2.4.2 Singular spectrum analysis

With SSA, the time series is first embedded into a p-dimensional space known as the trajectory matrix. Singular value decomposition is then applied to decompose the trajectory matrix into a sum of elementary matrices [32–34], each of which is associated with a process mode.

Subsequently, the elementary matrices that contribute to the norm of the original matrix are grouped, with each group giving an approximation of the original matrix. Finally, the smoothed approximations or modes of the time series are recovered by diagonal averaging of the elementary matrices obtained from decomposing the trajectory matrix. Although SSA is a linear method, it can readily be extended to nonlinear forms, such as kernel-based SSA or SSA with autoassociative neural networks. Nonetheless, it has not been used widely in statistical process monitoring as yet, although some studies have provided promising results [2, 35, 36].

Table 1 gives a summary of multiscale methods that have been considered in process monitoring schemes over the last two decades.
