sphere the data for the linear discriminant case (that iswhen sy for all )it is convenient to first "whitenor sphere the data as follows let be an invertible matrix such that bbobtainedfor examplevia the cholesky method we linearly transform each data point to : - and each mean uy to : - uy let the random vector be distributed according to the mixture pdf - ( -uy sy ( -uy ( th:ay ( pp |sy thenby the transformation theorem the vector - has density - ( thx ay - ( the ( -uy (bb ( -uy - | ( pp = - ay ( pp = ( -uy ( -uy - = ay ( pp kx -uy this is the pdf of mixture of standard -dimensional normal distributions the name "spheringderives from the fact that the contours of each mixture component are perfect spheres classification of the transformed data is now particularly easyclassify as : argminy kx uy ln ay note that this rule only depends on the prior probabilities and the distance from to the transformed means { this procedure can lead to significant dimensionality reduction of the data namelythe data can be projected onto the space spanned by the differences between the mean vectors { when there are classesthis is ( )-dimensional spaceas opposed to the -dimensional space of the original data we explain the precise ideas via an example example (classification after data reductionconsider an equal mixture of three -dimensional gaussian distributions with identical covariance matrices after sphering the datathe covariance matrices are all equal to the identity matrix suppose the mean vectors of the sphered data are [ - ] [ - ]and [ ]the left panel of figure shows the -dimensional (sphereddata from each of the three classes figure leftoriginal data rightprojected data the data are stored in three matrices and here is how the data was generated and plotted