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var
, cov
and cor
compute the variance of x
and the covariance or correlation of x
and y
if these
are vectors. If x
and y
are matrices then the
covariances (or correlations) between the columns of x
and the
columns of y
are computed.
cov2cor
scales a covariance matrix into the corresponding
correlation matrix efficiently.
var(x, y = NULL, na.rm = FALSE, use)cov(x, y = NULL, use = "everything",
method = c("pearson", "kendall", "spearman"))
cor(x, y = NULL, use = "everything",
method = c("pearson", "kendall", "spearman"))
cov2cor(V)
a numeric vector, matrix or data frame.
NULL
(default) or a vector, matrix or data frame with
compatible dimensions to x
. The default is equivalent to
y = x
(but more efficient).
logical. Should missing values be removed?
an optional character string giving a
method for computing covariances in the presence
of missing values. This must be (an abbreviation of) one of the strings
"everything"
, "all.obs"
, "complete.obs"
,
"na.or.complete"
, or "pairwise.complete.obs"
.
a character string indicating which correlation
coefficient (or covariance) is to be computed. One of
"pearson"
(default), "kendall"
, or "spearman"
:
can be abbreviated.
symmetric numeric matrix, usually positive definite such as a covariance matrix.
For r <- cor(*, use = "all.obs")
, it is now guaranteed that
all(abs(r) <= 1)
.
For cov
and cor
one must either give a matrix or
data frame for x
or give both x
and y
.
The inputs must be numeric (as determined by is.numeric
:
logical values are also allowed for historical compatibility): the
"kendall"
and "spearman"
methods make sense for ordered
inputs but xtfrm
can be used to find a suitable prior
transformation to numbers.
var
is just another interface to cov
, where
na.rm
is used to determine the default for use
when that
is unspecified. If na.rm
is TRUE
then the complete
observations (rows) are used (use = "na.or.complete"
) to
compute the variance. Otherwise, by default use = "everything"
.
If use
is "everything"
, NA
s will
propagate conceptually, i.e., a resulting value will be NA
whenever one of its contributing observations is NA
.
If use
is "all.obs"
, then the presence of missing
observations will produce an error. If use
is
"complete.obs"
then missing values are handled by casewise
deletion (and if there are no complete cases, that gives an error).
"na.or.complete"
is the same unless there are no complete
cases, that gives NA
.
Finally, if use
has the value "pairwise.complete.obs"
then the correlation or covariance between each pair of variables is
computed using all complete pairs of observations on those variables.
This can result in covariance or correlation matrices which are not positive
semi-definite, as well as NA
entries if there are no complete
pairs for that pair of variables. For cov
and var
,
"pairwise.complete.obs"
only works with the "pearson"
method.
Note that (the equivalent of) var(double(0), use = *)
gives
NA
for use = "everything"
and "na.or.complete"
,
and gives an error in the other cases.
The denominator NA
when there is only one
observation (whereas S-PLUS has been returning NaN
).
For cor()
, if method
is "kendall"
or
"spearman"
, Kendall's cov()
, a non-Pearson method is unusual but available for
the sake of completeness. Note that "spearman"
basically
computes cor(R(x), R(y))
(or cov(., .)
) where R(u)
:= rank(u, na.last = "keep")
. In the case of missing values, the
ranks are calculated depending on the value of use
, either
based on complete observations, or based on pairwise completeness with
reranking for each pair.
When there are ties, Kendall's
Scaling a covariance matrix into a correlation one can be achieved in
many ways, mathematically most appealing by multiplication with a
diagonal matrix from left and right, or more efficiently by using
sweep(.., FUN = "/")
twice. The cov2cor
function
is even a bit more efficient, and provided mostly for didactical
reasons.
Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988). The New S Language. Wadsworth & Brooks/Cole.
Kendall, M. G. (1938). A new measure of rank correlation, Biometrika, 30, 81--93. 10.1093/biomet/30.1-2.81.
Kendall, M. G. (1945). The treatment of ties in rank problems. Biometrika, 33 239--251. 10.1093/biomet/33.3.239
cor.test
for confidence intervals (and tests).
cov.wt
for weighted covariance computation.
sd
for standard deviation (vectors).
# NOT RUN {
var(1:10) # 9.166667
var(1:5, 1:5) # 2.5
## Two simple vectors
cor(1:10, 2:11) # == 1
## Correlation Matrix of Multivariate sample:
(Cl <- cor(longley))
## Graphical Correlation Matrix:
symnum(Cl) # highly correlated
## Spearman's rho and Kendall's tau
symnum(clS <- cor(longley, method = "spearman"))
symnum(clK <- cor(longley, method = "kendall"))
## How much do they differ?
i <- lower.tri(Cl)
cor(cbind(P = Cl[i], S = clS[i], K = clK[i]))
## cov2cor() scales a covariance matrix by its diagonal
## to become the correlation matrix.
cov2cor # see the function definition {and learn ..}
stopifnot(all.equal(Cl, cov2cor(cov(longley))),
all.equal(cor(longley, method = "kendall"),
cov2cor(cov(longley, method = "kendall"))))
##--- Missing value treatment:
# }
# NOT RUN {
<!-- % "everything", "all.obs", "complete.obs", "na.or.complete", "pairwise.complete.obs" -->
# }
# NOT RUN {
C1 <- cov(swiss)
range(eigen(C1, only.values = TRUE)$values) # 6.19 1921
## swM := "swiss" with 3 "missing"s :
swM <- swiss
colnames(swM) <- abbreviate(colnames(swiss), min=6)
swM[1,2] <- swM[7,3] <- swM[25,5] <- NA # create 3 "missing"
## Consider all 5 "use" cases :
(C. <- cov(swM)) # use="everything" quite a few NA's in cov.matrix
try(cov(swM, use = "all")) # Error: missing obs...
C2 <- cov(swM, use = "complete")
stopifnot(identical(C2, cov(swM, use = "na.or.complete")))
range(eigen(C2, only.values = TRUE)$values) # 6.46 1930
C3 <- cov(swM, use = "pairwise")
range(eigen(C3, only.values = TRUE)$values) # 6.19 1938
## Kendall's tau doesn't change much:
symnum(Rc <- cor(swM, method = "kendall", use = "complete"))
symnum(Rp <- cor(swM, method = "kendall", use = "pairwise"))
symnum(R. <- cor(swiss, method = "kendall"))
## "pairwise" is closer componentwise,
summary(abs(c(1 - Rp/R.)))
summary(abs(c(1 - Rc/R.)))
## but "complete" is closer in Eigen space:
EV <- function(m) eigen(m, only.values=TRUE)$values
summary(abs(1 - EV(Rp)/EV(R.)) / abs(1 - EV(Rc)/EV(R.)))
# }
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