Sangdon Lee wrote: > Dear All, > > Thanks for valuable information. Just one more question! > Could somebody explain to me the difference/similarity among > Mahalanobis distance, Hotelling's T-square and PCA ? I believe that > Hotelling's T-square is the sum of squares of the principal component > scores if all components are extracted as the number of input > variables. It seems to me that the Mahalanobis distance is the same as > the Hotelling's T-square.
You are correct. T-squared from PCA with all possible components is exactly equal to Mahalanobis distance. If fewer than the maximum number of components are used, then the two quantities would NOT be equal.
I think that the confusion here, which I missed before, is whether we're dealing with one or two groups. T^2 is the multivariate test for two groups, whereas Mahalanobis D^2 can be used with either one or two groups. For a single group, D^2 measures the deviation of a single observation from the group centroid. For two groups, D^2 estimates the difference between the group centroids. Thus the relation between T^2 and D^2 (and discriminant analysis) is for the case of two groups, while D^2 relates to PCA only for a single sample. For one group S represents the total sums-of-squares, while for two groups S represents the pooled within-group sums-of-squares.
Reference:
Jackson, J.E. (1991) "A User's Guide To Principal Components", John Wiley and Sons, New York. Jackson doesn't specifically refer to Mahalanobis, but his formula 1.7.3 shows the relationship between T-squared and x'Sx which is the Mahalanobis formula.
-- Paige Miller Eastman Kodak Company
============================================================= Dr. Richard E. Strauss (806) 742-2719 Biological Sciences (806) 742-2963 Fax Texas Tech University [EMAIL PROTECTED] Lubbock, TX 79409-3131 <http://www.biol.ttu.edu/Strauss/Strauss.html> =============================================================
. . ================================================================= Instructions for joining and leaving this list, remarks about the problem of INAPPROPRIATE MESSAGES, and archives are available at: . http://jse.stat.ncsu.edu/ . =================================================================
