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23
Fedorovich Anna Igorevna, Dnipropetrovsk National University named after O. Gonchar,
the Physic-Technical Faculty E-mail: [email protected]
Classification of facilities multi parameters experimental measurements of their parameters
Abstract: The problem of classification of objects that are described by many parameters. Evaluated the potential use of the entropy transformations for this task, as well as application of the combined test of nonparametric statistics Bush Wind, to identify objects belonging to the same class.
Keywords: sample measurements, nonparametric statistics, the criterion, entropy conversion.
measurements with unknown statistical laws and is a combination of unique test van der Waerden and criterion Klotz. First, each pair of samples Ll(k) and Lj(k) combined into one streamlined sample >£2 >... > £2n-1 > and determines the rank transformations Lm (k)
2n
R[Lm(k)] = Z(Lm(k)) = Rm(k) , and a rank value Rm(k)
i=i
calculated indicators proximity and scale of the formulas
In the manufacture and release of the first batch of products the problem arises of evaluating the technology of their production through the identification of products whose parameters do not meet the requirements of their tasks and scale. We consider this problem on an example of three-parameter classification of objects. Assume that the requirements set for them. This expectation a,, a2, a3, scale parameters D,, D2, D3 and correlation coefficients r12, r23, r13 . Information about them is contained in the measurements
x,, x2, x3. Statistical connections and differences between
<
S =
them are in the conditional distribution laws W
2(2 - -)-n
W (/J and W ((X ) .Assuming a normal distribution laws, we examine sample measurements of their entropic transformations of the form
k=1 r R (k) 1 . 2n +1 _
/ 2n Jç v 2 i n +1_
L (xlx 2x 3 ) = - ln
• Conditional
entropy transformations will be equal to
x. - a.
L
ln [D (1 - rj ))
- r.
f ^ x. - a
' y
2 (1 - < )
If you know the sample measurements of parameters M of objects, the way they get one-dimensional transformation
T_ =
... . . k , ,2n +1
here — W(z) function of the inverse Gaussian probability integral, for which you can apply an approximation of the formY(z) = 4,9l(z0,14 -(l-zf4).
Bush Wind test is a combination of criteria Sm and Tm
type W (m) = —2ln [2 (l - O * (|S„|)) - 2ln [2 (l - O * (\Tm\))]
entropy M samples Lm (k) = L m = 1,2,...,M, k = l,2,...,n, as the sum ofthe three conditional
. . J.ky
entropy transformations Lm (k ) = ^^ L
i=1 i*j
These M samples are unclassified. To solve this problem we use the statistical criterion proximity Bush Wind, to select objects whose parameters match the specified requirements. As a reference, take the sample L1 (k) and compare it with samples Lj (k), j = 2,3..,N, calculating M -1 indicators Bush Wind. They are random variables and sample sizes n > 30 is described by the chi-square distribution with four degrees of freedom, if the location and scale compared samples are identical. In this case, with a probability of 0.95, the inequality W < 9,5 [1]. Thus of the N objects can be identified anomalous class and use this knowledge in tasks of inspection control objects.
The Bush Wind test is used to test the hypothesis of equality of expectations and variances of two samples
here (z) =1 - 0,852exp
z +1,5774 2,0637
z > 0,
® * (-z ) = 1 *(z).
The threshold values in the tables are the criterion of mathematical statistics [2, 512].
Using the sequence of tests W(m) distinguish it from those who belong to the first class R* ) = sgn(W0 - W(m)).
Obviously, their relative numbers can serve as an estimate of the production technology objects
1 M
P = — I sgn (W, - W (m)).
M m=1
We investigate the effectiveness of entropy method using three generators normal random variables with zero mean and unit variance ^1(k), %2(k), %3(k), for the sampling of random variables are mutually correlated facilities 1st class conditional distribution laws and parameters a,,
«13 ; D11,
D
D • r r r
13 12 23 13
k/ _
--an ((k))),
2
x
Classification of facilities multi parameters experimental measurements of their parameters
X 2 ^ ) = ^ (Z k + (k)) ,
* 3 / ) = ai3 + VD3 // ) + .
Using these expressions for M the control objects define their one-dimensional sample entropy transformations
L (k) = L
m v ' m
xl(k)/
+ L
x2(k)y
+L
x 5(k)j
here
m = 1,2,..., M, k = l,2,...,n.
Assuming that the first reference sample, we calculate the criterion of the Bush-Wind for Wl three dimensions n = 10,25,50 and construct a histogram (M = 1000), as well as evaluate their expectations and variances. Data for the experiments shown in Table 1.
Table 1
«11 «12 «13 Du d12 DB II rB r23
3 4 5 0,5 1 1,5 0,6 0,7 0,8
Figure 1. — Histogram indicator Bush Wind entropy of transformations
The threshold values W0 for these samples are equal lengths of 8.65; 9.09 and 9.5. We define the number of criteria
M.
w (m
n! < Wo. The ratio —L is an estimate of the probability
x' M * m /
of making the right decisions classification: P1 = lA ^ - ^, p. _M/ m / /[ )
p _ /(M -1)
P =
M -1).
The experimental results are presented in Table 2. Table 2
n 10 25 50
P ' 0,637 0,982 0,993
W 3,552 5,435 8,863
0,684 1,071 1,427
Statistics indicator Bush Wind consistent with the
theoretical calculation of the authors, that is subject to the probability distribution chi-square with four degrees of freedom, with n > 30. Object recognition as belonging to class 1, increases with an increase in sample size measurements
Suppose that the known parameters of the objects of the 2nd class, differing from those of the 1st class a21, a22, a23; D21, D22, D23; r12, r23, r13. We form M the entropy of samples conversion to various embodiments of the changed parameters and spend the factor analysis of their impact on statistical regularities criterion W. Wind. The effect on the efficiency of the labeling only shifts and only the scale only correlation coefficients, as well as various combinations thereof. Table 3 shows the results of numerical experiments. They contain indicators of the probability of recognition of objects Class 2 as objects belonging to the first class, as well as statistics indicator Bush Wind. Table 3
«21 = 3 «22 = 4 «23 = 5 D21 = 1 D22 = 2 D23 = 3 rl2 = 0.6 r23 = 0.7 ra = 0.8 «21 = 3 «22 = 4 «23 = 5 D21 = 0.5 D22 = 1 D23 = 1.5 r12 = 0 r27 = 0 r13 = 0 «21 = 4 «22 = 5 «23 = 6 D21 = 1 D22 = 2 D23 = 3 r12 = 0.7 r27 = 0.8 r13 = 0.9
n 10 25 50 10 25 50 10 25 50
P ' 0,95 0,64 0,05 0,99 0,61 0,04 0,99 0,98 0,9
W 5,54 8,08 13,9 6,47 8,31 14,2 3,31 4,13 6,01
1,816 2,633 2,902 1,039 2,624 3,111 1,274 2,133 2,471
According to the results in Table 3 histogram (Figure 2) indicator Bush Wind samples for different lengths ofthe original
data in the case where objects of class 2 are different from the objects of class 1 and expectation and variance and correlation.
Figure 2. — Histogram indicator Bush Wind entropy of transformations with the recognition of objects of class 2, as objects of class 1.
From the analysis of Table 3 and Figure 2 it is clear that any change in the parameters initial values entails a change in the statistics of entropy change and performance criterion W. Wind. With increasing shift only increases the expectation index Bush Wind, an increase in the dispersion increases as the expectation. With the destruction of the correlation expectation index Bush Wind increases 3 times, and also increases the scale. In the event that changed all the statistics of the input data object recognition occurs with greater accuracy.
Assessing the impact of the volume of raw data on the probability of recognition of objects by class. We studied a sample volume n = 10,25,50. It was found that at low volumes measurements significantly worsens probability detection measurements of different classes Even if the total difference of all the statistical parameters of n = 10 of input data when probability making a wrong decision about 0.9. Therefore, entropy conversion are useful for the classification of objects with large volumes of measurement.
References:
1. Fedorovich A. I. Criterion Bush Wind in the tasks of monitoring of technical objects. - Dnepropetrovsk: Published by "System Technology", - 2010. 4 (69) - P. 36-44.
2. Kobzar A. I. Applied Mathematical Statistics - М.: ФИЗМАТ ЛИТ, - 2006. - P. 511-514.
3. Bush J. R., Wieand H. S. An asymptotically optimal nonparametric statistic for testing equally of two normal population means and variances - Commun. Stat. - Theor. Math. 1982. V. 11, № 1. Р. 1-12.
