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PART ONE An Introduction to the Theory and Methods of Time Series Analysis 3

Chapter 1．Theory of Stationary Time Series 3

1.1 The definition of stationary stochastic processes 3

1.2 The spectral representation of covariance function 12

1.3 The Hilbert space of second order processes 18

1.4 Stochastic integral and the isomorphic relationship between Hξ and the functional space L2(dFξ) 21

1.4.1 Orthogonal stochastic measure 21

1.4.2 Stochastic integral and the representation of stationary processes 22

1.4.3．Karhunen theorem 26

1.5 Strong law of large numbers for stationary series 28

1.6 Sampling theorem for stochastic stationary processes 33

Chapter 2．ARMA Model and Model Fitting 36

2.1 ARMA model and the Wold decomposition 36

2.2 Orthogonal basis in Hilbert space Hξ 41

2.3 The covariance function of ARMA model and Yule-Walker equation 47

2.4 Model fitting under the criterion of one-step ahead prediction error 53

2.5 M.E.model fitting for observed data 63

2.5.1 M.E.model fitting with sample covariance 63

2.5.2 Order selection problem 65

Chapter 3．Prediction,Filtering and Spectral Analysis of Time Series 72

3.1 Prediction of time series 72

3.1.1 The prediction formula for AR models 74

3.1.2 The prediction formula for ARMA models 78

3.2 The linear filtering of time series 81

3.3 Spectral analysis of time series 91

3.3.1 Theory and methods of hidden periodicities analysis 92

3.3.2 Theory and methods of spectral density estimations 100

PART TWO Case Studies in Time Series Analysis 113

Case Ⅰ．Digital Processing of a Dynamic Marine Gravity Meter 113

1．Problem statement and working diagram of a dynamic marine gravity meter 113

2．The first test for solving the problem 114

3．Design a new digital filter under Min-Max criterion 120

4．The frequency rectification by filtering 129

5．Practical checking in the prospecting field of the East Sea of China 132

Case Ⅱ．Digital Filters Design by Maximum Entropy Modelling 135

1．Problem statement 135

2．Design the filter by maximum entropy modelling 139

3．A practical filter design 144

Case Ⅲ．The Spectral Analysis of the Visual Evoked Potentials of Normal and Congenital Dull Children(Down's disease) 147

1．Introduction 147

2．Spectral analysis of VEP records for dull and normal children 148

3．Statistical analysis for detection of characteristics 153

4．Physiological interpretation 157

Appendix Ⅲ 159

Case Ⅳ．Statistical Analysis of VEP and AI by the Principal Component Analysis of Time Series in Frequency Domain 162

1．Introduction 162

2．Principal component analysis in frequency domain and its application in AI analysis 165

3．Practical checking 169

4．Discussion 170

Appendix Ⅳ 172

Case Ⅴ．Periodicity Analysis of LH Release in Isolated Pituitary Gland by Hidden Frequency Analysis 178

1．Introduction 178

2．Statistical analysis of LH release 179

3．Practical rhythm analysis of LH release 185

4．Discussion 187

Case Ⅵ．Statistical Detection of Uranian Ring Signals from the Light Curve of Photoelectric Observation 193

1．Introduction 193

2．Statistical detection of weak ring signals from the noise background 196

3．Discussion 204

Case Ⅶ．On the Forecasting of Freight Transportation by a New Model Fitting Procedure of Time Series 207

1．Introduction 207

2．A new model fitting procedure for freight transportation prediction 212

3．Forecasting for freight transportation of practical data 218

4．Dicussion 221

Appendix Ⅶ 226

A.1 On the X-11 processing procedure 226

A.2 Simple exponential smoothing predictor 231

A.3 Program for fitting a spline function 232

Case Ⅷ．The Water Flow Prediction in Xiang River 235

1．Introduction 235

2．Constructing a prediction formula based on the hidden periodicities by the quantile method 236

3．Comparison and discussion 241

Appendix Ⅷ 247

A.1 Quantile method for detecting the hidden periodicities 247

A.2 RMA forecasting method 248

Case Ⅸ．Miscellaneous Cases Study 250

Ⅸ.1 Long term weather forecasting by seasonal ARIMA model 250

Ⅸ.1.1 Some relevant knowledge 250

(1)Seasonal ARIMA model 250

(2)M.L.E.and M.S.S.E.under the normal distribution 252

(3)Powell's algorithm for seeking the extreme value of a convex function 254

(4)Roots identification of a polynomial by Jury's method 256

Ⅸ.1.2 Modelling and forecasting for the temperature in Shanghai 259

Ⅸ.2 Outlier analysis and interpolation of missing data in a measuring system 261

Ⅸ.2.1 Basic knowledge on outlier analysis 261

Ⅸ.2.2 Interpolation for missing data for AR(p)model 267

Ⅸ.2.3 Practical application for a range measuring system 269

Bibliography 273

Subject Index 277