26.4 Correlation and Regression Analysis

: cov (x)

: cov (x, opt)

: cov (x, y)

: cov (x, y, opt)

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

0:

normalize with N-1, provides the best unbiased estimator of the covariance [default]

1:

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.

: corr (x)

: corr (x, y)

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.

: r = corrcoef (x)

: r = corrcoef (x, y)

: r = corrcoef (…, param, value, …)

: [r, p] = corrcoef (…)

: [r, p, lci, hci] = corrcoef (…)

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.

param, value are optional pairs of parameters and values which modify the calculation. Valid options are:

"alpha"

Confidence level used for 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.

Output r is a matrix of Pearson’s product moment correlation coefficients for each pair of variables.

Output p is a matrix of pair-wise p-values testing for the null hypothesis of a correlation coefficient of zero.

Outputs lci and hci are matrices containing, respectively, the lower and higher bounds of the 95% confidence interval of each correlation coefficient.

See also: corr, cov.

: spearman (x)

: spearman (x, y)

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).

See also: ranks, kendall.

: kendall (x)

: kendall (x, y)

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).

See also: ranks, spearman.