Extrapolation, Interpolation, and Smoothing

It has been the opinion of many that Wiener will be remembered for his Extrapolation long after Cybernetics is forgotten. Indeed few computer-science students would know today what cybernetics is all about, while every communication student knows what Wiener's filter is. The work was circulated as a classified memorandum in 1942, as it was connected with sensitive war-time efforts to improve radar communication. This book became the basis for modern communication theory, by a scientist considered one of the founders of the field of artifical intelligence. Combining ideas from statistics and time-series analysis, Wiener used Gauss's method of shaping the characteristic of a detector to allow for the maximal recognition of signals in the presence of noise. This method came to be known as the "Wiener filter."

Norbert Wiener (1894–1964) served on the faculty in the Department of Mathematics at MIT from 1919 until his death. In 1963, he was awarded the National Medal of Science for his contributions to mathematics, engineering, and biological sciences. He was the author of many books, including Norbert Wiener—A Life in Cybernetics and the National Book Award-winning God & Golem, Inc.: A Comment on Certain Points Where Cybernetics Impinges on Religion (both published by the MIT Press).

Introduction

1

(1)

The Purpose of This Book

1

(1)

Time Series

1

(1)

Communication Engineering

2

(1)

Techniques of Time Series and Communication Engineering Contrasted

3

(5)

The Ensemble

4

(1)

Correlation

4

(2)

The Periodogram

6

(1)

Operational Calculus

7

(1)

The Fourier Integral; Need of the Complex Plane

8

(1)

Time Series and Communication Engineering---The Synthesis

8

(3)

Prediction

9

(1)

Filtering

9

(1)

Policy Problems

10

(1)

Permissible Operators: Translation Group in Time

11

(2)

Past and Future

12

(1)

Subclasses of Operators

12

(1)

Norms and Minimisation

13

(2)

The Calculus of Variations

14

(1)

Ergodic Theory

15

(6)

Brownian Motion

20

(1)

Summary of Chapters

21

(4)

Resume of Fundamental Mathematical Notions

25

(31)

Fourier Series

25

(12)

Orthogonal Functions

31

(3)

The Fourier Integral

34

(1)

Laguerre Functions

35

(1)

More on the Fourier Integral; Realizability of Filters

36

(1)

Generalized Harmonic Analysis

37

(7)

Discrete Arrays and Their Spectra

43

(1)

Multiple Harmonic Analysis and Coherency Matrices

44

(1)

Smoothing Problems

45

(1)

Ergodic Theory

46

(1)

Brownian Motion

47

(4)

Poisson Distributions

51

(1)

Harmonic Analysis in the Complex Domain

52

(4)

The Linear Predictor for a Single Time Series

56

(25)

Formulation of the Problem of the Linear Predictor

56

(1)

The Minimization Problem

57

(3)

The Factorization Problem

60

(4)

The Predictor Formula

64

(17)

Examples of Prediction

65

(3)

A Limiting Example of Prediction

68

(3)

The Prediction of Functions Whose Derivatives Possess Auto-correlation Coefficients

71

(3)

Spectrum Lines and Non-absolutely Continuous Spectra

74

(2)

Prediction by the Linear Combination of Given Operators

76

(2)

The Linear Predictor for a Discrete Time Series

78

(3)

The Linear Filter for a Single Time Series

81

(23)

Formulation of the General Filter Problem

81

(1)

Minimization Problem for Filters

82

(2)

The Factorization of the Spectrum

84

(2)

Prediction and Filtering

86

(2)

The Error of Performance of a Filter; Long-lag Filters

88

(2)

Filters and Ergodic Theory

90

(1)

Computation of Specific Filter Characteristics

91

(1)

Lagging Filters

92

(2)

The Determination of Lag and Number of Meshes in a Filter

94

(1)

Detecting Filters for High Noise Level

95

(9)

Filters for Pulses

95

(2)

Filters Having Characteristics Linearly Dependent on Given Characteristics

97

(4)

Computation of Filter: Resume

101

(3)

The Linear Predictor and Filter for Multiple Time Series

104

(13)

Symbolism and Definitions for Multiple Time Series

104

(1)

Minimization Problem for Multiple Time Series

105

(1)

Method of Undetermined Coefficients

106

(3)

Multiple Prediction

109

(1)

Special Cases of Prediction

110

(1)

A Discrete Case of Prediction

111

(1)

General Technique of Discrete Prediction

112

(5)

Miscellaneous Problems Encompassed by the Technique of This Book

117

(7)

The Problem of Approximate Differentiation

117

(2)

An Example of Approximate Differentiation

119

(1)

A Misleading Example of Approximate Differentiation

120

(1)

Interpolation and Extrapolation

121

(3)

Appendix A TABLE OF THE LAGUERRE FUNCTIONS

124

(5)

Appendix B THE WIENER RMS (ROOT MEAN SQUARE) ERROR CRITERION IN FILTER DESIGN AND PREDICTION

129

(20)

Norman Levinson

1. Linear Filters

130

(1)

2. Minimisation of RMS Error

131

(5)

3. Determination of the Weighting Function

136

(3)

4. Realization of Operator---Mathematical Formulation

139

(4)

5. RC Filter

143

(3)

6. Prediction and Lag with and without Noise

146

(3)

Appendix C A HEURISTIC EXPOSITION OF WIENER'S MATHEMATICAL THEORY OF PREDICTION AND FILTERING

149

(1)

Norman Levinson

1. The Auto-correlation Function

150

(2)

2. The Integral Equation

152

(1)

3. The Modified Integral Equation

153

(2)

4. The Factorization Problem

155

(2)

5. The Functions ψ1, ψ2, and ∞

157

(1)

6. The Prediction Operator

158

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Extrapolation, Interpolation, and Smoothing of Stationary Time Series With Engineering Applications

Summary

Author Biography

Table of Contents

Supplemental Materials