One of the simplifications is that we may assume that

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Unformatted text preview: A for Multi-class Problems - October 14, 2009 FDA method for Multi-clas s Problems For the k - class problem, we need to find a projection from d- dimensional space to a (k − 1)- dimensional space. (It is more reasonable to have at least 2 directions) Basically, the within class covariance matrix where is easily to obtain: and . However, the between class covariance matrix is not easy to obtain. One of the simplifications is that we may assume that the total covariance constant, since is easy to compute, we can get using the following relationship: Actually, there is another generation for Thus the total covariance matrix . Denote a total mean vector of the data is by is wikicour senote.com/w/index.php?title= Stat841&pr intable= yes 22/74 10/09/2013 Stat841 - Wiki Cour se Notes Thus we obtain Since the total covariance is the sum of the within class covariance class covariance matrix , thus we obtain and the between class covariance , we can denote the second term as the general between Therefore, Recall that in the two class case problem, we have From the general form, Apparently, they are very similar. Now, we are trying to find the optimal transformation. Basically, we have where is a vector, is a transformation matrix, i.e. , and is a column vector. Thus we obtain Similarly, we obtain wikicour senote.com/w/index.php?title= Stat841&pr intable= yes 23/74 10/09/2013 Stat841 - Wiki Cour se Notes Now, we use the determinant of the matrix, i.e. the product of the eigenvalues of the matrix, as our measure. The solution for this question is that the columns of the transformation matrix are exactly the eigenvectors that correspond to largest k − 1 eigenvalues with respect to Also, note that we can use as our measure. Recall that Thus we obtain that Similarly, we can get . Thus we have following criterion function Similar to the two class case problem, we have: max subject to To solve this optimization problem a Lagrange multiplier Λ, which actually is a Differen...
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