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Interpolation that leads to the narrowest intervals and its application to expert systems and intelligent control

Интерполяция, дающая самые узкие интервалы, и ее приложения к экспертным системам и интеллектуальному управлению

  • Mathematical Research
  • Published:
Reliable Computing

Abstract

In many real-life situations, we want to reconstruct the dependencyy=f(x 1,…, xn) from the known experimental resultsx (k)i , y(k). In other words, we want tointerpolate the functionf from its known valuesy (k)=f(x (k)1 ,…, x (k)n ) in finitely many points\(\bar x^{(k)} = (x_1^{(k)} , \ldots ,x_n^{(k)} )\), 1≤k≤N There are many functions that go through given points. How to choose one of them?

The main goal of findingf is to be able to predicty based onx i. If we getx i from measurements, then usually, we only getintervals that containx i. As a result of applyingf, we get an interval y of possible values ofy. It is reasonable to choosef for which the resulting interval is the narrowest possible. In this paper, we formulate this choice problem in mathematical terms, solve the corresponding problem for several simple cases, and describe the application of these solutions to intelligent control.

Abstract

Во многих практических задачах требуется воостановить зависимостьu=f(x 1,…, xn) на основании экспериментально полученных данныхx (k)i , y(k). Другими словами, нам нужноинмернолироєамь функциюf по ее известным значениямy (k)=f(x (k)1 ,…, x (k)n ) в конечном множестве точек\(\bar x^{(k)} = (x_1^{(k)} , \ldots ,x_n^{(k)} )\), 1≤k≤N Сушествует много функций, проходящих через заданные точки Как выбрать одну из них?

Основная цель поиска функцииf состоит в том, чтобы иметь возможность предсказывать значенияy на основанииx i Еслиx i получены в результате измерений, то, как правило, мы имеем не сами значенияx i, аинмерєлы, содержащие зти значения. Применив функциюf, мы получим интервал у возможных значенийy. Имеет смысл выбратьf таким, чтобы этот результирующий интервал был по возможности более узким. В работе эта проблема выбора формулируется математически и решается для нескольких простых случаев Описывается также применение этих решений в интеллектуальном управлении.

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Authors and Affiliations

  1. Department of Systems Science, Tokyo Institute of Technology, 4259 Nagatsuta-cho, Midori-ku, 227, Yokohama, Japan

    Hung T. Nguyen

  2. Computer Science Department, University of Texas at El Paso, 79968, El Paso, TX, USA

    Vladik Kreinovich

  3. NASA Johnson Space Center, 77058, Houston, TX, USA

    Bob Lea

  4. PRC. 1500 PRC Drive, 22102, McLean, VA, USA

    Dana Tolbert

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  1. Hung T. Nguyen
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  2. Vladik Kreinovich
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  3. Bob Lea
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  4. Dana Tolbert
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Nguyen, H.T., Kreinovich, V., Lea, B. et al. Interpolation that leads to the narrowest intervals and its application to expert systems and intelligent control. Reliable Comput 1, 299–315 (1995). https://doi.org/10.1007/BF02385260

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