prcomp
Principal Components Analysis
Performs a principal components analysis on the given data matrix
and returns the results as an object of class prcomp
.
- Keywords
- multivariate
Usage
prcomp(x, ...)## S3 method for class 'formula':
prcomp(formula, data = NULL, subset, na.action, \dots)
## S3 method for class 'default':
prcomp(x, retx = TRUE, center = TRUE, scale. = FALSE,
tol = NULL, \dots)
## S3 method for class 'prcomp':
predict(object, newdata, \dots)
Arguments
- formula
- a formula with no response variable, referring only to numeric variables.
- data
- an optional data frame (or similar: see
model.frame
) containing the variables in the formulaformula
. By default the variables are taken fromenvironment(formula)
. - subset
- an optional vector used to select rows (observations) of the
data matrix
x
. - na.action
- a function which indicates what should happen
when the data contain
NA
s. The default is set by thena.action
setting ofoptions
, and isna.fail
if that is unset. Thefactory-fresh default isna.omit
. - ...
- arguments passed to or from other methods. If
x
is a formula one might specifyscale.
ortol
. - 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 toscale
. - 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 ofx
can be supplied. The value is passed toscale
. - 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 betol = 0
ortol = sqrt(.Machine$double.eps)
, which would omit essentially constant components. - object
- Object of class inheriting from
"prcomp"
- newdata
- An optional data frame or matrix in which to look for
variables with which to predict. If omitted, the scores are used.
If the original fit used a formula or a data frame or a matrix with
column names,
newdata
must contain columns with the same names. Otherwise it must contain the same number of columns, to be used in the same order.
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 these objects prints the results in a nice
format and the plot
method produces a scree plot.
Unlike princomp
, variances are computed with the usual
divisor $N - 1$.
Note that scale = TRUE
cannot be used if there are zero or
constant (for center = TRUE
) variables.
Value
prcomp
returns a 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 elementloadings
.x if retx
is true the value of the rotated data (the centred (and scaled if requested) data multiplied by therotation
matrix) is returned. Hence,cov(x)
is the diagonal matrixdiag(sdev^2)
. For the formula method,napredict()
is applied to handle the treatment of values omitted by thena.action
.center, scale the centering and scaling used, or FALSE
.
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.
concept
PCA
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 (2002) Modern Applied Statistics with S, Springer-Verlag.
See Also
Examples
library(stats)
## signs are random
require(graphics)
## the variances of the variables in the
## USArrests data vary by orders of magnitude, so scaling is appropriate
prcomp(USArrests) # inappropriate
prcomp(USArrests, scale = TRUE)
prcomp(~ Murder + Assault + Rape, data = USArrests, scale = TRUE)
plot(prcomp(USArrests))
summary(prcomp(USArrests, scale = TRUE))
biplot(prcomp(USArrests, scale = TRUE))