pcares

Residuals from principal component analysis

Syntax

residuals = pcares(X,ndim)
[residuals,reconstructed] = pcares(X,ndim)

Description

residuals = pcares(X,ndim) returns the residuals obtained by retaining ndim principal components of the n-by-p matrix X. Rows of X correspond to observations, columns to variables. ndim is a scalar and must be less than or equal to p. residuals is a matrix of the same size as X. Use the data matrix, not the covariance matrix, with this function.

pcares does not normalize the columns of X. To perform the principal components analysis based on standardized variables, that is, based on correlations, use pcares(zscore(X), ndim). You can perform principal components analysis directly on a covariance or correlation matrix, but without constructing residuals, by using pcacov.

[residuals,reconstructed] = pcares(X,ndim) returns the reconstructed observations; that is, the approximation to X obtained by retaining its first ndim principal components.

Examples

This example shows the drop in the residuals from the first row of the Hald data as the number of component dimensions increases from one to three.

r1 = pcares(ingredients,1);
r2 = pcares(ingredients,2);
r3 = pcares(ingredients,3);

r11 = r1(1,:)
r11 =
2.0350  2.8304  -6.8378  3.0879

r21 = r2(1,:)
r21 =
-2.4037  2.6930  -1.6482  2.3425

r31 = r3(1,:)
r31 =
0.2008  0.1957  0.2045  0.1921

References

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

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

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

 Seber, G. A. F. Multivariate Observations. Hoboken, NJ: John Wiley & Sons, Inc., 1984.