02 MSSA

MSSA. Background.

Construction of a trajectory matrix

The first step in the MSSA is the same as for the SSA. One starts with the construction of the trajectory matrix. Although the construction procedure is quite similar, one should be aware of the size of the trajectory matrix. Let $s$ denote the number of the given time series, $N$ the length of each of $s$ time series, $L$ - window size and $2 \leq L \leq N/2$ and $L \in N$, $K=N-L+1$ The trajectory matrix for the MSSA results in a $L \times Ks$ matrix $X$. We simply stack the trajectory matrices $X^i$ for each of $s$ time series. The columns of the trajectory matrix $X^i$ for $i$-th time series comprise lagged versions of $i$ time series just like in SSA.

The $X^i$ matrices can stacked in a vertical or horizontal direction, however due to performance reasons for SVD, horizontal direction is often preferred.

Thus, each matrix $X^i$ for $i=0,1,2,...,s$ can be represented as follows:

$$X^i = \begin{bmatrix} f^i_0 & f^i_1 &f^i_2 & f^i_3 &\dots & f^i_{N-L}\\\\ f^i_1 & f^i_2 &f^i_3 & f^i_4 &\dots & f^i_{N-L+1}\\\\ f^i_2 & f^i_3 &f^i_4 & f^i_5 &\dots & f^i_{N-L+2}\\\\ \vdots & \vdots &\vdots & \vdots &\vdots & \vdots\\\\ f^i_{L-1}& f_{L} &f^i_{L+1} & f^i_{L+2} &\dots & f^i_{N-1} \end{bmatrix}$$

It is clear that the elements of the anti-diagonals are equal. This a Hankel matrix.

Finally, we obtain the resulting trajectory matrix $X$ as

$$X = \begin{bmatrix} X^0 & X^1 & X^2 & ... & X^s \end{bmatrix}$$

Eigentriple grouping and elementary matrix separation

Time Series Reconstruction

Forecasting