On improving the forecast accuracy of the hidden Markov model

130 Pages·2016·1.33 MB·English

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On improving the foTrecast

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accuracy of the hiddpen Markov
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model

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Thomas Rooney
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Dissertation submitteed in partial fulfilment of the requirements for the
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degree of Master of Commerce
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The Faculty of Commerce
Department of Actuarial Science
January 2016
The financial assistance of the National Research Foundation (NRF) towards
this research is hereby acknowledged. Opinions expressed and conclusions
arrived at are those of the author and are not necessarily to be attributed to
the NRF.
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The copyright of this thesis vests in the author. No
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quotation from it or information derived from it is to be

published without full acknowledgeement of the source.
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The thesis is to be used for private study or non-
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commercial research purposes only.

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Published by the Universit y of Cape Town (UCT) in terms
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of the non-exclusive license granted to UCT by the author.
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Declaration
I, Thomas Jeffrey Amhurst Rooney, hereby declare that the work on which
thisdissertationisbasedismyoriginalwork(exceptwhereacknowledgements
indicate otherwise) and that neither the whole work nor any part of it has
been, is being or is to be submitted for another degree in this or any other
University. I empower the University of Cape Town to reproduce for the
purpose of research either the whole or any portion of the contents in any
manner whatsoever.
15 November 2016
Signature removed
TJA Rooney Date

Abstract
The forecast accuracy of a hidden Markov model (HMM) may be low due
first, to the measure of forecast accuracy being ignored in the parameter-
estimation method and, second, to overfitting caused by the large number of
parameters that must be estimated.
A general approach to forecasting is described which aims to resolve these
two problems and so improve the forecast accuracy of the HMM. First, the
application of extremum estimators to the HMM is proposed. Extremum
estimators aim to improve the forecast accuracy of the HMM by minimising
an estimate of the forecast error on the observed data. The forecast accuracy
is measured by a score function and the use of some general classes of score
functions is proposed. This approach contrasts with the standard use of
a minus log-likelihood score function. Second, penalised estimation for the
HMM is described. The aim of penalised estimation is to reduce overfitting
and so increase the forecast accuracy of the HMM. Penalties on both the
state-dependent distribution parameters and transition probability matrix
areproposed. Inaddition, anumberofcross-validationapproachesfortuning
the penalty function are investigated.
Empirical assessment of the proposed approach on both simulated and
real data demonstrated that, in terms of forecast accuracy, penalised HMMs
fittedusingextremumestimatorsgenerallyoutperformedunpenalisedHMMs
fitted using maximum likelihood.
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Acknowledgements
Firstly, I would like express my sincere gratitude to my supervisor, Asso-
ciate Professor Iain L. MacDonald for his guidance, support, patience and
motivation for this dissertation.
Iwouldalsoliketothankmyparentsfortheirunconditionalsupportboth
generally and, in particular, throughout writing this dissertation.
The financial assistance of the National Research Foundation (NRF) to-
wards this research is hereby acknowledged. Opinions expressed and con-
clusions arrived at are those of the author and are not necessarily to be
attributed to the NRF.
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Contents
Abstract i
Acknowledgements iii
Notation and abbreviations viii
1 Introduction 1
2 Introduction to hidden Markov models 3
2.1 The basic HMM . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Likelihood functions and forecast distributions . . . . . . . . . 5
3 Forecasting and extremum estimators 7
3.1 The basic forecasting approach . . . . . . . . . . . . . . . . . 8
3.2 Extremum estimators for the HMM . . . . . . . . . . . . . . . 9
3.3 Consistency of extremum estimators . . . . . . . . . . . . . . 11
3.4 Model checking and measuring uncertainty of a forecast . . . . 11
3.4.1 Checking a forecasting model with pseudo-residuals . . 12
3.4.2 Forecast intervals and potential risk . . . . . . . . . . . 13
4 Two general forms of score functions 16
4.1 The minus log-likelihood score function . . . . . . . . . . . . . 16
4.2 Point-forecast based score functions . . . . . . . . . . . . . . . 17
4.2.1 Pairwise loss functions and optimal point predictors . . 18
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