20-Discriminant2

# number of discriminant functions there is one

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Unformatted text preview: centroids, one for each group. * Number of discriminant functions. There is one discriminant function for 2-group discriminant analysis, but for higher order DA, the number of functions (each with its own cut-off value) is the lesser of (g - 1), where g is the number of groups, or p,the number of discriminating (independent) variables. Each discriminant function is orthogonal to the others. . . . . . . Mahalonobis Distance Mahalanobis distances are used in analyzing cases in discriminant analysis. For instance, one might wish to analyze a new, unknown set of cases in comparison to an existing set of known cases. Mahalanobis distance is the distance between a case and the centroid for each group in attribute space (p-dimensional space deﬁned by p variables) taking into account the covariance of the variables. The population version: Suppose g groups, and p variable, and that the mean for group i is a vector µi = [µi1 , µi2 , . . . µip ], 1 ≤ i ≤ g, and call σ the variance-covariance matrix (which we suppose to be the same in all the groups). The Mahalanobis distance between group i and group j is: Dij = (µi − µj )′ Σ−1 (µi − µj ) . . . . . . The Mahalanobis distance is often used to compute the distance of a case x and the centre of the population as: D2 (x, µ) = (x − µ)′ Σ−1 (x − µ) When the distribution is multivariate normal the D2 follows a χ2 distribution. p Suppose now, we do not know the population variance-covariance, we estimate it by the pooled variance covariance matrix ∑g i=1 (ni − 1)Ci C= ∑ i (ni − 1) Then the Mahalanobis distance between the observation x and the group centroid i is: ¯ ¯ ¯ D2 (x, xi ) = (x...
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