SVD module

This module provides the Singular Values Decomposition of the selected data. The main utility of SVD is to inform us on the complexity of the data matrix.

For more details about the method, see SVD in Wikipedia.

The left panel contains two control inputs:

• SVD parameters: the Dimension selector enables to select the number of singular values used to build figures in the Data vs. Model, Residuals and Contributions tabs of the right panel.

• Glitch removal in kinetics enables to remove spikes in the data from the visualization of singular vectors in tab Vectors

  • Level is the index of the target delay vector from which the spike is to be removed.

  • the Clean button removes the spike. The code masks the point with the largest absolute value in the selected vector.

  • the Cancel button cancels the last spike removal

The right panel contains a set of tabs covering different aspects of the results:

• Singular Values contains two figures

  • the spectrum of singular values (golden dots, dotted blue line) and a baseline of noise estimated from the largest singular values (violet dashed lines). The number of species that can be identified in the data is given by the index of the smallest singular value standing out of the noise.

  • the lack-of-fit spectrum, which gives the percentage of the signal that is not represented depending on the number of singular vectors used in the signal recomposition. One can appreciate on this graph how adding a new species improves the model. For large indexes, on gets the noise-to-signal ratio.

  Rq: except for ideal data matrices, there is always an ambiguity of plus or minus one species (at best) on the cutting level from both figures. One gets rather a clear indication on the largest decomposition that would not be acceptable.

• Vectors presents the wavelength-wise and delay-wise singular vectors. The idea here is to discard vectors that contain pure noise. Here again, the step from signal to noise is often not clearcut.

  Spikes in the data matrix can create artificial signal, and one can remove the spikes in the decay-wise vectors by using the Glitch removal in kinetics tool in the left panel.

• Data vs. Model shows the SVD data recomposition and the original matrix side-by-side. The recomposition is driven by the Dimension parameter entered in the left panel.

  This for illustration, but in order to appreciate the effects of Dimension on the quality of the model, it is better to focus on the nex tab: Residuals

• Residuals shows the difference between the data matrix and its reconstruction from SVD vectors, controlled by Dimension in the left panel.

• Contributions shows the individual components of the SVD reconstruction.

• Statistics provides a table of results with the singular values, the lack-of-fit and the standard deviation of the residuals, versus the number of singular vectors.