Maximum Entropy Spectral Analysis Mesa

MESA is an outgrowth of the predictive deconvolution filtering techniques developed by geophysicists for oil exploration.2,3 Its specific goal is to obtain high resolution measurements from minimum length data—precisely the requirement for the identification of short-term cycles in the market.

If we have a frequency source followed by a filter, the output of the filter is just those frequency components of the filter that were allowed to pass through the filter. This is mathematically expressed as the product of the frequency source and the filter in the frequency domain. We can find the Fourier series (not to be confused with the Fourier transform) for both the source and the filter to obtain a description in the time domain. The process of taking the product of the two Fourier series is called convolution. Convolution is the equivalent of sliding the filter time response past the source time response, taking the product at each time increment, and summing all the products.

The maximum entropy approach to spectral analysis is a variation of deconvolution filtering techniques. A deconvolution filter whitens the spectrum of the signal on which it operates; that is, when convolved with the original signal it outputs a new signal with a constant spectrum. A constant spectrum signal is called white noise because it contains energy at all frequencies. This approach to spectral analysis is also known as the Markov spectrum or the autoregressive spectrum. Burg realized that this approach yields the spectrum having the "maximum entropy" of all possible spectra that are consistent with the measured autocorrelation function. Entropy is a term first used in thermodynamics to describe the degree of disorder and has more recently been used as a quantitative term in information theory. Therefore, "maximum entropy" is a case having the least amount of information, and deconvolution filtering produces an output having the least amount of information.

The advantage of deconvolution filtering is immediately obvious. Finding the frequency spectrum does not involve a convolution in the frequency domain with a cumbersome window spectrum (the FFT period) that unavoidably destroys spectral resolution. The convolution has already taken place in the time domain between the input signal and the digital filter. Therefore, no window sidelobes or serious end effects exist with the FFT. The truncation of the data set is important only to the extent that enough data must be available to allow the building of an efficient whitening filter that can reduce the output data to a random series. This is routinely done using only about one cycle's worth of input data.

The maximum entropy estimate is the optimal choice for measuring cycles because it is maximally noncommittal with regard to any missing data and is simultaneously const rained to be consistent with all available data. The "correct" length of data to be used for analysis is perhaps the most critical aspect of using MESA. In any event, the fact that MESA attains its high-resolution measurement with a short amount of data makes its use ideal for the market where current measurements are mandatory for relevant results.

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