Singular spectral analysis of periodic series with period L

[Adel Makram]

Let's have a time series with a period=L. Suppose we arbitrarily choose the window length of the trajectory matrix to be equal to L which is also equal to the period of a time series. Then the second column of the matrix will also start with the same entry as the first column, because all columns are of length L which is also equal to the period. But if we perform SVD of the matrix, we should get a reduced rank of 1, because all columns are alike. So what is the interpretation of that case?

 

[Adel Makram]

Ok, here is an attached image of the tarjectory matrix X, the column vector of length L which is the window length of the series. Now suppose that the time series that is represented by this matrix has a period which is just equal to the time between 2 successful Xs values. For example, the period of the time series is equal to the time between x1 and x2 which is also equal to the time between X2 and X3 and so forth ( sorry I mentioned, the period =L in the origial post). In other words, the time series has a constant value as a function of time if we only scan it with time intervals =the time difference between 2 successful Xs. Now the matrix surely degenerates into a rank one matrix on doing Singular Value Decomposition (SVD) operation. Then what is the interpretation of that case? And in general, what value of L should be used to grantee the non-reduction of the matrix into one rank?