Index: The Book of Statistical Proofs ▷ General Theorems ▷ Probability theory ▷ Correlation ▷ Sample correlation matrix

Definition: Let $x = \left\lbrace x_1, \ldots, x_n \right\rbrace$ be a sample from a random vector $X \in \mathbb{R}^{p \times 1}$. Then, the sample correlation matrix of $x$ is the matrix whose entries are the sample correlation coefficients between pairs of entries of $x_1, \ldots, x_n$:

\[\label{eq:corrmat-samp-v1} \mathrm{R}_{xx} = \begin{bmatrix} r_{x^{(1)},x^{(1)}} & \ldots & r_{x^{(1)},x^{(n)}} \\ \vdots & \ddots & \vdots \\ r_{x^{(n)},x^{(1)}} & \ldots & r_{x^{(n)},x^{(n)}} \end{bmatrix}\]

where the $r_{x^{(j)},x^{(k)}}$ is the sample correlation between the $j$-th and the $k$-th entry of $X$ given by

\[\label{eq:corrmat-samp-v2} r_{x^{(j)},x^{(k)}} = \frac{\sum_{i=1}^n (x_{ij}-\bar{x}^{(j)}) (x_{ik}-\bar{x}^{(k)})}{\sqrt{\sum_{i=1}^n (x_{ij}-\bar{x}^{(j)})^2} \sqrt{\sum_{i=1}^n (x_{ik}-\bar{x}^{(k)})^2}}\]

in which $\bar{x}^{(j)}$ and $\bar{x}^{(k)}$ are the sample means

\[\label{eq:mean-samp} \begin{split} \bar{x}^{(j)} &= \frac{1}{n} \sum_{i=1}^n x_{ij} \\ \bar{x}^{(k)} &= \frac{1}{n} \sum_{i=1}^n x_{ik} \; . \end{split}\]
 
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Metadata: ID: D169 | shortcut: corrmat-samp | author: JoramSoch | date: 2021-12-14, 07:45.