nLinearCoeffs - Number of nonzero linear coefficients in discriminant analysis

  classifier - MATLAB ([original](http://www.mathworks.com/help/stats/classificationdiscriminant.nlinearcoeffs.html)) ([raw](?raw))

Regularization is the process of finding a small set of predictors that yield an effective predictive model. For linear discriminant analysis, there are two parameters, γ and δ, that control regularization as follows. cvshrink helps you select appropriate values of the parameters.

Let Σ represent the covariance matrix of the data X, and let X^ be the centered data (the data X minus the mean by class). Define

The regularized covariance matrix Σ˜ is

Whenever γ ≥ MinGamma, Σ˜ is nonsingular.

Let μk be the mean vector for those elements of X in class k, and let _μ_0 be the global mean vector (the mean of the rows of X). Let C be the correlation matrix of the data X, and let C˜ be the regularized correlation matrix:

where I is the identity matrix.

The linear term in the regularized discriminant analysis classifier for a data point x is

The parameter δ enters into this equation as a threshold on the final term in square brackets. Each component of the vector [C˜−1D−1/2(μk−μ0)] is set to zero if it is smaller in magnitude than the threshold δ. Therefore, for class k, if component j is thresholded to zero, component j of x does not enter into the evaluation of the posterior probability.

The DeltaPredictor property is a vector related to this threshold. When δ ≥ DeltaPredictor(i), all classes k have

Therefore, when δ ≥ DeltaPredictor(i), the regularized classifier does not use predictor i.