# The tasks of processing simulation results., Pearson's consent...

## Tasks for processing simulation results.

When processing the results of a computer experiment with the model M m , the following problems most often arise: definition of the empirical law of distribution of a random variable, verification of homogeneity of distributions, comparison of mean values ​​and variances of variables, etc. These problems from the point of view of mathematical statistics are typical tasks for testing statistical hypotheses.

The problem of determining the empirical law of distribution of a random variable is the most common of those listed, but for a correct solution it requires a large number of realizations N. In this case, according to the results of a computer experiment, the values ​​of the selective distribution law T (y) (or density function / e ) and put forward the null hypothesis I 0 , that the obtained empirical distribution is consistent with some theoretical distribution. This hypothesis is checked using the statistical criteria of Kolmogorov's agreement, Pearson's, Smirnov's consent, etc., and the statistical processing of the results necessary in this case is carried out whenever possible in the process of modeling the system & pound; on a computer.

To accept or refute the hypothesis, select some random variable U, which characterizes the degree of divergence between the theoretical and empirical distribution, due to the insufficiency of the statistical material and other random factors. The law of distribution of this random variable depends on the distribution law of the random variable r and the number of realizations N under statistical modeling of the system. If the probability of the discrepancy between the theoretical and empirical distributions P {and t ^ and} is large in terms of the applied consent criterion, then the hypothesis about the form of the distribution of Ip is not refuted. The choice of the type of theoretical distribution r (y) (or/( y )) is carried out from the graphs (histograms) (y) (or, (y)), printed or on the display screen.

Let's consider the peculiarities of the use of a number of the consent criteria for processing the results of modeling the system 5 [7, 11, 18, 21, 25].

Kolmogorov's consent criterion. It is based on the choice as the measure of divergence I/ of the quantity f * (y)].

It follows from Kolmogorov's theorem that <5-.0 /A has a function

distribution If the value of <5 calculated on the basis of the experimental data is less than the tabulated value at the selected significance level y, then the hypothesis H 0 is accepted, otherwise the discrepancy between P (y) and P (y) is considered non-random / and the hypothesis # 0 is rejected.

Kolmogorov's criterion for processing simulation results is useful when all parameters of the theoretical distribution function are known. The lack of use of this criterion is associated with the need for fixation in the computer memory to determine) all statistical frequencies with the aim of ordering them in ascending order.

## Pearson's agreement criterion.

Based on the definition of and as a measure of variance  where - the number of values ​​of the random variable r y, falling into the 1-st sub-interval; P (- is the probability that the random variable y] hits the i-th sub-interval calculated from the theoretical distribution, (1 is the number of sub-intervals divided by the measurement interval in the machine experiment.

For the N-th law, the distribution law of the quantity , which is a measure of discrepancy, depends only on the number of subintervals and approaches the distribution law x (chi-square) with (& lt;/- 1) degrees of freedom, where r is the number of parameters of the theoretical distribution law.

It follows from Pearson's theorem that, for any distribution function f (y) of the random variable w, the distribution of the quantity x has the form

where T (k/ 2) is the gamma function; r is the value of the random variable x k ** (1-r - 1 is the number of degrees of freedom.) The distribution functions f * (r) are tabulated.

With the calculated value of and = x * and the number of degrees of freedom k with the help of tables, the probability P (x ^ x} is found. If this the probability exceeds a certain level of significance of y, then the hypothesis H 0 is not refuted by the results of the machine experiment.

The criterion of Smirnov's agreement. When assessing the adequacy of the machine model M m of the real system 5, it becomes necessary to test the hypothesis Я 0 , which consists in the fact that two samples belong to the same general population . If the samples are independent and the distribution laws of the sets P (u) in P (z), from which the samples are extracted, are continuous functions of their arguments V, then to test the hypothesis H 0 , one can use the Smirnov's test of agreement, the application of which reduces to the following. According to the available results, the empirical distribution functions f ^ (u) and f * (r) are calculated and determine Then, at a given level of significance, y find a tolerance where ЛГ, and Я 2 - the volumes of the samples being compared for/^ (и) and are carried out

comparison of values ​​/ & gt; and GK: if 0> in y , then the null hypothesis H 0 about the identity of the distribution laws r (i) and D (r) with a confidence probability of R = 1-y are rejected.

## Student consent criterion.

Comparison of the mean values ​​of two independent samples taken from normal populations with unknown but equal variances M = * [p]; [\$], reduces to testing the null hypothesis H 0 : A = u = 0 on the basis of the Student's consent criterion (/ -crytery). Verification by this criterion reduces to performing the following actions. Calculate the estimate where n N2 are the sample sizes for the estimation of and in 2 , respectively; 5? and a are estimates of the variances of the corresponding samples.

Then determine the number of degrees of freedom * I, + I 2 - 2, select the significance level y and find the value/ y from the tables. The calculated value of/is compared with the tabular/ y and if |/| & lt;/ г , the hypothesis H 0 is not refuted by the results of the machine experiment.

## Fisher's agreement criterion.

The problem of comparing variances reduces to testing the null hypothesis H 0 , which consists in the belonging of two samples to the same general population. Let's compare two variances d and o, obtained by processing the simulation results and having to x and to 2 of the degrees of freedom, respectively, with a> gt. In order to disprove the null hypothesis H 0 : it is necessary for the significance level y indicate the significance of the discrepancy between a and a. Given the independence of samples taken from normal collections, the Fisher distribution (^ -criteria) is used as a criterion of significance, which depends only on the number of degrees freedom

k l - 2 = I 2 > - 1, where λ, and λ 2 are the sample sizes for estimating a and a respectively.

The algorithm for applying the Fisher criterion is as follows: 1) the sample ratio P = d x 1d 2 is calculated 2) is determined purely by the degrees of freedom in k 2 I 2 - 1;

3) at a selected level of significance, у on the ^ -distribution tables contains

the values ​​of the boundaries of the critical region Р х = 1/[Л-у/2 (* 1 ^ 2)] b < sub> 2 );

4) the inequality P x <1>

2 is checked; if this inequality is satisfied, then

with confidence probability P null hypothesis I 0 : =

is accepted.

Although the considered estimates of the required characteristics of the process of the functioning of the 5 * system, obtained as a result of the computer experiment with the model M m , are the simplest, but cover the majority of cases encountered in the practice of processing the results of modeling the system for the purposes of its investigation and designing.

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