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Principal component analysis on covariance matrix


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


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 pca.

[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.


load hald
covx = cov(ingredients);
[COEFF,latent,explained] = pcacov(covx)
  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 =

explained =

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Tall Array Support

pcacov and factoran do not work directly on tall arrays. Instead, use C = gather(cov(X)) to compute the covariance matrix of a tall array. Then, you can use pcacov or factoran on the in-memory covariance matrix. Alternatively, you can use pca directly on a tall array.

For more information, see Tall Arrays.


[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.

Introduced before R2006a

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