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 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, …)
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. The ‘factory-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 (unlessrank.
is specified less thanmin(dim(x))
.). Other settings for tol could betol = 0
ortol = sqrt(.Machine$double.eps)
, which would omit essentially constant components.- rank.
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
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:
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
.
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 (2002) Modern Applied Statistics with S, Springer-Verlag.
See Also
Examples
library(stats)
# 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))
# }