Linear discriminant analysis covariance matrix

When all are considered equal, this means that they can be replaced by S, the pooled variance-covariance matrix, which is the case for linear discriminant analysis. The discrimination boundaries then are linear and is given by... 

Linear discriminant analysis (LDA) is a classification method that uses the distance between the incoming sample and the class centroid to classify the sample. For LDA using Mahalanobis distances, the classification metric uses the pooled variance-covariance matrix to weight the Mahalanobis distance ) between the incoming... 

The particular Bayesian classifier that we consider in this chapter is Bayesian linear discriminant analysis (BLDA). For BLDA one assumes that the class covariance matrices Sr are equal. A pooled covariance matrix is constructed as follows... 

For most applications in the "omics" fields, even the most simple multivariate techniques such as Linear Discriminant Analysis (LDA) cannot be applied directly. From Equation 2 it is clear that an inverse of the the covariance matrix 2 needs to be calculated, which is impossible in cases where the number of variables exceeds the number of samples. In practice, the number of samples is nowhere near the number of variables. For QDA, the situation is even worse to allow a stable matrix inversion, every single class should have at least as many samples as variables (and preferably quite a bit more). A common approach is to compress the information in the data into a low number of latent variables (LVs), either using PCA (leading... 

ECVA consists of several steps that finally lead to the actual class membership prediction. First there is a matrix compression step using SVD (if number of variables exceed number of samples n>m)). This is followed by the calculation of covariance matrices, both for the within class variatiOTi and between class variation. Subsequently, a PLS is performed between these matrices and the class relationship and finally a linear discriminant analysis is performed on the PLS results. This indicates that ECVA is a very different classification technique than PLS-DA described above, which is why we have chosen to show how it performs in addition to the above-menti(Mied techniques. 

Another parametric routine implements a discriminant function by the method commonly called linear discriminant function analysis. It is nearly identical to the linear Bayesian discriminant, except that instead of using the covariance matrix, the sum of cross-products matrix is used. Results obtained with the routine are ordinarily very similar to those obtained using the linear Bayes routine. The routine implemented as LDFA is a highly modified version of program BMD04M taken from the Biomedical Computer Programs Package (47). 

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