The property of Moore Penrose Generalized Inverse And The. Correlativity Between Two Random Variables 1
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1 ISSN : Volume Issue The property of Moore Penrose Generalized Inverse nd The Correlativity Between Two Random Variables Baoguang Tian, Chunyun Sheng (Department of mathematics, Qingdao university of science and technology,qingdao,66) bstract. It is an important problem to study the correlativity of two random variables. The correlation coefficient can be used to measure the correlativity of two random variables and the generalized correlation coefficient can be used to measure the correlativity of two random vectors. In this paper, The property of Moore Penrose generalized inverse has been proved and we make use of it to mainly discuss the property of generalized correlation coefficient of two random vectors and obtain some meaningful results. Keywords: Moore Penrose Generalized Inverse, Generalized Correlation Coefficient, Latent Root, Canonical Correlation Coefficient. Introduction Let x and y respectively be p-dimension and q-dimension random vectors, V is covariance matrix of x and y, That is Cov x, y, V y x V x V Vxy V V y Cov y x Vyx V () Because non-zero eigenvalues of VVyxV Vxy is the same as yx xy V V V V V, the non-zero eigenvalues of VVyxV Vxy is not negative. Here expresses Moore Penrose generalized inverse matrix of matrix. When the rank of VVyxV Vxy is r, it can be proved that the eigenvalues of VVyxV Vxy,, r, is not larger than number. Let i i, i,,, r, then,,, i i r is called canonical correlation coefficient of x and y. The correlativity of random vectors x and y can be measured by function of canonical correlation coefficient and five kinds of generalized correlation coefficient are defined by using This work is supported by the Science and Technology Program of Shandong Universities of China(JL57,JL4) and the Base Research Program of Science and Technology of Qingdao City( jch) Corresponding author: mhwang@yeah.net 65
![ISSN : 77-698 Volume Issue canonical correlation coefficient in article []. Hoetelling[] has defined another two kind generalized correlation coefficient.](/docs-images/89/99381406/images/2-0.jpg "References [3-6] discuss the properties and applications of generalized correlation coefficient. In this paper,we use rz( x, y ) to stand for generalized correlation coefficient of x and y.")
2 ISSN : Volume Issue canonical correlation coefficient in article []. Hoetelling[] has defined another two kind generalized correlation coefficient. References [3-6] discuss the properties and applications of generalized correlation coefficient. In this paper,we use rz( x, y ) to stand for generalized correlation coefficient of x and y. The Main Results Theorem If C C, then Let be n n square matrix and C be n n nonnegative symmetric matrix. So, C C Proof. s matrix C, there is an orthogonal square P to make C can be written r C P P Multiplying both sides of (3) on the left by Let P P C (), r diag,, r r r P P P P. (3) P and on the right by P, we have r r P P P P, then (4) That is, So we can get and the more, r r r r r r r r r r (5) (6) r r, r r. (7) Yet, 66
3 ISSN : Volume Issue r r P P P P P P r r r r r r P P P P P P r r r r r r P P P P By the above, we know that r r P P P P ccording to the property of Moor Penrose generalized matrix, there is Therefore Theorem is true. Theorem and Then r r C P P P P r r C P P P P C. Let and B respectively be p p and q q square matrix which exist the inverse, V BV V B. V, (8) (9) rz( x, By) rz( x, y) () Proof. Because the covariance of X and BY is 67
4 ISSN : Volume Issue Covx, By, V By x V x V By Cov By x V VxyB BVyx BVB () The canonical correlation coefficient of x and By satisfies the following equation yx xy i i BV B BV V V B I () By Theorem and V V, BV VB, we can get that V V, BV V B, So there is In fact V V, BV B B V B. (3) V V V V V V V V V V VV VV VV V V V V V V V V V V. Here we make use of V V, V V and VV VV. ccording to the definition of Moore Penrose matrix we know that is correct and we can use the same method to testify Substituting (3) into (), we have 68 V V BV B B V B. yx xy i B V B BV V V B I.
5 That is International Journal of Mathematical Engineering and Science ISSN : Volume Issue V V V V yx xy i I (4) This means that the canonical correlation coefficient of x and By Equals the canonical correlation coefficient of x and y, so their generalized correlation coefficient equals too. The above result can be generalized. Let and B respectively be n p and m q matrix with full column rank. They have the singular value decomposition P Q and B P Q, where P, Q, P and Q are orthogonal matrix, and are diagonal matrix. Theorem 3 For the above matrix and B, when Q V V Q, QV V Q, we have the following result rz(x,by)=rz(x,y) (5) Proof. let be the canonical correlation coefficient of x and By, then i i satisfies the following equation yx xy i BV B BV V V B I. Substituting P Q and B P Q into the above equation, we can get yx xy i Q V Q Q V Q QV Q Q V Q I (6) QV V Q, QV V Q. Note ccording to Theorem, there is a result By the proving procedure of Theorem, we have To substitute them into (6), we can get This completes the proof of theorem 3. QV V Q, QV V Q QV Q Q V Q Q V Q Q V Q V V V V yx xy i I (7) 69
6 ISSN : Volume Issue References. Zhang, Y.T., Fang K.T.: Introduction to multivariate statistical analysis. M. Science Press Beijing (983).. Hoetelling. Relations between two sets of variables. J. Biometrika, 36, (936). 3. Dong Xiaomeng. Sampling Distribution and Hypothesis Testing of Non-linear Generalized Correlation Coeficient and its R-program with R-language. J. Science Technology and Engineering,, () 4. Zhu Xianhai Yang Xuefeng Measures of Multivariate ssociation and Robustness of BLUE Journal of Northeast Normal University,, -5(995) 5. Zhang Wenwen. Under linear transform the generalized correlation coefficient of the random vector and relevant problems Mathematical Physics Journal,, 8-87(99) 6. Ding Yong. dditivity of verage Mutual Communication and Inequality for General Correlation Coefficients Chinese Journal of Engineering Mathematics, 4, 8-86(7) 7
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