nLinearCoeffs

Number of nonzero linear coefficients in discriminant analysis classifier

Syntax

ncoeffs = nLinearCoeffs(Mdl)

ncoeffs = nLinearCoeffs(Mdl,delta)

Description

ncoeffs = nLinearCoeffs(Mdl) returns the number of nonzero linear coefficients in the linear discriminant analysis classifier Mdl.

ncoeffs = nLinearCoeffs(Mdl,delta) returns the number of nonzero linear coefficients for the threshold parameter delta.

Examples

Find the Number of Nonzero Coefficients in a Discriminant Analysis Classifier

Find the number of nonzero coefficients in a discriminant analysis classifier for various Delta values.

Create a discriminant analysis classifier from the fishseriris data.

load fisheriris
obj = fitcdiscr(meas,species);

Find the number of nonzero coefficients in obj.

ncoeffs = nLinearCoeffs(obj)

ncoeffs = 
4

Find the number of nonzero coefficients for delta = 1, 2, 4, and 8.

delta = [1 2 4 8];
ncoeffs = nLinearCoeffs(obj,delta)

ncoeffs = 4×1

     4
     4
     3
     0

The DeltaPredictor property gives the values of delta where the number of nonzero coefficients changes.

ncoeffs2 = nLinearCoeffs(obj,obj.DeltaPredictor)

ncoeffs2 = 4×1

     4
     3
     1
     2

Input Arguments

Mdl — Trained discriminant analysis classifier

ClassificationDiscriminant model object | CompactClassificationDiscriminant model object

Trained discriminant analysis classifier, specified as a ClassificationDiscriminant model object trained with fitcdiscr, or a CompactClassificationDiscriminant model object created with compact.

delta — Threshold

numeric scalar | numeric vector

Threshold, specified as a numeric scalar or numeric vector. See Gamma and Delta.

Example: delta = [1 2 3]

Data Types: single | double

Output Arguments

ncoeffs — Number of nonzero coefficients in discriminant analysis classifier

nonnegative integer

Number of nonzero coefficients in the discriminant analysis classifier Mdl, returned as a nonnegative integer.

If you call nLinearCoeffs with the delta argument, ncoeffs is the number of nonzero linear coefficients for the threshold parameter delta. If delta is a vector, ncoeffs is a vector with the same number of elements.

If Mdl is a quadratic discriminant analysis classifier, ncoeffs is the number of predictors in Mdl.

More About

Gamma and Delta

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

$$D=\text{diag}\left({\widehat{X}}^T*\widehat{X}\right).$$

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

$$\tilde{\Sigma }=\left(1-\gamma \right)\Sigma +\gamma D.$$

Whenever γ ≥ MinGamma, $$\tilde{\Sigma }$$ is nonsingular.

Let μ_k be the mean vector for those elements of X in class k, and let μ₀ 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:

$$\tilde{C}=\left(1-\gamma \right)C+\gamma I,$$

where I is the identity matrix.

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

$${\left(x-{\mu }_0\right)}^T{\tilde{\Sigma }}^{-1}\left({\mu }_k-{\mu }_0\right)=\left[{\left(x-{\mu }_0\right)}^T{D}^{-1/2}\right]\left[{\tilde{C}}^{-1}{D}^{-1/2}\left({\mu }_k-{\mu }_0\right)\right].$$

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

$$\left|{\tilde{C}}^{-1}{D}^{-1/2}\left({\mu }_k-{\mu }_0\right)\right|\le \delta .$$

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

Version History

Introduced in R2011b