Partial autocorrelation function: Difference between revisions

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Given a time series <math>z_t</math>, the partial autocorrelation of lag ''k'', denoted <math>\alpha(k)</math>, is the [[autocorrelation]] between <math>z_t</math> and <math>z_{t+k}</math> with the linear dependence of <math>z_t</math> on <math>z_{t+1}</math> through <math>z_{t+k-1}</math> removed; equivalently, it is the autocorrelation between <math>z_{t+1}</math> and <math>z_{t+k+1}</math> that is not accounted for by lags 1 to ''k''&nbsp;, inclusive.

: <math>\alpha(1) = \operatorname{Corcorr}(z_{2}, z_1),\text{ for }k= 1,</math>

: <math>\alpha(k) = \operatorname{Corcorr}(z_{t+k+1} - P_{t,k}(z_{t+k+1}),\, z_{t+1} - P_{t,k}(z_{t+1})),\text{ for }k\geq 2,</math>

where <math>P_{t,k}(x)</math> is surjective operator of orthogonal projection of <math>x</math> onto the linear subspace of Hilbert space spanned by <math> x_{t+1}, \dots, x_{t+k}</math>.