Self-consistent significance level as a statistical predictor of epilepsy attack

In this paper we discuss one aspect of kinetic approach to the time series modeling in terms of dynamical system. The main idea of non-stationary analysis is based on the interpretation of kinetic equation for empirical distribution function density as a reduced description of statistical mechanics for appropriate dynamical system. For example, if distribution function density is satisfied to Liouville equation, then the average velocity of corresponding mechanical system can be obtained from this equation, writing in the form of continuity Euler equation. The inverse problem is to approximate some sample trajectory of a random process by an appropriate dynamical system, which could be constructed as dynamical basis of empirical kinetic equation for the sample distribution function density. Theoretical problem of this approach is that the joined distribution function of two or more samples of data corresponds to the weighted sum of distribution functions of separate samples, but the sum of semigroup generators in general case is not a generator of any equivalent semigroup. Nevertheless, there is a technique, using so-called Chernoff theorem from the group theory, which allows to construct some iteration procedure to obtain semigroup as an asymptotical state, which is equivalent in some sense to average shift operator over the trajectory of appropriate dynamical system. This method enables us to construct a strict approach to nonstationary time series modeling with non-parametric estimation of statistical properties of corresponding sample distribution function. The illustration of this method is given on the example of electroencephalogram time series analysis with the aim of construction special predictor of an attack of epilepsy. This predictor is so-called self-consistent stationary level, which is defined as stationary point of significance level of distribution function of distances between sample distribution function of investigated process.