pcares

Residuals from principal component analysis

Syntax

residuals = pcares(X,NumComponents)

[residuals,reconstructed] = pcares(X)

Description

residuals = pcares(X,NumComponents) returns the residuals obtained by retaining NumComponents principal components of the data matrix X.

pcares does not normalize the columns of X. You can perform principal component analysis based on standardized variables using pcares(zscore(X),NumComponents). To perform principal component analysis directly on a covariance or correlation matrix, but without computing residuals, use pcacov.

[residuals,reconstructed] = pcares(X) additionally returns an approximation to X obtained by retaining its first NumComponents principal components. reconstructed is equal to X minus residuals.

Examples

Compute Principal Component Residuals

Load the hald data set, which measures the effect of cement composition on its hardening heat.

load hald

The matrix ingredients contains the percent composition of four chemicals present in the cement. Compute the residuals after retaining one, two, and three of the principal components, respectively.

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

Create a variable containing the residuals for the first row of the ingredients data.

firstrow = [r1(1,:); r2(1,:); r3(1,:)]

firstrow = 3×4

    2.0350    2.8304   -6.8378    3.0879
   -2.4037    2.6930   -1.6482    2.3425
    0.2008    0.1957    0.2045    0.1921

Display a bar graph of the residual magnitudes for each chemical.

bar(abs(firstrow))
xlabel("Number of PCA Components")
ylabel("Magnitude of Residual")
legend(["Tricalcium silicate" "Dicalcium silicate" ...
"Tricalcium aluminate" "Tetracalcium alumino ferrite"])

The graph shows that the residuals decrease as the number of principal components increases from one to three.

Compute Reconstructed Data from Principal Components

Load the cereal data set, which contains nutritional information for 77 cereals.

load cereal

Create a variable X containing the Potass, Fiber, Carbo, and Protein data, and remove any observations that contain zeros.

X = [Potass Fiber Carbo Protein];
X = X(~any(X==0,2),:);

Compute a reconstructed version of X using two principal components. Store the residuals in the variable r and the reconstructed data in the variable X2.

[r,X2] = pcares(X,2);

Compute the mean percentage difference between the reconstructed data and the original data.

meandiff = 100*mean((X2-X)./X)

meandiff = 1×4

   -0.1427    0.5480   -0.0683   10.5382

The function reconstructs the Potass, Fiber, and Carbo data well using two principal components, but the reconstructed Protein values are only within 10.5% of the original data.

Input Arguments

X — Input data

numeric matrix

Input data, specified as a numeric matrix. The rows of X contain observations, and the columns contain variables.

Data Types: single | double

NumComponents — Number of principal components to retain

positive integer

Number of principal components to retain, specified as a positive integer. NumComponents must be less than or equal to the number of columns in X.

Data Types: single | double

Output Arguments

residuals — Residuals

numeric matrix

Residuals, returned as a numeric matrix with the same dimensions as X.

reconstructed — Reconstructed observations

numeric matrix

Reconstructed observations, returned as a numeric matrix with the same dimensions as X. The reconstructed observations are an approximation to X obtained by retaining its first NumComponents principal components. reconstructed is equal to X minus residuals.

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: A User's Perspective. New York: Oxford University Press, 1988.

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

Version History

Introduced before R2006a