Definition:Autocorrelation Matrix

Definition
Let $S$ be a strictly stationary stochastic process giving rise to a time series $T$.

Let $\sequence {s_n}$ be a sequence of $n$ successive values of $T$:
 * $\sequence {s_n} = \tuple {z_1, z_2, \dotsb, z_n}$

The autocorrelation matrix associated with $S$ for $\sequence {s_n}$ is:


 * $\mathbf P_n = \begin {pmatrix}

1 & \rho_1 & \rho_2 & \cdots & \rho_{n - 1} \\ \rho_1 & 1 & \rho_1 & \cdots & \rho_{n - 2} \\ \rho_2 & \rho_1 & 1 & \cdots & \rho_{n - 3} \\ \vdots  & \vdots   & \vdots   & \ddots & \vdots \\ \rho_{n - 1} & \rho_{n - 2} & \rho_{n - 3} & \cdots & 1 \end {pmatrix}$

where $\rho_k$ is the autocorrelation of $S$ at lag $k$.

That is, such that:
 * $\sqbrk {P_n}_{i j} = \rho_{\size {i - j} }$

Also see

 * Definition:Autocovariance Matrix