Extrapolation of Functions of Many Variables by Means of Metric Analysis

The paper considers a problem of extrapolating functions of several variables. It is assumed that the values of the function of m variables at a finite number of points in some domain D of the m-dimensional space are given. It is required to restore the value of the function at points outside the domain D. The paper proposes a fundamentally new method for functions of several variables extrapolation. In the presented paper, the method of extrapolating a function of many variables developed by us uses the interpolation scheme of metric analysis. To solve the extrapolation problem, a scheme based on metric analysis methods is proposed. This scheme consists of two stages. In the first stage, using the metric analysis, the function is interpolated to the points of the domain D belonging to the segment of the straight line connecting the center of the domain D with the point M, in which it is necessary to restore the value of the function. In the second stage, based on the auto regression model and metric analysis, the function values are predicted along the above straight-line segment beyond the domain D up to the point M. The presented numerical example demonstrates the efficiency of the method under consideration.


Introduction
One of the main problems of data processing in many areas is the problem of functions of several variables extrapolation.In the presented paper, the method of a function of many variables extrapolation, developed by us, uses the interpolation scheme of metric analysis.Below is a brief description of the metric analysis of interpolation of the values of functions of several variables and its application [3].

Extrapolation scheme
At the first and second stages of the scheme of extrapolation of functions of many variables presented in the article, the interpolation of metric analysis is used.The extrapolation scheme uses the interpolation method for the functional dependence: where the function F( X) is unknown and is subject to recovery, either at one point X * or in a set of given points on the basis of known values of the function According to the method of interpolation, based on metric analysis, interpolation values are found as solutions of problems of minimum of the measure of metric uncertainty with respect to the point where W is a matrix of the metric uncertainty and the interpolation value is determined by a linear combination and is given by .
The matrix of the metric uncertainty is defined by where n, where w k , k = 1, . . ., m are metric weights (see [3]).
Next, a metric analysis scheme is used to predict (extrapolate) the function of a single variable using an autoregressive model of metric analysis.
Let us consider a function y = f (x) of one variable x with known values The problem of finding the extrapolated value Y n+1 is reduced to the problem of interpolation of functions of several variables by means of a nonlinear autoregressive model [3,4]: Then the extrapolation of the function y = f (x) is reduced to the interpolation of the function of l variables Y = F(y 1 , y 2 , . . ., y l ) with values in n − l points The extrapolated value Y ext = Y n+1 is defined as an interpolation value of the function Y = F(y 1 , y 2 , . . ., y l ) at the point X * : where The number l determines the dimension of the space of vectors, and its value in the test was found as a solution of extremal problem [1,2] where minimization is done in l, Y re is the vector of realized values and Y ext is the vector of extrapolated values.
The extrapolation scheme for the function of several variables (1) at a given point X * = (X * 1 , . . ., X * m ) T consists of two stages.At the first stage, the point X 0 = (X 01 , . . ., X 0m ) T is selected inside the cluster of the realized values of the function Y = F( X).Then the points X 0 and X * are connected by a straight line segment which is divided into L equal segments with nodes In the points belonging to the cluster and rectilinear segment (10), the interpolation of the values of the function (1) is performed using the scheme ( 4)-(5) on the set of known values of the function Y i , i = 1, . . ., n at the points X i = (X i1 , . . ., X im ) T belonging to the cluster.At the second stage, the interpolated values, which were calculated at the first step Y = (Y 1 , . . ., Y l ) T at the points (12), are successively extrapolated to the remaining nodes S k = (S k1 , . . ., S km ) T , k = l + 1, . . ., L + 1, where By using the autoregressive scheme (6)-(9) at the nodes one can obtain the extrapolated value

Conclusion
In this paper a fundamentally new method of the extrapolation for functions of several variables is proposed.The obtained numerical results of extrapolation of functions of several variables, taken from different areas, show that the presented scheme can provide a reliable accuracy of extrapolation results.
The publication was prepared under the support of the "PFUR University Program 5-100".

W − 1
is the inverse matrix for the (n − l) × (n − l) matrix of metric uncertainty, Y = (Y l+1 , . . ., Y n ) T is the (n − l)-dimensional vector of the values of the extrapolated function.