Threshold strategy of an estimation in a problem of choice of the best variant

Task: It is necessary to choose the most suitable variant from some set of objects by those or other criteria.

Coding and decoding in transfer systems of the discrete information. Example 1. Coding and decoding in transfer systems of the discrete information. By means of a choice of an appropriate method of coding it is necessary to reduce probability of erroneous acceptance as much as possible.

Choice of the best supplier. Example 2. Choice of the best supplier.

Adaptive control methods It is required to solve a problem not for concrete, certain object, and for any object from some set . The algorithm should "adapt", should "be arranged" to an object and after "training" should provide purpose achievement.

(controlled random process) The control theory Object of control (controlled random process) Control purpose Control algorithm (strategy)

Models of controlled objects and operating systems The object O is described by an element x of phase space Х. Y = {y} – set of variants. Dynamics of phase space: control strategy is characterized by a rule:

Controlled random processes Definition. Controlled random process is the pair formed by object of control (family of conditional probabilities ) and a class of admissible strategy , where - family of the distributions having a final measure - family of random variables with values in space Х.

The control purpose - optimization of value of some functional which depending on a condition of controlled process accepts this or that value. The most typical kinds of functionals: the average income received for time of control till the moment t inclusive. 1) the average loss of the income in comparison with highest possible expected income at the moment of time t. 2) - highest possible expected income at a concrete choice of parameter of environment. where

It is necessary to choose thus control strategy , that for any admissible parameter one of conditions was carried out: or where - the expectation value sign provided that strategy is chosen and concrete parameter of environment is fixed.

Threshold strategy of control We use (k-1)-stage strategy of rejecting of the worst variant according to the results of estimation. Let are available: phase space Х={x}, class of admissible strategy and set of variants Y={1,2,..К}. - incomes of applied at present moment of time variants Random variables , t=1,…,T

The control purpose consists in minimization of the guaranteed size of an expectation value of the full income losses, that is in minimization of the functional We assume that all probabilities are unknown.

Rejection procedure Let M variants (1≤M≤K) with numbers and with initial incomes be available. Let's apply variants cyclically, at accumulating relevant full incomes until then time of modeling yet won't end or for some variable n 0 the inequality won't be executed

Function of losses If the object of control is known it is possible to receive maximum expectation income equal to . You should apply a variant corresponding to the greatest value of . If object of control is unknown, the real expected income for application of the threshold strategy is equal to . Function of losses is equal to The goal is to determine

Property of invariancy for threshold strategy of estimation: Let then the equality holds:

On the other hand the aproximate equality holds where

Optimization of calculation of value of the minimum losses at a choice of the best variant. Let K=2 and The goak e consists in a value finding: In the game formulation this size corresponds to the top price of game. Here it is necessary to make calculations of values , then to find

it is possible to present function of losses in the form: On the other hand, at К=2 it is possible to present function of losses in the form: Here only the first composed needs modeling, and the second one is calculated directly. Therefore it becomes possible at once, for one run of algorithm to calculate sequence of values of function for some increasing set of values using earlier found sums , corresponding to the previous value of . and Therefore at once we find value The problem dares for steps .

In the latter case calculations are reduced to a value finding , that corresponds to the bottom price of game.

Thanks for attention! ! Shelonina Tatyana, the associate professor of the applied mathematics department Novgorod State University of a name of Yaroslav the Wise Thanks for attention! !