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PART ONE An Introduction to the Theory and Methods of Time Series Analysis3

Chapter 1．Theory of Stationary Time Series3

1.1 The definition of stationary stochastic processes3

1.2 The spectral representation of covariance function12

1.3 The Hilbert space of second order processes18

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

1.4.1 Orthogonal stochastic measure21

1.4.2 Stochastic integral and the representation of stationary processes22

1.4.3．Karhunen theorem26

1.5 Strong law of large numbers for stationary series28

1.6 Sampling theorem for stochastic stationary processes33

Chapter 2．ARMA Model and Model Fitting36

2.1 ARMA model and the Wold decomposition36

2.2 Orthogonal basis in Hilbert space Hξ41

2.3 The covariance function of ARMA model and Yule-Walker equation47

2.4 Model fitting under the criterion of one-step ahead prediction error53

2.5 M.E.model fitting for observed data63

2.5.1 M.E.model fitting with sample covariance63

2.5.2 Order selection problem65

Chapter 3．Prediction,Filtering and Spectral Analysis of Time Series72

3.1 Prediction of time series72

3.1.1 The prediction formula for AR models74

3.1.2 The prediction formula for ARMA models78

3.2 The linear filtering of time series81

3.3 Spectral analysis of time series91

3.3.1 Theory and methods of hidden periodicities analysis92

3.3.2 Theory and methods of spectral density estimations100

PART TWO Case Studies in Time Series Analysis113

Case Ⅰ．Digital Processing of a Dynamic Marine Gravity Meter113

1．Problem statement and working diagram of a dynamic marine gravity meter113

2．The first test for solving the problem114

3．Design a new digital filter under Min-Max criterion120

4．The frequency rectification by filtering129

5．Practical checking in the prospecting field of the East Sea of China132

Case Ⅱ．Digital Filters Design by Maximum Entropy Modelling135

1．Problem statement135

2．Design the filter by maximum entropy modelling139

3．A practical filter design144

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

1．Introduction147

2．Spectral analysis of VEP records for dull and normal children148

3．Statistical analysis for detection of characteristics153

4．Physiological interpretation157

Appendix Ⅲ159

Case Ⅳ．Statistical Analysis of VEP and AI by the Principal Component Analysis of Time Series in Frequency Domain162

1．Introduction162

2．Principal component analysis in frequency domain and its application in AI analysis165

3．Practical checking169

4．Discussion170

Appendix Ⅳ172

Case Ⅴ．Periodicity Analysis of LH Release in Isolated Pituitary Gland by Hidden Frequency Analysis178

1．Introduction178

2．Statistical analysis of LH release179

3．Practical rhythm analysis of LH release185

4．Discussion187

Case Ⅵ．Statistical Detection of Uranian Ring Signals from the Light Curve of Photoelectric Observation193

1．Introduction193

2．Statistical detection of weak ring signals from the noise background196

3．Discussion204

Case Ⅶ．On the Forecasting of Freight Transportation by a New Model Fitting Procedure of Time Series207

1．Introduction207

2．A new model fitting procedure for freight transportation prediction212

3．Forecasting for freight transportation of practical data218

4．Dicussion221

Appendix Ⅶ226

A.1 On the X-11 processing procedure226

A.2 Simple exponential smoothing predictor231

A.3 Program for fitting a spline function232

Case Ⅷ．The Water Flow Prediction in Xiang River235

1．Introduction235

2．Constructing a prediction formula based on the hidden periodicities by the quantile method236

3．Comparison and discussion241

Appendix Ⅷ247

A.1 Quantile method for detecting the hidden periodicities247

A.2 RMA forecasting method248

Case Ⅸ．Miscellaneous Cases Study250

Ⅸ.1 Long term weather forecasting by seasonal ARIMA model250

Ⅸ.1.1 Some relevant knowledge250

(1)Seasonal ARIMA model250

(2)M.L.E.and M.S.S.E.under the normal distribution252

(3)Powell's algorithm for seeking the extreme value of a convex function254

(4)Roots identification of a polynomial by Jury's method256

Ⅸ.1.2 Modelling and forecasting for the temperature in Shanghai259

Ⅸ.2 Outlier analysis and interpolation of missing data in a measuring system261

Ⅸ.2.1 Basic knowledge on outlier analysis261

Ⅸ.2.2 Interpolation for missing data for AR(p)model267

Ⅸ.2.3 Practical application for a range measuring system269

Bibliography273

Subject Index277