Acta Geodaetica, Geophysica et Montanistica Hungarica 24. (1989)

3-4. szám - Hajagos B.–Steiner F.: Measure of the linear dependence

Acta Geod. Geoph. Mont. Hung., Vol. 24 (3-4), pp. 417-439 (1989) MEASURE OF THE LINEAR DEPENDENCE В Hajagos^ and F Steiner^ University for Heavy Industry, Chair of Geophysicg, H-3515 Miskolc, Egyetemváros, Hungary [Manuscript received July 14, 1988] Taking as basis the supermodel f^(x) which includes all symmetric stable probability distributions and properly models many cases in earth physics, the generation of the linear dependence and of the determination of the value r characterizing the measure of the dependence can be carried out easily, well arranged and without additional conditions. The paper presents the exact formulas for its determination together with simple ap­proximative methods and draws attention to the distortions in the tradi­tional method of determination of the correlation coefficient, e.g. that it is not resistant, further that irrealistically high coefficients near to 1 may be obtained by the traditional method even in case of medium depen­dences if the distribution has heavy flanks, i.e. if the actual distribu­tion is far from the Gaussian one. Keywords: correlation; Gaussian distribution; linear dependence; robust estimation 1. INTRODUCTION There is a lot of practical problems in every field of science which are connected - according to the terminology of probability - with the dependent or independent character of random variables. If a deterministic dependence is evident, moreover its analytic form is also known (from physical prin­ciples or empirically) and a random variable appears only so to say secondarily in the connections as an additive component, then an adjustment method is to be used (e.g. based on the least squares method, using the L^-norm or the method of the most frequent value (Steiner 1988)). It may be superfluous in such cases to ask about the measure of the dependence of a random variable ^ from an other random variable Ç which question can be answered practically only by introducing Acta Geod. Geoph. Mont. Hung. 24, 1989 Akadémiai Kiadó, Budapest

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