Performs a principal components analysis on the given data matrix
and returns the results as an object of class prcomp.
prcomp(x, …)# S3 method for formula
prcomp(formula, data = NULL, subset, na.action, …)
# S3 method for default
prcomp(x, retx = TRUE, center = TRUE, scale. = FALSE,
tol = NULL, rank. = NULL, …)
# S3 method for prcomp
predict(object, newdata, …)
a formula with no response variable, referring only to numeric variables.
an optional data frame (or similar: see
model.frame) containing the variables in the
formula formula. By default the variables are taken from
environment(formula).
an optional vector used to select rows (observations) of the
data matrix x.
arguments passed to or from other methods. If x is
a formula one might specify scale. or tol.
a numeric or complex matrix (or data frame) which provides the data for the principal components analysis.
a logical value indicating whether the rotated variables should be returned.
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.
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.
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 (unless rank. is specified
less than min(dim(x)).). Other settings for tol could be
tol = 0 or tol = sqrt(.Machine$double.eps), which
would omit essentially constant components.
optionally, a number specifying the maximal rank, i.e.,
maximal number of principal components to be used. Can be set as
alternative or in addition to tol, useful notably when the
desired rank is considerably smaller than the dimensions of the matrix.
object of class inheriting from "prcomp"
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.
prcomp returns a list with class "prcomp"
containing the following components:
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).
the matrix of variable loadings (i.e., a matrix
whose columns contain the eigenvectors). The function
princomp returns this in the element loadings.
if retx is true the value of the rotated data (the
centred (and scaled if requested) data multiplied by the
rotation matrix) is returned. Hence, cov(x) is the
diagonal matrix diag(sdev^2). For the formula method,
napredict() is applied to handle the treatment of values
omitted by the na.action.
the centering and scaling used, or FALSE.
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.
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.
# NOT RUN {
C <- chol(S <- toeplitz(.9 ^ (0:31))) # Cov.matrix and its root
all.equal(S, crossprod(C))
set.seed(17)
X <- matrix(rnorm(32000), 1000, 32)
Z <- X %*% C ## ==> cov(Z) ~= C'C = S
all.equal(cov(Z), S, tol = 0.08)
pZ <- prcomp(Z, tol = 0.1)
summary(pZ) # only ~14 PCs (out of 32)
## or choose only 3 PCs more directly:
pz3 <- prcomp(Z, rank. = 3)
summary(pz3) # same numbers as the first 3 above
stopifnot(ncol(pZ$rotation) == 14, ncol(pz3$rotation) == 3,
all.equal(pz3$sdev, pZ$sdev, tol = 1e-15)) # exactly equal typically
# }
# NOT RUN {
## 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))
# }