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 $$\widehat{X}$$ be the centered data (the data *X* minus
the mean by class). Define

The regularized covariance matrix $$\tilde{\Sigma}$$ is

Whenever *γ* ≥ `MinGamma`

, $$\tilde{\Sigma}$$ is nonsingular.

Let *μ*_{k} 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 $$\tilde{C}$$ 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 $$\left[{\tilde{C}}^{-1}{D}^{-1/2}\left({\mu}_{k}-{\mu}_{0}\right)\right]$$ 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`

.