DESCRIPTION
OF N-PLS METHOD
History
Multilinear PLS (nPLS) comes as a straightforward extension to the more established PLS model [1] with non orthogonal loadings, as introduced by Martens & Nęs [2].
Its initial formulation was made by Bro [3,4]. The actual computation of the model, and specifically of the regression coefficient underwent some improvements thanks to various contributions in more recent years by Smilde AK[5] and De Jong S [6]. In particular the latter showed how the model can be computed equivalently even when the deflation step is removed from the algorithm.
The initial PARAFAC-like structure of the model of
X has been
recently [7] changed into a Tucker structure. I.e., for the three-way case:
where the W identifies the weights in the third (K) and second (J) mode
G is the core and
E are the residuals. (IxJK)
refers to the way the array is matricised [4]; namely as a (two-way) matrix with
I rows and JK columns. This modification improved the prediction
capability on the X array, in the calibration phase as well as for
independent sets of data.
This model has been
successfully employed in many fields in particular in the field of analytical
chemistry as well as batch process monitoring [8,9].
Aim:
The main purpose is to fit a multivariate (namely multi-way) calibration model
on a multi-way array X of dimensions I x J x K
x ...
to predict the one or more variables present in a Y array or
matrix of dimensions I x L x M x ...
In the nPLS1 case, such as
for the model implemented in CuBatch, L and M are equal to 1 and
therefore Y is a vector (thus, henceforth, identified with y).
For a three-way array X and a predicted variable y, the
model is then stated as:
where W are the weights (the letters J and K refer to the mode), G is the
matricised core. Ex are the residuals for X, B is the matrix with
the internal regression coefficients and eY are the
residuals for the y vector.
Criterion:
While for PARAFAC and other decomposition models, the aim is to maximise the
captured variance (that is, to minimise the distance between the data and the
model used to describe them), in the case of multilinear PLS1 the objective
function is to maximise the covariance between the scores t for one
component (as they are extracted one at a time) and the
y not yet accounted for.
For a three way X and a predicted variable
y the problem to solve is then the following:
Algorithm:
Originally [3,4] the algorithm required deflation on the X after
each component was fitted, modifications introduced by de Jong removed this
necessity and, albeit the components are still extracted one at a time,
deflation is applied to the y.
For the three-way case the algorithm is the following
| i. | f = 1, e = y |
| ii |
 |
| iii. | reshape e into Z of
dimensions (J x K) |
| iv. | compute
and
(i.e. the f-th column of the weights)
as the left and right first singular vectors of Z |
| v. |
 |
| vi. | compute the regression coefficients |
| vii. | update the residuals |
| viii. | Repeat from ii. until the desired number of components have been extracted |
| ix. | Compute the core: where + is the Moore-Penrose inverse |
If X is four or more dimensional step iii., v. and ix. are straightforwardly extended to the new case
E.g. for four-ways X
iii. v. ix. The weights vectors are computed (step iv.) as loading vectors of a 1 component PARAFAC model computed on the Z array:
where all weights are subsequently normalised to length 1
Code:
The code for nPLS is the one from the N-way toolbox [10] version (2.1) of
which is included in the software.
Updates (but CuBatch is not
guaranteed to work with them) can be downloaded at
www.models.kvl.dk
Applications:
Multivariate calibration, batch process monitoring.
Dataset reference:
Fluorescence.mat
References:
[1] Wold S et al, Siam Journal on
Scientific and Statistical Computing, 5 (1984),
735-743
[2] Martens H, Nęs T, "Methods for calibration" in Multivariate
Calibration, John Wiley & Sons, Chicester, 1989
[3] Bro R, Journal of Chemometrics, 10 (1996), 47-61
[4] Bro R, PhD dissertation, University of Amsterdam (1998)
[5] Smilde AK, Journal of Chemometrics, 11 (1997), 367-377
[6] de Jong S, Journal of Chemometrics, 12 (1998), 77-81
[7] Bro R et al, Chemometrics and Intelligent Laboratory Systems
58 (2001), 3-13
[8] Bro R, Heimdal H, Chemometrics and Intelligent Laboratory Systems
34 (1996), 85-102
[9] Gurden SP et al, Chemometrics and Intelligent Laboratory Systems
59 (2001), 121-136
[10] Andersson CA, Bro R,
Chemometrics and Intelligent Laboratory Systems, 52 (2000), 1-4
