Book contents
- Frontmatter
- Contents
- Preface
- Donors
- Bayesian Methods: General Background
- Monkeys, Kangaroos, and N
- The Theory and Practice of the Maximum Entropy Formalism
- Bayesian Non-Parametric Statistics
- Generalized Entropies and the Maximum Entropy Principle
- The Probability of a Probability
- Prior Probabilities Revisited
- Band Extensions, Maximum Entropy and the Permanence Principle
- Theory of Maximum Entropy Image Reconstruction
- The Cambridge Maximum Entropy Algorithm
- Maximum Entropy and the Moments Problem: Spectroscopic Applications
- Maximum-Entropy Spectrum from a Non-Extendable Autocorrelation Function
- Multichannel Maximum Entropy Spectral Analysis Using Least Squares Modelling
- Multichannel Relative-Entropy Spectrum Analysis
- Maximum Entropy and the Earth's Density
- Entropy and Some Inverse Problems in Exploration Seismology
- Principle of Maximum Entropy and Inverse Scattering Problems
- Index
Maximum-Entropy Spectrum from a Non-Extendable Autocorrelation Function
Published online by Cambridge University Press: 04 May 2010
- Frontmatter
- Contents
- Preface
- Donors
- Bayesian Methods: General Background
- Monkeys, Kangaroos, and N
- The Theory and Practice of the Maximum Entropy Formalism
- Bayesian Non-Parametric Statistics
- Generalized Entropies and the Maximum Entropy Principle
- The Probability of a Probability
- Prior Probabilities Revisited
- Band Extensions, Maximum Entropy and the Permanence Principle
- Theory of Maximum Entropy Image Reconstruction
- The Cambridge Maximum Entropy Algorithm
- Maximum Entropy and the Moments Problem: Spectroscopic Applications
- Maximum-Entropy Spectrum from a Non-Extendable Autocorrelation Function
- Multichannel Maximum Entropy Spectral Analysis Using Least Squares Modelling
- Multichannel Relative-Entropy Spectrum Analysis
- Maximum Entropy and the Earth's Density
- Entropy and Some Inverse Problems in Exploration Seismology
- Principle of Maximum Entropy and Inverse Scattering Problems
- Index
Summary
INTRODUCTION
In the first “published” reference on maximum entropy spectra, an enormously influential and seminal symposium reprint, Burg (1967) announced his new method based upon exactly known, error free autocorrelation samples. Burg showed that if the first n samples were indeed the beginning of a legitimate autocorrelation function (ACF) that the next sample (n+1) was restricted to lie in a very small range. If that (n+1) sample were chosen to be in the center of the allowed range, then the sample number (n+2) would have the greatest freedom to be chosen, again in a small range. In the same paper, Burg also showed that the extrapolation to the center point of the permissable range corresponds to a maximum entropy situation in which the available data were fully utilized, while no unwarranted assumptions were made about unavailable data. In fact unmeasured data were to be as random as possible subject to the constraint that the power spectral density produce ACF values in agreement with the known, exact ACF.
Some time later it was recognized that the realization of exactly known ACF values rarely if ever occurs in practice and that the ACF is usually estimated from a few samples of the time series or from some other experimental arrangement and is therefore subject to measurement error. Thus the concept of exact matching of given ACF values was weakened to approximate matching: up to the error variance.
- Type
- Chapter
- Information
- Maximum Entropy and Bayesian Methods in Applied StatisticsProceedings of the Fourth Maximum Entropy Workshop University of Calgary, 1984, pp. 207 - 211Publisher: Cambridge University PressPrint publication year: 1986