Burg entropy

This is a principle of minimum empirical entropy (first sum) plus maximum weighted-Burg entropy (second sum). Once again, differentiation does not always yield the solution. Taking the second derivative of Eq. (49a) shows that the first derivative attains a maximum if and only if... 

Shannon-type entropy — E nm nm Burg-type entropy In nm... 

Other constrained methods have also been applied. Beatham and Orchard (1976) experimented with Biraud s method but experienced only limited success. Vasquez et al (1981) found that maximum entropy is capable of yielding excellent results on simulated ESCA data. The authors, who used Burg s method, cite its freedom from need for trial-and-error optimization. They did, however, have to develop methods of dealing with problems of instability and lack of an order-selecting criterion. 

In the limit of all Qm = 1/M, because X nm obeys normalization, the net principle is maximization of the first term in Eq. (43). This is a weighted form of Burg s entropy, with weights zm — 1. 

We can summarize this section as follows. The ML object obeys maximum entropy of type H (Jaynes) or type H1 (Burg) when the white object definition (31) of maximum ignorance is used and when the df sites are sparsely or highly populated, respectively. 

Burg, J. P. (1967). Maximum entropy spectral analysis. Paper presented at the 37th Annual Society of Exploration Geophysicists Meeting, Oklahoma City. 

Unfortunately, there is great scope for confusion, as two distinct techniques include the phrase maximum entropy in their names. The first technique, due to Burg,135 uses the autocorrelation coefficients of the time series signal, and is effectively an alternative means of calculating linear prediction coefficients. It has become known as the maximum-entropy method (MEM). The second technique, which is more directly rooted in information theory, estimates a spectrum with the maximum entropy (i.e. assumes the least about its form) consistent with the measured FID. This second technique has become known as maximum-entropy reconstruction (MaxEnt). The two methods will be discussed only briefly here. Further details can be found in references 24, 99, 136 and 137. Note that Laue et a/.136 describe the MaxEnt technique although they refer to it as MEM. 

J.P. Burg, Maximum Entropy Spectral Analysis, Dissertation, Stanford University, Stanford, 1975. 

A practical route to an initial esiimate of the Lagrange multipliers X/ s is via the Burg algorithm (104,105). This generates a spectrum of maximum entropy provided the times tr are equally spaced, tr = rht. In many applications (see Sec. IV.B), there are specific reasons to preferring unequally spaced time intervals. One option is to use the... 

Brawn DR (1989) A maximum entropy approach to underconstraint and inconsistency in the seismic source inverse problem Finding and interpreting seismic source moments, Ph.D. Thesis, University of the Witwatersrand, Johaimes-burg... 

FIGURE 2 Plots of different maximum entropy functions and their derivatives. Shown are the Burg (logp), Gull and Daniell (-p log p), maximum emptiness (-log(cosh p)). and square root (yp) measures. 

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