Local Information Algorithm for Estimating the Spectra *

Radu Neagu & Igor Zurbenko

Frequency analysis of a given signal is needed when we want to determine the major driving forces that combine and generate the signal. Most of the time, these major driving forces are certain phenomenons that happen over regular (or irregular) time intervals, therefore generating specific waves of certain fixed (or floating) periods and fixed (or floating) amplitudes. To this, generally a certain amount of random fluctuations (what we call noise) gets added. All this information about the composition of a signal is reflected in its spectral density function. In this presentation, our concern is to build an optimal estimate of the spectral density function that has a natural construction and outperforms the existing methods of estimating spectra. We define optimal in terms of entropy: we choose the estimate that gives maximum entropy. Since maximizing entropy is a special case of minimizing cross-entropy, which is a measure of the ‘information dissimilarity’ between the real spectral density function and its estimate, we construct a nonparametric adaptive spectral estimate that asymptotically minimizes the cross-entropy. The method is based on linearly approximating the local information present in our process. Performance of the method is illustrated on a simulated example.