Análise estatística de séries de contagem com estrutura periódica

Doutoramento em MatemáticaOs modelos autoregressivos de valores inteiros multivariados (MINAR) desempenham um papel central na análise estatística de séries temporais de contagem. Dentro do razoavelmente grande espectro de modelos MINAR propostos na literatura, muito poucos focam a análise de séries de contagem com estrutura periódica. A análise dos processos de contagem multivariados apresenta muitos desafios que vão desde a especificação do modelo até à estimação de parâmetros. Esta tese tem como objetivo dar uma contribuição nessa direção. Especificamente, o objetivo deste trabalho é duplo: primeiro, introduzimos o processo multivariado periódico de ordem um, PMINAR(1). As propriedades probabilísticas e estatísticas do modelo são estudadas em detalhe. Para superar as dificuldades computacionais decorrentes da utilização do método da máxima verosimilhança introduzimos uma abordagem baseada na verosimilhança composta. O desempenho do método proposto e outros métodos concorrentes na estimação dos parâmetros é comparado através de um estudo de simulação. A previsão também é abordada. Uma aplicação de dados reais relacionados com a análise de fogos é apresentada. Em segundo lugar, propomos dois modelos INAR (univariado e bivariado) com estrutura periódica, S-PINAR(1) e BS-PINAR(1), respetivamente. Ambos os modelos são baseados no operador signed thinning permitindo contagens de valores positivos e negativos. Apresentamos as propriedades probabilísticas básicas e estatísticas dos modelos periódicos. As inovações são modeladas através das distribuições Skellam univariada e bivariada, respetivamente. Para avaliar o desempenho dos estimadores dos mínimos quadrados condicionais e da máxima verosimilhança condicional, foi realizado um estudo de simulação para o modelo S-PINAR(1).Multivariate INteger–valued AutoRegressive (MINAR) processes play a central role in the statistical analysis of integer-valued time series. Within the reasonably large spectrum of MINAR models proposed in the literature, however, only a few focus on the analysis of time series of count data with periodic structure. The analysis of multivariate counting processes presents many challenging problems ranging from model specification to parameter estimation. This thesis aims at giving a contribution towards this direction. Specifically, the purpose of this research is two-fold: first, we introduce the periodic multivariate process of order one (PMINAR(1) in short). The probabilistic and also the statistical properties of the model are studied in detail. To overcome the computational difficulties arising from the use of the maximum likelihood method we introduce a composite likelihood-based approach. The performance of the proposed method and other competitors methods of estimation is compared through a simulation study. Forecasting is also addressed. An application to a real data set related with the analysis of fire activity is presented. Secondly, we propose two INAR (univariate and bivariate) models with periodic structure, S-PINAR(1) and BS-PINAR(1), respectively. Both models are based on the signed thinning operator allowing for positive and negative counts. We examine the basic probabilistic and also the statistical properties of the periodic models. Innovations are modeled by univariate and bivariate Skellam distributions, respectively. To study the performance of the conditional least squares and conditional maximum likelihood estimators, a simulation study is conducted for the S-PINAR(1) model

Sistemas de alarme ótimos e sua aplicação a séries financeiras

Doutoramento em MatemáticaThis thesis focuses on the application of optimal alarm systems to non linear time series models. The most common classes of models in the analysis of real-valued and integer-valued time series are described. The construction of optimal alarm systems is covered and its applications explored. Considering models with conditional heteroscedasticity, particular attention is given to the Fractionally Integrated Asymmetric Power ARCH, FIAPARCH(p; d; q) model and an optimal alarm system is implemented, following both classical and Bayesian methodologies. Taking into consideration the particular characteristics of the APARCH(p; q) representation for financial time series, the introduction of a possible counterpart for modelling time series of counts is proposed: the INteger-valued Asymmetric Power ARCH, INAPARCH(p; q). The probabilistic properties of the INAPARCH(1; 1) model are comprehensively studied, the conditional maximum likelihood (ML) estimation method is applied and the asymptotic properties of the conditional ML estimator are obtained. The final part of the work consists on the implementation of an optimal alarm system to the INAPARCH(1; 1) model. An application is presented to real data series.Esta tese centra-se na aplicação de sistemas de alarme ótimos a modelos de séries temporais não lineares. As classes de modelos mais comuns na análise de séries temporais de valores reais e de valores inteiros são descritas com alguma profundidade. É abordada a construção de sistemas de alarme ótimos e as suas aplicações são exploradas. De entre os modelos com heterocedasticidade condicional é dada especial atenção ao modelo ARCH Fraccionalmente Integrável de Potência Assimétrica, FIAPARCH(p; d; q), e é feita a implementação de um sistema de alarme ótimo, considerando ambas as metodologias clássica e Bayesiana. Tomando em consideração as características particulares do modelo APARCH(p; q) na aplicação a séries de dados financeiros, é proposta a introdução do seu homólogo para a modelação de séries temporais de contagens: o modelo ARCH de valores INteiros e Potência Assimétrica, INAPARCH(p; q). As propriedades probabilísticas do modelo INAPARCH(1; 1) são extensivamente estudadas, é aplicado o método da máxima verosimilhança (MV) condicional para a estimação dos parâmetros do modelo e estudadas as propriedades assintóticas do estimador de MV condicional. Na parte final do trabalho é feita a implementação de um sistema de alarme ótimo ao modelo INAPARCH(1; 1) e apresenta-se uma aplicação a séries de dados reais

Time Series Modelling

The analysis and modeling of time series is of the utmost importance in various fields of application. This Special Issue is a collection of articles on a wide range of topics, covering stochastic models for time series as well as methods for their analysis, univariate and multivariate time series, real-valued and discrete-valued time series, applications of time series methods to forecasting and statistical process control, and software implementations of methods and models for time series. The proposed approaches and concepts are thoroughly discussed and illustrated with several real-world data examples

Doprinos analizi vremenskih nizova sa celobrojnim vrednostima

New thinning operators are defined as differences of two negative binomial thinning operators. On the basis of such defined operators, time series with discrete Laplace marginals are defined. Some important features of all introduced models are determined. Estimators of unknown parameters are derived and their asimptotic behaviour are discussed. All models are checked on simulated data and compared with some of existing models. An application in real-life situations are presented. Also, a method for identification of latent components of the models are give

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