Automatic Autocorrelation and Spectral Analysis

Transcription

1 Piet M.T. Broersen Automatic Autocorrelation and Spectral Analysis With 104 Figures Sprin ger

2 1 Introduction Time Series Problems 1 2 Basic Concepts Random Variables Normal Distribution Conditional Densities Functions of Random Variables Linear Regression General Estimation Theory Exercises 26 3 Periodogram and Lagged Product Autocorrelation Stochastic Processes Autocorrelation Function Spectral Density Function Estimation of Mean and Variance Autocorrelation Estimation Periodogram Estimation Summary of Nonparametric Methods Exercises 56 4 ARMA Theory Time Series Models White Noise Moving Average Processes MA(1) Process with Zero Outside the Unit Circle Autoregressive Processes AR(1) Processes AR(1) Processes with a Pole Outside the Unit Circle AR(2) Processes AR( p) Processes ARMA( p,q) Processes Harmonie Processes with Poles on the Unit Circle 78

3 x 4.7 Spectra of Time Series Models Some Examples Exercises 86 Relations for Time Series Models Time Series Estimation Yule-W alker Relations and the Levinson-Durbin Recursion Additional AR Representations Additional AR Relations The Relation between the Variances of x and e n for an AR( p) Process Parameters from Reflection Coefficients Reflection Coefficients from Parameters Autocorrelations from Reflection Coefficients Autocorrelations from Parameters Relation for MA Parameters Accuracy Measures for Time Series Models Prediction Error Model Error Power Gain Spectral Distortion More Relative Measures Absolute and Squared Measures Cepstrum as a Measure for Autocorrelation Functions ME and the Triangulär Bias Computational Rules for the ME Exercises 113 Estimation of Time Series Models Historical Remarks About Spectral Estimation Are Time Series Models Generally Applicable? Maximum Likelihood Estimation AR ML Estimation MA ML Estimation ARMA ML Estimation AR Estimation Methods Yule-Walker Method Forward Least-squares Method Forward and Backward Least-squares Method Burg's Method Asymptotic AR Theory Finite-sample Practice for Burg Estimates of White Noise Finite-sample Practice for Burg Estimates of an AR(2) Process Model Error (ME) of Burg Estimates of an AR(2) Process MA Estimation Methods 135

4 xi 6.6 ARMA Estimation Methods ARMA( p,q) Estimation, First-stage ARMA( p,q) Estimation, First-stage Long AR ARMA( p,q) Estimation, First-stage Long MA ARMA( p,q) Estimation, First-stage Long COV ARMA(/?,<?) Estimation, First-stage Long Rinv ARMA( p,q) Estimation, Second-stage ARMA( p,q) Estimation, Simulations Covariance Matrix of ARMA Parameters The Covariance Matrix of Estimated AR Parameters The Covariance Matrix of Estimated MA Parameters The Covariance Matrix of Estimated ARMA Parameters Estimated Autocovariance and Spectrum Estimators for the Mean and the Variance Estimation of the Autocorrelation Function The Residual Variance The Power Spectral Density Exercises AR Order Selection Overview of Order Selection Order Selection in Linear Regression Asymptotic Order-selection Criteria Relations for Order-selection Criteria Finite-sample Order-selection Criteria Kullback-Leibler Discrepancy The Penalty Factor Finite-sample AR Criterion CIC Order-selection Simulations Subset Selection Exercises MA and ARMA Order Selection Introduction Intermediate AR Orders for MA and ARMA Estimation Reduction of the Number of ARMA Candidate Models Order Selection for MA Estimation Order Selection for ARMA Estimation Exercises ARMASA Toolbox with Applications Introduction Selection of the Model Type The Language of Random Data Reduced-statistics Order Selection Accuracy of Reduced-statistics Estimation ARMASA Applied to Harmonie Processes 233

5 xii 9.7 ARMASA Applied to Simulated Random Data ARMASA Applied to Real-life Data Turbulence Data Radar Data Satellite Data Lung Noise Data River Data Exercises 248 ARMASA Toolbox Advanced Topics in Time Series Estimation Accuracy of Lagged Product Autocovariance Estimates Generation of Data Subband Spectral Analysis Missing Data Irregulär Data Multishift, Slotted, Nearest-neighbour Resampling ARMAsel for Irregulär Data Performance of ARMAsel for Irregulär Data Exercises 286 Bibliography 287 Index 295