Mathematical Considerations in the Estimation of Spectra
About: This article is published in Technometrics.The article was published on 1961-05-01. It has received 283 citations till now.
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TL;DR: In this paper, the problem of the estimation of a probability density function and of determining the mode of the probability function is discussed. Only estimates which are consistent and asymptotically normal are constructed.
Abstract: : Given a sequence of independent identically distributed random variables with a common probability density function, the problem of the estimation of a probability density function and of determining the mode of a probability function are discussed. Only estimates which are consistent and asymptotically normal are constructed. (Author)
10,114 citations
01 Jan 1978
TL;DR: A comprehensive catalog of data windows along with their significant performance parameters from which the different windows can be compared is included, and an example demonstrates the use and value of windows to resolve closely spaced harmonic signals characterized by large differences in amplitude.
Abstract: This paper makes available a concise review of data windows and their affect on the detection of harmonic signals in the presence of broad-band noise, and in the presence of nearby strong harmonic interference. We also call attention to a number of common errors in the application of windows when used with the fast Fourier transform. This paper includes a comprehensive catalog of data windows along with their significant performance parameters from which the different windows can be compared. Finally, an example demonstrates the use and value of windows to resolve closely spaced harmonic signals characterized by large differences in amplitude.
7,130 citations
01 Sep 1982
TL;DR: In this article, a local eigenexpansion is proposed to estimate the spectrum of a stationary time series from a finite sample of the process, which is equivalent to using the weishted average of a series of direct-spectrum estimates based on orthogonal data windows to treat both bias and smoothing problems.
Abstract: In the choice of an estimator for the spectrum of a stationary time series from a finite sample of the process, the problems of bias control and consistency, or "smoothing," are dominant. In this paper we present a new method based on a "local" eigenexpansion to estimate the spectrum in terms of the solution of an integral equation. Computationally this method is equivalent to using the weishted average of a series of direct-spectrum estimates based on orthogonal data windows (discrete prolate spheroidal sequences) to treat both the bias and smoothing problems. Some of the attractive features of this estimate are: there are no arbitrary windows; it is a small sample theory; it is consistent; it provides an analysis-of-variance test for line components; and it has high resolution. We also show relations of this estimate to maximum-likelihood estimates, show that the estimation capacity of the estimate is high, and show applications to coherence and polyspectrum estimates.
3,921 citations
3,494 citations
01 Nov 1981
TL;DR: In this paper, a summary of many of the new techniques developed in the last two decades for spectrum analysis of discrete time series is presented, including classical periodogram, classical Blackman-Tukey, autoregressive (maximum entropy), moving average, autotegressive-moving average, maximum likelihood, Prony, and Pisarenko methods.
Abstract: A summary of many of the new techniques developed in the last two decades for spectrum analysis of discrete time series is presented in this tutorial. An examination of the underlying time series model assumed by each technique serves as the common basis for understanding the differences among the various spectrum analysis approaches. Techniques discussed include the classical periodogram, classical Blackman-Tukey, autoregressive (maximum entropy), moving average, autotegressive-moving average, maximum likelihood, Prony, and Pisarenko methods. A summary table in the text provides a concise overview for all methods, including key references and appropriate equations for computation of each spectral estimate.
2,941 citations
References
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Book•
14 Jul 2012
TL;DR: This account attempts to provide and relate the necessary ideas and techniques in reasonable detail to develop the insight necessary to plan both the acquisition of adequate data and sound procedures for its reduction to meaningful estimates.
Abstract: The measurement of power spectra is a problem of steadily increasing importance which appears to some to be primarily a problem in statistical estimation. Others may see it as a problem of instrumentation, recording and analysis which vitally involves the ideas of transmission theory. Actually, ideas and techniques from both fields are needed. When they are combined, they provide a basis for developing the insight necessary (i) to plan both the acquisition of adequate data and sound procedures for its reduction to meaningful estimates and (ii) to interpret these estimates correctly and usefully. This account attempts to provide and relate the necessary ideas and techniques in reasonable detail — Part I of this article appeared in the January, 1958 issue of THE BELL SYSTEM TECHNICAL JOURNAL.
1,353 citations
Book•
01 Jan 1957
TL;DR: In this article, the spectrum is estimated by using a regression spectrum and the regression spectrum is then used to estimate the spectral density of a time series with respect to the spectrum of the time series.
Abstract: Stationary Stochastic Processes and Their Representations: 1.0 Introduction 1.1 What is a stochastic process? 1.2 Continuity in the mean 1.3 Stochastic set functions of orthogonal increments 1.4 Orthogonal representations of stochastic processes 1.5 Stationary processes 1.6 Representations of stationary processes 1.7 Time and ensemble averages 1.8 Vector processes 1.9 Operations on stationary processes 1.10 Harmonizable stochastic processes Statistical Questions when the Spectrum is Known (Least Squares Theory): 2.0 Introduction 2.1 Preliminaries 2.2 Prediction 2.3 Interpolation 2.4 Filtering of stationary processes 2.5 Treatment of linear hypotheses with specified spectrum Statistical Analysis of Parametric Models: 3.0 Introduction 3.1 Periodogram analysis 3.2 The variate difference method 3.3 Effect of smoothing of time series (Slutzky's theorem) 3.4 Serial correlation coefficients for normal white noise 3.5 Approximate distributions of quadratic forms 3.6 Testing autoregressive schemes and moving averages 3.7 Estimation and the asymptotic distribution of the coefficients of an autoregressive scheme 3.8 Discussion of the methods described in this chapter Estimation of the Spectrum: 4.0 Introduction 4.1 A general class of estimates 4.2 An optimum property of spectrograph estimates 4.3 A remark on the bias of spectrograph estimates 4.4 The asymptotic variance of spectrograph estimates 4.5 Another class of estimates 4.6 Special estimates of the spectral density 4.7 The mean square error of estimates 4.8 An example from statistical optics Applications: 5.0 Introduction 5.1 Derivations of spectra of random noise 5.2 Measuring noise spectra 5.3 Turbulence 5.4 Measuring turbulence spectra 5.5 Basic ideas in a statistical theory of ocean waves 5.6 Other applications Distribution of Spectral Estimates: 6.0 Introduction 6.1 Preliminary remarks 6.2 A heuristic derivation of a limit theorem 6.3 Preliminary considerations 6.4 Treatment of pure white noise 6.5 The general theorem 6.6 The normal case 6.7 Remarks on the nonnormal case 6.8 Spectral analysis with a regression present 6.9 Alternative estimates of the spectral distribution function 6.10 Alternative statistics and the corresponding limit theorems 6.11 Confidence band for the spectral density 6.12 Spectral analysis of some artificially generated time series Problems in Linear Estimation: 7.0 Preliminary discussion 7.1 Estimating regression coefficients 7.2 The regression spectrum 7.3 Asymptotic expression for the covariance matrices 7.4 Elements of the spectrum 7.5 Polynomial and trigonometric regression 7.6 More general trigonometric and polynomial regression 7.7 Some other types of regression 7.8 Detection of signals in noise 7.9 Confidence intervals and tests Assorted Problems: 8.0 Introduction 8.1 Prediction when the conjectured spectrum is not the true one 8.2 Uniform convergence of the estimated spectral density to the true spectral density 8.3 The asymptotic distribution of an integral of a spectrograph estimate 8.4 The mean square error of prediction when the spectrum is estimated 8.5 Other types of estimates of the spectrum 8.6 The zeros and maxima of stationary stochastic processes 8.7 Prefiltering of a time series 8.8 Comments on tests of normality Problems Appendix on complex variable theory Bibliography Index.
902 citations
01 Jan 1946
775 citations
Book•
01 Jan 1960
TL;DR: Probability Theory as the study of Mathematical Models of Random Phenomena as mentioned in this paper is a generalization of probability theory for the study and analysis of statistical models of random variables.
Abstract: Probability Theory as the Study of Mathematical Models of Random Phenomena. Basic Probability Theory. Independence and Dependence. Numerical-Valued Random Phenomena. Mean and Variance of a Probability Law. Normal, Poisson, and Related Probability Laws. Random Variables. Expectation of a Random Variable. Sums of Independent Random Variables. Sequences of Random Variables. Tables. Answers to Odd-Numbered Exercises. Index.
766 citations