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By Giudici

Utilized information Mining for company 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

Example text

It can be shown that both τY |X and UY |X take values in the [0,1] interval. Note, in particular, that: τY |X = UY |X if and only if the variables are independent; τY |X = UY |X = 1 if and only if Y has maximum dependence on X. The indexes described have a simple operational interpretation regarding specific aspects of the dependence link between the variables. In particular, both τY |X and UY |X represent alternative quantifications of the reduction of the Y heterogeneity that can be explained through the dependence of Y on X.

Ap2  .  , Yn2 xn1 xn2 xnp that is, in matrix terms, p Y2 = aj 2 Xj = Xa2 , j =1 where the vector of the coefficients a2 = (a12 , . . , ap2 ) is chosen in such a way that max Var(Y2 ) = max(a2 , Sa2 ), under the constraints a 2 a2 = 1 and a 2 a1 = 0. Note the second constraint, which requires the two vectors a2 and a1 orthogonal. This means that the first and second components will be uncorrelated. 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.

Let X be a data matrix with n rows and p columns. The main summary measures can be expressed directly in terms of matrix operations on X. For example, the arithmetic mean of the variables, described by a p-dimensional vector X, can be obtained directly from the data matrix as X= 1 1 X, n where 1 indicates a (row) vector of length n with all elements equal to 1. As previously mentioned, it is often better to standardise the variables in X. To achieve this aim, we first need to subtract the mean from each variable.

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