Definition: Sample covariance matrix
Index:
The Book of Statistical Proofs ▷
General Theorems ▷
Probability theory ▷
Covariance ▷
Sample covariance matrix
Sources:
Metadata: ID: D153 | shortcut: covmat-samp | author: JoramSoch | date: 2021-05-20, 07:46.
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 covariance matrix of $x$ is given by
\[\label{eq:covmat-samp} \hat{\Sigma} = \frac{1}{n} \sum_{i=1}^{n} (x_i - \bar{x}) (x_i - \bar{x})^\mathrm{T}\]and the unbiased sample covariance matrix of $x$ is given by
\[\label{eq:covmat-samp-unb} S = \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x}) (x_i - \bar{x})^\mathrm{T}\]where $\bar{x}$ is the sample mean.
- Wikipedia (2021): "Sample mean and covariance"; in: Wikipedia, the free encyclopedia, retrieved on 2020-05-20; URL: https://en.wikipedia.org/wiki/Sample_mean_and_covariance#Definition_of_sample_covariance.
Metadata: ID: D153 | shortcut: covmat-samp | author: JoramSoch | date: 2021-05-20, 07:46.