pcacov

Principal component analysis on covariance matrix

Description

example

coeff = pcacov(V) performs principal component analysis on the square covariance matrix V and returns the principal component coefficients, also known as loadings.

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

example

[coeff,latent] = pcacov(V) also returns a vector containing the principal component variances, meaning the eigenvalues of V.

example

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

Examples

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Create a covariance matrix from the hald dataset.

load hald
covx = cov(ingredients);

Perform principal component analysis on the covx variable.

[coeff,latent,explained] = pcacov(covx)
coeff = 4×4

   -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 = 4×1

  517.7969
   67.4964
   12.4054
    0.2372

explained = 4×1

   86.5974
   11.2882
    2.0747
    0.0397

The first component explains over 85% of the total variance. The first two components explain nearly 98% of the total variance.

Input Arguments

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Covariance matrix, specified as a square, symmetric, positive semidefinite matrix.

Data Types: single | double

Output Arguments

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Principal component coefficients, returned as a matrix the same size as V. Each column of coeff contains coefficients for one principal component. The columns are in order of decreasing component variance.

Principal component variances, returned as a vector with length equal to size(coeff,1). The vector latent contains the eigenvalues of V.

Percentage of the total variance explained by each principal component, returned as a vector the same size as latent. The entries in explained range from 0 (none of the variance is explained) to 100 (all of the variance is explained).

References

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

[2] Jolliffe, I. T. Principal Component Analysis. 2nd ed. New York: Springer-Verlag, 2002.

[3] Krzanowski, W. J. Principles of Multivariate Analysis: A User's Perspective. New York: Oxford University Press, 1988.

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

Extended Capabilities

Introduced before R2006a