Statistical correlation

In probability theory and statistics, correlation, also called correlation coefficient, indicates both strength and direction of the relationship between two variables. It is a numeric measure of the strength of linear relationship between two random variables. In general statistical usage, correlation or co-relation refers to the departure of two variables from independence. In this broad sense there are several coefficients, measuring the degree of correlation, adapted to the nature of data.

A number of different coefficients are used for different situations. The best known is the Pearson product-moment correlation coefficient, which is obtained by dividing the covariance of the two variables by the product of their standard deviations. Despite its name, it was first introduced by Francis Galton.

Mathematical properties
The correlation &rho;X, Y between two random variables X and Y with expected values &mu;X and &mu;Y and standard deviations &sigma;X and &sigma;Y is defined as:



\rho_{X,Y}={\mathrm{cov}(X,Y) \over \sigma_X \sigma_Y} ={E((X-\mu_X)(Y-\mu_Y)) \over \sigma_X\sigma_Y}.$$

Since &mu;X = E(X), &sigma;X2 = E(X2) &minus; E2(X) and likewise for Y, we may also write


 * $$\rho_{X,Y}=\frac{E(XY)-E(X)E(Y)}{\sqrt{E(X^2)-E^2(X)}~\sqrt{E(Y^2)-E^2(Y)}}$$

The correlation is defined only if both standard deviations are finite and both of them are nonzero. It is a corollary of the Cauchy-Schwarz inequality that the correlation cannot exceed 1 in absolute value.

The correlation is 1 in the case of an increasing linear relationship, &minus;1 in the case of a decreasing linear relationship, and some value in between in all other cases, indicating the degree of linear dependence between the variables. The closer the coefficient is to either &minus;1 or 1, the stronger the correlation between the variables.

If the variables are independent then the correlation is 0, but the converse is not true because the correlation coefficient detects only linear dependencies between two variables. Here is an example: Suppose the random variable X is uniformly distributed on the interval from &minus;1 to 1, and Y = X2. Then Y is completely determined by X, so that X and Y are dependent, but their correlation is zero; they are uncorrelated. However, in the special case when X and Y are jointly normal, independence is equivalent to uncorrelatedness.

The sample correlation
If we have a series of n measurements of X  and Y  written as xi  and yi  where i = 1, 2, ..., n, then the Pearson product-moment correlation coefficient can be used to estimate the correlation of X  and Y. The Pearson coefficient is also known as the "sample correlation coefficient". It is especially important if X and Y  are both normally distributed. The Pearson correlation coefficient is then the best estimate of the correlation of X and Y. The Pearson correlation coefficient is written:



r_{xy}=\frac{\sum (x_i-\bar{x})(y_i-\bar{y})}{(n-1) s_x s_y} $$

where $$\bar{x}$$ and $$\bar{y}$$ are the sample means of xi and yi, sx  and sy  are the sample standard deviations of xi  and yi  and the sum is from i = 1 to  n. As with the population correlation, we may rewrite this as



r_{xy}=\frac{n\sum x_iy_i-\sum x_i\sum y_i} {\sqrt{n\sum x_i^2-(\sum x_i)^2}~\sqrt{n\sum y_i^2-(\sum y_i)^2}}. $$

Again, as is true with the population correlation, the absolute value of the sample correlation must be less than or equal to 1.

The sample correlation coefficient is the fraction of the variance in yi that is accounted for by a linear fit of xi  to yi. This is written


 * $$r_{xy}^2=1-\frac{\sigma_{y|x}^2}{\sigma_y^2}$$

where &sigma;y2 is the square of the error of a linear fit of yi  to xi  by the equation y = a + bx.


 * $$\sigma_{y|x}^2=\sum_{i=1}^n (y_i-a-bx_i)^2$$

and &sigma;y2 is just the variance of y


 * $$\sigma_y^2=\sum_{i=1}^n (y_i-\bar{y})^2$$

Note that since the sample correlation coefficient is symmetric in xi and yi, we will get the same value for a fit of xi  to yi :


 * $$r_{xy}^2=1-\frac{\sigma_{x|y}^2}{\sigma_x^2}$$

This equation also gives an intuitive idea of the correlation coefficient for higher dimensions. Just as the above described sample correlation coefficient is the fraction of variance accounted for by the fit of a 1-dimensional linear submanifold to a set of 2-dimensional vectors (xi, yi ), so we can define a correlation coefficient for a fit of an m-dimensional linear submanifold to a set of n-dimensional vectors. For example, if we fit a plane z = a + bx + cy to a set of data (xi, yi , zi ) then the correlation coefficient of z  to x  and y  is


 * $$r^2=1-\frac{\sigma_{z|xy}^2}{\sigma_z^2}.\,$$

Non-parametric correlation coefficients
Pearson's correlation coefficient is a parametric statistic, and it may be less useful if the underlying assumption of normality is violated. Non-parametric correlation methods, such as Spearman's &rho; and Kendall's &tau; may be useful when distributions are not normal; they are a little less powerful than parametric methods if the assumptions underlying the latter are met, but are less likely to give distorted results when the assumptions fail.

Other measures of dependence among random variables
To get a measure for more general dependencies in the data (also nonlinear) it is better to use the correlation ratio which is able to detect almost any functional dependency, or mutual information which detects even more general dependencies.

Copulas and correlation
Most people erroneously believe that the information given by a correlation coefficient is enough to define the dependence structure between random variables. But to fully capture the dependence between random variables we must consider the copula between them. The correlation coefficient completely defines the dependence structure only in very particular cases, for example when the cumulative distribution functions are elliptic (as with, for example, the multivariate normal distribution).

Correlation matrices
The correlation matrix of n random variables X1, ..., Xn is the n &times;  n matrix whose i,j entry is corr(Xi, Xj). If the measures of correlation used are product-moment coefficients, the correlation matrix is the same as the covariance matrix of the standardized random variables Xi /SD(Xi) for i = 1, ..., n. Consequently it is necessarily a non-negative definite matrix.

The correlation matrix is symmetrical (the correlation between $$X_i$$ and $$X_j$$ is the same as the correlation between $$X_j$$ and $$X_i$$).

"Correlation does not imply causation"
The conventional dictum that "correlation does not imply causation" is treated in the article titled spurious relationship. See also correlation implies causation (logical fallacy). However, correlations are not presumed to be acausal, though the causes may not be known.