Maximum Entropy Spectral Codes

In order to sort out a few things, I am going to hand wave the Burg algorithm.  The Burg algorithm came out of Stanford in the 70s, with the idea of encoding a time series into a constant bit stream, thus fitting a time varying signal into a fixed size channel,  it is going to be very close to following Shannon Law. 

The simplest way to think of the algorithm is to take the time series and run a moving window across the series to make it minimum phase then compute the large but finite dimensional Fourier series.  In economic terms the windowing function used  means no inversion in the yield (spectral) curve.  So the large but finite dimensional spectral estimate is upward and symmetrical divided.

From there, the easiest trick is to successively lower the resolution of the spectral intensities and run the result through a Huffman encoder until the resulting bit rate meets the channel requirement.  With a little extra side information one can reconstruct the original spectrum, perform the inverse Fourier transform and get a copy of the signal that meets a constant SNR.

This is a cheap, short cut explanation,  one should go read the actual paper.  Why do I bring this up? Because once one decides to code a data series to a fixed SNR, then a number of spectral functions, like Fourier, and various polynomials, even Laplace)  work pretty well as long as they capture the original data.  The data loss comes from the entropy encoding.

So, one could perform maximum entropy encoding with a Fourier transform or a series of polynomials, like the Fibonacci.  I pick the Fibonacci series because it is likely the natural polynomials that describe the bounded functions in a quantized production system.  The Burg algorithm assumes an infinite, or highly sampled data rate in the series, but the data we start with is usually already bound by short order polynomials.  But using the Burg algorithm on raw economic time series is still a good idea, and the spectral components should again be duals of the yield curve.

Regardless of the modeling functions, we still get the rank concept,  the number of components in the channel. Example: When we deal with a series of bond trades over different terms, we are dealing with data that has already had phase 1 of the Burgs algorithm completed, by the bond traders themselves.  The assumption is that the bond traders have already performed a large but finite Fourier (or other functional) decomposition of the total GDP flow.  So running these trading sequences through the Huffman encoder allows us to predict the number of spectral components in the next state change, that is, how  bond traders will reorganize to increase or decrease rank.  We are not doing the first part, the modeling part, traders do that.

Using maximum entropy analysis of economic data is always going to have this format, we assume the spectral decomposition has been done by the firm or household or agent, the economy itself.  Some monthss back I assumed a Dual Norm system, first a minimum variance part then a minimum entropy part.  In the market we reduce entropy, in the firm we reduce variance. This basic formulation still holds, I think.

Hopefully this clears thing up a bit.  I have to stop, reset  and correct errors or assumptions now and thin, this is an ongoing discovery.