
Time Series: A Biostatistical Introduction
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Content
- Cover
- Title Page
- Copyright Page
- Preface
- Preface to the first edition
- Contents
- 1 Introduction
- 1.1 Examples
- 1.2 Separating deterministic and stochastic components of variation
- 1.3 Objectives of time-series analysis
- 1.4 Notation
- 1.5 Trend, serial dependence, and stationarity
- 1.6 Duality between trend and serial dependence
- 1.7 Software
- 2 Simple descriptive methods of analysis
- 2.1 Time plots
- 2.2 Smoothing
- 2.2.1 Moving averages
- 2.2.2 Kernel smoothing
- 2.2.3 Polynomial regression
- 2.2.4 Regression splines
- 2.2.5 Smoothing splines
- 2.2.6 Splines, kernels, and moving averages
- 2.2.7 Non-linear smoothers
- 2.2.8 Additive models
- 2.2.9 Summary
- 2.3 Differencing
- 2.4 The autocovariance and autocorrelation functions
- 2.5 Estimating the autocovariance and autocorrelation functions
- 2.5.1 Equally spaced series: the correlogram
- 2.5.2 Unequally spaced series: the variogram
- 2.6 Impact of trend removal on autocorrelation structure
- 2.7 The periodogram
- 2.8 The connection between the correlogram and the periodogram
- 2.9 Exercises
- 3 Theory of stationary processes
- 3.1 Notation and definitions
- 3.2 The spectrum of a stationary process
- 3.3 Linear filters
- 3.4 Autoregressive moving average processes
- 3.4.1 The backward shift operator: a convenient notational shorthand
- 3.4.2 Second-order properties of the moving average process
- 3.4.3 Second-order properties of the autoregressive process: AR(p)
- 3.4.4 Second-order properties of the autoregressive moving average process: ARMA(p, q)
- 3.4.5 Invertibility
- 3.4.6 Why are ARMA processes useful?
- 3.5 Sampling and integration of stationary random functions
- 3.6 Impact of autocorrelation on elementary statistical methods
- 3.7 Hierarchical model specification: signal-plus-noise
- 3.8 Exercises
- 4 Discrete-time models for unreplicated series
- 4.1 ARIMA processes as empirical models for non-stationary time series
- 4.2 Fitting an ARIMA model
- 4.2.1 Identification
- 4.2.2 Estimation
- 4.2.3 Diagnostic checking
- 4.2.4 Fitting an ARIMA model to Covid-19 test positivity rates
- 4.3 State-space models
- 4.4 Further reading
- 4.5 Exercises
- 5 Continuous-time linear models for unreplicated series
- 5.1 Introduction
- 5.2 The Mat"00E9rn class of correlation functions
- 5.3 Fitting a Mat"00E9rn process model
- 5.3.1 Exploring residual correlation using the empirical variogram
- 5.3.2 Estimation
- 5.4 Some other constructions for continuous-time stochastic process models: Brownian motion, Wiener, and Ornstein-Uhlenbeck processes
- 5.5 A case study in renal failure monitoring
- 5.6 A conditional re-interpetation of the Mat"00E9rn model
- 5.7 Further reading
- 5.8 Exercises
- 6 Generalized linear models for unreplicated series
- 6.1 Generalized linear mixed models
- 6.2 Exploratory analysis: testing for temporal correlation
- 6.3 Parameter estimation
- 6.4 Forecasting
- 6.5 Exercises
- 7 Replicated series
- 7.1 Introduction
- 7.2 Exploratory analysis
- 7.3 Parameter estimation
- 7.4 Extension to generalized linear models
- 7.4.1 Random-effects models
- 7.4.2 Marginal models
- 7.5 Exercises
- 8 Spectral analysis
- 8.1 Introduction
- 8.2 The periodogram revisited
- 8.3 Periodogram-based test of white noise
- 8.4 A periodogram-based goodness-of-fit test
- 8.5 The fast Fourier transform
- 8.6 Smoothing the periodogram
- 8.6.1 Simple moving averages
- 8.6.2 Weighted moving averages
- 8.6.3 Spectral windows
- 8.6.4 Regression spline estimates
- 8.6.5 Autoregressive spectral estimates
- 8.6.6 Summary
- 8.7 Adjusting spectral estimates for the effects of filtering
- 8.8 Harmonics
- 8.9 Leakage
- 8.10 Spectral checking
- 8.11 Combining and comparing spectral estimates
- 8.11.1 Comparing spectral estimates from time series collected under different experimental conditions
- 8.11.2 Combining spectral estimates from replicated time series
- 8.12 Periodogram analysis as generalized linear modelling
- 8.12.1 A log-linear gamma model
- 8.12.2 Fitting parametric models
- 8.12.3 Random effects
- 8.13 Discussion: strengths and weaknesses of spectral analysis
- 8.14 Further reading
- 8.15 Exercises
- 9 Bivariate time-series analysis
- 9.1 Introduction
- 9.2 The cross-covariance and cross-correlation functions
- 9.3 Estimating the cross-covariance function
- 9.4 Bivariate spectral analysis
- 9.4.1 The spectrum of a bivariate process
- 9.4.2 Estimating the cross-spectrum
- 9.5 Further reading
- 9.6 Exercises
- Appendix A Background statistical theory
- A.1 Notational conventions
- A.2 Some widely used probability distributions
- A.2.1 The binomial distribution
- A.2.2 The Poisson distribution
- A.2.3 The normal distribution
- A.3 Multivariate distributions
- A.3.1 Independent and dependent random variables: joint, marginal, and conditional distributions
- A.3.2 The multivariate normal distribution
- A.4 Modes of inference
- A.4.1 Parameter estimation
- A.4.2 Significance testing
- A.4.3 Prediction
- A.5 Likelihood-based inference
- A.5.1 The likelihood and log-likelihood functions
- A.5.2 Parameter estimation
- A.5.3 Likelihood-based significance testing
- A.5.4 Prediction
- A.6 A closer look at the Maten process with ?=1/2
- A.7 Further reading
- Appendix B Matrix representation of linear models
- B.1 Definition of the general linear model
- B.2 Ordinary least squares (OLS) estimation
- B.3 Properties of ordinary least squares estimators
- B.4 Generalized least squares
- B.5 Further reading
- Appendix C Complex numbers
- C.1 Definitions
- C.2 Polar representation of complex numbers
- C.3 The complex exponential function
- C.4 Further reading
- References
- Index
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