R: Principal Components Analysis

prcomp {stats}

R Documentation

Principal Components Analysis

Description

Performs a principal components analysis on the given data matrix and returns the results as an object of class prcomp.

Usage

prcomp(x, retx = TRUE, center = TRUE, scale. = FALSE, tol = NULL)

Arguments

`x`	a numeric or complex matrix (or data frame) which provides the data for the principal components analysis.
`retx`	a logical value indicating whether the rotated variables should be returned.
`center`	a logical value indicating whether the variables should be shifted to be zero centered. Alternately, a vector of length equal the number of columns of `x` can be supplied. The value is passed to `scale`.
`scale.`	a logical value indicating whether the variables should be scaled to have unit variance before the analysis takes place. The default is `FALSE` for consistency with S, but in general scaling is advisable. Alternatively, a vector of length equal the number of columns of `x` can be supplied. The value is passed to `scale`.
`tol`	a value indicating the magnitude below which components should be omitted. (Components are omitted if their standard deviations are less than or equal to `tol` times the standard deviation of the first component.) With the default null setting, no components are omitted. Other settings for tol could be `tol = 0` or `tol = sqrt(.Machine$double.eps)`, which would omit essentially constant components.

Details

The calculation is done by a singular value decomposition of the (centered and possibly scaled) data matrix, not by using eigen on the covariance matrix. This is generally the preferred method for numerical accuracy. The print method for the these objects prints the results in a nice format and the plot method produces a scree plot.

Value

prcomp returns an list with class "prcomp" containing the following components:

`sdev`	the standard deviations of the principal components (i.e., the square roots of the eigenvalues of the covariance/correlation matrix, though the calculation is actually done with the singular values of the data matrix).
`rotation`	the matrix of variable loadings (i.e., a matrix whose columns contain the eigenvectors). The function `princomp` returns this in the element `loadings`.
`x`	if `retx` is true the value of the rotated data (the centred (and scaled if requested) data multiplied by the `rotation` matrix) is returned.

Note

The signs of the columns of the rotation matrix are arbitrary, and so may differ between different programs for PCA, and even between different builds of R.

References

Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988) The New S Language. Wadsworth & Brooks/Cole.

Mardia, K. V., J. T. Kent, and J. M. Bibby (1979) Multivariate Analysis, London: Academic Press.

Venables, W. N. and B. D. Ripley (1997, 9) Modern Applied Statistics with S-PLUS, Springer-Verlag.

Examples

## the variances of the variables in the
## USArrests data vary by orders of magnitude, so scaling is appropriate
data(USArrests)
prcomp(USArrests)  # inappropriate
prcomp(USArrests, scale = TRUE)
plot(prcomp(USArrests))
summary(prcomp(USArrests, scale = TRUE))
biplot(prcomp(USArrests, scale = TRUE))

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