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Compute the covariance matrix.
If each row of x and y is an observation, and each column is
a variable, then the (i, j)-th entry of
cov (x, y) is the covariance between the i-th
variable in x and the j-th variable in y.
cov (x) = 1/(N-1) * SUM_i (x(i) - mean(x)) * (y(i) - mean(y))
where N is the length of the x and y vectors.
If called with one argument, compute cov (x, x), the
covariance between the columns of x.
The argument opt determines the type of normalization to use. Valid values are
normalize with N-1, provides the best unbiased estimator of the covariance [default]
normalize with N, this provides the second moment around the mean
Compatibility Note:: Octave always treats rows of x and y
as multivariate random variables.
For two inputs, however, MATLAB treats x and y as two
univariate distributions regardless of their shapes, and will calculate
cov ([x(:), y(:)]) whenever the number of elements in
x and y are equal. This will result in a 2x2 matrix.
Code relying on MATLAB’s definition will need to be changed when
running in Octave.
See also: corr.
Compute matrix of correlation coefficients.
If each row of x and y is an observation and each column is
a variable, then the (i, j)-th entry of
corr (x, y) is the correlation between the
i-th variable in x and the j-th variable in y.
corr (x,y) = cov (x,y) / (std (x) * std (y))
If called with one argument, compute corr (x, x),
the correlation between the columns of x.
See also: cov.
Compute a matrix of correlation coefficients.
x is an array where each column contains a variable and each row is an observation.
If a second input y (of the same size as x) is given then calculate the correlation coefficients between x and y.
r is a matrix of Pearson’s product moment correlation coefficients for each pair of variables.
p is a matrix of pair-wise p-values testing for the null hypothesis of a correlation coefficient of zero.
lci and hci are matrices containing, respectively, the lower and higher bounds of the 95% confidence interval of each correlation coefficient.
param, value are pairs of optional parameters and values. Valid options are:
"alpha"Confidence level used for the definition of the bounds of the confidence interval, lci and hci. Default is 0.05, i.e., 95% confidence interval.
"rows"Determine processing of NaN values. Acceptable values are "all",
"complete", and "pairwise". Default is "all".
With "complete", only the rows without NaN values are considered.
With "pairwise", the selection of NaN-free rows is made for each
pair of variables.
Compute Spearman’s rank correlation coefficient rho.
For two data vectors x and y, Spearman’s rho is the correlation coefficient of the ranks of x and y.
If x and y are drawn from independent distributions,
rho
has zero mean and variance
1 / (N - 1),
where N is the length of the x and y vectors, and is
asymptotically normally distributed.
spearman (x) is equivalent to
spearman (x, x).
Compute Kendall’s tau.
For two data vectors x, y of common length N, Kendall’s tau is the correlation of the signs of all rank differences of x and y; i.e., if both x and y have distinct entries, then
1
tau = ------- SUM sign (q(i) - q(j)) * sign (r(i) - r(j))
N (N-1) i,j
in which the q(i) and r(i) are the ranks of x and y, respectively.
If x and y are drawn from independent distributions,
Kendall’s
tau
is asymptotically normal with mean 0 and variance
(2 * (2N+5)) / (9 * N * (N-1)).
kendall (x) is equivalent to kendall (x,
x).
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