Fisher's criterion is an objective function used to choose a linear transformation to one dimension that separates two classes of multivariate observations.
Let the class sample means be 
and 
, and let 
be the sum of the two within-class scatter matrices.
For a nonzero vector 
, the criterion is the ratio
![J(w)=([w^T(mu_1-mu_2)]^2)/(w^TS_Ww).](/images/equations/FishersCriterion/NumberedEquation1.svg) |
The numerator is the squared separation between the class means after projection,
and the denominator is the within-class scatter after
projection. The criterion is unchanged when 
is multiplied by a nonzero scalar. When 
is a nonsingular matrix,
every maximizing direction is proportional to
 |
Using any such direction for the projection gives a Fisher linear discriminant.
