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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
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
Suppose now, we do not know the population
variance-covariance, we estimate it by the pooled variance
i=1 (ni − 1)Ci
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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- Fall '14