Definition:Autocovariance Matrix
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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 autocovariance matrix associated with $S$ for $\sequence {s_n}$ is:
- $\boldsymbol \Gamma_n = \begin {pmatrix} \gamma_0 & \gamma_1 & \gamma_2 & \cdots & \gamma_{n - 1} \\ \gamma_1 & \gamma_0 & \gamma_1 & \cdots & \gamma_{n - 2} \\ \gamma_2 & \gamma_1 & \gamma_0 & \cdots & \gamma_{n - 3} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ \gamma_{n - 1} & \gamma_{n - 2} & \gamma_{n - 3} & \cdots & \gamma_0 \end {pmatrix}$
where $\gamma_k$ is the autocovariance of $S$ at lag $k$.
That is, such that:
- $\sqbrk {\Gamma_n}_{i j} = \gamma_{\size {i - j} }$
Also see
- Results about autocovariance matrices can be found here.
Sources
- 1994: George E.P. Box, Gwilym M. Jenkins and Gregory C. Reinsel: Time Series Analysis: Forecasting and Control (3rd ed.) ... (previous) ... (next):
- Part $\text {I}$: Stochastic Models and their Forecasting:
- $2$: Autocorrelation Function and Spectrum of Stationary Processes:
- $2.1$ Autocorrelation Properties of Stationary Models:
- $2.1.3$ Positive Definiteness and the Autocovariance Matrix
- $2.1$ Autocorrelation Properties of Stationary Models:
- $2$: Autocorrelation Function and Spectrum of Stationary Processes:
- Part $\text {I}$: Stochastic Models and their Forecasting: