Recursive relations for multistep prediction of a stationary time series (Q2744932)

For a zero mean weakly stationary time series \(X_n\) with covariance \(\gamma_n\) the coefficients \(a^h_{n,i}\) of best (in error variance) linear prediction for \(h\) steps are considered. I.e., the forecast of \(X_{n+h}\) is \(\hat X_{n+h}=\sum_{i=1}^n a^h_{n,i}X_{n+1-i}\). The author derives some recursive relations for \(a^h_{n,i}\) and the variance of the error \(v_n^h\), e.g. \(a^h_{n,i}=a^h_{n-1,i}-a^h_{n,n}a^1_{n-1,n-i}\) for \(i=1\),\dots,\(n-1\) and NEWLINE\[NEWLINE a^h_{n,n}=\left[ \gamma_{n+h-1}-\sum_{i=1}^{n-1} a_{n-1,i}^1\gamma_{n+h-i-1} \right] (v^1_{n-1})^{-1}, NEWLINE\]NEWLINE \(v_n^h=v^h_{n-1}-(a^h_{n,n})^2v^1_{n-1}\), with \(v^h_0=\gamma_0\). It is proposed to use these relations in speech recognition algorithms based on multistep predictors.