Estimation of spectrum and chaos of nonstationary processes
by maximum entropy method
V.T. Sarychev
Siberian Phisical and Technical Institute, Tomsk, Russia
Traditionally, begining wiht Boltsman, entropy is used as a measure of chaos. But the utilization of this measure in the study of real processes present some difficulties.1) As a rule, statistical features of processes are described by continuous functions of probability density;corresponding to such descripti on differential entropy is defined to within some constant. This constant is a significant obstacle in the comparison of differential entropies of various processes. One of the possible ways of constant pussing away - to estimate two values of entropy: one value corresponds to real process, the other - to the same process under the condition of its complete chaotization, subtracting the second value from the first one we get the measure of chaos not dependent on the constant.
Different number of freedom degrees corresponds to different processes. To compare these processes as for chaotization degree we are to calculate entropy values per one degree of freedom. So you are to estimate the number of degrees of freedom of processes examined.
The assessment of density probability predisposses the presence
of ansambles of realization of the process studied. Often only one
realization is at the dispoosal of the investigator. The method of
the estimation of spectrum, entropy values and the number of
degrees of freedom of process according to its one realization is
desoribed in the report. The essense of the method is as following.
The process is presented as the superposition of accidental
number of oscillations with accidental values of frequency 
and
complex amplitude A. To describe the process the function of
density of oscilations` distribution (FDOD) 
is initiated.The
parametric view of this function discovered using the principle
of entropy maximum is as
where 
- trigonometric polynom the coefficient s of which are
Lagrange multipliers, the values of which are as
estimation according to the sample. The values of parameters C
and 
are estimated according to these date too.
The number of oscillations g studied as the number of degrees of
freedom of process is defined by the integral from
to A and .
Differential entropy per one oscillation is defined by the
expression :
The corresponding value of entropy under the condition of complete chaotization of process is defined as
Complex spectrum of process was estimated as the first moment
using amplitude FDOD . In general case the dependence of
Lagrange multipliers on the initial data is nonlinear, and
their estimate is performed quantatively by the search in the
minimum functional of descrepancy of initial and modelled signal.
The method was approved on modelled signals and on natural signals of a medical, biological and physical character.
This research was supported by Russian Fundamental Research Fund (Grant 93-012-1065).