## Interval linear regression analysis based on Minkowski difference - a bridge between traditional and interval linear regression models.(English)Zbl 1249.65012

Summary: In this paper we extend the traditional linear regression methods to the (numerical input)-(interval output) data case assuming both the observation/measurement error and the indeterminacy of the input-output relationship. We propose three different models based on three different assumptions of interval output data. In each model the errors are defined as intervals by solving the interval equation representing the relationship among the interval output, the interval function and the interval error. We formalize the estimation problem of parameters of the interval function so as to minimize the sum of square/absolute interval errors. Introducing suitable interpretation of minimization of an interval function, each estimation problem is well-formulated as a quadratic or linear programming problem. It is shown that the proposed methods have close relation to both traditional and interval linear regression methods which are formulated in different manners.

### MSC:

 65C60 Computational problems in statistics (MSC2010) 65G40 General methods in interval analysis 62J05 Linear regression; mixed models 65K05 Numerical mathematical programming methods 90C05 Linear programming 90C20 Quadratic programming
Full Text:

### References:

  Aubin J.-P., Frankowska H.: Set-Valued Analysis. Birkhäuser, Boston 1990 · Zbl 1168.49014  Diamond P.: Fuzzy least squares. Inform. Sci. 46 (1988), 141-157 · Zbl 0663.65150  Diamond P., Tanaka H.: Fuzzy regression analysis. Fuzzy Sets in Decision Analysis, Operations Research and Statistics (R. Słowinski, Kluwer, Boston 1988, pp. 349-387 · Zbl 0922.62058  Dubois D., Prade H.: Fuzzy numbers: An overview. Analysis of Fuzzy Information, Vol. I: Mathematics and Logic (J. C. Bezdek, CRC Press, Boca Raton 1987, pp. 3-39 · Zbl 0663.94028  Huber P. J.: Robust statistics. Ann. Math. Statist. 43 (1972), 1041-1067 · Zbl 0254.62023  Ignizio J. P.: Linear Programming in Single- & Multiple-Objective Systems. Prentice-Hall, Englewood Cliffs, NJ 1982 · Zbl 0484.90068  Inuiguchi M., Kume Y.: Goal programming problems with interval coefficients and target ontervals. European J. Oper. Res. 52 (1991), 345-360 · Zbl 0734.90056  Moore R. E.: Methods and Applications of Interval Analysis. SIAM, Philadelphia 1979 · Zbl 0417.65022  Shape W. F.: Mean-absolute-deviation characteristic lines for securities and portfolios. Management Sci. 18 (1971), 2, B1-B13 · Zbl 0225.90009  Tanaka H., Hayashi, I., Nagasaka K.: Interval regression analysis by possibilistic measures (in Japanese). Japan. J. Behaviormetrics 16 (1988), 1, 1-7 · Zbl 04538823  Tanaka H., Lee H.: Interval regression analysis by quadratic programming approach. IEEE Trans. Fuzzy Systems 6 (1998), 4, 473-481  Tanaka H., Uejima, S., Asai K.: Linear regression analysis with fuzzy model. IEEE Trans. Systems Man Cybernet. 12 (1982), 903-907 · Zbl 0501.90060  Tanaka H., Watada J.: Possibilistic linear systems and their application to the linear regression model. Fuzzy Sets and Systems 27 (1988), 275-289 · Zbl 0662.93066
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.