Eine Plattform für die Wissenschaft: Bauingenieurwesen, Architektur und Urbanistik
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Reliability of the estimations of the correlation coefficient
Abstract On the basis upon n corresponding value-pairs (xi,yi), i = 1,…,n, the closeness of correspondence between the random variables ξ and η is customarily characterized by the classical correlation coefficient r (see Eq. (2) in the present paper), equally in the geosciences and in the everyday life. It is shown in the present paper the lack of the robustness of Eq. (2) (r has even no meaning for circa 40% of the types occurring in the geosciences), and the lack of the resistance (one single outlying value-pair can distort the r-value in an incredible degree). The modern correlation coefficient $ r_{rob} $ (see Eq. (9) in this paper) is completely resistant against outliers, and in the same time also robust; Eq. (9) is applicable even if ξ and η are of Cauchy type, very far lying from the Gaussian distribution and even from the most frequently occurring so-called statistical distribution (see Eq. 6). For the Gauchy distribution neither the scatter (variance) nor the expected value exist therefore for this distribution-type even the classical theoretical value (see Eq. 3) does not exist; the calculation of r according to Eq. (2) gives in this case an “estimation” of a not existing quantity. In the paper are presented the results of a time consuming series of Monte Carlo calculations made equally for the statistical and Gaussian distributions and for n = 10; 30 and 100; the errors characterized by the semi-interquartile and semi-intersextile ranges of the modern $ r_{rob} $ (Eq. 9) were calculated and tabulated for rt = 0; 0.1; 0.2; … 0.7 and 0.8. An approximate method is also given (see the simple Eqs 16 and 17) to determine that value of n which assures a prescribed accuracy of the modern $ r_{rob} $.
Reliability of the estimations of the correlation coefficient
Abstract On the basis upon n corresponding value-pairs (xi,yi), i = 1,…,n, the closeness of correspondence between the random variables ξ and η is customarily characterized by the classical correlation coefficient r (see Eq. (2) in the present paper), equally in the geosciences and in the everyday life. It is shown in the present paper the lack of the robustness of Eq. (2) (r has even no meaning for circa 40% of the types occurring in the geosciences), and the lack of the resistance (one single outlying value-pair can distort the r-value in an incredible degree). The modern correlation coefficient $ r_{rob} $ (see Eq. (9) in this paper) is completely resistant against outliers, and in the same time also robust; Eq. (9) is applicable even if ξ and η are of Cauchy type, very far lying from the Gaussian distribution and even from the most frequently occurring so-called statistical distribution (see Eq. 6). For the Gauchy distribution neither the scatter (variance) nor the expected value exist therefore for this distribution-type even the classical theoretical value (see Eq. 3) does not exist; the calculation of r according to Eq. (2) gives in this case an “estimation” of a not existing quantity. In the paper are presented the results of a time consuming series of Monte Carlo calculations made equally for the statistical and Gaussian distributions and for n = 10; 30 and 100; the errors characterized by the semi-interquartile and semi-intersextile ranges of the modern $ r_{rob} $ (Eq. 9) were calculated and tabulated for rt = 0; 0.1; 0.2; … 0.7 and 0.8. An approximate method is also given (see the simple Eqs 16 and 17) to determine that value of n which assures a prescribed accuracy of the modern $ r_{rob} $.
Reliability of the estimations of the correlation coefficient
Steiner, F. (Autor:in) / Hajagos, B. (Autor:in)
2001
Aufsatz (Zeitschrift)
Englisch
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