Length Reduction of Singular Spectrum Analysis With Guarantee Exact Perfect Reconstruction via Block Sliding Approach

The conventional singular spectrum analysis is to divide a signal into segments where there is only one non-overlapping point between two consecutive segments. By putting these segments into the columns of a matrix and performing the singular value decomposition on the matrix, various one dimensional singular spectrum analysis vectors are obtained. Since different one dimensional singular spectrum analysis vectors represent different parts of the signal such as the trend part, the oscillation part and the noise part of the signal, the singular spectrum analysis plays a very important role in the nonlinear and adaptive signal analysis. However, as the length of each one dimensional singular spectrum analysis vector is the same as that of the original signal, there is a redundancy among these one dimensional singular spectrum analysis vectors. In order to reduce the required computational power and the required units for the memory storage for performing the singular spectrum analysis, this article proposes a method to reduce the total number of the elements of all the one dimensional singular spectrum analysis vectors. In particular, the length of the shift block for generating the trajectory matrix is increased from one to a positive integer greater than one under a certain criterion. In this case, the total number of the columns of the trajectory matrix is reduced. As a result, the total number of the off-diagonals of all the two dimensional singular spectrum analysis matrices is reduced. Hence, the total number of the elements of all the one dimensional singular spectrum analysis vectors is reduced. In order to guarantee exact perfect reconstruction, this article reformulates the de-Hankelization process. In particular, the first element of the off-diagonals of all the two dimensional singular spectrum analysis matrices is taken as the elements of the one dimensional singular spectrum analysis vectors. Exact perfect reconstruction condition is derived. Simulations show that exact perfect reconstruction can be achieved while the total number of the elements of all the one dimensional singular spectrum analysis vectors is significantly reduced.