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By Giudici

Utilized info Mining for enterprise and by way of Giudici, Paolo, Figini, Silvia [Wiley,2009] (Paperback) 2d variation [Paperback]

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The expression for the second principal component can be obtained through the method of Lagrange multipliers, and a2 is the eigenvector (normalised and orthogonal to a1 ) corresponding to the second largest eigenvalue of S. This process can be used recursively to define the kth principal component, with k less than the number of variables p. In general, the vth principal component, for v = 1, . . , k, is given by the linear combination p Yv = aj v Xj = Xav j =1 in which the vector of the coefficients av is the eigenvector of S corresponding to the vth largest eigenvalue.

We remark that the covariance is an absolute index. That is, with the covariance it is possible to identify the presence of a relationship between two quantities but little can be said about the degree of such relationship. 5 X1 ... Xj ... Xh Variance–covariance matrix. X1 ... Xj ... Xh Var(X1 ) ... Cov(Xj , X1 ) ... Cov(Xh , X1 ) ... Cov(X1 , Xj ) ... Var(Xj ) ... ... ... ... Cov(X1 , Xh ) ... ... Var(Xh ) ... order to use the covariance as an exploratory index it is necessary to normalise it, so that it becomes a relative index.

In the general case, such a measure is defined by I J X = 2 (nij − n∗ij ) i=1 j =1 where n∗ij = ni+ n+j , n n∗ij 2 , i = 1, 2, . . , I ; j = 1, 2, . . , J. Note that X2 = 0 if the X and Y variables are independent. In fact in such a case, the factors in the numerator are all zero. We note that the X2 statistic can be written in the equivalent form   I J 2 n ij − 1 X2 = n  ni+ n+j i=1 j =1 which emphasizes the dependence of the statistic on the number of observations, n; this is a potential problem since the value of X2 increases with the sample size n.

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