Processing data, dimensionality reduction - linear discriminant analysis (linear discriminant analysis LDA)

Recommended to walk PCA this section. In the face of the following data types (data label for supervised), PCA can not find a good projection direction so that the maximum variance of the data after conversion.

So the introduction of LDA.

Defining an original data set is $ x = (x_ {1}, x_ {2} ,, x_ {n}, y_ {i}) $, m th

Sample, n features, i class label (The formula concise, i = 2)

Mean data set y1 u1, scatter matrix corresponding to Sl, Y ₂ is U ₂ , S2.          Hereinafter plus ~ represents a projection later.

purpose:

After the projection data within the maximum variance based , inter-class variance minimum

The definition of a parameter optimization requires:

$J=\frac{\left | \widetilde{\mu _{1}}-\widetilde{\mu _{2}} \right |^{2}}{\widetilde{S_{1}^{2}}+\widetilde{S_{2}^{2}}}$

(Complicated formula, photographs make up the numbers)