@PHDTHESIS\{IMM1999-01703, author = "H. A. Nielsen", title = "Parametric and Non-Parametric System Modelling", year = "1999", school = "Informatics and Mathematical Modelling, Technical University of Denmark, {DTU}", address = "Richard Petersens Plads, Building 321, {DK-}2800 Kgs. Lyngby", type = "", url = "http://www2.compute.dtu.dk/pubdb/pubs/1703-full.html", abstract = "The present thesis consists of ten research papers published in the period 1996-1999 together with a summary report. The thesis deals with different aspects of mathematical modelling of systems using data and, if possible, partial knowledge about the systems. In the first part of the thesis the focus is on combinations of parametric and non-parametric methods of regression. This combination can be in terms of additive models where e.g. one or more non-parametric term is added to a linear regression model. It can also be in terms of conditional parametric models where the coefficients of a linear model are estimated as functions of some explanatory variable(s). Also, software for handling the estimation is presented. The software runs under {S-PLUS} and R and contains also a number of tools useful when doing model diagnostics or interpreting the results. Adaptive estimation is also considered. It is shown that adaptive estimation in conditional parametric models can be performed by combining the well known methods of local polynomial regression and recursive least squares with exponential forgetting. The approach used for estimation in conditional parametric models also highlights how recursive least squares with exponential forgetting can be generalized and improved by approximating the time-varying parameters with polynomials locally in time. In one of the papers well known tools for structural identification of linear time series are generalized to the non-linear time series case. For this purpose non-parametric methods together with additive models are suggested. Also, a new approach specifically designed to detect non-linearities is introduced. Confidence intervals are constructed by use of bootstrapping. As a link between non-parametric and parametric methods a paper dealing with neural networks is included. In this paper, neural networks are used for predicting the electricity production of a wind farm. The results are compared with results obtained using an adaptively estimated {ARX-}model. Finally, two papers on stochastic differential equations are included. In the first paper, among other aspects, the properties of a method for parameter estimation in stochastic differential equations is considered within the field of heat dynamics of buildings. In the second paper a lack-of-fit test for stochastic differential equations is presented. The test can be applied to both linear and non-linear stochastic differential equations. Some applications are presented in the papers. In the summary report references are made to a number of other applications. Resum\'{e} p{\aa} dansk: N{\ae}rv{\ae}rende afhandling best{\aa}r af ti artikler publiceret i perioden 1996-1999 samt et sammendrag og en perspektivering heraf. I afhandlingen behandles aspekter af matematisk modellering af systemer vha. data og, s{\aa}fremt det er muligt, delvis viden om disse systemer. Den f{\o}rste del af artiklerne fokuserer p{\aa} kombinationer af parametriske og ikke-parametriske regressionsmetoder. S{\aa}danne kombinationer kan v{\ae}re additive, hvor f.eks. et eller flere ikke-parametriske led adderes til en line{\ae}r regressionsmodel. En anden mulighed er betinget parametriske modeller, hvor koefficienterne i en line{\ae}r model estimeres som funktioner af en eller flere forklarende variable. Endvidere pr{\ae}senteres et {EDB-}program til h{\aa}ndtering af og estimation i s{\aa}danne modeller. Programmet er en udvidelse til {S-PLUS} og R. Programmet inkluderer ogs{\aa} en r{\ae}kke v{\ae}rkt{\o}jer, der er nyttige i forbindelse med diagnostik og fortolkning af resultater. Endvidere behandles adaptiv estimation. Det vises, at der ved at kombinere adaptiv estimation i line{\ae}re modeller med lokal polynomiel regression, som begge er velkendte metoder, f{\aa}s en metode, der kan h{\aa}ndtere adaptiv estimation i betinget parametriske modeller. Den anvendte metode til estimation i betinget parametriske modeller tydeligg{\o}r ogs{\aa}, hvorledes den rekursive mindste kvadraters metode med eksponentiel glemsel kan generaliseres og forbedres ved at approksimere de tidsvarierende parametre med polynomier, der er lokale i tid. Ikke-parametriske metoder bruges sammen med additive modeller til at generalisere velkendte metoder til strukturel identifikation af ine{\ae}re tidsr{\ae}kker. S{\aa}ledes opn{\aa}s nye metoder, der kan h{\aa}ndtere ikke-line{\ae}re tidsr{\ae}kker. Endvidere introduceres et nyt v{\ae}rkt{\o}j specifikt designet til at detektere ikke-lineariteter. Konfidensintervaller konstrueres vha. bootstrapping. Som et forbindelsesled mellem ikke-parametriske og parametriske metoder er der inkluderet en artikel vedr. neurale netv{\ae}rk. Her bruges neurale netv{\ae}rk til at forudsige elproduktionen i en vindm{\o}llepark, og der sammenlignes med pr{\ae}diktioner p{\aa} basis af en adaptivt estimeret {ARX-}model. Til slut er to artikler vedr. stokastiske differentialligninger inkluderet. Den f{\o}rste artikel vedr{\o}rer varmedynamik for bygninger. Her unders{\o}ges bl.a. egenskaberne for en metode til estimation af parametrene i stokastiske differentialligninger. I den anden artikel pr{\ae}senteres et test for lack-of-fit af stokastiske differentialligninger. Metoden kan benyttes i forbindelse med b{\aa}de line{\ae}re og ikke-line{\ae}re systemer. I den f{\o}rste del af afhandlingen refereres der til en r{\ae}kke anvendelser. Desuden pr{\ae}senteres andre anvendelser i artiklerne." }