# SVD is not applicable to time series analysis?

I am studying a deconvolution problem for which observations include noise. I am considering using one of several common regularization methods, including Tikhonov's solution, truncated Singular Value Decomposition (SVD), or Wiener filter.

A manuscript I am reading, Riel and Berkhout, "Resolution in seismic trace inversion by parameter estimation" in Geophysics, 50(9), pp. 1440-1455, 1985, compares analysis in the Fourier domain with general SVD analysis:

Examples where SVD-based techniques may be used while Fourier techniques are no longer applicable are: time-invariant filtering, space-variant wave field extrapolation [...], and linear parameter estimation problems. Furthermore, in many nonlinear problems linearization is meaningful, again allowing application of the SVD. The nonlinear problem treated is such an example. In our opinion many problem areas in seismic exploration stand to benefit from an analysis involving the SVD. However, there are some limitations compared to Fourier analysis. For instance, SVD is not applicable to time-series analysis. Furthermore, the eigenvectors and eigenvalues do not, in general, have the clear physical meaning the sinusoids and their amplitudes in Fourier analysis do. For these reasons SVD-based methods should, when possible, not replace Fourier analysis methods, but rather be used in conjunction with them.

I was surprised to read that "SVD is not applicable to time-series analysis." It is not clear to me that this is the case, and indeed the authors use SVD to construct an inverse filter for their convoluted time series.

I wonder if there is common knowledge in the DSP community regarding SVD's unsuitability for time series analysis in general or specific aspects, as this is the first time I have come across this statement. (It is also surprising to find no SVD or eigenanalysis tag.) Thanks in advance.

• Maybe it has a different meaning when taken in context, but I would agree; the SVD can be used in certain time series analysis applications. – Jason R Dec 13 '13 at 12:42
• Thanks @Jason, I added more context around that quote, though not sure if changes the meaning or provides any justification for the statement. I guess a question that remains for me is where SVD is deficient in comparison to Fourier analysis, except in the interpretability of the orthogonal basis and coefficients. – hatmatrix Dec 13 '13 at 13:23
• Computing the SVD of the covariance matrix of time series is the basis of well-known parametric methods like ESPRIT or MUSIC. So I disagree that the SVD has no applications for time-series analysis. – pichenettes Dec 13 '13 at 13:49

## 1 Answer

SVD of multi-variate time-series is of course very common. The point I think the author is making is that SVD cannot meaningfully be directly applied to a univariate time-series, instead it is commonly used indirectly to explore the main eigenvalues of a transition matrix, estimated from observed trajectories of a certain length. see http://en.wikipedia.org/wiki/Singular_spectrum_analysis . SSA assumes the time-series is stationary and can be modeled as Markov.

• Ah, so it means that you cannot extract eigenvalues from univariate time series in the way that you can get a frequency spectrum. But for deconvolution problems, the SVD is applied to the deconvolution matrix (as is the Fourier matrix) so it is indirectly applicable. This interpretation makes sense! The blurb on the relation between SSA and Fourier analysis was separately helpful. (Sorry, it seems I do not have enough points to upvote you also.) – hatmatrix Dec 13 '13 at 17:16
• But why cant is be used directly to a univariate time-series? Also, what -exactly- is meant by univariate? Does that mean from one sensor only? – TheGrapeBeyond Dec 13 '13 at 19:48
• Univariate time series only has one variable (I guess "single sensor" would be one interpretation) so the time series is a vector and not a matrix. So you could construct an autocorrelation matrix, but not a variance-covariance matrix, in my understanding. – hatmatrix Dec 15 '13 at 2:01
• but singular spectrum analysis could be used also for non stationary time series – dato datuashvili Apr 13 '14 at 10:17