Bayesian spectrum analysis and parameter estimation
This book is primarily a research document on the application of probability theory to the parameter estimation problem. The people who will be interested in this material are physicists, chemists, economists, and engineers who have to deal with data on a daily basis; consequently, we have included a great deal of introductory and tutorial material. Any person with the equivalent of the mathematics background required for the graduate-level study of physics should be able to follow the material contained in this book, though not without effort. In this work we apply probability theory to the problem of estimating parameters in rather general models. In particular when the model consists of a single stationary sinusoid we show that the direct application of probability theory will yield frequency estimates an order of magnitude better than a discrete Fourier transform in signal-to-noise of one. Latter, we generalize the problem and show that probability theory can separate two close frequencies long after the peaks in a discrete Fourier transform have merged.
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SINGLE STATIONARY SINUSOID PLUS NOISE
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absorption spectrum accuracy estimates aliasing amplitudes analyzed ANGULAR FREQUENCY approximately assume averaged data Bayesian Bessel inequality calculation channel Chapter chirp data set data values decay rate derived direct probability discrete Fourier transform equation estimated frequency estimated the frequency evidence example expansion order expectation value factor fast Fourier transform Figure frequency estimates frequency model frequency problem Gaussian given improper prior indicates integrals Jaynes Jeffreys prior joint analysis low frequency matrix multiple frequencies noise variance a2 nonuniform nonuniformly sampled nuisance parameters obtain ORDER TREND orthogonal orthonormal model functions oscillation parameter estimates peak phase plot posterior probability density power carried power spectral density prior information prior probability probability distribution probability theory procedure relative sunspot numbers residuals Schuster periodogram separated signal signal-to-noise ratio simple single frequency single harmonic frequency sinusoid square standard deviation Student t-distribution sufficient statistic term true frequency two-frequency model uniformly sampled vectors zero