# ACCIDENTAL PROCESSES

ACCIDENTAL PROCESSES — processes which course in time partially of a pla is completely unpredictable. S.'s theory of the item serves for creation of quantitative models of real processes, including for forecasting of their future values on the basis of the current information and the priori data, for allocation of useful information in the presence of hindrances, assessment of not measured parameters, etc.

In medicine the large number of processes (e.g., process of reproduction of tumor cells, number of calls of ambulance, etc.) is defined by so large number of uncontrollable factors that it is reasonable to carry out their adequate description and the analysis within S.'s theory of the item.

Mathematically S. items represent such functions of time, value to-rykh at every moment is a variate (see. Probabilities theory ). To each simple accidental event at the same time there corresponds nek-paradise the certain nonrandom function of time called by implementation, or a trajectory, S. the item. Properties of implementations serve as the main object of research of the theory of S. of the item. These properties are expressed by probability of nek-ry events (e.g., an exit of trajectories for the fixed level, hit to the set area, existence or lack of jumps in the set time slice, etc.). Within S.'s theory of the item also some statistical problems are solved (e.g., problems of filtering, ekstero-and interopolyation).

It is generally considered to be that accidental process is set (i.e. the task is formulated) when all joint functions of distribution of values of process for any final set of timepoints are defined; functions of distribution carry the name of finite-dimensional functions of distribution.

Other nonrandom functions connected with S. the item are m { t) — the mathematical expectation of S. of the item characterizing an average on a set of observations value C. of the item, and R (ti, t2) — the correlation function characterizing degree of dependence of values C. of the item in different timepoints (see. Correlation analysis ).

Main classes of accidental processes. Considering a big variety of S. of the item, from all their set separate classes are allocated and the methods of a research are developed for each class.

Stationary S. the item is S. the item, in to-rykh all finite-dimensional functions of distribution iye change at shift of time for a fixed value. Stationary S. the item have a number of characteristic properties: the average value of stationary S. of the item at every moment same, and correlation function R (t1, t2), depends only on a difference between timepoints of t1 and t2. Pages of the item of this type can be presented by the sum, or integral, harmonic oscillations, amplitudes and phases to-rykh are variates. Intensity of harmonious components form S.'s range of the item. A special case of stationary S. of the item is ergodic stationary S. of the item: within this method on the unique implementation of S. of the item it is possible to recover all its probabilistic characteristics. In particular. for each trajectory of ergodic accidental process an average on time to equally mathematical expectation of S. of the item.

Gaussian S. the item is S. the item, in to-rykh all finite-dimensional functions of distributions are Gaussian. Only two functions — mathematical expectation of m (t) and correlation function R are necessary for its task (t1, t2).

Markov S. the item have the following property: for any timepoint the future of process depends only on its condition of time at present and does not depend on its background. For Markov S.'s task of the item the nobility only one-dimensional functions of distribution and probability of transition from one state to another is enough. Markov S. items form a big class of processes, to-ry includes Markov S. of the item with independent increments, diffusion S. of the item, spasmodic Markov S. of the item, the branching S. of the item, etc.

The quantity of various classes C. of the item applied at mathematical modeling of the real phenomena constantly increases according to requirements of practice. In a medico-biol. the practician S. of the item are used generally in theoretical researches that is connected with complexity of the mathematical apparatus applied in S.'s analysis of the item. The founder of cybernetics N. Wiener by means of stationary S.' theory of the item in 1961 investigated rhythms of biocurrents of a brain. Later S. items found application at quantitative researches in neurophysiology and cardiology (stationary and diffusion S. for the item), oncology (Markov accidental processes of reproduction and death), in epidemiology and health care. 