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Approximation of misclassification probabilities in linear discriminant analysis with repeated measurements
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering. Department of Mathematics, University of Dar Es Salaam Tanzania.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering. Department of Energy and Technology, Uppsala, Sweden.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering.ORCID iD: maroh70 0000-0001-9896-4438
2020 (English)Report (Other academic)
Abstract [en]

In this paper, we propose approximations for the probabilities of misclassification in linear discriminant analysis when means follow a growth curve structure. The discriminant function can classify a new observation vector of p repeated measurements into one of two multivariate normal populations with equal covariance matrix. We derive certain relations of the statistics under consideration in order to obtain approximations of the misclassification errors. Finally, we perform Monte Carlo simulations to evaluate the performance of proposed results.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2020. , p. 21
Series
LiTH-MAT-R, ISSN 0348-2960 ; 2020:05
Keywords [en]
Approximation; Growth Curve model; linear discriminant function; probability of misclassification
National Category
Probability Theory and Statistics
Identifiers
URN: urn:nbn:se:liu:diva-165700Libris ID: v6g4v547s6l4wclbOAI: oai:DiVA.org:liu-165700DiVA, id: diva2:1430067
Available from: 2020-05-13 Created: 2020-05-13 Last updated: 2020-05-28Bibliographically approved
In thesis
1. Contributions to linear discriminant analysis with applications to growth curves
Open this publication in new window or tab >>Contributions to linear discriminant analysis with applications to growth curves
2020 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis concerns contributions to linear discriminant analysis with applications to growth curves.

Firstly, we present the linear discriminant function coefficients in a stochastic representation using random variables from the standard univariate distributions. We apply the characterized distribution in the classification function to approximate the classification error rate. The results are then extended to large dimension asymptotics under assumption that the dimension p of the parameter space increases together with the sample size n to infinity such that the ratio  converges to a positive constant c  (0, 1).

Secondly, the thesis treats repeated measures data which correspond to multiple measurements that are taken on the same subject at different time points. We develop a linear classification function to classify an individual into one out of two populations on the basis of the repeated measures data that when the means follow a growth curve structure. The growth curve structure we first consider assumes that all treatments (groups) follows the same growth profile. However, this is not necessarily true in general and the problem is extended to linear classification where the means follow an extended growth curve structure, i.e., the treatments under the experimental design follow different growth profiles.

At last, a function of the inverse Wishart matrix and a normal distribution finds its application in portfolio theory where the vector of optimal portfolio weights is proportional to the product of the inverse sample covariance matrix and a sample mean vector. Analytical expressions for higher order moments and non-central moments of the portfolio weights are derived when the returns are assumed to be independently multivariate normally distributed. Moreover, the expressions for the mean, variance, skewness and kurtosis of specific estimated weights are obtained. The results are complemented using a Monte Carlo simulation study, where data from the multivariate normal and t-distributions are discussed.

Abstract [sv]

Den här avhandlingen studerar diskriminantanalys, klassificering av tillväxtkurvor och portföljteori.

Diskriminantanalys och klassificering är flerdimensionella tekniker som används för att separera olika mängder av objekt och för att tilldela nya objekt till redan definierade grupper (så kallade klasser). En klassisk metod är att använda Fishers linjära diskriminantfunktion och när alla parametrar är kända så kan man enkelt beräkna sannolikheterna för felklassificering. Tyvärr är så sällan fallet, utan parametrarna måste skattas från data, och då blir Fishers linjära diskriminantfunktion en funktion av en Wishartmatris och multivariat normalfördelade vektorer. I den här avhandlingen studerar vi hur man kan approximativt beräkna sannolikheten för felklassificering under antagande att dimensionen på parameterrummet ökar tillsammans med antalet observationer genom att använda en särskild stokastisk representation av diskriminantfunktionen.

Upprepade mätningar över tiden på samma individ eller objekt går att modellera med så kallade tillväxtkurvor. Vid klassificering av tillväxtkurvor, eller rättare sagt av upprepade mätningar för en ny individ, bör man ta tillvara på både den spatiala- och temporala informationen som finns hos dessa observationer. Vi vidareutvecklar Fishers linjära diskriminantfunktion att passa för upprepade mätningar och beräknar asymptotiska sannolikheter för felklassificering.

Till sist kan man notera att snarlika funktioner av Wishartmatriser och multivariat normalfördelade vektorer dyker upp när man vill beräkna de optimala vikterna i portföljteori. Genom en stokastisk representation studerar vi egenskaperna hos portföljvikterna och gör dessutom en simuleringsstudie för att förstå vad som händer när antagandet om normalfördelning inte är uppfyllt.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2020. p. 47
Series
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 2071
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:liu:diva-165558 (URN)10.3384/diss.diva-165558 (DOI)9789179298562 (ISBN)
Public defence
2020-06-08, Online through Zoom (to register: https://bit.ly/36fuupt) and Hopningspunkten, B Building, Campus Valla, Linköping, 15:15 (English)
Opponent
Supervisors
Available from: 2020-05-06 Created: 2020-05-06 Last updated: 2020-05-19Bibliographically approved

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Ngailo, Edwardvon Rosen, DietrichSingull, Martin

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