Utilized info Mining for enterprise and by way of Giudici, Paolo, Figini, Silvia [Wiley,2009] (Paperback) 2d variation [Paperback]
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Additional resources for Applied Data Mining for Business and Industry, 2nd edition
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.