The LDA Module

The LDA Object

Topics:

What is LDA?

How to use LDA.

References

What is LDA?

Linear Discriminat Analysis (LDA) [1] is a feature extraction method that provides a linear transformation of n-dimensional feature vectors (or samples) into a m-dimensional space (m < n), so that samples belonging to the same class are close together but samples from different classes are far apart from each other.

If we define the total scatter matrix Tm (or covariance matrix) and the within class scatter matrix Wm in the m space, we try to minimize Det(Wm) and maximize Det(Tm) simultaneously or 

      | Tm |                   !
     -------- = | Wm^-1 Tm |   =   max
      | Wm |
The solution for the transformation matrix A are the eigenvectors of Wn^-1 Tn with the m largest eigenvalues: 

     (e1 e2 .. en) = eigenvec( Wn^-1 Tn )

               A'  = (e1 e2 .. em)

    new feature vector:    y = A x
The scatter matrices for the new data samples Tm and Wm are then both diogonal which means the coefficients are uncorrelated and they can be calculated as 
     Tm = A Tn A'
     Wm = A Wn A'
The Janus LDA object consists of

How to use LDA

A LDA object is created similar to other JANUS objects by using the type name LDA and a user definded name. The user also has to give the name of a FeatureSet, the name of a Feature from this FeatureSet and its dimension. The FeatureSet must already exist, the Feature need not to. It is only locked to make sure it will have the correct dimension.

% FeatureSet fs
fs
% LDA lda fs feat 16
lda
Now classes can be added with the add method. The classes could be triphones, phonemes or anything else. Using just the name of the LDA object gives a list of the existing classes:

% lda add c1
% lda add c2
% lda
c1 c2
Later we will use a Path object to fill the mean vectors and scatter matrices of the LDA object. This path contains (senone) indices that we can map into our LDA classes. We can map more than one index into one LDA class but not the other way round:

% lda map 0 -class c1
% lda map 1 -class c1
% lda map 5 -class c1
The indices we haven't used will be ignored. That means the according sample won't be taken into account for any class mean or the within scatter matrix. Using the map method without the optional flag -class will give us the LDA class index (Don't mix it up with the (senone) index!). This class index can be converted into the class name using the name method:

% lda map 0          
0
% lda map 4
-1
% lda name [lda map 0]
c1
% lda name [lda map 4]
(null)
After setting all means and scatter matrices to 0 (methods clearMeans and clearScatters) which is already the case when you just defined a LDA object, 2 passes over the whole speech data base must be run. In a 1st pass the total mean and means for each LDA class are obtained taking samples from the FeatureSet and indices and weighting factors from a Path object. This indices are mapped to the LDA classes as defined before. In the 2nd pass the total scatter matrix (covariance matrix) and the within class scatter matrix are calculated using the same input. For the total scatter matrix it is necessary to have the total mean, for the within class scatter we need all class means. 
% lda clearMeans
% foreach utterance $utts {
>   ...
>   lda accuMeans path
> }

% lda clearScatters
% foreach utterance $utts {
>   ...
>   lda accuScatters path
> }
If you want to look at the means or the scatter matrices here is how you can do that: 
% lda.mean                         ;# look at the total mean
% lda:c1.mean                      ;# look at mean of class c1
% lda:c1.mean configure -count     ;# look at the count
% lda.matrixT                      ;# look at total scatter
% lda.matrixW configure -count     ;# look at count of within class scatter
The whole reason we did this, was to find a linear transformation (each feature vector or speech vector multiplied with a matrix A) that minimizes the within class scatter in the new feature space and maximizes the total scatter. A way to find this matrix A is the simultaneous diagonalisation [1]. It is implemented in JANUS as a DMatrix method since the LDA scatter matrices are of type DMatrix. 
% DMatrix K
% [DMatrix A] simdiag K lda.matrixT lda.matrixW
Now we can transform the feature feat into a new feature newfeat with equal or less coefficients. 
% [FMatrix B] DMatrix A  ;# convert to FMatrix
% fs matmul newfeat feat B -cut 10
Here is a small test script you can run with JANUS.

Extracting the means and scatter matrices with more than one machine is possible using LDA I/O methods.

References:

maier@ira.uka.de