The covariance of every pair of columns of a numeric matrix or data frame, computed by the compiled core. Covariances describe how variables move together; the matrix is the input to principal components, Mahalanobis distances and multivariate drift checks. Column names are carried through to both dimensions of the result.
Value
A symmetric ncol(x) by ncol(x) numeric matrix, with the
column names of x (if any) as both row and column names; entry
units are the product of the two columns' units.
Details
Entry \((j, k)\) is \(\frac{1}{n - 1} \sum_i (x_{ij} - \bar
x_j)(x_{ik} - \bar x_k)\), the n - 1 denominator of
stats::cov(), with which it agrees to rounding. The
diagonal holds the column variances.
Missing values are not removed: an NA in a column makes every covariance
involving that column NA (pairwise or complete-case deletion is not done;
use stats::cov() with use = for those). A single
row gives NaN throughout. Refusals: a vector that is not a matrix
("must be a matrix or data frame") and a data frame with a non-numeric
column ("every column of x must be numeric"). Logical and integer matrices
are converted to double. The cost is \(O(n p^2)\) for \(n\) rows and
\(p\) columns.
Examples
X <- cbind(a = c(1, 2, 3, 4), b = c(2, 4, 7, 8), c = c(5, 3, 2, 1))
ccovm(X)
#> a b c
#> a 1.666667 3.500000 -2.166667
#> b 3.500000 7.583333 -4.583333
#> c -2.166667 -4.583333 2.916667
# Agrees with stats::cov().
all.equal(ccovm(X), stats::cov(X))
#> [1] TRUE
# The diagonal is the column variances.
all.equal(diag(ccovm(X)), apply(X, 2, stats::var))
#> [1] TRUE
# Data frames are accepted.
ccovm(data.frame(u = 1:5, v = c(2, 1, 4, 3, 6)))
#> u v
#> u 2.5 2.5
#> v 2.5 3.7
# A plain vector is not a matrix of variables.
try(ccovm(1:5))
#> Error : `x` must be a matrix or data frame