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Krasnoshtanov A. P., Kupriyanova E. V., Protsykova M. A., Rakovskaia C.A., Slonova L.A.

Siberian Aerospace Academy

Krasnoyarsk State University

Krasnoyarsk State Trade-economic Institute

Russia, Krasnoyarsk


Annotation. The problems of interconnected stochastic systems identification in the broad sense are considered in the work. In addition the priory information about the parametric model structure is absent. Under these conditions the nonparametric modeling occurs to be perspective.


Красноштанов А. П., Куприянова E. В., Процыкова M. А., Раковская С. А., Слонова Л. А.

Сибирская Аэрокосмическая Академия

Красноярский Государственный Университет

Красноярский Государственный Торгово-экономический Институт

Россия, Красноярск


Аннотация. В работе рассматриваются задачи идентификации взаимосвязанных стохастических систем в широком смысле. При этом априорная информация о параметрической структуре модели отсутствует. В этих условиях перспективным оказывается непараметрическое моделирование.

Let be the vectors of observed random sequences representing accordingly output and input variables of the analyzed process. Let's define the function of solutions as some function representing the output of the adaptive system in the influence.

For the open scheme, shown in the figure 1, following variables are introduced

which are observed in discrete moments of time of finite or infinite size.

Further index in observed variables is omitted because of the simplicity reason. The noise and, are such, that Let's form the optimum criterion where is some convex function. The problem of searching of the best amounts to the minimization with respect to that is the search if such that It's clear that the form of will be determined be the form of function. If then


We also may consider other types of

Further the problem is in estimating by the observations which are formed out of the initial samples The sign denotes the time vector.

With the passive information accumulation, i.e. in the presence of the sample the nonparametric estimating algorithm (1) is :


where the matrices are such, that

Where T- transpose sign.

While constructing the model of some object the problem is in finding the estimate of function describing the object's behavior. This estimate is usually found by using the mean square optimum criterion.

It's clear that the estimate error mean square will be minimal if to take as the mean of the random value in the given i.e. represents the regression.

The construction of the regression models gives a good result in case of symmetric conditional probability densities, as since the mean value of practically coincides the most probable one. If the probability distribution has the brightly expressed asymmetric character then it is better to use the probability models. As the function is the optimum criterion we'll take where , -the Dirac delta-function. After the minimization of this criterion with respect to we get the mode of conditional probability density, i.e.


As it is known the nonparametric estimate of conditional probability density is the estimate


Where the kernels are symmetric, delta-like for functions with properties of probability density at the set

It's important from the practical point of view to analyze the regression models in nonstationary (unstable) conditions. It's assumed that the probabilistic characteristics of an object are changing during the time in unknown way. The nonparametric estimate of regression function which looks like:


and it should be modified by putting the additional weights - some kind of "memory functions", decreasing with the increase. Ibis fact corresponds to "forgetting" the old information.

Essential interest, from the practical point of view, represents the case when for some relations between the vector components and the parametric relation is known up to the set of parameters , and other relations are unknown. In this situation we collide with the problem of identification which doesn't correspond neither to the parametric nor to the nonparametric level of priory information. These models arise when describing the complex interconnected processes (often this situation arises while modeling of manufacturing processes with the continuous technology) and has a form of an iterconnected system:

, (6)

where the index at the arguments denotes the definite (the -th) set of these components out of all components, -th parametric model, - nonparametric (the -th) model, -the vector of the output variables of the model, - the vector of the input variables, -the parameters estimate.

The combined nonparametric models in the class of H-models look like


where the sample of statistically independent observations of size at the domain of the set. While using the H-models (7) the defects which are possible while using the models (6) are excluded. It's clear that if then the models (7) transfer to the models (6). The set is called the domain of technological rule.

Let the manufacturing process be some aggregate of interconnected technological devices (objects), which models we'll denote as where -accordingly output and input variables of the -st object, -the parametric relation on , up to parameters and т - the number of objects. In case if all models are constructed (some models can be the models of nonparametric type), then the problem of model construction of the whole complex often arises. It's natural to use for mis purpose the models of local objects of the type (6), (7). But arithmetic "assembling" of these models as the rule doesn't provide the desired accuracy. In techniques the analog of this fact is the assembling of some article of separate blocks each one of which satisfies the technical conditions but the functioning of the whole article doesn't satisfy the according technical specifications and that's why the necessity of the adjustment of the whole article arises, which represents the simultaneous adjustment of all or some local blocks. The process of "assembling" of the macro-model out of local ones and their following " adjustment" we called the models macro-synthesis.


1. Medvedev A.V. Nonparametric adaptation systems. Novosibirsk: Nauka 1983

2. Tarasenko F.P. Nonparametric statistic. Tomsk: TGU, 1976.

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