On One Algorithm in the Problem of Interpolation of Multidimensional Time Series

To reconstruct one of the missing values $t+\tau$ of time series $x(t)$ that is a realization of a multidimensional stationary process with discrete time, the authors propose the following linear interpolation form:
$$ \hat x(t+\tau)=\sum^p_{j+1}[\alpha_j x(t-1)+\beta_j x(t+T+j)], $$
where $\alpha_j$ and $\beta_j$ are numerical matrices to be determined; $T$ is the length of the omission $(0\leq\tau\leq T)$. A system of normal equations in $\alpha_j$ and $\beta_j$ is constructed in accordance with the least-squares principle. Provided that the real time series is modeled by an autoregression process of order $p$, the authors develop an iterative solution procedure for the system, written in terms of the forward and backward prediction coefficients, and demonstrate that the procedure converges for an extensive class of random processes.