Time Series A Biostatistical Introduction

Time series analysis is one of several branches of statistics whose practical importance has increased with the availability of powerful computing tools. Methodology originally developed for specialized applications, for example in business forecasting or geophysical signal processing, is nowwidely available in general statistical packages. These computing developments have helped to bring the subject closer to the mainstream of applied statistics. This book is an introductory account of time-series analysis, written from the perspective of an applied statistician with a particular interest in biological applications. Separate chapters cover exploratory methods, the theory of stationary random processes, spectral analysis, repeatedmeasurements, ARIMA modelling, forecasting, and bivariate time-series analysis. Throughout, analyses of data-sets drawn from the biological and medical sciences are integrated with the methodological development. The book is unique in its emphasis on biological and medical applications of time-series analysis. Nevertheless, its methodological content is more widely applicable, and it should be useful to both students and practitioners of applied statistics, whatever their specialization.

Peter J. Diggle: Department of Mathematics and Statistics Lancaster University

Introduction	p. 1
Definitions, examples, and notation	p. 1
Objectives of time-series analysis	p. 10
More notation	p. 11
Trend, serial dependence and stationarity	p. 13
Duality between trend and serial dependence	p. 16
Software	p. 18
Simple descriptive methods of analysis	p. 19
Time plots	p. 19
Smoothing	p. 21
Differencing	p. 31
The autocovariance and autocorrelation functions	p. 32
Estimating the autocorrelation function	p. 39
Impact of trend removal on autocorrelation structure	p. 45
The periodogram	p. 47
The connection between the correlogram and the periodogram	p. 55
Theory of stationary processes	p. 58
Notation and definitions	p. 58
The spectrum of a stationary random process	p. 59
Linear filters	p. 66
The autoregressive moving average process	p. 71
Sampling and accumulation of stationary random functions	p. 85
Implications of autocorrelation for elementary statistical methods	p. 87
Further reading	p. 92
Spectral analysis	p. 94
Introduction	p. 94
The periodogram revisited	p. 95
Periodogram-based tests of white noise	p. 98
The fast Fourier transform	p. 101
Periodogram averages	p. 103
Other smooth estimates of the spectrum	p. 108
Adjusting spectral estimates for the effects of filtering	p. 113
Combining and comparing spectral estimates	p. 117
Fitting parametric models	p. 124
Discussion: strengths and weaknesses of spectral analysis	p. 127
Further reading	p. 133
Repeated measurements	p. 134
Introduction	p. 134
Repeated measurements as multivariate data	p. 135
Incorporating time-series structure	p. 138
Formulating the model: time plots and the variogram	p. 142
Fitting the model: maximum likelihood estimation and testing	p. 152
Case-study: analysis of data on protein content of milk samples	p. 157
Further reading	p. 163
Fitting autoregressive moving average processes to data	p. 165
Introduction: ARIMA processes as models for non-stationary time series	p. 165
Identification	p. 167
Estimation	p. 172
Diagnostic checking	p. 177
Case studies	p. 182
Further reading	p. 187
Forecasting	p. 188
Preamble	p. 188
Forecasting by extrapolation of polynomial trends	p. 190
Exponential smoothing	p. 192
The Box-Jenkins approach to forecasting	p. 194
Further reading	p. 201
Elements of bivariate time-series analysis	p. 202
Introduction	p. 202
The cross-covariance and cross-correlation functions	p. 202
Estimating the cross-correlation function	p. 204
The spectrum of a bivariate process	p. 212
Estimating the cross-spectrum	p. 215
Further reading	p. 220
References	p. 222
Data sets	p. 228
Levels of luteinizing hormone in blood samples	p. 228
Wool prices at weekly markets	p. 229
UK deaths from bronchitis, emphysema, and asthma	p. 238
Growth of colonies of paramecium aurelium	p. 239
Body weights of rats under three different experimental treatments	p. 240
Protein content of milk samples	p. 241
Annual numbers of lynx trapped in the Mackenzie River district of north-west Canada from 1821 to 1934	p. 244
Matrix approach to the general linear model	p. 245
Definition of the general linear model	p. 245
The multivariate Normal distribution	p. 247
Least squares estimation	p. 248
Properties of least squares estimates under a Normality assumption	p. 248
Generalized least squares	p. 249
Further reading	p. 250
Complex numbers	p. 251
Definitions	p. 251
Polar representation of complex numbers	p. 252
The complex exponential function	p. 252
Further reading	p. 253
Index	p. 254
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Time Series A Biostatistical Introduction

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