Durbin–Levinson forecast recursion

Input: An integer $m\ge 1$ and the lags $\gamma (0),\dots ,\gamma (m)$ of the autocovariance function of a weakly stationary real process. Set $\gamma (-r)=\gamma (r)$. Require every leading covariance matrix ${\mathrm{\Gamma }}_k=(\gamma (i-j){)}_{i,j=1}^k$ to be positive definite for $1\le k\le m$.

Output: Coefficient vectors ${\varphi }_k=({\varphi }_{k,1},\dots ,{\varphi }_{k,k}{)}^T$ and error-variance values ${\nu }_k$ for $1\le k\le m$.

Instructions: Set ${\nu }_0=\gamma (0)$, ${\kappa }_1=\gamma (1)\mathrm{/}{\nu }_0$, ${\varphi }_{1,1}={\kappa }_1$, and ${\nu }_1={\nu }_0(1-{\kappa }_1^2)$. For each $k=2,\dots ,m$, set ${\kappa }_k=[\gamma (k)-{\sum }_{j=1}^{k-1}{\varphi }_{k-1,j}\gamma (k-j)]\mathrm{/}{\nu }_{k-1}$, ${\varphi }_{k,j}={\varphi }_{k-1,j}-{\kappa }_k{\varphi }_{k-1,k-j}$ for $1\le j<k$, ${\varphi }_{k,k}={\kappa }_k$, and ${\nu }_k={\nu }_{k-1}(1-{\kappa }_k^2)$. Return all vectors and variances. The positive-definite leading-block assumption makes each division well-defined; this is proved in Durbin–Levinson solves the successive prediction equations ↗. The procedure operates on covariance input and makes no Gaussian or conditional-expectation claim.