Multivariate Sample Properties

Correlation in Paired Sample

Sample correlation coefficient \(r\) measures the strength and direction of the linear relationship between two paired samples \(X = (X_1, X_2, \ldots, X_n)\) and \(Y = (Y_1, Y_2, \ldots, Y_n)\). Note that the order of components is important and the sample cannot be independently permuted if the correlation is of interest. The correlation coefficient between \(X\) and \(Y\) can be computed as:

\[ r = \frac{\sum^n_{i = 1} (X_i - \bar{X}) (Y_i - \bar{Y})}{\sqrt{\sum^n_{i = 1} (X_i - \bar{X})^2 \cdot \sum^n_{i = 1}(Y_i - \bar{Y})^2}}. \]

The sample covariance is defined as:

\[ \begin{align} s_{XY} &= \frac{1}{n - 1} \sum^n_{i = 1} (X_i - \bar{X}) (Y_i - \bar{Y}) \\ &= \frac{1}{n - 1} \left( \sum^n_{i = 1} X_i Y_i - n \bar{X} \bar{Y} \right). \end{align} \]

The correlation coefficient can be expressed as:

\[ r = \frac{s_{XY}}{s_X s_Y}, \]

where \(s_X\) and \(s_Y\) are sample standard deviations of samples \(X\) and \(Y\).

The correlation ranges between -1 and 1, which are two ideal cases of decreasing and increasing linear trends. Zero correlation does not, in general, imply independence but signfies the lack of any linear relationship between samples.