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EXTRAPOLATION, INTERPOLATION, AND SMOOTHING OF STATIONARY TIME SERIES

Technology Press Books PRINCIPLES OF ELECTRICAL ENGINEERING SERIES By Members of the Staff of the Department of Electrical Engineering Massachusetts Institute of Technology ELECTRIC CIRCUITS A First Course in Circuit Analysis for Electrical Engineers MAGNETIC CIRCUITS AND TRANSFORMERS A First Course for Power and Communication Engineers APPLIED ELECTRONICS A First Course in Electronics, Electron Tubes, and Associated Circuits THE MATHEMATICS OF CIRCUIT ANALYSIS Extensions to the Mathematical Training of Electrical Engineers By E. A. Guillemin SCIENT1FIC SOCIETIES IN THE UNITED STATES By R. S. Bates Q. E. D., M.I. T. in World War II By John Burchard WAVELENGTH TABLES Measured and compiled under the direction of George R. Harrison THE MOVEMENT OF FACTORY WORKERS By C. A. Myers and W. R. Maclaurin INDEX FOSSILS OF NORTH AMERICA By H. W. Shimer and R. R. Shrock CYBERNETICS or Control and Communication in the Animal and the Machine By Norbert Wiener EXTRAPOLATION, INTERPOLATION, AND SMOOTHING OF STATIONARY TIME SERIES with Engineering Applications By Norbert Wiener

EXTRAPOLATION, INTERPOLATION, AND SMOOTHING OF STATIONARY TIME SERIES With Engineering ApplicCltions by Norbert Wiener PROfESSOR Of MATHEMATICS MASSACHUSETTS INSTITUTE Of TECHNOLOGY Published jOintly by THE TECHNOLOGY PRESS OF THE MASSACHUSETTS INSTITUTE OF TECHNOLOGY end JOHN WILEY & SONS, INC., NEW YORK CHAPMAN & HALL, LIMITED, LONDON

PREFACE Largely because of the impetus gained during World War II, com munication and control engineering have reached a very high level of development today. Many perhaps do not realize that the present age is ready for a significant turn in the development toward far greater heights than we have ever anticipated. The point of departure may well be the recasting and unifying of the theories of control and com munication in the machine and in the animal on a statistical basis. The philosophy of this subject is contained in my book entitled Cybernetics. * The present monograph represents one phase of the new theory pertaining to the methods and techniques in the design of com munica tion systems; it was first published during the war as a classified report to Section D 2, National Defense Research Committee, and is now released for general use. In order to supplement the present text by less complete but simpler engineering methods two notes by Professor Norman Levinson, in which he develops some of the main ideas in a simpler mathematical form, have been added as Appendixes Band C. This material, which first appeared in the Journal of M athe matics and Physics, is reprinted by permission. In the main, the mathematical developments here presented are new. However, they are along the lines suggested by A. KoImogoroff (Interpolation und Extrapolation von stationaren zufalligen Folgen, Bulletin de l'academie des sciences de U.R.S.S., Ser. Math. 5, pp. 3-14, 1941; cf. also P. A. Kosulajeff, Sur les problemes d'interpolation et d'extrapolation des suites stationnaires, Comptes rendus de l'academie des sciences de U.R.S.S., Vol. 30, pp. 13-17, 1941.) An earlier note of Kolmogoroff appears in the Paris Comptes rendus for 1939. To the several colleagues who have helped me by their criticism, and in particular to President Karl T. Compton, Professor H. M. James, Dr. Warren Weaver, Mr. Julian H. Bigelow, and Professor Norman Levinson, I wish to express my gratitude. Norbert Wiener Cambridge, MassachuseHs March,1949 ... Published by John Wiley & Sons, Inc., New York. v

INTRODUCTION CONTENTS The Purpose of This Book. 0.1 0.2 Time Series 0.3 Co'mmumcation Engine,ering 0.4 Techniques of Time Series and Communication Engineering Con- . trasted . 0.41 The Ensemble 0.42 Correlation . 0.43 The Periodogram 0.44 Operational Calcullls 0.45 The Fourier Integral; Need of the Complex Plane. 0.5 Time Series and Communication Engineering-The Synthesis 0.51 Prediction. 0.52 Filtering . 0.53 Policy Problems . 0.6 0.61 Past and Future. 0.62 Subclasses of Operators. 0.7 Norms and Minimization 0.71 The Calculus of Variations. 0.8 Ergodic Theory . 0.81 Brownian Motion 0.9 Permissible Operators: Translation Group in Time. Summary of Chapters . 1 1 2 3 4 4 6 7 8 8 9 9 10 11 12 12 13 14 15 20 21 Chapter I RESUME OF FUNDAMENTAL MATHEMATICAL NOTIONS 25 1.00 Fourier Series 1.01 Orthogonal Functions 1.02 The Fourier Integral 1.03 Laguerre Functions . 1.04 More on the Fourier Integral; Realizability of Filters . 1.1 Generalized Harmonic Analysis 1.18 Discrete Arrays and Their Spectra 1.2 Multiple Harmonic Analysis and Coherency Matrices 1.3 1.4 Ergodic Theory . 1.5 Brownian Motion 1.6 1.1 Harmonic AnalysiS in the Complex Domain . Poisson Distributions Smoothing Problems Chapter II THE LINEAR PREDICTOR FOR A SINGLE TIME SERIES 2.01 Formulation of the Problem of the Linear Predictor 2.02 The Minimization Problem. 2.03 The Factorization Problem. vii 25 31 34 35 36 37 43 44 45 46 41 51 52 56 56 57 60

viii CONTENTS 2.04 The Predictor Formula . 2.1 Examples of Prediction. 2.2 A Limiting Example of Prediction 2.3 The Prediction of Functions Whose Derivatives Possess Auto-correla- tion Coefficients . . Spectrum Lines and Non-absolutely Continuous Spectra Prediction by the Linear Combination of Given Operators 2.4 2.5 2.6 The Linear Predictor for a Discrete Time Series Chapter III THE LINEAR FILTER FOR A SINGLE TIME SERIES Prediction and Filtering Filters and Ergodic Theory Formulation of the General Filter Problem 3.0 3.1 Minimization Problem for Filters . 3.2 The Factorization of the Spectrum 3.3 3.4 The Error of Performance of a Filter; Long-lag Filters 3.5 3.6 Computation of Specific Filter Characteristics 3.7 Lagging Filters 3.8 The Determination of Lag and Number of Meshes in a Filter. 3.9 Detecting Filters for High Noise Level 3.91 Filters for Pulses. 3.92 Filters Having Characteristics Linearly Dependent on Given Charac- . . . teristics . 3.93 Computation of Filter: Resume . 64 65 68 71 74 76 78 81 81 82 84 86 88 90 91 92 94 95 95 97 101 Chapter IV THE LINEAR PREDICTOR AND FILTER FOR MULTIPLE TIME 104 SERIES Symbolism and Definitions for Multiple Time Series 4.0 4.1 Minimization Problem for Multiple Time Series. 4.2 Method of Undetermined Coefficients. 4.3 Multiple Prediction . 4.4 4.5 A Discrete Case of Prediction . 4.6 General Technique of Discrete Prediction Special Cases of Prediction. 104 105 106 109 110 111 112 Chapter V MISCELLANEOUS PROBLEMS ENCOMPASSED BY THE 117 TECHNIQUE OF THIS BOOK 5.0 The Problem of Approximate Differentiation 5.1 An Example of Approximate Differentiation. . 5.2 A Misleading Example of Approximate Differentiation. 5.3 Interpolation and Extrapolation . Appendix A TABLE OF THE LAGUERRE FUNCTIONS 117 119 120 121 124 Appendix B . THE WIENER RMS (ROOT MEAN SQUARE) ERROR CRITE- 129 RION IN FILTER DESIGN AND PREDICTION (by Norman Levinson) • . 1. 2. Linear Filters. Minimization of RMS Error . 130 131

CONTENTS 3. 4. 5. 6. Determination of the Weighting Function Realization of Operator-Mathematical Formulation RC Filter. Prediction ahd Lag with and without Noise. . . . . . . . . ix 136 139 143 146 Appendix C A HEURISTIC EXPOSITION OF WIENER'S MATHEMATICAL 149 THEORY OF PREDICTION AND FILTERING (by Norman Levinson) l. 2. 3. 4. 5. 6. The Auto-correlation Function. The Integral Equation . The Modified Integral Equation The Factorization Problem The Functions "'1, "'2, and ex The Prediction Operator 150 152 153 155 157 158

INTRODUCTION 0.1 The Purpose of This Book This book represents an attempt to unite the theory and practice of two fields of work which are of vital importance in the present emer gency, and which have a complete natural methodological unity, but which have up to the present drawn their inspiration from two entirely distinct traditions, and which are widely different in their vocabulary and the training of their personnel. These two fields are those of time series in statistics and of communication engineering. 0.2 Time Series Time series are sequences, discrete or continuous, of quantitative data assigned to specific moments in time and studied with respect to the statistics of their distribution in time. They may be simple, in which case they consist of a single numerically given observation at each moment of the discrete or continuous base sequence; Or multiple, in which case they consist of a number of separate quantities tabulated according to a time common to all. The closing price of wheat at Chicago, tabulated by days, is a simple time series. The closing prices of all grains constitute a multiple time series. The fields of statistical practice in which time series arise divide themselves rougbly into two categories: the statistics of economic, sociologic, and short-time biological data, on the one hand; and the statistics of astronomical, meteorological, geophysical, and physical data, on the other. In the first category our time series are relatively short under anything like comparable basic conditions. These short runs forbid the drawing of conclusions involving the variable or variables at a distant future time to any high degree of precision. The whole emphasis is on the drawing of some sort of conclusion with a reasonable expecta tion that it be significant and accurate within a very liberal error. On the other hand, since the quantities measured are often subject to human control, questions of policy and of the effect of a change of policy on the statistical character of the time series assume much importance. In the second category of time series, typified by series of meteoro logical data, long runs of accurate data taken under substantially uniform external conditions are the rule rather than the exception. Accordingly 1

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