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Gaussian Copula Modelling for Integer-Valued Time Series

Lennon, Hannah

[Thesis]. Manchester, UK: The University of Manchester; 2016.

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Abstract

This thesis is concerned with the modelling of integer-valued time series. The data naturally occurs in various areas whenever a number of events are observed over time. The model considered in this study consists of a Gaussian copula with autoregressive-moving average (ARMA) dependence and discrete margins that can be specified, unspecified, with or without covariates. It can be interpreted as a 'digitised' ARMA model. An ARMA model is used for the latent process so that well-established methods in time series analysis can be used.Still the computation of the log-likelihood poses many problems because it is the sum of 2^N terms involving the Gaussian cumulative distribution function when N is the length of the time series. We consider an Monte Carlo Expectation-Maximisation (MCEM) algorithm for the maximum likelihood estimation of the model which works well for small to moderate N. Then an Approximate Bayesian Computation (ABC) method is developed to take advantage of the fact that data can be simulated easily from an ARMA model and digitised. A spectral comparison method is used in the rejection-acceptance step. This is shown to work well for large N. Finally we write the model in an R-vine copula representation and use a sequential algorithm for the computation of the log-likelihood. We evaluate the score and Hessian of the log-likelihood and give analytic solutions for the standard errors. The proposed methodologies are illustrated using simulation studies and highlight the advantages of incorporating classic ideas from time series analysis into modern methods of model fitting. For illustration we compare the three methods on US polio incidence data (Zeger, 1988) and we discuss their relative merits.

Bibliographic metadata

Type of resource:
Content type:
Administered thesis
Form of thesis:
Traditional
Type of submission:
Doctoral level ETD - final
Thesis title:
Gaussian Copula Modelling for Integer-Valued Time Series
Degree type:
Doctor of Philosophy
Degree programme:
PhD Mathematical Sciences
Publication date:
2016-09-20T16:45:38
Institution:
The University of Manchester
Location:
Manchester, UK
Total pages:
228
Abstract:
This thesis is concerned with the modelling of integer-valued time series. The data naturally occurs in various areas whenever a number of events are observed over time. The model considered in this study consists of a Gaussian copula with autoregressive-moving average (ARMA) dependence and discrete margins that can be specified, unspecified, with or without covariates. It can be interpreted as a 'digitised' ARMA model. An ARMA model is used for the latent process so that well-established methods in time series analysis can be used.Still the computation of the log-likelihood poses many problems because it is the sum of 2^N terms involving the Gaussian cumulative distribution function when N is the length of the time series. We consider an Monte Carlo Expectation-Maximisation (MCEM) algorithm for the maximum likelihood estimation of the model which works well for small to moderate N. Then an Approximate Bayesian Computation (ABC) method is developed to take advantage of the fact that data can be simulated easily from an ARMA model and digitised. A spectral comparison method is used in the rejection-acceptance step. This is shown to work well for large N. Finally we write the model in an R-vine copula representation and use a sequential algorithm for the computation of the log-likelihood. We evaluate the score and Hessian of the log-likelihood and give analytic solutions for the standard errors. The proposed methodologies are illustrated using simulation studies and highlight the advantages of incorporating classic ideas from time series analysis into modern methods of model fitting. For illustration we compare the three methods on US polio incidence data (Zeger, 1988) and we discuss their relative merits.
Additional digital content not deposited electronically:
N/A
Non-digital content not deposited electronically:
N/A
Keyword(s):
Thesis main supervisor(s):
Thesis co-supervisor(s):
Degree grantor:
Language:
en

Institutional metadata

University researcher(s):
Academic department(s):

Record metadata

Manchester eScholar ID:
uk-ac-man-scw:303983
Created by:
Lennon, Hannah
Created:
20th September, 2016, 15:45:39
Last modified by:
Lennon, Hannah
Last modified:
2nd November, 2016, 10:13:47

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