Empirical reconstruction of fuzzy model of experiment in the Euclidean metric

Abstract

In this paper we introduce a method for the fuzzy model reconstruction and a method for measurements reduction on the basis of test signals by maximization a posteriori possibility. It ensures the maximum accuracy of the measurements reduction. It is used the model of measurement errors with fuzzy constraints on its Euclidean norms. z 2 Rn;

Keywords

• mathematical modeling
• fuzzy sets
• decision making
• analysis and interpretation of data
• measurement and computing systems

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( 1 ) The measurement result of the Euclidean space Rn is accompanied by a additive noise . By a result of of measure ( 1 ) it is required to estimate the value of the parameter vector = U ', where U 2 (RN ! RM ) is de ned linear operator [4].

In the paper [1] it is shown that if ( 1 ) vector ' 2 RN and a linear operator priori arbitrary, and 2 Rn is the fuzzy vector in Rn with distribution possibilities ( ) [2,3]. And for the determination of the model of the measuring device there are involved the measurement results j = fj + j ; j = 1; : : : ; m; of known test signals, f1; : : : ; fm, where the error 1; : : : ; m 2 Rn of test measurements are fuzzy elements of Rn with the given distribution of possibilities. The estimates of the maximum possibility Ab and fb are values of fuzzy elements and ' respectively, as a solution of the maximin problem (Ab; fb) = arg max min(

A;f (x

Af ); N (X

AF )); ( 2 ) and the estimate ub is value of the fuzzy element of given by ub = U fb. Here, x; A; f; X are implementation of fuzzy elements of ; ; '; , respectively. The same scheme of test measurements in matrix form is given by = F + N .

Consider the solution of the problem ( 2 ), where the operator and element ' are a priori arbitrary, so that (A) = 1 for every A 2 (RN ! Rn) and '(f ) = 1 for any f 2 RN , and distribution possibilities of measurement error is given by

N (Z) = ( 3 ) ( 4 ) where 0( ) : [0; 1) ! [0; 1] is strictly decreasing function, 0(0) = 1, lim z!1 0.

Then the problem ( 2 ) leads to the following minimax problem: min max(kx A;f

Af k2; kX

AF k22): Let us denote J1(A; f ) = kx Af k2, J2(A) = kX AF k22, then J (A; f ) = max(J1(A; f ); J2(A)), and the problem ( 3 ) can be rewritten as

J (A; f ) = max(J1(A; f ); J2(A)) min : A;f Depending on which of the minimum values J1(Ab0; fb(Ab0)) or J2(Ab0), is less it is selected di erent methods for solving the problem ( 3 ), they are considered in [1].

Example 1. Let the unknown operator A is de ned by the matrix of size 2 2, A = a11 a22 , given m test signals, f1 = f11 ; : : : ; fm = f1m , forming a21 a22 f21 f2m columns of the matrix F 2 (Rm ! R2), and test results are given as vectors x1 = x11 ; : : : ; xm = x1m , forming columns of the matrix X 2 (Rm ! x21 x2m R2). Then the scheme of tests is given by

= AF + N; where the j-th column of N de nes the error of j-th test measure, j = 1; : : : ; m. For the error matrix N that de ned by the distribution of possibilities of its values by the form N (X) = 0(kZk22), where 0( ) : R+ ! [0; 1] is a monotonically decreasing function, 0(0) = 1, 0(+1) = 0.

Let the rank of F is equal to two. The signal f is measured according to the scheme = Af + . The result x 2 R2 of this measurement is known. The distribution is (z) = 0(kzk2) of possibility of fuzzy vector 2 R2 for measurement error Af . It is required to determine the reduction of the vector to the form which would be a measurement of the signal f by the instrument I 2 (R2 ! R2).

Let us write the problem of calculating the reduction as a minimax problem xk2; kAF Minimum of J2(A) = kA F Xk22 is achieved on a single matrix Ab0 = XF , since the rank of F is equal to two and therefore holds (I F F ) = 0. If this matrix is nonsingular, then J1(Ab0; fb(Ab0)) = 0 J2(Ab0) and (A ; f ) = (XF ; (XF ) 1x), a result of the reduction is f = (XF ) 1x. Example 2. Let the unknown operator A each the number of f associates a two-dimensional vector Af = a1f , i.e. A is de ned by the matrix of size a2f

Let us consider the minimax problem xk2; kAF In this case, the operator A such that its value space is the dimensional linear subspace of R2, containing the results of test and reduction measurement. The location of the points x and x1; : : : ; xm such that inequality k(I Ab0Ab0 )xk2 kX(I F F )k22 is not satis ed and the point (Ab0; fb0) is not the point at which is achieved the minimax in (5). To specify the value space of the operator A from the geometric point view means to specify the line through the origin of coordinates. For any such line value of the vector a, which determines the action operator A with a given space of values, given length vector a along a given line.

To achieve the minimax we have to change the direction of the vector a, specifying the space of values of the operator A. Calculating J1(A; f ) as the square of the distance from x to the line with direction vector a, and J2(A) are making such disposition of the space values of A, at which the equality J1(A; f ) = J2(A). The length of the projection of x on a one-dimensional subspace divided by the length of a vector a, yields a reduction of measurement x. This ensures a compromise between being able to test and reduction measurements. Conclusions. In this paper we consider a method of empirical reconstruction and reduction of measurements for fuzzy model in restrictions on Euclidean norms of signals and the operator of the model. The information about the model is contained in a series of test experiments and reduction measurements. This work was supported by the Russian Foundation for Basic Research (project no. 11-07-00338-a, 09-01-96508 and 09-07-00505-a).

References

1. T.A. Kopit , A.I. Chulichkov , D.M. Ustinin , Empirical reconstruction of fuzzy model of the experiment and the reduction of measurements in the Euclidean metric , Computational Methods and Programming , vol. 12 , pp. 220 { 226 , 2011 .[in Russian].

2. L. A. Zadeh , Fuzzy Sets as a Basis for a Theory of Possibility, Fuzzy Sets Syst ., no. 1 , pp. 3 - 28 , 1978 .

3. Yu . P. Pytev, Possibility as an Alternative of Probability , Fizmatlit, Moscow, 2007 . [in Russian].

4. Yu . P. Pytev, Methods of Mathematical Simulation of Measuring-Computing Systems, Fizmatlit , Moscow, 2002 . [in Russian].