Fisher's linear discriminant
Linear discriminant analysis (LDA) and the related Fisher's linear discriminant are methods used in statistics, pattern recognition and machine learning to find a linear combination of features which characterizes or separates two or more classes of objects or events. The resulting combination may be used as a linear classifier, or, more commonly, for dimensionality reduction before later classification.
Fisher's linear discriminant[edit]
The terms Fisher's linear discriminant and LDA are often used interchangeably, although Fisher's original article[1] actually describes a slightly different discriminant, which does not make some of the assumptions of LDA such as normally distributed classes or equal class covariances.
Suppose two classes of observations have means 
and covariances 
. Then the linear combination of features 
will have means 
and variances 
for 
. Fisher defined the separation between these two distributions to be the ratio of the variance between the classes to the variance within the classes:
This measure is, in some sense, a measure of the signal-to-noise ratio for the class labelling. It can be shown that the maximum separation occurs when
When the assumptions of LDA are satisfied, the above equation is equivalent to LDA.
Be sure to note that the vector 
is the normal to the discriminant hyperplane. As an example, in a two dimensional problem, the line that best divides the two groups is perpendicular to .
Generally, the data points to be discriminated are projected onto ; then the threshold that best separates the data is chosen from analysis of the one-dimensional distribution. There is no general rule for the threshold. However, if projections of points from both classes exhibit approximately the same distributions, a good choice would be the hyperplane between projections of the two means, 
and 
. In this case the parameter c in threshold condition 
can be found explicitly:
.