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);