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Matrices whose rows (or columns) consists of monomials of sequential powers are called Vandermonde matrices and can be used to describe several useful concepts and have properties that can be helpful for solving many kinds of problems. In this thesis we will discuss this matrix and some of its properties as well as a generalization of it and how it can be applied to curve fitting discharge current for the purpose of ensuring electromagnetic compatibility.

In the first chapter the basic theory for later chapters is introduced. This includes the Vandermonde matrix and some of its properties, history, applications and generalizations, interpolation and regression problems, optimal experiment design and modelling of electrostatic discharge currents with the purpose to ensure electromagnetic compatibility.

The second chapter focuses on finding the extreme points for the determinant for the Vandermonde matrix on various surfaces including spheres, ellipsoids, cylinders and tori. The extreme points are analysed in three dimensions or more.

The third chapter discusses fitting a particular model called the p-peaked Analytically Extended Function (AEF) to data taken either from a standard for electromagnetic compatibility or experimental measurements. More specifically the AEF will be fitted to discharge currents from the IEC 62305-1 and IEC 61000-4-2 standards for lightning protection and electrostatic discharge immunity as well as some experimentally measured data of similar phenomena.

Multi-peaked analytically extended function (AEF), previously applied by the authors to modelling of lightning discharge currents, is used in this paper for representation of the electrostatic discharge (ESD) 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. ESD currents’ modelling is illustrated through two examples: one corresponding to an approximation of the IEC Standard 61000-4-2 waveshape, and the other to representation of some measured ESD current. 

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.