonas ¨
Osterberg and Asaph Keikara Muhumuza (with support from his super-visors Dr. John M. Mango and Dr. Godwin Kakuba) made important contributions to the research on the Vandermonde determinant and Samya Suleiman’s understanding of mortality rate forecasting and other aspects of actuarial mathematics was necessary for the work to progress.
I am also glad that I had the opportunity to take part in the supervision of talented master students Andromachi Boulogari and Belinda Strass and use the foundations they laid in their degree projects for further research.
Many thanks to all my coworkers at M¨alardalen University, especially to Dr. Christopher Engstr¨om, Dr. Johan Richter and Docent Linus Carls-son for managing the bachelor’s and master’s programmes in Engineering mathematics together with me.
Perhaps most importantly, I thank my family for all the support, en-couragement and assistance you have given me. A special mention to my sister for help with translating from 18th century French, it is perfectly un-derstandable that you decided to move to the other side of the Earth after that. I will wonder my whole life how my father, whose entire mathematics career consisted of unsuccessfully solving a single problem on the blackboard in 9th grade, would have reacted to this dissertation if he were still with us. Fortunately my mother continues to be an endless source of support and encouragement. I am continually surprised and delighted over how much of her work ethics, sense of quality and unhealthy work habits I seem to have inherited from her.
Without the ideas, requests, remarks, questions, encouragements and patience of those around me this work would not have been completed.
Karl Lundeng˚ard V¨aster˚as, September, 2019
Acknowledgements
Many thanks to all my coauthors and supervisors. My main supervisor, Pro-fessor Sergei Silvestrov, introduced me to the Vandermonde matrix and fre-quently suggested new problems and research directions throughout my time as a doctoral student. I have learned many lessons about mathematics and academia from him and my co-supervisor Professor Anatoliy Malyarenko. My other co-supervisor Dr. Milica Ranˇci´c played a crucial role and she is a role model with regards to conscientiousness, work ethic, communication and patience. I have learned invaluable lessons about interdisciplinary re-search, communication and time and resource management from her. I also want to thank Dr. Vesna Javor for her regular input that improved the research on electromagnetic compatibility considerably.
Cooperating with other doctoral students was very valuable. Jonas ¨
Osterberg and Asaph Keikara Muhumuza (with support from his super-visors Dr. John M. Mango and Dr. Godwin Kakuba) made important contributions to the research on the Vandermonde determinant and Samya Suleiman’s understanding of mortality rate forecasting and other aspects of actuarial mathematics was necessary for the work to progress.
I am also glad that I had the opportunity to take part in the supervision of talented master students Andromachi Boulogari and Belinda Strass and use the foundations they laid in their degree projects for further research.
Many thanks to all my coworkers at M¨alardalen University, especially to Dr. Christopher Engstr¨om, Dr. Johan Richter and Docent Linus Carls-son for managing the bachelor’s and master’s programmes in Engineering mathematics together with me.
Perhaps most importantly, I thank my family for all the support, en-couragement and assistance you have given me. A special mention to my sister for help with translating from 18th century French, it is perfectly un-derstandable that you decided to move to the other side of the Earth after that. I will wonder my whole life how my father, whose entire mathematics career consisted of unsuccessfully solving a single problem on the blackboard in 9th grade, would have reacted to this dissertation if he were still with us. Fortunately my mother continues to be an endless source of support and encouragement. I am continually surprised and delighted over how much of her work ethics, sense of quality and unhealthy work habits I seem to have inherited from her.
Without the ideas, requests, remarks, questions, encouragements and patience of those around me this work would not have been completed.
Popul¨
arvetenskaplig sammanfattning
Det finns m˚anga f¨oreteelser i v¨arlden som det ¨ar ¨onskv¨art att beskriva med en matematisk modell. I b¨asta fall kan modellen h¨arledas ifr˚an l¨amplig grundl¨aggande teori men ibland ¨ar det inte m¨ojligt att g¨ora det, antingen d¨arf¨or att det inte finns n˚agon v¨al utvecklad teori eller f¨or att den teori som finns kr¨aver information som inte ¨ar tillg¨anglig. I detta fall s˚a beh¨ovs en modell som, i n˚agon m˚an, st¨ammer ¨overens med teori och empiriska observa-tioner men som inte ¨ar h¨arledd fr˚an den grundl¨aggande teorin. S˚adana mod-eller kallas f¨or fenomenologiska modeller. I denna avhandling konstrueras fenomenologiska modeller av tv˚a olika fenomen, str¨ommen i elektrostatiska urladdningar och d¨odsrisk.
Elektrostatiska urladdningar sker n¨ar laddning snabbt fl¨odar fr˚an ett objekt till ett annat. V¨albekanta exempel ¨ar blixtnedslag eller sm˚a st¨otar orsakade av statisk elektricitet. F¨or ingenj¨orer ¨ar det viktigt att kunna beskriva denna typ av elektriska str¨ommar f¨or att se till att elektroniska system inte ¨ar f¨or k¨ansliga f¨or elektromagnetisk p˚averkan utifr˚an och att de inte st¨or andra system d˚a de anv¨ands.
D¨odsrisken beskriver sannolikheten f¨or d¨od vid en viss ˚alder. Den kan anv¨andas f¨or att uppskatta livskvaliteten i ett land eller andra demografiska eller f¨ors¨akringsrelaterade ¨andam˚al.
En egenskap hos b˚ade elektrostatiska urladdningar och d¨odsrisk som kan vara utmanande att modellera ¨ar omr˚aden d¨ar en brant ¨okning f¨oljs av en l˚angsam s¨ankning. S˚adana m¨onster f¨orekommer ofta i elektrostatiska urladdningar och i m˚anga l¨ander ¨okar d¨odsrisken kraftigt vid ¨overg˚angen fr˚an barn till vuxen och f¨or¨andras sedan l˚angsamt fram till tidig medel˚alder. I denna avhandling anv¨ands en matematisk funktion som kallas poten-sexponentialfunktionen som en byggsten f¨or att konstruera fenomenologiska modeller av str¨ommen i elektrostatiska urladdningar samt d¨odsrisk utifr˚an empiriska data f¨or respektive fenomen. F¨or elektrostatiska urladdningar f¨oresl˚as en metod som kan konstruera modeller med olika noggrannhet och komplexitet. F¨or d¨odsrisker f¨oresl˚as n˚agra enkla modeller som sedan j¨amf¨ors med tidigare f¨oreslagna modeller.
I avhandlingen diskuteras ocks˚a extrempunkterna hos Vandermonde de-terminanten. Detta ¨ar ett matematiskt problem som f¨orekommer inom flera olika omr˚aden men f¨or avhandlingen ¨ar den mest relevanta till¨ampningen att extrempunkterna kan hj¨alpa till att v¨alja l¨ampliga data att anv¨anda n¨ar man konstruerar modeller med hj¨alp av en teknik som kallas f¨or optimal design. N˚agra allm¨anna resultat f¨or hur extrempunkterna kan hittas p˚a di-verse ytor, t.ex. sf¨arer och kuber, presenteras och det ges exempel p˚a hur resultaten kan till¨ampas.
Popul¨
arvetenskaplig sammanfattning
Det finns m˚anga f¨oreteelser i v¨arlden som det ¨ar ¨onskv¨art att beskriva med en matematisk modell. I b¨asta fall kan modellen h¨arledas ifr˚an l¨amplig grundl¨aggande teori men ibland ¨ar det inte m¨ojligt att g¨ora det, antingen d¨arf¨or att det inte finns n˚agon v¨al utvecklad teori eller f¨or att den teori som finns kr¨aver information som inte ¨ar tillg¨anglig. I detta fall s˚a beh¨ovs en modell som, i n˚agon m˚an, st¨ammer ¨overens med teori och empiriska observa-tioner men som inte ¨ar h¨arledd fr˚an den grundl¨aggande teorin. S˚adana mod-eller kallas f¨or fenomenologiska modeller. I denna avhandling konstrueras fenomenologiska modeller av tv˚a olika fenomen, str¨ommen i elektrostatiska urladdningar och d¨odsrisk.
Elektrostatiska urladdningar sker n¨ar laddning snabbt fl¨odar fr˚an ett objekt till ett annat. V¨albekanta exempel ¨ar blixtnedslag eller sm˚a st¨otar orsakade av statisk elektricitet. F¨or ingenj¨orer ¨ar det viktigt att kunna beskriva denna typ av elektriska str¨ommar f¨or att se till att elektroniska system inte ¨ar f¨or k¨ansliga f¨or elektromagnetisk p˚averkan utifr˚an och att de inte st¨or andra system d˚a de anv¨ands.
D¨odsrisken beskriver sannolikheten f¨or d¨od vid en viss ˚alder. Den kan anv¨andas f¨or att uppskatta livskvaliteten i ett land eller andra demografiska eller f¨ors¨akringsrelaterade ¨andam˚al.
En egenskap hos b˚ade elektrostatiska urladdningar och d¨odsrisk som kan vara utmanande att modellera ¨ar omr˚aden d¨ar en brant ¨okning f¨oljs av en l˚angsam s¨ankning. S˚adana m¨onster f¨orekommer ofta i elektrostatiska urladdningar och i m˚anga l¨ander ¨okar d¨odsrisken kraftigt vid ¨overg˚angen fr˚an barn till vuxen och f¨or¨andras sedan l˚angsamt fram till tidig medel˚alder. I denna avhandling anv¨ands en matematisk funktion som kallas poten-sexponentialfunktionen som en byggsten f¨or att konstruera fenomenologiska modeller av str¨ommen i elektrostatiska urladdningar samt d¨odsrisk utifr˚an empiriska data f¨or respektive fenomen. F¨or elektrostatiska urladdningar f¨oresl˚as en metod som kan konstruera modeller med olika noggrannhet och komplexitet. F¨or d¨odsrisker f¨oresl˚as n˚agra enkla modeller som sedan j¨amf¨ors med tidigare f¨oreslagna modeller.
I avhandlingen diskuteras ocks˚a extrempunkterna hos Vandermonde de-terminanten. Detta ¨ar ett matematiskt problem som f¨orekommer inom flera olika omr˚aden men f¨or avhandlingen ¨ar den mest relevanta till¨ampningen att extrempunkterna kan hj¨alpa till att v¨alja l¨ampliga data att anv¨anda n¨ar man konstruerar modeller med hj¨alp av en teknik som kallas f¨or optimal design. N˚agra allm¨anna resultat f¨or hur extrempunkterna kan hittas p˚a di-verse ytor, t.ex. sf¨arer och kuber, presenteras och det ges exempel p˚a hur resultaten kan till¨ampas.
Popular science summary
There are many phenomena in the world that it is desirable to describe using a mathematical model. Ideally the mathematical model is derived from the appropriate fundamental theory but sometimes this is not feasible, either because the fundamental theory is not well understood or because the theory requires a lot of information to be applicable. In these cases it is necessary to create a model that, to some degree, matches the fundamental theory and the empirical observations, but is not derived from the fundamental theory. Such models are called phenomenological models. In this thesis phenomenological models are constructed for two phenomena, electrostatic discharge currents and mortality rates.
Electrostatic discharge currents are rapid flows of electric charge from one object to another. Well-known examples are lightning strikes or small electric chocks caused by static electricity. Describing such currents is im-portant when engineers want to ensure that electronic systems are not dis-turbed too much by external electromagnetic disturbances or disturbs other systems when used.
Mortality rate describes the probability of a dying at certain age. It can be used to assess the quality of life in a country or for other demographical or actuarial purposes.
For electrostatic discharge currents and mortality rates an important feature that can be challenging to model is a steep increase followed by a slower decrease. This pattern is often observed in electrostatic discharge currents and in many countries the mortality rate increases rapidly in the transition from childhood to adulthood and then changes slowly until the beginning of middle age.
In this thesis a mathematical function called the power-exponential func-tion is used as a building block to construct phenomenological models of electrostatic discharge currents and mortality rates based on empirical data for the respective phenomena. For electrostatic discharge currents a method-ology for constructing models with different accuracy and complexity is pro-posed. For the mortality rates a few simple models are suggested and com-pared to previously suggested models.
The thesis also discusses the extreme points of the Vandermonde deter-minant. This is a mathematical problem that appears in many areas but for this thesis the most relevant application is that it helps choosing the appropriate data to use when constructing a model using a technique called optimal design. Some general results for finding the extreme points of the Vandermonde determinant on various surfaces, e.g. spheres or cubes, and applications of these results are discussed.
Popular science summary
There are many phenomena in the world that it is desirable to describe using a mathematical model. Ideally the mathematical model is derived from the appropriate fundamental theory but sometimes this is not feasible, either because the fundamental theory is not well understood or because the theory requires a lot of information to be applicable. In these cases it is necessary to create a model that, to some degree, matches the fundamental theory and the empirical observations, but is not derived from the fundamental theory. Such models are called phenomenological models. In this thesis phenomenological models are constructed for two phenomena, electrostatic discharge currents and mortality rates.
Electrostatic discharge currents are rapid flows of electric charge from one object to another. Well-known examples are lightning strikes or small electric chocks caused by static electricity. Describing such currents is im-portant when engineers want to ensure that electronic systems are not dis-turbed too much by external electromagnetic disturbances or disturbs other systems when used.
Mortality rate describes the probability of a dying at certain age. It can be used to assess the quality of life in a country or for other demographical or actuarial purposes.
For electrostatic discharge currents and mortality rates an important feature that can be challenging to model is a steep increase followed by a slower decrease. This pattern is often observed in electrostatic discharge currents and in many countries the mortality rate increases rapidly in the transition from childhood to adulthood and then changes slowly until the beginning of middle age.
In this thesis a mathematical function called the power-exponential func-tion is used as a building block to construct phenomenological models of electrostatic discharge currents and mortality rates based on empirical data for the respective phenomena. For electrostatic discharge currents a method-ology for constructing models with different accuracy and complexity is pro-posed. For the mortality rates a few simple models are suggested and com-pared to previously suggested models.
The thesis also discusses the extreme points of the Vandermonde deter-minant. This is a mathematical problem that appears in many areas but for this thesis the most relevant application is that it helps choosing the appropriate data to use when constructing a model using a technique called optimal design. Some general results for finding the extreme points of the Vandermonde determinant on various surfaces, e.g. spheres or cubes, and applications of these results are discussed.
Notation
Matrix and vector notation
v, M - Bold, roman lower- and uppercase letters denote vectors and matrices respectively. Mi,j - Element on the ith row and jth column of M.
M·,j, Mi,· - Column (row) vector containing all elements
from the jth column (ith row) of M. [aij]nmij - n × m matrix with element aij in
the ith row and jth column. Vnm, Vn= Vnn - n × m Vandermonde matrix.
Gnm, Gn= Gnn - n × m generalized Vandermonde matrix.
Anm, An= Ann - n × m alternant matrix.
Standard sets
Z, N, R, C - Sets of all integers, natural numbers (including 0), real numbers and complex numbers.
Snp, Sn= Sn2 - The n-dimensional sphere defined by the p - norm, Snp(r) = ( x ∈ Rn+1 n+1 X k=1 |xk|p= r ) . Ck
[K] - All functions on K with continuous kth derivative. Special functions
Definitions can be found in standard texts.
Suggested sources use notation consistent with thesis.
Hn, Pn(α,β) - Hermite and Jacobi polynomials, see [2].
Γ(x), γ(x, y), ψ(x) - The Gamma-, incomplete Gamma and Digamma functions, see [2].
2F2(a, b; c; x) - The hypergeometric function, see [2].
Gm,np,q z a b
- The Meijer G-function, see [236]. Ei(x) - The exponential integral, see [2].
Notation
Matrix and vector notation
v, M - Bold, roman lower- and uppercase letters denote vectors and matrices respectively. Mi,j - Element on the ith row and jth column of M.
M·,j, Mi,· - Column (row) vector containing all elements
from the jth column (ith row) of M. [aij]nmij - n × m matrix with element aij in
the ith row and jth column. Vnm, Vn= Vnn - n × m Vandermonde matrix.
Gnm, Gn= Gnn - n × m generalized Vandermonde matrix.
Anm, An= Ann - n × m alternant matrix.
Standard sets
Z, N, R, C - Sets of all integers, natural numbers (including 0), real numbers and complex numbers.
Snp, Sn= Sn2 - The n-dimensional sphere defined by the p - norm, Snp(r) = ( x ∈ Rn+1 n+1 X k=1 |xk|p= r ) . Ck
[K] - All functions on K with continuous kth derivative. Special functions
Definitions can be found in standard texts.
Suggested sources use notation consistent with thesis.
Hn, Pn(α,β) - Hermite and Jacobi polynomials, see [2].
Γ(x), γ(x, y), ψ(x) - The Gamma-, incomplete Gamma and Digamma functions, see [2].
2F2(a, b; c; x) - The hypergeometric function, see [2].
Gm,np,q z a b
- The Meijer G-function, see [236]. Ei(x) - The exponential integral, see [2].
Probability theory and statistics Pr[A] - Probability of event A.
Pr[A|B] - Conditional probability of event A given B. EX[Y ] - Expected value of quantity Y with respect to X. Var(X) - Variance of X.
AIC - Akaike Information Criterion, see Definition 1.14. AICC - Second order correction of the AIC, see Remark 1.9.
I(f, g) - Kullback–Leibler divergence, see Definition 1.15. Mortality rate
Sx(∆x) - Survival function, see Definition 1.19.
Tx - Remaining lifetime for an individual of age x.
µ(x) - Mortality rate at age x, see Definition 1.20.
mx,t - Central mortality rate at age x, year t, see page 66.
Other df dx = f
0(x) - Derivative of the function f with respect to x.
dkf
dxk = f
(k)(x) - kth derivative of the function f with respect to x.
∂f ∂x= f
0(x) - Partial derivative of the function f with respect to x.
a¯b - Rising factorial a¯b= a(a + 1) · · · (a + b − 1).
Probability theory and statistics Pr[A] - Probability of event A.
Pr[A|B] - Conditional probability of event A given B. EX[Y ] - Expected value of quantity Y with respect to X. Var(X) - Variance of X.
AIC - Akaike Information Criterion, see Definition 1.14. AICC - Second order correction of the AIC, see Remark 1.9.
I(f, g) - Kullback–Leibler divergence, see Definition 1.15. Mortality rate
Sx(∆x) - Survival function, see Definition 1.19.
Tx - Remaining lifetime for an individual of age x.
µ(x) - Mortality rate at age x, see Definition 1.20.
mx,t - Central mortality rate at age x, year t, see page 66.
Other df dx = f
0(x) - Derivative of the function f with respect to x.
dkf
dxk = f
(k)(x) - kth derivative of the function f with respect to x.
∂f ∂x= f
0(x) - Partial derivative of the function f with respect to x.
Contents
List of Papers 13
1 Introduction 15
1.1 The Vandermonde matrix . . . 19
1.1.1 Who was Vandermonde? . . . 19
1.1.2 The Vandermonde determinant . . . 21
1.1.3 Inverse of the Vandermonde matrix . . . 25
1.1.4 The alternant matrix . . . 26
1.1.5 The generalized Vandermonde matrix . . . 29
1.1.6 The Vandermonde determinant in systems with Coulombian interactions . . . 30
1.1.7 The Vandermonde determinant in random matrix theory . . . 33
1.2 Curve fitting . . . 37
1.2.1 Linear interpolation . . . 37
1.2.2 Generalized divided differences and interpolation . . . 42
1.2.3 Least squares fitting . . . 45
1.2.4 Linear least squares fitting . . . 45
1.2.5 Non-linear least squares fitting . . . 46
1.2.6 The Marquardt least squares method . . . 47
1.3 Analysing how well a curve fits . . . 50
1.3.1 Regression . . . 50
1.3.2 Quantile-Quantile plots . . . 52
Contents
List of Papers 13 1 Introduction 15 1.1 The Vandermonde matrix . . . 191.1.1 Who was Vandermonde? . . . 19
1.1.2 The Vandermonde determinant . . . 21
1.1.3 Inverse of the Vandermonde matrix . . . 25
1.1.4 The alternant matrix . . . 26
1.1.5 The generalized Vandermonde matrix . . . 29
1.1.6 The Vandermonde determinant in systems with Coulombian interactions . . . 30
1.1.7 The Vandermonde determinant in random matrix theory . . . 33
1.2 Curve fitting . . . 37
1.2.1 Linear interpolation . . . 37
1.2.2 Generalized divided differences and interpolation . . . 42
1.2.3 Least squares fitting . . . 45
1.2.4 Linear least squares fitting . . . 45
1.2.5 Non-linear least squares fitting . . . 46
1.2.6 The Marquardt least squares method . . . 47
1.3 Analysing how well a curve fits . . . 50
1.3.1 Regression . . . 50
1.3.3 The Akaike information criterion . . . 53
1.4 D -optimal experiment design . . . 57
1.5 Electromagnetic compatibility and electrostatic discharge currents . . . 60
1.5.1 Electrostatic discharge modelling . . . 62
1.6 Modelling mortality rates . . . 65
1.6.1 Lee–Carter method for forecasting . . . 68
1.7 Summaries of papers . . . 71
2 Extreme points of the Vandermonde determinant 75 2.1 Extreme points of the Vandermonde determinant and related determinants on various surfaces in three dimensions . . . 77
2.1.1 Optimization of the generalized Vandermonde deter-minant in three dimensions . . . 77
2.1.2 Extreme points of the Vandermonde determinant on the three-dimensional unit sphere . . . 81
2.1.3 Optimisation using Gr¨obner bases . . . 82
2.1.4 Extreme points on the ellipsoid in three dimensions . 83 2.1.5 Extreme points on the cylinder in three dimensions . . 85
2.1.6 Optimizing the Vandermonde determinant on a sur-face defined by a homogeneous polynomial . . . 87
2.2 Extreme points of the Vandermonde determinant on the sphere 89 2.2.1 The extreme points on the sphere given by roots of a polynomial . . . 89
2.2.2 Further visual exploration on the sphere . . . 96
2.3 Extreme points of the Vandermonde determinant on some surfaces implicitly defined by a univariate polynomial . . . . 103
2.3.1 Critical points on surfaces given by a first degree uni-variate polynomial . . . 104
2.3.2 Critical points on surfaces given by a second degree univariate polynomial . . . 105
2.3.3 Critical points on the sphere defined by a p-norm . . . 107
2.3.4 The case p = 4 and n = 4 . . . 107
1.3.3 The Akaike information criterion . . . 53
1.4 D -optimal experiment design . . . 57
1.5 Electromagnetic compatibility and electrostatic discharge currents . . . 60
1.5.1 Electrostatic discharge modelling . . . 62
1.6 Modelling mortality rates . . . 65
1.6.1 Lee–Carter method for forecasting . . . 68
1.7 Summaries of papers . . . 71
2 Extreme points of the Vandermonde determinant 75 2.1 Extreme points of the Vandermonde determinant and related determinants on various surfaces in three dimensions . . . 77
2.1.1 Optimization of the generalized Vandermonde deter-minant in three dimensions . . . 77
2.1.2 Extreme points of the Vandermonde determinant on the three-dimensional unit sphere . . . 81
2.1.3 Optimisation using Gr¨obner bases . . . 82
2.1.4 Extreme points on the ellipsoid in three dimensions . 83 2.1.5 Extreme points on the cylinder in three dimensions . . 85
2.1.6 Optimizing the Vandermonde determinant on a sur-face defined by a homogeneous polynomial . . . 87
2.2 Extreme points of the Vandermonde determinant on the sphere 89 2.2.1 The extreme points on the sphere given by roots of a polynomial . . . 89
2.2.2 Further visual exploration on the sphere . . . 96
2.3 Extreme points of the Vandermonde determinant on some surfaces implicitly defined by a univariate polynomial . . . . 103
2.3.1 Critical points on surfaces given by a first degree uni-variate polynomial . . . 104
2.3.2 Critical points on surfaces given by a second degree univariate polynomial . . . 105
2.3.3 Critical points on the sphere defined by a p-norm . . . 107
CONTENTS
2.3.5 Some results for even n and p . . . 110 2.3.6 Some results for cubes and intersections of planes . . . 118 2.3.7 Optimising the probability density function of the
eigenvalues of the Wishart matrix . . . 120 3 Approximation of electrostatic discharge currents using the
analytically extended function 123 3.1 The analytically extended function (AEF) . . . 125 3.1.1 The p-peak analytically extended function . . . 126 3.2 Approximation of lightning discharge current functions . . . . 133 3.2.1 Fitting the AEF . . . 133 3.2.2 Estimating parameters for underdetermined systems . 134 3.2.3 Fitting with data points as well as charge flow and
specific energy conditions . . . 135 3.2.4 Calculating the η-parameters from the β-parameters . 138 3.2.5 Explicit formulas for a single-peak AEF . . . 139 3.2.6 Fitting to lightning discharge currents . . . 140 3.3 Approximation of electrostatic discharge currents using the
AEF by interpolation on a D-optimal design . . . 143 3.3.1 D -optimal approximation for exponents given by a
class of arithmetic sequences . . . 145 3.3.2 D -optimal interpolation on the rising part . . . 146 3.3.3 D -optimal interpolation on the decaying part . . . 148 3.3.4 Examples of models from applications and experiments 150 3.3.5 Modelling of ESD currents . . . 150 3.3.6 Modelling of lightning discharge currents . . . 152 3.3.7 Summary of ESD modelling . . . 159 4 Comparison of models of mortality rate 161 4.1 Modelling and forecasting mortality rates . . . 162 4.2 Overview of models . . . 162 4.3 Power-exponential mortality rate models . . . 164
CONTENTS
2.3.5 Some results for even n and p . . . 110 2.3.6 Some results for cubes and intersections of planes . . . 118 2.3.7 Optimising the probability density function of the
eigenvalues of the Wishart matrix . . . 120 3 Approximation of electrostatic discharge currents using the
analytically extended function 123 3.1 The analytically extended function (AEF) . . . 125 3.1.1 The p-peak analytically extended function . . . 126 3.2 Approximation of lightning discharge current functions . . . . 133 3.2.1 Fitting the AEF . . . 133 3.2.2 Estimating parameters for underdetermined systems . 134 3.2.3 Fitting with data points as well as charge flow and
specific energy conditions . . . 135 3.2.4 Calculating the η-parameters from the β-parameters . 138 3.2.5 Explicit formulas for a single-peak AEF . . . 139 3.2.6 Fitting to lightning discharge currents . . . 140 3.3 Approximation of electrostatic discharge currents using the
AEF by interpolation on a D-optimal design . . . 143 3.3.1 D -optimal approximation for exponents given by a
class of arithmetic sequences . . . 145 3.3.2 D -optimal interpolation on the rising part . . . 146 3.3.3 D -optimal interpolation on the decaying part . . . 148 3.3.4 Examples of models from applications and experiments 150 3.3.5 Modelling of ESD currents . . . 150 3.3.6 Modelling of lightning discharge currents . . . 152 3.3.7 Summary of ESD modelling . . . 159 4 Comparison of models of mortality rate 161 4.1 Modelling and forecasting mortality rates . . . 162 4.2 Overview of models . . . 162 4.3 Power-exponential mortality rate models . . . 164
4.3.1 Multiple humps . . . 165
4.3.2 Single hump model . . . 165
4.3.3 Split power-exponential model . . . 166
4.3.4 Adjusted power-exponential model . . . 166
4.4 Fitting and comparing models . . . 167
4.4.1 Some comments on fitting . . . 168
4.4.2 Results and discussion . . . 174
4.5 Comparison of parametric models applied to mortality rate forecasting . . . 178
4.5.1 Comparison of models . . . 180
4.5.2 Results, discussion and further work . . . 180
References 185 Index 209 List of Figures 211 List of Tables 215 List of Definitions 216 List of Theorems 217 List of Lemmas 218 4.3.1 Multiple humps . . . 165
4.3.2 Single hump model . . . 165
4.3.3 Split power-exponential model . . . 166
4.3.4 Adjusted power-exponential model . . . 166
4.4 Fitting and comparing models . . . 167
4.4.1 Some comments on fitting . . . 168
4.4.2 Results and discussion . . . 174
4.5 Comparison of parametric models applied to mortality rate forecasting . . . 178
4.5.1 Comparison of models . . . 180
4.5.2 Results, discussion and further work . . . 180
References 185 Index 209 List of Figures 211 List of Tables 215 List of Definitions 216 List of Theorems 217 List of Lemmas 218
List of Papers
Paper A Karl Lundeng˚ard, Jonas ¨Osterberg and Sergei Silvestrov.
Extreme points of the Vandermonde determinant on the sphere and some limits involving the generalized Vandermonde determinant. Accepted for publication in Algebraic structures and Applications. SPAS2017, V¨aster˚as and Stockholm, Sweden, October 4 – 6, 2017, Sergei Silvestrov, Anatoliy Malyarenko, Milica Ranˇci´c (Eds), Springer International Publishing, 2019.
Paper B Karl Lundeng˚ard, Jonas ¨Osterberg and Sergei Silvestrov. Optimization of the determinant of the Vandermonde matrix on the sphere and related surfaces.
Methodology and Computing in Applied Probability, Volume 20, Issue 4, pages 1417 – 1428, 2018.
Paper C Asaph Keikara Muhumuza, Karl Lundeng˚ard, Jonas ¨Osterberg, Sergei Silvestrov, John Magero Mango and Godwin Kakuba. Extreme points of the Vandermonde determinant on surfaces implicitly determined by a univariate polynomial.
Accepted for publication in Algebraic structures and Applications. SPAS2017, V¨aster˚as and Stockholm, Sweden, October 4 – 6, 2017, Sergei Silvestrov, Anatoliy Malyarenko, Milica Ranˇci´c (Eds), Springer International Publishing, 2019.
Paper D Asaph Keikara Muhumuza, Karl Lundeng˚ard, Jonas ¨Osterberg, Sergei Silvestrov, John Magero Mango and Godwin Kakuba. Optimization of the Wishart joint eigenvalue probability density distribution based on the Vandermonde determinant.
Accepted for publication in Algebraic structures and Applications. SPAS2017, V¨aster˚as and Stockholm, Sweden, October 4 – 6, 2017, Sergei Silvestrov, Anatoliy Malyarenko, Milica Ranˇci´c (Eds), Springer International Publishing, 2019.
Paper E Karl Lundeng˚ard, Milica Ranˇci´c, Vesna Javor and Sergei Silvestrov. On some properties of the multi-peaked analytically extended function for approximation of lightning discharge currents. Chapter 10 in Engineering Mathematics I: Electromagnetics, Fluid Mechanics, Material Physics and Financial Engineering,
Volume 178 of Springer Proceedings in Mathematics & Statistics, Sergei Silvestrov and Milica Ranˇci´c (Eds),
Springer International Publishing, pages 151–176, 2016.
List of Papers
Paper A Karl Lundeng˚ard, Jonas ¨Osterberg and Sergei Silvestrov.
Extreme points of the Vandermonde determinant on the sphere and some limits involving the generalized Vandermonde determinant. Accepted for publication in Algebraic structures and Applications. SPAS2017, V¨aster˚as and Stockholm, Sweden, October 4 – 6, 2017, Sergei Silvestrov, Anatoliy Malyarenko, Milica Ranˇci´c (Eds), Springer International Publishing, 2019.
Paper B Karl Lundeng˚ard, Jonas ¨Osterberg and Sergei Silvestrov. Optimization of the determinant of the Vandermonde matrix on the sphere and related surfaces.
Methodology and Computing in Applied Probability, Volume 20, Issue 4, pages 1417 – 1428, 2018.
Paper C Asaph Keikara Muhumuza, Karl Lundeng˚ard, Jonas ¨Osterberg, Sergei Silvestrov, John Magero Mango and Godwin Kakuba. Extreme points of the Vandermonde determinant on surfaces implicitly determined by a univariate polynomial.
Accepted for publication in Algebraic structures and Applications. SPAS2017, V¨aster˚as and Stockholm, Sweden, October 4 – 6, 2017, Sergei Silvestrov, Anatoliy Malyarenko, Milica Ranˇci´c (Eds), Springer International Publishing, 2019.
Paper D Asaph Keikara Muhumuza, Karl Lundeng˚ard, Jonas ¨Osterberg, Sergei Silvestrov, John Magero Mango and Godwin Kakuba. Optimization of the Wishart joint eigenvalue probability density distribution based on the Vandermonde determinant.
Accepted for publication in Algebraic structures and Applications. SPAS2017, V¨aster˚as and Stockholm, Sweden, October 4 – 6, 2017, Sergei Silvestrov, Anatoliy Malyarenko, Milica Ranˇci´c (Eds), Springer International Publishing, 2019.
Paper E Karl Lundeng˚ard, Milica Ranˇci´c, Vesna Javor and Sergei Silvestrov. On some properties of the multi-peaked analytically extended function for approximation of lightning discharge currents. Chapter 10 in Engineering Mathematics I: Electromagnetics, Fluid Mechanics, Material Physics and Financial Engineering,
Volume 178 of Springer Proceedings in Mathematics & Statistics, Sergei Silvestrov and Milica Ranˇci´c (Eds),
Paper F Karl Lundeng˚ard, Milica Ranˇci´c, Vesna Javor and Sergei Silvestrov. Estimation of parameters for the multi-peaked AEF current functions.
Methodology and Computing in Applied Probability, Volume 19, Issue 4, pages 1107 – 1121, 2017.
Paper G Karl Lundeng˚ard, Milica Ranˇci´c, Vesna Javor and Sergei Silvestrov. Electrostatic discharge currents representation using the
analytically extended function with p peaks by interpolation on a D-optimal design.
Facta Universitatis Series: Electronics and Energetics, Volume 32, Issue 1, pages 25 – 49, 2019.
Paper H Karl Lundeng˚ard, Milica Ranˇci´c and Sergei Silvestrov. Modelling mortality rates using power-exponential functions. Submitted to journal, 2019.
Paper I Andromachi Boulougari, Karl Lundeng˚ard, Milica Ranˇci´c, Sergei Silvestrov, Belinda Strass and Samya Suleiman. Application of a power-exponential function based model to mortality rates forecasting.
Communications in Statistics: Case Studies, Data Analysis and Applications, Volume 5, Issue 1, pages 3 – 10, 2019.
Parts of the thesis have been presented at the following international conferences: • ASMDA 2015 - 16th Applied Stochastic Models and Data Analysis
In-ternational Conference with 4th Demographics 2015 Workshop, Piraeus, Greece, June 30 – July 4, 2015.
• SPLITECH 2017 - 2nd International Multidisciplinary Conference on Computer and Energy Science, Split, Croatia, July 12 – 14, 2017. • EMC+SIPI 2017 - IEEE International Symposium on Electromagnetic
Compatibility, Signal and Power Integrity, Washington DC, USA, August 7 – 11, 2017.
• SPAS 2017 - International Conference on Stochastic Processes and Alge-braic Structures, V¨aster˚as, Sweden, October 4 – 6, 2017.
• SMTDA 2018 - 5th Stochastic Modelling Techniques and Data Analysis International Conference, Chania, Crete, Greece, June 12 – 15, 2018. • IWAP 2018 - 9th International Workshop on Applied Probability,
Bu-dapest, Hungary, June 18–21, 2018.
Summaries of papers A-I with a brief description of the thesis authors contributions to each paper can be found in Section 1.7.
Paper F Karl Lundeng˚ard, Milica Ranˇci´c, Vesna Javor and Sergei Silvestrov. Estimation of parameters for the multi-peaked AEF current functions.
Methodology and Computing in Applied Probability, Volume 19, Issue 4, pages 1107 – 1121, 2017.
Paper G Karl Lundeng˚ard, Milica Ranˇci´c, Vesna Javor and Sergei Silvestrov. Electrostatic discharge currents representation using the
analytically extended function with p peaks by interpolation on a D-optimal design.
Facta Universitatis Series: Electronics and Energetics, Volume 32, Issue 1, pages 25 – 49, 2019.
Paper H Karl Lundeng˚ard, Milica Ranˇci´c and Sergei Silvestrov. Modelling mortality rates using power-exponential functions. Submitted to journal, 2019.
Paper I Andromachi Boulougari, Karl Lundeng˚ard, Milica Ranˇci´c, Sergei Silvestrov, Belinda Strass and Samya Suleiman. Application of a power-exponential function based model to mortality rates forecasting.
Communications in Statistics: Case Studies, Data Analysis and Applications, Volume 5, Issue 1, pages 3 – 10, 2019.
Parts of the thesis have been presented at the following international conferences: • ASMDA 2015 - 16th Applied Stochastic Models and Data Analysis
In-ternational Conference with 4th Demographics 2015 Workshop, Piraeus, Greece, June 30 – July 4, 2015.
• SPLITECH 2017 - 2nd International Multidisciplinary Conference on Computer and Energy Science, Split, Croatia, July 12 – 14, 2017. • EMC+SIPI 2017 - IEEE International Symposium on Electromagnetic
Compatibility, Signal and Power Integrity, Washington DC, USA, August 7 – 11, 2017.
• SPAS 2017 - International Conference on Stochastic Processes and Alge-braic Structures, V¨aster˚as, Sweden, October 4 – 6, 2017.
• SMTDA 2018 - 5th Stochastic Modelling Techniques and Data Analysis International Conference, Chania, Crete, Greece, June 12 – 15, 2018. • IWAP 2018 - 9th International Workshop on Applied Probability,
Bu-dapest, Hungary, June 18–21, 2018.
Summaries of papers A-I with a brief description of the thesis authors contributions to each paper can be found in Section 1.7.
Chapter 1
Introduction
This chapter is partially based on Papers D, E, H, and I
Paper D Asaph Keikara Muhumuza, Karl Lundeng˚ard, Jonas ¨Osterberg, Sergei Silvestrov, John Magero Mango and Godwin Kakuba. Optimization of the Wishart joint eigenvalue probability density distribution based on the Vandermonde determinant.
Accepted for publication in Algebraic structures and Applications. SPAS2017, V¨aster˚as and Stockholm, Sweden, October 4 – 6, 2017, Sergei Silvestrov, Anatoliy Malyarenko, Milica Ranˇci´c (Eds), Springer International Publishing, 2019.
Paper E Karl Lundeng˚ard, Milica Ranˇci´c, Vesna Javor and Sergei Silvestrov. On some properties of the multi-peaked analytically extended function for approximation of lightning discharge currents. Chapter 10 in Engineering Mathematics I: Electromagnetics, Fluid Mechanics, Material Physics and Financial Engineering,
Volume 178 of Springer Proceedings in Mathematics & Statistics, Sergei Silvestrov and Milica Ranˇci´c (Eds),
Springer International Publishing, pages 151–176, 2016. Paper H Karl Lundeng˚ard, Milica Ranˇci´c and Sergei Silvestrov.
Modelling mortality rates using power-exponential functions. Submitted to journal, 2019.
Paper I Andromachi Boulougari, Karl Lundeng˚ard, Milica Ranˇci´c, Sergei Silvestrov, Belinda Strass and Samya Suleiman. Application of a power-exponential function based model to mortality rates forecasting.
Communications in Statistics: Case Studies, Data Analysis and Applications, Volume 5, Issue 1, pages 3 – 10, 2019.
Chapter 1
Introduction
This chapter is partially based on Papers D, E, H, and I
Paper D Asaph Keikara Muhumuza, Karl Lundeng˚ard, Jonas ¨Osterberg, Sergei Silvestrov, John Magero Mango and Godwin Kakuba. Optimization of the Wishart joint eigenvalue probability density distribution based on the Vandermonde determinant.
Accepted for publication in Algebraic structures and Applications. SPAS2017, V¨aster˚as and Stockholm, Sweden, October 4 – 6, 2017, Sergei Silvestrov, Anatoliy Malyarenko, Milica Ranˇci´c (Eds), Springer International Publishing, 2019.
Paper E Karl Lundeng˚ard, Milica Ranˇci´c, Vesna Javor and Sergei Silvestrov. On some properties of the multi-peaked analytically extended function for approximation of lightning discharge currents. Chapter 10 in Engineering Mathematics I: Electromagnetics, Fluid Mechanics, Material Physics and Financial Engineering,
Volume 178 of Springer Proceedings in Mathematics & Statistics, Sergei Silvestrov and Milica Ranˇci´c (Eds),
Springer International Publishing, pages 151–176, 2016. Paper H Karl Lundeng˚ard, Milica Ranˇci´c and Sergei Silvestrov.
Modelling mortality rates using power-exponential functions. Submitted to journal, 2019.
Paper I Andromachi Boulougari, Karl Lundeng˚ard, Milica Ranˇci´c, Sergei Silvestrov, Belinda Strass and Samya Suleiman. Application of a power-exponential function based model to mortality rates forecasting.
Communications in Statistics: Case Studies, Data Analysis and Applications, Volume 5, Issue 1, pages 3 – 10, 2019.
INTRODUCTION
Two topics are discussed in this thesis, finding the extreme points of the Vandermonde determinant and phenomenological modelling using power-exponential functions. Several of the methods and approaches that are discussed are also applied to modelling of electrical current for use in elec-tromagnetic compatibility, or to modelling of mortality rate of humans for actuarial or demographical purposes. The topics are related since the ex-treme points of the Vandermonde determinant is relevant for certain curve fitting problems that can appear in the construction of the phenomenologi-cal models. An overview of the major relations between the different parts of the thesis are illustrated in Figure 1.1. The relations are of many kinds, common definitions and dependent results, conceptual connections as well as similarities in proof techniques and problem formulations.
This thesis is based on the nine papers listed on pages 13–14. The contents of the papers have been rearranged (and in some cases parts have been omitted) to avoid repetition and improve cohesion, but the original text and structure of the papers have been largely preserved. Significant parts of Chapters 1-3 have also appeared in [180]. If a section is based on a paper this is specified at the beginning of the section and unless otherwise specified any subsections are from the same source. A section that is based on a paper contains text from the paper that is unchanged except for modifications to correct misprints and ensure consistency within the thesis.
Chapter 1 introduces concepts used in later chapters. The Vandermonde matrix, its history, applications, generalizations and some of its proper-ties are introduced in Section 1.1. Section 1.2 discusses a few different ap-proaches to curve fitting. Section 1.3 discusses a few methods for evaluating the result. Basic optimal design is discussed in Section 1.4. Sections 1.5 and 1.6 introduce electromagnetic compatibility and mortality rate modelling.
Chapter 2 discusses the optimisation of the Vandermonde determinant over various surfaces. First the extreme points on a few different surfaces in three dimensions are examined, see Section 2.1. In Section 2.2 the determi-nant is optimised on the sphere in higher dimensions and some results for surfaces defined by a univariate polynomial are discussed in Section 2.3.
Chapter 3 discusses fitting a piecewise non-linear regression model to data. The particular model is introduced in Section 3.1 and a general frame-work for fitting it to data using the Marquardt least squares method is de-scribed in Sections 3.2.1–3.2.5. The framework is then applied to lightning discharge currents in Section 3.2.6. An alternate curve fitting method based on D-optimal interpolation (found analogously to the results in Section 2.2) is described and applied to electrostatic discharge currents in Section 3.3.
Chapter 4 compares several different mathematical models of mortality rate for humans. The comparison is done by fitting the models to central mortality rates from several different countries and then analysing how well the model fits and what happens when the results of the fitting is used for mortality rate forecasting (using the so called Lee–Carter method).
INTRODUCTION
Two topics are discussed in this thesis, finding the extreme points of the Vandermonde determinant and phenomenological modelling using power-exponential functions. Several of the methods and approaches that are discussed are also applied to modelling of electrical current for use in elec-tromagnetic compatibility, or to modelling of mortality rate of humans for actuarial or demographical purposes. The topics are related since the ex-treme points of the Vandermonde determinant is relevant for certain curve fitting problems that can appear in the construction of the phenomenologi-cal models. An overview of the major relations between the different parts of the thesis are illustrated in Figure 1.1. The relations are of many kinds, common definitions and dependent results, conceptual connections as well as similarities in proof techniques and problem formulations.
This thesis is based on the nine papers listed on pages 13–14. The contents of the papers have been rearranged (and in some cases parts have been omitted) to avoid repetition and improve cohesion, but the original text and structure of the papers have been largely preserved. Significant parts of Chapters 1-3 have also appeared in [180]. If a section is based on a paper this is specified at the beginning of the section and unless otherwise specified any subsections are from the same source. A section that is based on a paper contains text from the paper that is unchanged except for modifications to correct misprints and ensure consistency within the thesis.
Chapter 1 introduces concepts used in later chapters. The Vandermonde matrix, its history, applications, generalizations and some of its proper-ties are introduced in Section 1.1. Section 1.2 discusses a few different ap-proaches to curve fitting. Section 1.3 discusses a few methods for evaluating the result. Basic optimal design is discussed in Section 1.4. Sections 1.5 and 1.6 introduce electromagnetic compatibility and mortality rate modelling.
Chapter 2 discusses the optimisation of the Vandermonde determinant over various surfaces. First the extreme points on a few different surfaces in three dimensions are examined, see Section 2.1. In Section 2.2 the determi-nant is optimised on the sphere in higher dimensions and some results for surfaces defined by a univariate polynomial are discussed in Section 2.3.
Chapter 3 discusses fitting a piecewise non-linear regression model to data. The particular model is introduced in Section 3.1 and a general frame-work for fitting it to data using the Marquardt least squares method is de-scribed in Sections 3.2.1–3.2.5. The framework is then applied to lightning discharge currents in Section 3.2.6. An alternate curve fitting method based on D-optimal interpolation (found analogously to the results in Section 2.2) is described and applied to electrostatic discharge currents in Section 3.3.
Chapter 4 compares several different mathematical models of mortality rate for humans. The comparison is done by fitting the models to central mortality rates from several different countries and then analysing how well the model fits and what happens when the results of the fitting is used for mortality rate forecasting (using the so called Lee–Carter method).
Curve fitting Linear interpolation Section 1.2.1 Least squares method Section 1.2.3 Non-linear least squares fitting
Section 1.2.5
D-optimal design Section 1.4
Linear least squares fitting
Section 1.2.4
The Marquardt least squares method
Section 1.2.6 Extreme points of the Vandermonde determinant Vandermonde matrix Section 1.1 Extreme points on various surfaces in 3D Section 2.1 Optimization on a sphere Section 2.2 Optimization on a surface defined by a univariate polynomial Section 2.3
Phenomenological modelling with power-exponential functions
Power exponential function
Electromagnetic compatibility Section 1.5 The Analytically Extended Function Section 3.1 Lightning discharge current modelling Section 3.2 Interpolation on a D-optimal design Section 3.3 Evaluation of curve fit Section 1.3 Mortality rate modelling Section 1.6 Mortality rate models fitted to data Section 4.1 Mortality rate models applied to forecasting Section 4.5 Figure 1.1: Illustration of the most significant connections in the thesis.
Curve fitting Linear interpolation Section 1.2.1 Least squares method Section 1.2.3 Non-linear least squares fitting
Section 1.2.5
D-optimal design Section 1.4
Linear least squares fitting
Section 1.2.4
The Marquardt least squares method
Section 1.2.6 Extreme points of the Vandermonde determinant Vandermonde matrix Section 1.1 Extreme points on various surfaces in 3D Section 2.1 Optimization on a sphere Section 2.2 Optimization on a surface defined by a univariate polynomial Section 2.3
Phenomenological modelling with power-exponential functions
Power exponential function
Electromagnetic compatibility Section 1.5 The Analytically Extended Function Section 3.1 Lightning discharge current modelling Section 3.2 Interpolation on a D-optimal design Section 3.3 Evaluation of curve fit Section 1.3 Mortality rate modelling Section 1.6 Mortality rate models fitted to data Section 4.1 Mortality rate models applied to forecasting Section 4.5 Figure 1.1: Illustration of the most significant connections in the thesis.
1.1. THE VANDERMONDE MATRIX
1.1
The Vandermonde matrix
The Vandermonde matrix is a well-known matrix with a very special form that appears in many different circumstances, a few examples are polynomial interpolation (see Sections 1.2.1 and 1.2.2), least squares curve fitting (see Section 1.2.3), optimal experiment design (see Section 1.4), construction of error-detecting and error-correcting codes (see [31, 124, 242] as well as more recent work such as [28]), determining if a market with a finite set of traded assets is complete [62], calculation of the discrete Fourier transform [241] and related transforms such as the fractional discrete Fourier transform [215], the quantum Fourier transform [70], and the Vandermonde transform [11, 12], solving systems of differential equations with constant coefficients [213], various problems in mathematical physics [283], nuclear physics [51], and quantum physics [249, 271], systems of Coulombian interactions (see Section 1.1.6) and describing properties of the Fisher information matrix of stationary stochastic processes [158] and in various places in random matrix theory (see Sections 1.1.7 and 2.3.7).
In this section we will review some of the basic properties of the Van-dermonde matrix, starting with its definition.
Definition 1.1. A Vandermonde matrix is an n × m matrix of the form
Vmn(xn) = h xi−1j im,n i,j = 1 1 · · · 1 x1 x2 · · · xn .. . ... . .. ... xm−11 xm−12 · · · xm−1 n (1)
where xi ∈ C, i = 1, . . . , n. If the matrix is square, n = m, the notation
Vn= Vnmwill be used.
Remark 1.1. Note that in the literature the term Vandermonde matrix is often used for the transpose of the matrix given above.
1.1.1 Who was Vandermonde?
The matrix is named after Alexandre Th´eophile Vandermonde (1735–1796) who had a varied career that began with law studies and some success as a concert violinist, transitioned into work in science and mathematics in the beginning of the 1770s that gradually turned into administrative and leadership positions at various Parisian institutions as well as work in politics and economics in the end of the 1780s [86]. His entire mathematical career consisted of four published papers, first presented to the French Academy of Sciences in 1770 and 1771 and published a few years later.
The first paper, M´emoire sur la r´esolution des ´equations [279], discusses some properties of the roots of polynomial equations, more specifically for-mulas for the sum of the roots and a sum of symmetric functions of the
pow-1.1. THE VANDERMONDE MATRIX
1.1
The Vandermonde matrix
The Vandermonde matrix is a well-known matrix with a very special form that appears in many different circumstances, a few examples are polynomial interpolation (see Sections 1.2.1 and 1.2.2), least squares curve fitting (see Section 1.2.3), optimal experiment design (see Section 1.4), construction of error-detecting and error-correcting codes (see [31, 124, 242] as well as more recent work such as [28]), determining if a market with a finite set of traded assets is complete [62], calculation of the discrete Fourier transform [241] and related transforms such as the fractional discrete Fourier transform [215], the quantum Fourier transform [70], and the Vandermonde transform [11, 12], solving systems of differential equations with constant coefficients [213], various problems in mathematical physics [283], nuclear physics [51], and quantum physics [249, 271], systems of Coulombian interactions (see Section 1.1.6) and describing properties of the Fisher information matrix of stationary stochastic processes [158] and in various places in random matrix theory (see Sections 1.1.7 and 2.3.7).
In this section we will review some of the basic properties of the Van-dermonde matrix, starting with its definition.
Definition 1.1. A Vandermonde matrix is an n × m matrix of the form
Vmn(xn) = h xi−1j im,n i,j = 1 1 · · · 1 x1 x2 · · · xn .. . ... . .. ... xm−11 xm−12 · · · xm−1 n (1)
where xi ∈ C, i = 1, . . . , n. If the matrix is square, n = m, the notation
Vn= Vnmwill be used.
Remark 1.1. Note that in the literature the term Vandermonde matrix is often used for the transpose of the matrix given above.
1.1.1 Who was Vandermonde?
The matrix is named after Alexandre Th´eophile Vandermonde (1735–1796) who had a varied career that began with law studies and some success as a concert violinist, transitioned into work in science and mathematics in the beginning of the 1770s that gradually turned into administrative and leadership positions at various Parisian institutions as well as work in politics and economics in the end of the 1780s [86]. His entire mathematical career consisted of four published papers, first presented to the French Academy of Sciences in 1770 and 1771 and published a few years later.
The first paper, M´emoire sur la r´esolution des ´equations [279], discusses some properties of the roots of polynomial equations, more specifically for-mulas for the sum of the roots and a sum of symmetric functions of the
pow-ers of the roots. This paper has been mentioned as important since it con-tains some of the fundamental ideas of group theory (see for instance [168]), but generally this work is overshadowed by the works of the contempo-rary Joseph Louis Lagrange (1736–1813) [166]. He also notices the equality a2b + b2c + ac2− a2c − ab2− bc2= (a − b)(a − c)(b − c), which is a special
case of the formula for the determinant of the Vandermonde matrix, but this connection is not discussed in the paper.
The second paper, Remarques sur des probl`emes de situation [280], dis-cusses the problem of the knight’s tour (what sequence of moves allows a knight to visit all squares on a chessboard exactly once). This paper is con-sidered the first mathematical paper that uses the basic ideas of what is now called knot theory [237].
The third paper, M´emoire sur des irrationnelles de diff´erents ordres avec une application au cercle [281], is a paper on combinatorics and the most well-known result from the paper is the Chu–Vandermonde identity,
n X k=1 k Y j=1 r + 1 − j j n−k Y j=1 s + 1 − j j = n Y j=1 r + s + 1 − j j , where r, s ∈ R and n ∈ Z. The identity was first found by Chu Shih-Chieh
ca 1260 – ca 1320, traditional chinese: 朱世傑 in 1303 in The precious mirror of the four elements 四元玉
and was rediscovered (apparently independently) by Vandermonde [8, 223].
In the fourth paper M´emoire sur l’´elimination [282] Vandermonde dis-cusses some ideas for what we today call determinants, which are functions that can tell us if a linear equation system has a unique solution or not. The paper predates the modern definitions of determinants but Vander-monde discusses a general method for solving linear equation systems using alternating functions, which has strong relation to determinants. He also notices that exchanging exponents for indices in a class of expressions from his first paper will give a class of expressions that he discusses in his fourth paper [300]. This relation is mirrored in the relationship between the deter-minant of the Vandermonde matrix and the deterdeter-minant of a general matrix described in Theorem 1.3.
While Vandermonde’s papers can be said to contain many important ideas they do not bring any of them to maturity and he is therefore usu-ally considered a minor scientist and mathematician compared to well-known contemporary mathematicians such as ´Etienne B´ezout (1730–1783) and Pierre-Simon de Laplace (1749–1827) or scientists such as the chemist Antoine Lavoisier (1743–1794) that he worked with for some time after his mathematical career. The Vandermonde matrix does not appear in any of Vandermonde’s published works, which is not surprising considering that the modern matrix concept did not really take shape until almost a hundred years later in the works of Sylvester and Cayley [43, 268]. It is therefore
ers of the roots. This paper has been mentioned as important since it con-tains some of the fundamental ideas of group theory (see for instance [168]), but generally this work is overshadowed by the works of the contempo-rary Joseph Louis Lagrange (1736–1813) [166]. He also notices the equality a2b + b2c + ac2− a2c − ab2− bc2= (a − b)(a − c)(b − c), which is a special
case of the formula for the determinant of the Vandermonde matrix, but this connection is not discussed in the paper.
The second paper, Remarques sur des probl`emes de situation [280], dis-cusses the problem of the knight’s tour (what sequence of moves allows a knight to visit all squares on a chessboard exactly once). This paper is con-sidered the first mathematical paper that uses the basic ideas of what is now called knot theory [237].
The third paper, M´emoire sur des irrationnelles de diff´erents ordres avec une application au cercle [281], is a paper on combinatorics and the most well-known result from the paper is the Chu–Vandermonde identity,
n X k=1 k Y j=1 r + 1 − j j n−k Y j=1 s + 1 − j j = n Y j=1 r + s + 1 − j j , where r, s ∈ R and n ∈ Z. The identity was first found by Chu Shih-Chieh
ca 1260 – ca 1320, traditional chinese: 朱世傑 in 1303 in The precious mirror of the four elements 四元玉
and was rediscovered (apparently independently) by Vandermonde [8, 223].
In the fourth paper M´emoire sur l’´elimination [282] Vandermonde dis-cusses some ideas for what we today call determinants, which are functions that can tell us if a linear equation system has a unique solution or not. The paper predates the modern definitions of determinants but Vander-monde discusses a general method for solving linear equation systems using alternating functions, which has strong relation to determinants. He also notices that exchanging exponents for indices in a class of expressions from his first paper will give a class of expressions that he discusses in his fourth paper [300]. This relation is mirrored in the relationship between the deter-minant of the Vandermonde matrix and the deterdeter-minant of a general matrix described in Theorem 1.3.
While Vandermonde’s papers can be said to contain many important ideas they do not bring any of them to maturity and he is therefore usu-ally considered a minor scientist and mathematician compared to well-known contemporary mathematicians such as ´Etienne B´ezout (1730–1783) and Pierre-Simon de Laplace (1749–1827) or scientists such as the chemist Antoine Lavoisier (1743–1794) that he worked with for some time after his mathematical career. The Vandermonde matrix does not appear in any of Vandermonde’s published works, which is not surprising considering that the modern matrix concept did not really take shape until almost a hundred years later in the works of Sylvester and Cayley [43, 268]. It is therefore
1.1. THE VANDERMONDE MATRIX
strange that the Vandermonde matrix was named after him, a thorough discussion on this can be found in [300], but a possible reason is the simple formula for the determinant that Vandermonde briefly discusses in his fourth paper can be generalized to a Vandermonde matrix of any size. One of the main reasons that the Vandermonde matrix has become known is that it has an exceptionally simple expression for its determinant that in turn has a surprisingly fundamental relation to the determinant of a general matrix. We will be taking a closer look at the determinant of the Vandermonde ma-trix and related matrices several times in this thesis so the next section will introduce it and some of its properties.
1.1.2 The Vandermonde determinant
Often it is not the Vandermonde matrix itself that is useful, instead it is the multivariate polynomial given by its determinant that is examined and used. The determinant of the Vandermonde matrix is usually called the Vander-monde determinant (or VanderVander-monde polynomial or Vandermondian [283]) and can be written using an exceptionally simple formula. But before we discuss the Vandermonde determinant we will disuss the general determi-nant.
Definition 1.2. The determinant is a function of square matrices over a field F to the field F, det : Mn×n(F) → F such that if we consider the
determinant as a function of the columns
det(M) = det(M·,1, M·,2, . . . , M·,n)
of the matrix the determinant must have the following properties • The determinant must be multilinear
det(M·,1, . . . , aM·,k+ bN·,k, . . . , M·,n)
= a det(M·,1, . . . , M·,k, . . . , M·,n) + b det(M·,1, . . . , N·,k, . . . , M·,n).
• The determinant must be alternating, that is if M·,i= M·,j for some
i 6= j then det(M) = 0.
• If I is the identity matrix then det(I) = 1.
Remark 1.2. Defining the multilinear and alternating properties from the rows of the matrix will give the same determinant. The name of the alter-nating property comes from the fact that it combined with multilinearity implies that switching places between two columns changes the sign of the determinant.
This definition of the determinant is quite abstract but it is sufficient to define a unique function.
1.1. THE VANDERMONDE MATRIX
strange that the Vandermonde matrix was named after him, a thorough discussion on this can be found in [300], but a possible reason is the simple formula for the determinant that Vandermonde briefly discusses in his fourth paper can be generalized to a Vandermonde matrix of any size. One of the main reasons that the Vandermonde matrix has become known is that it has an exceptionally simple expression for its determinant that in turn has a surprisingly fundamental relation to the determinant of a general matrix. We will be taking a closer look at the determinant of the Vandermonde ma-trix and related matrices several times in this thesis so the next section will introduce it and some of its properties.
1.1.2 The Vandermonde determinant
Often it is not the Vandermonde matrix itself that is useful, instead it is the multivariate polynomial given by its determinant that is examined and used. The determinant of the Vandermonde matrix is usually called the Vander-monde determinant (or VanderVander-monde polynomial or Vandermondian [283]) and can be written using an exceptionally simple formula. But before we discuss the Vandermonde determinant we will disuss the general determi-nant.
Definition 1.2. The determinant is a function of square matrices over a field F to the field F, det : Mn×n(F) → F such that if we consider the
determinant as a function of the columns
det(M) = det(M·,1, M·,2, . . . , M·,n)
of the matrix the determinant must have the following properties • The determinant must be multilinear
det(M·,1, . . . , aM·,k+ bN·,k, . . . , M·,n)
= a det(M·,1, . . . , M·,k, . . . , M·,n) + b det(M·,1, . . . , N·,k, . . . , M·,n).
• The determinant must be alternating, that is if M·,i= M·,j for some
i 6= j then det(M) = 0.
• If I is the identity matrix then det(I) = 1.
Remark 1.2. Defining the multilinear and alternating properties from the rows of the matrix will give the same determinant. The name of the alter-nating property comes from the fact that it combined with multilinearity implies that switching places between two columns changes the sign of the determinant.
This definition of the determinant is quite abstract but it is sufficient to define a unique function.
