Als

Purpose

Alternating Least Squares computational engine for multivariate curve resolution (MCR).

Synopsis

[c,s] = als(x,c0,options);

Description

ALS decomposes a matrix X as CS such that X = CS + E where E is minimized in a least squares sense.

Inputs are the matrix to be decomposed x (size m by n), and the initial guess c0. If c0 is size m by k, where k is the number of factors, then it is assumed to be the initial guess for C. If c0 is size k by n then it is assumed to be the initial guess for S (If m=n then, c0 is assumed to be the initial guess for C).

An optional input options is described below.

The outputs are the estimated matrix c (m by k) and s (k by n). Usually c is a matrix of contributionss and s is a matrix of spectra. The function

[c,s] = als(x,c0)

will decompose x using an non-negatively constrained alternating least squares calculation. To include other constraints, use the options described below.

Note that if no non-zero equality constraints are imposed on a factor the spectra are normalized to unit length. This can lead to significant scaling differences between factors that have non-zero equality constraints and those that do not.

Options

• display: [ 'off' | {'on'} ] governs level of display to command window,

• plots: [ 'none' | {'final'} ] governs level of plotting,

• ccon: [ 'none' | 'reset' | {'fastnnls'} ] non-negativity on contributionss, (fastnnls = true least-squares solution)

• scon: [ 'none' | 'reset' | {'fastnnls'} ] non-negativity on spectra, (fastnnls = true least-squares solution)

• cc: [ ] contributions equality constraints, must be a matrix with M rows and up to K columns with NaN where equality constraints are not applied and real value of the constraint where they are applied. If fewer than K columns are supplied, the missing columns will be filled in as unconstrained,

• ccwts: [inf] a scalar value or a 1xK vector with elements corresponding to weightings on constraints (0, no constraint, 0<wt<inf imposes constraint "softly", and inf is hard constrained). If a scalar value is passed for ccwts, that value is applied for all K factors,

• sc: [ ] spectra equality constraints, must be a matrix with N columns and up to K rows with NaN where equality contraints are not applied and real value of the constraint where they are applied. If fewer than K rows are supplied, the missing rows will be filled in as unconstrained.

• scwts: [inf] weighting for spectral equality constraints (see ccwts)

• sclc: [ ] contributions scale axis, vector with M elements otherwise 1:M is used,

• scls: [ ] spectra scale axis, vector with N elements otherwise 1:N is used,

• condition: [{'none'}| 'norm' ] type of conditioning to perform on S and C before each regression step. 'norm' conditions each spectrum or contributions to its own norm. Conditioning can help stabilize the regression when factors are significantly different in magnitude.

• tolc: [ {1e-5} ] tolerance on non-negativity for contributionss,

• tols: [ {1e-5} ] tolerance on non-negativity for spectra,

• ittol: [ {1e-8} ] convergence tolerance,

• itmax: [ {100} ] maximum number of iterations,

• timemax: [ {3600} ] maximum time for iterations,

• rankfail: [ 'drop' |{'reset'}| 'random' | 'fail' ] how are rank deficiencies handled:

• drop - drop deficient components from model

• reset - reset deficient components to initial guess

• random - replace deficient components with random vector

• fail - stop analysis, give error

Examples

To decompose a matrix x without non-negativity constraints use:

options = als('options');

options.ccon = 'none';

options.scon = 'none';

[c,s] = als(x,c0,options);

The following shows an example of using soft-constraints on the second spectral component of a three-component solution assuming that the variable softs contains the spectrum to which component two should be constrained.

[m,n] = size(x);

options = als('options');

options.sc = NaN\*ones(3,n); %all 3 unconstrained

options.sc(2,:) = softs; %constrain component 2

options.scwts = 0.5; %consider as 1/2 of total signal in X

[c,s] = als(x,c0,options);

See Also

mcr, parafac, pca