ALS module

The Alternated Least Squares (ALS) algorithms factorizes the data matrix given a number of species and some constraints (e.g., positivity of kinetics…).

The left panel contains the controls, and the right panel displays the outputs.

Controls

• Dimension is the number of species. This should be consistent with the results of the SVD analysis.

• Max # Iter. is the maximal number of iterations allowed before stopping the optimizer.

• Log convergence threshold controls the logarithm of the stopping convergence threshold of the optimizer.

• The Run button starts the ALS optimization.

Several tabs enable to fine tune the ALS analysis:

• Options tab

  • Initialization enables to select a starting point

    • |SVD| takes the absolute values of the SVD vectors

    • NMF takes the solution of a Non-negative Matrix Factorization algorithm

    • Sequential performs a series of ALS optimizations with increasing dimension

    • Restart enables to restart from a previous run. It does not work if Dimension is changed.

  • Use SVD-filtered matrix uses the noise filtering ability of thee SVD reconstruction. The matrix is computed with the dimension specified in the SVD tab.

  • Opt S first start by optimizing the spectra vectors, instead of the kinetics vectors by default.

• S const. tab: constraints on the spectra vectors

  • S > 0: positivity constraint

  • S unimodal: unimodality constraint

  • Normalize: normalize the spectra:

    • SUM(S) = 1 normalizes the area of the spectra. The default is to normalize the intensities.

  • Smooth: a smoothing factor to get less noisy spectra, used as the span parameter in the loess function.

  • Fix spectral shape(s) Select a .csv file with spectra to be constrained. The file should be in the same format as the data files (separator, decimal mark) and contain a header with the names of the species. The first column should contain the wavelength. It is possible to load several files. Example: (https://github.com/ppernot/SK-Ana/blob/master/data/spectrum_ABC_Kinet.csv)

    The spectra are interpolated on the wavl grid of the data matrix. By default, the spectra are used as such (hard constraint) which is often too strong and results in poor solutions. Activating Soft constraint enables to input a weight for the similarity constraint in the loss function of the ALS. The logWeight slider enables to tune this weight.

• C const. tab: constraints on the kinetics vectors

  • C > 0: positivity constraint

  • Closure: imposes the conservation of matter by normalizing the sum of the kinetics to 1 (at each delay).

    Warning: This works only for unimolecular processes and could be in conflict with normalization constrains on the spectra.

  • Presence matrix enables to specify the occurrence of individual species in different experiments when datasets have been delay-tiled. By default the matrix is filled with ones (1). Put 0 where a species is not expected to occur. When finished, press Done.

Outputs

• Alternated Least Squares tab

  This shows the convergence message of the ALS code.

  Tip: For a succesful fit, the lack-of-fit should be versy close to the lack-of-fit statistics of a SVD with the same dimension.

• Diagnostics tab

  This gives access to several results:

  • Data vs. Model whichh compares side by side the best fit model to the data matrix

  • Residualswhich shows the residuals map and an histogram of the residuals compared to the histogram of the data.

  • SVD of residuals which provides the Singular Values Decomposition of the residuals matrix.In the ideal case, all vectors should be featureless and appear as pure noise. A normal Q-Q plot is provided to assess the normality of the residuals distribution.

• Kinetics and Spectra tab

  This tab provides zoomable plots of the spectra and the associated kinetics, identified by color code. The data can be saved to disk.

• Contributions tab

  This tab shows the contribution matrix of each species with its weight.

• Ambiguity tab

  In most cases, the ALS decomposition is not unique and subject to rotational ambiguity. If one transforms/combines the spectra, the inverse transformation applied to the kinetics will leave their combination unchanged. The range of eligible transformations is limited by the varoius constraints on the ALS solutions (positivity…).

  The algorithm performs a brute force exploration of transformation matrices and might require a long time to finish. It returns a subset of valid spectra and kinetics, from which one can appreciate the level of ambiguity and/or search for better behaved solutions than the ones returned by the ALS.

  Several controls are available:

  • Pick 2 or 3 vectors: according to the dimension of the ALS decomposition, one has to choose a set of vectors. This might be the full set for dimensions 2 and 3, but the algorithm does not allow explorations of more than three-vectors transformations.

  • Relative positivity threshold: because of the noise in the data, one has to enable some level on non-positivity in the transformed vectors. The slider enables to pick a level.

  • Exploration step: the step amplitude for the exploration of the transformation matrix elements. The smaller the better and more accurate, but very small steps might incur very long calculations.

  • Start: click to start process when all other parameters have been chosen.

  • Stop: early stop of the process (it works sometimes…)

  • Save: save the subset of transformed spectra and kinetics to disk