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Durham University

Department of Mathematical Sciences

Academic Staff

Publication details for Peter Craig

Craig, Peter S. (1988). Time Series Analysis for Directional Data. Statistics. University of Dublin, Trinity College. PhD: 1-165.

Author(s) from Durham

Abstract

This thesis is an account of some aspects of time series analysis for directional data (or, more strictly, circular data), which is an almost totally unexplored area of statistics. The thesis is in four chapters.

The first concerns a family of models for directional time series which is naturally derived from the ARMA family of time series models. The identification problem for the family is discussed and an analogue of the auto-correlation function is defined. The remainder of the chapter is devoted to estimation of that analogue with major attention being given to showing that the estimators used are both consistent and asymptotically normal.

The second chapter examines the estimation problem in detail for the simplest model from the family introduced in the first chapter. A form of moment estimation is described and its asymptotic properties derived. The majority of the chapter is devoted to maximum likelihood estimation. Maximum likelihood estimation is shown to to be consistent and asymptotically normal. The asymptotic properties are quantified and shown to be superior to those for moment estimation, and the chapter closes with a discussion of the
computational problems involved in performing maximum likelihood estimation for the model.

The third chapter deals with a number of aspects of Markov models for directional time series. Most of the chapter is given to a discussion of the various bivariate circular distributions to be found in the literature, while stationarity, higher order models, and estimation properties are also considered.

The final chapter is a trial data analysis for a sequence of wind directions. Two useful diagnostic techniques are introduced. The analysis proceeds from the models of the first chapter to the Markov models of the third chapter and the chapter concludes with an attempt to model some of the seasonal behaviour apparent in the data.