=Paper=
{{Paper
|id=None
|storemode=property
|title=Empirical Reconstruction of Fuzzy Model of Experiment in the Euclidean Netric
|pdfUrl=https://ceur-ws.org/Vol-758/paper_6.pdf
|volume=Vol-758
}}
==Empirical Reconstruction of Fuzzy Model of Experiment in the Euclidean Netric==
https://ceur-ws.org/Vol-758/paper_6.pdf
Empirical reconstruction of fuzzy model of
experiment in the Euclidean metric
Tatiana Kopit and Alexey Chulichkov
Department of Computer Methods of Physics, Faculty of Physics, Moscow State
University, Moscow, 119991 Russia
kopit tanya@mail.ru, achulichkov@gmail.com
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.
Keywords: mathematical modeling, fuzzy sets, decision making, anal-
ysis and interpretation of data, measurement and computing systems
Introduction. In this paper we consider a fuzzy experiment conducted by the
scheme
ξ = Λϕ + ν. (1)
The measurement result ξ of the Euclidean space Rn is accompanied by a addi-
tive noise ν. By a result of ξ of measure (1) it is required to estimate the value of
the parameter vector η = U ϕ, where U ∈ (RN → RM ) is defined linear operator
[4].
In the paper [1] it is shown that if (1) vector ϕ ∈ RN and a linear operator Λ
priori arbitrary, and ν ∈ Rn is the fuzzy vector in Rn with distribution possibil-
ities π ν (·) [2,3]. And for the determination of the model of the measuring device
Λ there are involved the measurement results ξj = Λfj + νj , j = 1, . . . , m, of
known test signals, f1 , . . . , fm , where the error ν1 , . . . , νm ∈ Rn of test measure-
ments are fuzzy elements of Rn with the given distribution of possibilities. The
estimates of the maximum possibility A b and fb are values of fuzzy elements Λ
and ϕ respectively, as a solution of the maximin problem
b fb) = arg max min(π ν (x − Af ), π N (X − AF )),
(A, (2)
A,f
and the estimate u b is value of the fuzzy element of η given by ub = U fb. Here,
x, A, f, X are implementation of fuzzy elements of ξ, Λ, ϕ, Ξ, respectively. The
same scheme of test measurements in matrix form is given by Ξ = ΛF + N .
Consider the solution of the problem (2), where the operator Λ and element
ϕ are a priori arbitrary, so that π Λ (A) = 1 for every A ∈ (RN → Rn ) and
π ϕ (f ) = 1 for any f ∈ RN , and distribution possibilities of measurement error
is given by
π ν (z) = µ0 (kzk2 ), z ∈ Rn ; π N (Z) = µ0 (kZk22 ), Z ∈ (Rm → Rn ),
where µ0 (·) : [0, ∞) → [0, 1] is strictly decreasing function, µ0 (0) = 1, lim µ0 (z) =
z→∞
0.
Then the problem (2) leads to the following minimax problem:
min max(kx − Af k2 , kX − AF k22 ). (3)
A,f
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 . (4)
A,f
Depending on which of the minimum values J1 (A b0 , fb(A b0 )) or J2 (A
b0 ), is less it
is selected different methods for solving the problem (3), they are considered in
[1].
Example
1. Let
the unknown operator A isdefined by the matrix ofsize 2 × 2,
a11 a22 f11 f1m
A= , given m test signals, f1 = , . . . , fm = , forming
a21 a22 f21 f2m
columns of the matrix F∈ (R m → R2 ), and test results are given as vectors
x11 x1m
x1 = , . . . , xm = , forming columns of the matrix X ∈ (Rm →
x21 x2m
R2 ). Then the scheme of tests is given by
Ξ = AF + N,
where the j-th column of N defines the error of j-th test measure, j = 1, . . . , m.
For the error matrix N that defined by the distribution of possibilities of its val-
ues by the form π N (X) = µ0 (kZk22 ), where µ0 (·) : R+ → [0, 1] is a monotonically
decreasing function, µ0 (0) = 1, µ0 (+∞) = 0.
Let the rank of F is equal to two. The signal f is measured according to
the scheme ξ = Af + ν. The result x ∈ R2 of this measurement is known.
The distribution is π ν (z) = µ0 (kzk2 ) of possibility of fuzzy vector ν ∈ R2 for
measurement error Af . It is required to determine the reduction of the vector
ξ to the form which would be a measurement of the signal f by the instrument
I ∈ (R2 → R2 ).
Let us write the problem of calculating the reduction as a minimax problem
min(max kAf − xk2 , kAF − Xk22 ).
A,f
Minimum of J2 (A) = kA∗ F − Xk22 is achieved on a single matrix A b0 = XF − ,
since the rank of F is equal to two and therefore holds (I − F F − ) = 0. If
this matrix is nonsingular, then J1 (A b0 )) = 0 ≤ J2 (A
b0 , fb(A b0 ) and (A∗ , f∗ ) =
− − −1 − −1
(XF , (XF ) x), a result of the reduction is f∗ = (XF ) x.
Example 2. Let the unknownoperator A each the number of f associates a
a1 f
two-dimensional vector Af = , i.e. A is defined by the matrix of size
a2 f
a1
1 × 2, A = ; is given m of test scalar signals f1 , . . . , fm , forming a matrix
a2
F ∈ Rm →R1 , consisting of a single line, and test results are given as vectors
x11 x1m
x1 = , . . . , xm = , forming columns of X ∈ Rm → R2 . Scheme
x21 x2m
and model test and measurement reducible are the same as in previous Example
1.
Let us consider the minimax problem
min(max kAf − xk2 , kAF − Xk22 ). (5)
A,f
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 − A b− )xk2 ≤
b0 A
0
kX(I − F − F )k22 is not satisfied and the point (A b0 , 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 Sys-
tems, Fizmatlit, Moscow, 2002. [in Russian].