On the extremes of randomly sub-sampled time series

In this paper, we investigate the extremal properties of randomly sub-sampled stationary sequences. Motivation comes from the need to account for the effect of missing values on the analysis of time series and the comparison of schemes for monitoring systems with breakdowns or systems with automatic replacement of devices in case of failures

Optimal alarm systems for count processes

In many phenomena described by stochastic processes, the implementation of an alarm system becomes fundamental to predict the occurrence of future events. In this work we develop an alarm system to predict whether a count process will upcross a certain level and give an alarm whenever the upcrossing level is predicted. We consider count models with parameters being functions of covariates of interest and varying on time. This article presents classical and Bayesian methodology for producing optimal alarm systems. Both methodologies are illustrated and their performance compared through a simulation study. The work finishes with an empirical application to a set of data concerning the number of sunspot on the surface of the sun

Stationary solutions for integer-valued autoregressive processes

The purpose of this paper is to introduce and develop a family of Z+-valued autoregressive processes of order p (INAR(p)) by using the generalized multiplication ⊙F of van Harn and Steutel (1982). We obtain various distributional and regression properties for these models. A number of stationary INAR(p) processes with specific marginals are presented and are shown to generalize several existing models. 1

Contributos para a análise estatística de séries de contagem

Doutoramento em MatemáticaA análise das séries temporais de valores inteiros tornou-se, nos últimos anos, uma área de investigação importante, não só devido à sua aplicação a dados de contagem provenientes de diversos campos da ciência, mas também pelo facto de ser uma área pouco explorada, em contraste com a análise séries temporais de valores contínuos. Uma classe que tem obtido especial relevo é a dos modelos baseados no operador binomial thinning, da qual se destaca o modelo auto-regressivo de valores inteiros de ordem p. Esta classe é muito vasta, pelo que este trabalho tem como objectivo dar um contributo para a análise estatística de processos de contagem que lhe pertencem. Esta análise é realizada do ponto de vista da predição de acontecimentos, aos quais estão associados mecanismos de alarme, e também da introdução de novos modelos que se baseiam no referido operador. Em muitos fenómenos descritos por processos estocásticos a implementação de um sistema de alarmes pode ser fundamental para prever a ocorrência de um acontecimento futuro. Neste trabalho abordam-se, nas perspectivas clássica e bayesiana, os sistemas de alarme óptimos para processos de contagem, cujos parâmetros dependem de covariáveis de interesse e que variam no tempo, mais concretamente para o modelo auto-regressivo de valores inteiros não negativos com coeficientes estocásticos, DSINAR(1). A introdução de novos modelos que pertencem à classe dos modelos baseados no operador binomial thinning é feita quando se propõem os modelos PINAR(1)T e o modelo SETINAR(2;1). O modelo PINAR(1)T tem estrutura periódica, cujas inovações são uma sucessão periódica de variáveis aleatórias independentes com distribuição de Poisson, o qual foi estudado com detalhe ao nível das suas propriedades probabilísticas, métodos de estimação e previsão. O modelo SETINAR(2;1) é um processo auto-regressivo de valores inteiros, definido por limiares auto-induzidos e cujas inovações formam uma sucessão de variáveis independentes e identicamente distribuídas com distribuição de Poisson. Para este modelo estudam-se as suas propriedades probabilísticas e métodos para estimar os seus parâmetros. Para cada modelo introduzido, foram realizados estudos de simulação para comparar os métodos de estimação que foram usados.The analysis of count processes has become an important area of research in the last two decades partially because of its wide applicability in different fields of science. Among the most successful integer-valued time series models proposed in the literature, we mention the binomial thinning based models class, which includes the autoregressive integer valued process of order p as a special case. This work aims to contribute to the statistical analysis of counting processes. In particular, the purpose of this thesis is two-folded: firstly, it explores the issue of event prediction associated with alarm mechanisms and secondly, it introduces two new models based on the binomial thinning operator. In many phenomena described by stochastic processes, the implementation of an alarm system becomes fundamental to predict the occurrence of future events. In this work we develop an alarm system to predict whether a count process will upcross a certain level and give an alarm whenever the upcrossing level is predicted. We consider count models with parameters being functions of covariates of interest and varying on time. Classical and Bayesian methodologies are applied for producing optimal alarm systems. Both methodologies are illustrated and their performance compared through a simulation study. As an example an empirical application to a set of data concerning the number of sunspot on the surface of the sun is presented. Within the binomial thinning based models class two new models are proposed and studied. The periodic integer-valued autoregressive model of order one with period T, driven by a periodic sequence of independent Poissondistributed random variables, is studied in some detail. Basic probabilistic and statistical properties of this model are discussed. Moreover, parameter estimation and prediction are topics also addressed. A class of self-exciting threshold integer-valued autoregressive models, driven by independent Poisson-distributed random variables, is also introduced. Basic probabilistic and statistical properties of this class of models are discussed. Moreover, parameter estimation is also addressed. Specifically, the methods of estimation under analysis are the least squares-type and likelihood-based ones. Their performances are compared through a simulation study

Intermittent demand forecasting with integer autoregressive moving average models

April 2009 This PhD thesis focuses on using time series models for counts in modelling and forecasting a special type of count series called intermittent series. An intermittent series is a series of non-negative integer values with some zero values. Such series occur in many areas including inventory control of spare parts. Various methods have been developed for intermittent demand forecasting with Croston’s method being the most widely used. Some studies focus on finding a model underlying Croston’s method. With none of these studies being successful in demonstrating an underlying model for which Croston’s method is optimal, the focus should now shift towards stationary models for intermittent demand forecasting. This thesis explores the application of a class of models for count data called the Integer Autoregressive Moving Average (INARMA) models. INARMA models have had applications in different areas such as medical science and economics, but this is the first attempt to use such a model-based method to forecast intermittent demand. In this PhD research, we first fill some gaps in the INARMA literature by finding the unconditional variance and the autocorrelation function of the general INARMA(p,q) model. The conditional expected value of the aggregated process over lead time is also obtained to be used as a lead time forecast. The accuracy of h-step-ahead and lead time INARMA forecasts are then compared to those obtained by benchmark methods of Croston, Syntetos-Boylan Approximation (SBA) and Shale-Boylan-Johnston (SBJ). The results of the simulation suggest that in the presence of a high autocorrelation in data, INARMA yields much more accurate one-step ahead forecasts than benchmark methods. The degree of improvement increases for longer data histories. It has been shown that instead of identification of the autoregressive and moving average order of the INARMA model, the most general model among the possible models can be used for forecasting. This is especially useful for short history and high autocorrelation in data. The findings of the thesis have been tested on two real data sets: (i) Royal Air Force (RAF) demand history of 16,000 SKUs and (ii) 3,000 series of intermittent demand from the automotive industry. The results show that for sparse data with long history, there is a substantial improvement in using INARMA over the benchmarks in terms of Mean Square Error (MSE) and Mean Absolute Scaled Error (MASE) for the one-step ahead forecasts. However, for series with short history the improvement is narrower. The improvement is greater for h-step ahead forecasts. The results also confirm the superiority of INARMA over the benchmark methods for lead time forecasts

Intermittent demand forecasting with integer autoregressive moving average models

This PhD thesis focuses on using time series models for counts in modelling and forecasting a special type of count series called intermittent series. An intermittent series is a series of non-negative integer values with some zero values. Such series occur in many areas including inventory control of spare parts. Various methods have been developed for intermittent demand forecasting with Croston’s method being the most widely used. Some studies focus on finding a model underlying Croston’s method. With none of these studies being successful in demonstrating an underlying model for which Croston’s method is optimal, the focus should now shift towards stationary models for intermittent demand forecasting. This thesis explores the application of a class of models for count data called the Integer Autoregressive Moving Average (INARMA) models. INARMA models have had applications in different areas such as medical science and economics, but this is the first attempt to use such a model-based method to forecast intermittent demand. In this PhD research, we first fill some gaps in the INARMA literature by finding the unconditional variance and the autocorrelation function of the general INARMA(p,q) model. The conditional expected value of the aggregated process over lead time is also obtained to be used as a lead time forecast. The accuracy of h-step-ahead and lead time INARMA forecasts are then compared to those obtained by benchmark methods of Croston, Syntetos-Boylan Approximation (SBA) and Shale-Boylan-Johnston (SBJ). The results of the simulation suggest that in the presence of a high autocorrelation in data, INARMA yields much more accurate one-step ahead forecasts than benchmark methods. The degree of improvement increases for longer data histories. It has been shown that instead of identification of the autoregressive and moving average order of the INARMA model, the most general model among the possible models can be used for forecasting. This is especially useful for short history and high autocorrelation in data. The findings of the thesis have been tested on two real data sets: (i) Royal Air Force (RAF) demand history of 16,000 SKUs and (ii) 3,000 series of intermittent demand from the automotive industry. The results show that for sparse data with long history, there is a substantial improvement in using INARMA over the benchmarks in terms of Mean Square Error (MSE) and Mean Absolute Scaled Error (MASE) for the one-step ahead forecasts. However, for series with short history the improvement is narrower. The improvement is greater for h-step ahead forecasts. The results also confirm the superiority of INARMA over the benchmark methods for lead time forecasts.EThOS - Electronic Theses Online ServiceGBUnited Kingdo