### 26.4 Correlation and Regression Analysis

— Function File: cov (x)
— Function File: cov (x, opt)
— Function File: cov (x, y)
— Function File: 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))

> 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

> See also: corr.

— Function File: corr (x)
— Function File: 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.

— Function File: spearman (x)
— Function File: 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)`, and is asymptotically normally distributed.

> `spearman (`x`)` is equivalent to `spearman (`x`,` x`)`.

> See also: ranks, kendall.

— Function File: kendall (x)
— Function File: 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 * (2` n`+5)) / (9 *` n`* (`n`-1))`.

> `kendall (`x`)` is equivalent to `kendall (`x`,` x`)`.

> See also: ranks, spearman.

— Function File: [theta, beta, dev, dl, d2l, p] = logistic_regression (y, x, print, theta, beta)

> Perform ordinal logistic regression.

> Suppose y takes values in k ordered categories, and let `gamma_i (`x`)` be the cumulative probability that y falls in one of the first i categories given the covariate x. Then

>               [theta, beta] = logistic_regression (y, x)

> fits the model

>               logit (gamma_i (x)) = theta_i - beta' * x,   i = 1 ... k-1

> The number of ordinal categories, k, is taken to be the number of distinct values of `round (`y`)`. If k equals 2, y is binary and the model is ordinary logistic regression. The matrix x is assumed to have full column rank.

> Given y only, `theta = logistic_regression (y)` fits the model with baseline logit odds only.

> The full form is

>               [theta, beta, dev, dl, d2l, gamma]
>                  = logistic_regression (y, x, print, theta, beta)

> in which all output arguments and all input arguments except y are optional.

> Setting print to 1 requests summary information about the fitted model to be displayed. Setting print to 2 requests information about convergence at each iteration. Other values request no information to be displayed. The input arguments theta and beta give initial estimates for theta and beta.

> The returned value dev holds minus twice the log-likelihood.

> The returned values dl and d2l are the vector of first and the matrix of second derivatives of the log-likelihood with respect to theta and beta.

> p holds estimates for the conditional distribution of y given x.