About an Unbiased Estimate of the Gradient with Minimum Variance in the Planning of the Experiment

Sidnyaev N.I,Razgulyaev S.V,Shafikova S.E

doi:10.13189/ujam.2014.020503

Abstract

This paper considers algorithms of search of an extremum, to solve the problem of planning the experiment using the gradient method. A feature of the algorithm is that when searching for the motion (when the maximum) does not occur in the direction of the gradient, which is unknown to us, but its estimate. Estimate of the gradient at the point when this factor space is based on the results of measurements carried out in the neighborhood. Researcher's task is to build a sensible plan with center to determine the estimate of the gradient in it. considered the most complete. Research issues of statistical estimation of the gradient when searching, is of great importance for the understanding of the use of gradient methods for planning the experiment. In general, the method of Box and Wilson is to repeat the procedure:  build - factorial experiment (3) in the neighborhood of a point;  estimating the gradient - calculation at this point, the results of the experiment;  finding estimates of the maximum (minimum) of the response function in this direction. As in the method of Box and Wilson , as well as other methods for searching the extremes of the response function , which are based on gradient method is used not the gradient , and its evaluation . We present a general formulation of the gradient estimation problem and its solution. Let the response function

Highlights

The task of finding the extremum of the mean characteristics, usually the task of finding the extremum of the response function η=f(х1,х2,...,хк)
Search extremum response function is performed by the response surface study
One of the most famous classes of gradient methods for searching the extremes of the response function in the practice of experimental plan can be considered a method developed by Box and Wilson [1, 2]

Summary

Introduction

The task of finding the extremum of the mean characteristics, usually the task of finding the extremum of the response function η=f(х1,х2,...,хк). 1. One of the most famous classes of gradient methods for searching the extremes of the response function in the practice of experimental plan can be considered a method developed by Box and Wilson [1, 2]. One of the most famous classes of gradient methods for searching the extremes of the response function in the practice of experimental plan can be considered a method developed by Box and Wilson [1, 2] The idea behind it is to use the method of steep climbing (steepest ascent) in conjunction with the series planned factorial experiment to find the gradient estimation. The transition to the new variables means the transfer origin to the point X 0 and compression (tension) on the coordinate axes [1]

The Task of Estimating the Gradient

Examples of Specific Implementations

Conclusions