Multivariate state estimation technique is described as follows:
$$X_e = D \cdot (D^T \otimes D)^{-1} \cdot (D^T \otimes X_o)$$
where $D$ is $n \times m$ process memory matrix, each column represents a monitoring state of the $n$ variables at a certain moment, and each row represents the monitoring states of a certain variable at $m$ moments
$D_{n \times m} = [X_{(t_1)}, X_{(t_2)}, ...X_{(t_m)}] = {\begin{bmatrix}x_1{(t_1)} & x_1{(t_2)} & ... & x1_{(t_m)}\\x_2{(t_1)} & x_2{(t_2)} &... & x_2{(t_m)}\\\vdots & \vdots & \vdots & \vdots\\x_n{(t_1)} & x_n{(t_2)} & ... & x_n{(t_m)}\end{bmatrix}}_{n \times m}$
supposed I have a $D$ matrix from this data:
\begin{array}{|c|c|c|c|} \hline Time& sensor1 & sensor2 & sensor3 & sensor4 \\ \hline 07:00 & 0.23 & 0.25 & 0.22 & 0.22 \\ \hline 07:10 & 0.21 & 0.23 & 0.24 & 0.21 \\ \hline 07:15 & 0.20 & 0.22 & 0.21 & 0.19 \\ \hline \end{array}
$D = \begin{bmatrix}0.23 & 0.21 & 0.20\\0.25 & 0.23 & 0.22\\0.22 & 0.24 & 0.21\\0.22 & 0.21 & 0.19\end{bmatrix}$
$D^T = \begin{bmatrix}0.23 & 0.25 & 0.22 & 0.22 \\0.21 & 0.23 & 0.24 & 0.21\\0.20 & 0.22 & 0.21 & 0.19\end{bmatrix}$
then how to calculate $D^T \otimes D$ ?
a paper said the $\otimes$ is nonlinear matrix operator (euclidean distance) :
$ \otimes(X,Y) = \sqrt{\sum_{i=1}^n (x_i-y_i)^2} $