Sample Covariance Matrix

Suppose \(X_1, \dots, X_p\) are random variables and \(X:=(X_1, \dots, X_p)\) is the random vector of said random variables. Given a sample \(\mathbf x_1, \dots \mathbf x_N\) of \(X\), how do we obtain an estimate for the covariance matrix \(\text{Cov}(X)\)?

To this end, let \(\mathbf X=( \mathbf X_{ij})\) denote the \(N \times p\) data matrix whose \(k\)th row is \(\mathbf x_k\) and let \(\bar x_i\) denote the sample mean of \(X_i\). That is, \(\bar x_i\) is the mean of the \(i\)th column of \(\mathbf X\). Then the sample covariance matrix \(Q = (Q_{ij})\) is defined by

\[\begin{equation*} Q_{ij} := \frac{1}{N-1} \sum_{k=1}^{N} (\mathbf X_{ki} - \bar x_i)(\mathbf X_{kj} - \bar x_j). \end{equation*}\]

We can write this a bit more compactly if we introduce yet more notation. Let \(\bar{\mathbf x}:=(\bar x_1, \dots, \bar x_p)\) denote the sample mean vector and \(M\) the \(N \times p\) matrix with all rows equal to \(\bar{\mathbf x}\). The matrix \(N:=\mathbf X - M\) is the data matrix that has been centered about the sample means, and we quickly see that

\[\begin{equation*} Q = N^T N. \end{equation*}\]