Some Forecast Methods in Regression Models for Categorical Time Series

We are dealing with the prediction of forthcoming outcomes of a categorical time series. We will assume that the evolution of the time series is driven by a covariate process and by former outcomes and that the covariate process itself obeys an autoregressive law. Two forecasting methods are presented. The first is based on an integral formula for the probabilities of forthcoming events and by a Monte Carlo evaluation of this integral. The second method makes use of an approximation formula for conditional expectations. The procedures proposed are illustrated by an application to data on forest damages

Statistical inference for detrended point processes

We consider a multivariate point process with a parametric intensity process which splits into a stochastic factor bt and a trend function at of a squared polynomial form with exponents larger than . Such a process occurs in a situation where an underlying process with intensity bt can be observed on a transformed time scale only. On the basis of the maximum likelihood estimator for the unknown parameter a detrended (or residual) process is defined by transforming the occurrence times via integrated estimated trend function. It is shown that statistics (mean intensity, periodogram estimator) based on the detrended process exhibit the same asymptotic properties as they do in the case of the underlying process (without trend function). Thus trend removal in point processes turns out to be an appropriate method to reveal properties of the (unobservable) underlying process - a concept which is well established in time series. A numerical example of an earthquake aftershock sequence illustrates the performance of the method.multivariate point process intensity process trend component detrending residual process periodogram estimator

Statistical inference for detrended point processes

AbstractWe consider a multivariate point process with a parametric intensity process which splits into a stochastic factor bt and a trend function at of a squared polynomial form with exponents larger than −12. Such a process occurs in a situation where an underlying process with intensity bt can be observed on a transformed time scale only. On the basis of the maximum likelihood estimator for the unknown parameter a detrended (or residual) process is defined by transforming the occurrence times via integrated estimated trend function. It is shown that statistics (mean intensity, periodogram estimator) based on the detrended process exhibit the same asymptotic properties as they do in the case of the underlying process (without trend function). Thus trend removal in point processes turns out to be an appropriate method to reveal properties of the (unobservable) underlying process – a concept which is well established in time series. A numerical example of an earthquake aftershock sequence illustrates the performance of the method

Statistical analysis of climate series : analyzing, plotting, modeling, and predicting with R

The book presents the application of statistical methods to climatological data on temperature and precipitation. It provides specific techniques for treating series of yearly, monthly and daily records. The results� potential relevance in the climate context is discussed. The methodical tools are taken from time series analysis, from periodogram and wavelet analysis, from correlation and principal component analysis, and from categorical data and event-time analysis. The applied models are - among others - the ARIMA and GARCH model, and inhomogeneous Poisson processes. The book also deal with a number of special statistical topics, e.g. the problem of trend-, season- and autocorrelation-adjustment, and with simultaneous statistical inference

Functions of event variables of a random system with complete connections

In applications it often occurs that the experimenter is faced with functions of random processes. Suppose, for instance, that he only can draw partial or incomplete information about the underlying process or that he has to classify events for the sake of efficiency. We assume that the underlying process is a random system with complete connections (which contains the Markovian case as a special one) satisfying some basic properties, and that a mapping operates on the event space. With these two elements we construct in Section 2 a new random system with complete connections which inherits the properties of the old one (Theorem 2.2.3). In Section 3 we prove a weak convergence theorem (Theorem 3.4.4) in the theoretical framework of the so-called distance diminishing models, which gives a straightforward application in Section 4 to conditional probabilities related to partially observed events (Theorems 4.1.3). Finally we prove a Shannon-McMillan-type theorem (Theorem 4.2.3) finding application to classification procedures.Random systems with complete connections learning models functions of random processes partially observed Markov chains