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Extreme points of the Vandermonde determinant and phenomenological modelling with power exponential functions
Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. (MAM)ORCID iD: 0000-0003-3204-617X
2019 (English)Doctoral thesis, comprehensive summary (Other academic)
##### Abstract [en]

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.

##### Place, publisher, year, edition, pages
Västerås: Mälardalen University , 2019.
##### Series
Mälardalen University Press Dissertations, ISSN 1651-4238 ; 293
Mathematics
##### Research subject
Mathematics/Applied Mathematics
##### Identifiers
ISBN: 978-91-7485-431-2 (print)OAI: oai:DiVA.org:mdh-44579DiVA, id: diva2:1329454
##### Public defence
2019-09-26, Delta, Mälardalens högskola, Västerås, 13:15 (English)
##### Supervisors
Available from: 2019-06-24 Created: 2019-06-24 Last updated: 2019-08-22Bibliographically approved
##### List of papers
1. Extreme points of the Vandermonde determinant on the sphere and some limits involving the generalized Vandermonde determinant
Open this publication in new window or tab >>Extreme points of the Vandermonde determinant on the sphere and some limits involving the generalized Vandermonde determinant
2013 (English)Report (Other academic)
##### Abstract [en]

The values of the determinant of Vandermonde matrices with real elements are analyzed both visually and analytically over the unit sphere in various dimensions. For three dimensions some generalized Vandermonde matrices are analyzed visually. The extreme points of the ordinary Vandermonde determinant on finite-dimensional unit spheres are given as the roots of rescaled Hermite polynomials and a recursion relation is provided for the polynomial coefficients. Analytical expressions for these roots are also given for dimension three to seven. A transformation of the optimization problem is provided and some relations between the ordinary and generalized Vandermonde matrices involving limits are discussed.

##### Place, publisher, year, edition, pages
Mälardalen University, 2013. p. 28
##### Keywords
Vandermonde matrix, Determinants, Extreme points, Unit sphere, Generalized Vandermonde matrix
##### National Category
Mathematical Analysis
##### Research subject
Mathematics/Applied Mathematics
##### Identifiers
urn:nbn:se:mdh:diva-23914 (URN)
##### Funder
Swedish Research Council Available from: 2013-12-27 Created: 2013-12-21 Last updated: 2019-06-24Bibliographically approved
2. Optimization of the Determinant of the Vandermonde Matrix and Related Matrices
Open this publication in new window or tab >>Optimization of the Determinant of the Vandermonde Matrix and Related Matrices
2018 (English)In: Methodology and Computing in Applied Probability, ISSN 1387-5841, E-ISSN 1573-7713, Vol. 20, no 4, p. 1417-1428Article in journal (Refereed) Published
##### Abstract [en]

The value of the Vandermonde determinant is optimized over various surfaces, including the sphere, ellipsoid and torus. Lagrange multipliers are used to find a system of polynomial equations which give the local extreme points in its solutions. Using Grobner basis and other techniques the extreme points are given either explicitly or as roots of polynomials in one variable. The behavior of the Vandermonde determinant is also presented visually in some interesting cases.

SPRINGER, 2018
##### Keywords
Vandermonde determinant, Optimization, Grobner basis, Orthogonal polynomials, Ellipsoid, Optimal experiment design, Homogeneous polynomials, 33C45, 11C20, 15B99, 08B99
##### National Category
Computational Mathematics Algebra and Logic
##### Research subject
Mathematics/Applied Mathematics
##### Identifiers
urn:nbn:se:mdh:diva-41818 (URN)10.1007/s11009-017-9595-y (DOI)000449431800019 ()2-s2.0-85033696033 (Scopus ID)
Available from: 2018-12-27 Created: 2018-12-27 Last updated: 2019-06-24Bibliographically approved
3. Extreme points of the Vandermonde determinant on surfaces implicitly determined by a univariate polynomial
Open this publication in new window or tab >>Extreme points of the Vandermonde determinant on surfaces implicitly determined by a univariate polynomial
(English)Manuscript (preprint) (Other academic)
##### Abstract [en]

In this paper some results on optimising the Vandermonde determinanton a few different surfaces defined by univariate polynomials are discussed. The coordinates of the extreme points are given as roots of polynomials. Applications in curve-fitting and electrostatics are also briefly discussed.

##### Keywords
Vandermonde Matrix, Vandermonde Determinant, Orthogonal Polynomials, n-Sphere, p-Norm, D-Optimal Design, Electrostatics
##### National Category
Mathematical Analysis
##### Research subject
Mathematics/Applied Mathematics
##### Identifiers
urn:nbn:se:mdh:diva-44569 (URN)
Available from: 2019-06-24 Created: 2019-06-24 Last updated: 2019-10-12Bibliographically approved
4. Optimization of the Wishart Joint Eigenvalue Probability Density Distribution Based on the Vandermonde Determinant.
Open this publication in new window or tab >>Optimization of the Wishart Joint Eigenvalue Probability Density Distribution Based on the Vandermonde Determinant.
(English)Manuscript (preprint) (Other academic)
##### Abstract [en]

A number of models from mathematics, physics, probability theory and statistics can be described in terms of Wishart matrices and their eigenvalues. The most prominent example being the Laguerre ensembles of the spectrum of Wishart matrix. We aim to express extreme points of the joint eigenvalue probability densitydistribution of a Wishart matrix using optimisation techniques for the Vandermondedeterminant over certain surfaces implicitly defined by univariate polynomials.

##### Keywords
Vandermonde determinant · Orthogonal Ensembles, Gaussian Ensembles, Wishart Ensembles, Eigenvalue Density optimization
##### National Category
Probability Theory and Statistics Mathematical Analysis
##### Research subject
Mathematics/Applied Mathematics
##### Identifiers
urn:nbn:se:mdh:diva-44568 (URN)
Available from: 2019-06-24 Created: 2019-06-24 Last updated: 2019-10-12Bibliographically approved
5. On Some Properties of the Multi-Peaked Analytically Extended Function for Approximation of Lightning Discharge Currents
Open this publication in new window or tab >>On Some Properties of the Multi-Peaked Analytically Extended Function for Approximation of Lightning Discharge Currents
2016 (English)In: Engineering Mathematics I: Electromagnetics, Fluid Mechanics, Material Physics and Financial Engineering, Series: Springer Proceedings in Mathematics & Statistics / [ed] Sergei Silvestrov, Milica Rančić, Heidelberg: Springer, 2016, p. 151-172Chapter in book (Refereed)
##### Abstract [en]

According to experimental results for lightning discharge currents, they are classified in the IEC 62305 Standard into waveshapes representing the first positive, first and subsequent negative strokes, and long-strokes. These waveshapes, especially shot-term pulses, are approximated with a few mathematical functions in literature, in order to be used in lightning discharge models for calculations of electromagnetic field and lightning induced effects. An analytically extended function (AEF) is presented in this paper and used for lightning currents modeling. The basic properties of this function with a finite number of peaks are examined. A general framework for estimating the parameters of the AEF using the Marquardt least-squares method (MLSM) for a waveform with an arbitrary (finite) number of peaks as well as for the given charge transfer and specific energy is described. This framework is used to find parameters for some common single-peak wave-forms and some advantages and disadvantages of the approach are also discussed.

##### Place, publisher, year, edition, pages
Heidelberg: Springer, 2016
##### Series
Springer Proceedings in Mathematics & Statistics, ISSN 2194-1009 ; 178
##### National Category
Computational Mathematics Other Electrical Engineering, Electronic Engineering, Information Engineering
##### Research subject
Mathematics/Applied Mathematics
##### Identifiers
urn:nbn:se:mdh:diva-33097 (URN)10.1007/978-3-319-42082-0_10 (DOI)2-s2.0-85015267484 (Scopus ID)978-3-319-42082-0 (ISBN)978-3-319-42081-3 (ISBN)
Available from: 2016-09-08 Created: 2016-09-08 Last updated: 2019-06-24Bibliographically approved
6. Estimation of Parameters for the Multi-peaked AEF Current Functions
Open this publication in new window or tab >>Estimation of Parameters for the Multi-peaked AEF Current Functions
2017 (English)In: Methodology and Computing in Applied Probability, ISSN 1387-5841, E-ISSN 1573-7713, Vol. 19, no 4, p. 1107-1121Article in journal (Refereed) Published
##### Abstract [en]

An examination of how the analytically extended function (AEF) can be used to approximate multi-peaked lightning current waveforms, is presented in the paper. A general framework for estimating the parameters of the AEF using the Marquardt least-squares method (MLSM) for a waveform with an arbitrary (finite) number of peaks is presented. This framework is used to find parameters for some common waveforms with a single peak, such as Standard IEC 62305 lightning currents. Illustration of fitting a p-peak AEF to recorded lightning current data is also presented.

##### Keywords
Analytically extended function, Electromagnetic compatibility, Lightning discharge, Marquardt least-squares method
##### National Category
Computational Mathematics Other Electrical Engineering, Electronic Engineering, Information Engineering
##### Research subject
Mathematics/Applied Mathematics
##### Identifiers
urn:nbn:se:mdh:diva-33096 (URN)10.1007/s11009-016-9501-z (DOI)000413792200007 ()2-s2.0-84973100963 (Scopus ID)
##### Conference
15th Applied Stochastic Models and Data Analysis International Conference (ASMDA), Univ Piraeus, Piraeus, GREECE, JUN 30-JUL 04, 2015
Available from: 2016-09-08 Created: 2016-09-08 Last updated: 2019-06-24Bibliographically approved
7. Electrostatic discharge currents representation using the analytically extended function with P peaks by interpolation on a D-optimal design
Open this publication in new window or tab >>Electrostatic discharge currents representation using the analytically extended function with P peaks by interpolation on a D-optimal design
2019 (English)In: Facta Universitatis Series: Electronics and Energetics, ISSN 0353-3670, Vol. 32, no 1, p. 25-49Article in journal (Refereed) Published
##### Abstract [en]

In this paper the Analytically Extended Function (AEF) with p peaks is used for representation of the electrostatic discharge (ESD) currents and lightning discharge currents. The fitting to data is achieved by interpolation of certain data points. In order to minimize unstable behaviour, the exponents of the AEF are chosen from a certain arithmetic sequence and the interpolated points are chosen according to a D-optimal design. The method is illustrated using several examples of currents taken from standards and measurements.

UNIV NIS, 2019
##### Keywords
Analytically extended function, electrostatic discharge (ESD) current, lightning discharge current, D-optimal design
##### National Category
Computational Mathematics Electrical Engineering, Electronic Engineering, Information Engineering
##### Research subject
Mathematics/Applied Mathematics
##### Identifiers
urn:nbn:se:mdh:diva-42695 (URN)10.2298/FUEE1901025L (DOI)000457549900002 ()
Available from: 2019-02-15 Created: 2019-02-15 Last updated: 2019-08-13Bibliographically approved
8. Modelling mortality rates using the powerexponential function
Open this publication in new window or tab >>Modelling mortality rates using the powerexponential function
2017 (English)In: Booklet of abstracts of SPAS2017 - International Conference on Stochastic Processes and Algebraic Structures– From Theory Towards Applications, 2017Conference paper, Oral presentation with published abstract (Refereed)
##### Abstract [en]

There are many models for the mortality rates of humans and other organisms. A phenomenon that complicates the modelling of human mortality rates is a rapid increase in mortality rate for young adults (in western Europe this is especially pronounced at the age of 25). We will examine models for mortality rates based on power-exponential functions, compare them to empirical data for mortality rates and other models.

##### National Category
Probability Theory and Statistics
##### Research subject
Mathematics/Applied Mathematics
##### Identifiers
urn:nbn:se:mdh:diva-36534 (URN)
##### Conference
SPAS2017 - International Conference on Stochastic Processes and Algebraic Structures– From Theory Towards Applications, Västerås and Stockholm, Sweden, October 4 – 6, 2017.
Available from: 2017-09-28 Created: 2017-09-28 Last updated: 2019-06-24Bibliographically approved
9. Application of a power-exponential function-based model to mortality rates forecasting
Open this publication in new window or tab >>Application of a power-exponential function-based model to mortality rates forecasting
2019 (English)In: Communications in Statistics: Case Studies, Data Analysis and Applications, E-ISSN 2373-7484, Vol. 5, no 1, p. 3-10Article in journal (Refereed) Published
##### Abstract [en]

There are many models for mortality rates. A well-known problem that complicates modeling of human mortality rates is the “accident hump” occurring in early adulthood. Here, two models of mortality rate based on power-exponential functions are presented and compared to a few other models. The models will be fitted to known data of measured death rates from several different countries using numerical techniques for curve-fitting with the nonlinear least-squares method. The properties of the model with respect to forecasting with the Lee–Carter method will be discussed.

##### Keywords
mortality rates modeling, power-exponential function, nonlinear curve fitting, Lee-Carter method
##### National Category
Probability Theory and Statistics Computational Mathematics
##### Research subject
Mathematics/Applied Mathematics
##### Identifiers
urn:nbn:se:mdh:diva-41090 (URN)10.1080/23737484.2019.1578705 (DOI)978-618-5180-27-0 (ISBN)978-618-5180-29-4 (ISBN)
##### Conference
SMTDA2018 International Conference and Demographics Workshop (12-15 June, 2018), Crete, Greece
##### Note

Originally presented at the SMTDA2018 conference.

Available from: 2018-09-30 Created: 2018-09-30 Last updated: 2019-10-14Bibliographically approved

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Lundengård, Karl

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Cite
Citation style
• apa
• ieee
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