als: alternating least squares multivariate curve resolution (MCR-ALS)

Description

This is an implementation of alternating least squares multivariate curve resolution (MCR-ALS). Given a dataset in matrix form d1, the dataset is decomposed as d1=C %*% t(S) where the columns of C and S represent components contributing to the data in each of the 2-ways that the matrix is resolved. In forming the decomposition, the components in each way many be constrained with e.g., non-negativity, uni-modality, selectivity, normalization of S and closure of C. Note that if more than one dataset is to be analyzed simultaneously, then the matrix S is assumed to be the same for every dataset in the bilinear decomposition of each dataset into matrices C and S.

Usage

als(CList, PsiList, S=matrix(), WList=list(),
thresh =.001, maxiter=100, forcemaxiter = FALSE,
optS1st=TRUE, x=1:nrow(CList[[1]]), x2=1:nrow(S),
baseline=FALSE, fixed=vector("list", length(PsiList)),
uniC=FALSE, uniS=FALSE, nonnegC = TRUE, nonnegS = TRUE,
normS=0, closureC=list())

Value

A list with components:

CList: A list with the same length as the number of datasets, containing the optimized matrix C at termination scaled by the optimized amplitudes for that dataset from AList.
S: The matrix S given as input.
rss: The residual sum of squares at termination.
resid: A list with the same length as the number of datasets, containing the residual matrix for each dataset
iter: The number of iterations performed before termination.

Arguments

CList: list with the same length as PsiList where each element is a matrix of dimension m by comp and represents the matrix C for each dataset
PsiList: list of datasets, where each dataset is a matrix of dimension m by n
S: matrix with n rows and comp columns, often representing (mass) spectra
WList: An optional list with the same length as PsiList, where each element is a matrix of dimension m by n giving the weight of that datapoint; note that if closure or normalization constraints are applied, then both are applied after the application of weights.
thresh: numeric value that defaults to .001; if ((oldrss - rss) / oldrss) < thresh then the optimization stops, where oldrss is the residual sum of squares at iteration x-1 and rss is the residual sum of squares at iteration x
maxiter: The maximum number of iterations to perform (where an iteration is optimization of either AList and C)
forcemaxiter: Logical indicating whether maxiter iterations should be performed even if the residual difference drops below thresh.
optS1st: logical indicating whether the first constrained least squares regression should estimate S or CList.
x: optional vector of labels for the rows of C, which are used in the application of unimodality constraints.
x2: optional vector of labels for the rows of S, which are used in the application of unimodality constraints.
baseline: logical indicating whether a baseline component is present; if baseline=TRUE then this component is exempt from constraints unimodality or non-negativity
fixed: list with the same length as PsiList in which each element is a vector of the indices of the components to fix to zero in each dataset
nonnegS: logical indicating whether the components (columns) of the matrix S should be constrained to non-negative values
nonnegC: logical indicating whether the components (columns) of the matrix C should be constrained to non-negative values
uniC: logical indicating whether unimodality constraints should be applied to the columns of C
uniS: logical indicating whether unimodality constraints should be applied to the columns of S
normS: numeric indicating whether the spectra are normalized; if normS>0, the spectra are normalized. If normS==1 the maximum of the spectrum of each component is constrained to be equal to one; if normS > 0 && normS!=1 then the norm of the spectrum of each component is constrained to be equal to one.
closureC: list; if the length is zero, then no closure constraints are applied. If the length is not zero, it should be equal to the number of datasets in the analysis, and contain numeric vectors consisting of the desired value of the sum of each row of the concentration matrix.

References

Garrido M, Rius FX, Larrechi MS. Multivariate curve resolution alternating least squares (MCR-ALS) applied to spectroscopic data from monitoring chemical reactions processes. Journal Analytical and Bioanalytical Chemistry 2008; 390:2059-2066.

Jonsson P, Johansson A, Gullberg J, Trygg J, A J, Grung B, Marklund S, Sjostrom M, Antti H, Moritz T. High-throughput data analysis for detecting and identifying differences between samples in GC/MS-based metabolomic analyses. Analytical Chemistry 2005; 77:5635-5642.

Tauler R. Multivariate curve resolution applied to second order data. Chemometrics and Intelligent Laboratory Systems 1995; 30:133-146.

Tauler R, Smilde A, Kowalski B. Selectivity, local rank, three-way data analysis and ambiguity in multivariate curve resolution. Journal of Chemometrics 1995; 9:31-58.

Examples

Run this code

## load 2 matrix datasets into variables d1 and d2
## load starting values for elution profiles
## into variables Cstart1 and Cstart2
## load time labels as x, m/z values as x2
data(multiex)

## starting values for elution profiles
matplot(x,Cstart1,type="l")
matplot(x,Cstart2,type="l",add=TRUE)

## using MCR-ALS, improve estimates for mass spectra S and the two
## matrices of elution profiles
## apply unimodality constraints to the elution profile estimates
## note that the starting estimates for S just contain a dummy matrix

test0 <- als(CList=list(Cstart1,Cstart2),S=matrix(1,nrow=400,ncol=2),
PsiList=list(d1,d2), x=x, x2=x2, uniC=TRUE, normS=0)

## plot the estimated mass spectra 
plotS(test0$S,x2)

## the known mass spectra are contained in the variable S
## can compare the matching factor of each estimated spectrum to
## that in S
matchFactor(S[,1],test0$S[,1])
matchFactor(S[,2],test0$S[,2])
 
## plot the estimated elution profiles
## this shows the relative abundance of the 2nd component is low 
matplot(x,test0$CList[[1]],type="l")
matplot(x,test0$CList[[2]],type="l",add=TRUE)