nLinearCoeffs

Number of nonzero linear coefficients

Syntax

ncoeffs = nLinearCoeffs(obj)
ncoeffs = nLinearCoeffs(obj,delta)


Description

ncoeffs = nLinearCoeffs(obj) returns the number of nonzero linear coefficients in the linear discriminant model obj.

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

Input Arguments

obj	Discriminant analysis classifier, produced using fitcdiscr.
delta	Scalar or vector value of the Delta parameter. See Gamma and Delta.

Output Arguments

ncoeffs

Nonnegative integer, the number of nonzero coefficients in the discriminant analysis model obj. If you call nLinearCoeffs with a delta argument, ncoeffs is the number of nonzero linear coefficients for threshold parameter delta. If delta is a vector, ncoeffs is a vector with the same number of elements. If obj is a quadratic discriminant model, ncoeffs is the number of predictors in obj.

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

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.