Statistics Toolbox™

pcacov - Principal component analysis on covariance matrix

Syntax

COEFF = pcacov(V)
[COEFF,latent] = pcacov(V)
[COEFF,latent,explained] = pcacov(V)

Description

COEFF = pcacov(V) performs principal components analysis on the p-by-p covariance matrix V and returns the principal component coefficients, also known as loadings. COEFF is a p-by-p matrix, with each column containing coefficients for one principal component. The columns are in order of decreasing component variance.

pcacov does not standardize V to have unit variances. To perform principal components analysis on standardized variables, use the correlation matrix R = V./(SD*SD')), where SD = sqrt(diag(V)), in place of V. To perform principal components analysis directly on the data matrix, use princomp.

[COEFF,latent] = pcacov(V) returns latent, a vector containing the principal component variances, that is, the eigenvalues of V.

[COEFF,latent,explained] = pcacov(V) returns explained, a vector containing the percentage of the total variance explained by each principal component.

Example

load hald
covx = cov(ingredients);
[COEFF,latent,explained] = pcacov(covx)
COEFF =
  0.0678 -0.6460  0.5673 -0.5062
  0.6785 -0.0200 -0.5440 -0.4933
 -0.0290  0.7553  0.4036 -0.5156
 -0.7309 -0.1085 -0.4684 -0.4844

latent =
  517.7969
  67.4964
  12.4054
  0.2372

explained =
  86.5974
  11.2882
  2.0747
  0.0397

References

[1] Jackson, J. E., A User's Guide to Principal Components, John Wiley and Sons, 1991.

[2] Jolliffe, I. T., Principal Component Analysis, 2nd Edition, Springer, 2002.

[3] Krzanowski, W. J., Principles of Multivariate Analysis, Oxford University Press, 1988.

[4] Seber, G. A. F., Multivariate Observations, Wiley, 1984.

See Also

barttest, biplot, factoran, pcares, princomp , rotatefactors

partialcorr

pcares