An Introduction to ChemoSpec2D
This vignette is based upon ChemoSpec2D version
0.5.0.
Background
ChemoSpec2D is designed to analyze 2D spectroscopic data
such as COSY and HSQC NMR spectra using appropriate chemometric
techniques. It deploys methods aimed primarily at classification of
samples and the identification of spectral features which are important
in distinguishing samples from each other. ChemoSpec2D
stores and manipulates each spectrum as a matrix of data, and hence a
data set is a collection of 2D spectra. Thus the entire data set is
naturally visualized as a 3D array with dimensions:
F2 × F1 × no. samples
or
2D Spectrum × no. samples
where F2 and F1 are NMR-speak for the x- and y-axes/dimensions. We will refer to this array as $\underline{\mathbf{X}}$. See Figure @ref(fig:array).
Configuration of the data array. I, J and K are array indices; F2 and F1 are standard terms for NMR dimensions. The mode terminology is typically used in the PARAFAC literature. This structure is also sometimes referred to as a data cube.
ChemoSpec2D treats each spectrum/matrix as the unit of
observation, and thus the physical sample that went into the
spectrometer corresponds to the sample from a statistical perspective.
Keeping this natural unit intact during analysis is referred to as a
strong multi-way analysis. In a weak analysis, the 3D data set
is unfolded into a series of contiguous 2D matrices and analyzed using
methods suitable for any 2D data set (such methods are fundamentally
bilinear) (Huang et al. (2003)). In the weak approach, each
slice of a 2D spectrum becomes just another 1D spectrum, and the
relationship between the slices in a single 2D spectrum is lost. Oddly
enough, the trilinear/strong analysis has fewer parameters to estimate
so it is simpler, but computationally more demanding. The interpretation
is also more straight-forward. These techniques are akin to PCA, and
seek to reduce the data to a limited number components represented
by scores and loadings. As with PCA, noise in the data set is
reduced, and correlating variables are collapsed.
The literature on the topics of chemometric analysis of 3D data sets
uses a wide variety of terminology to describe analysis options, and the
same mathematical analysis can be found under different guises in
different fields. Here are some of the terms relevant to
ChemoSpec2D:
- Multivariate Image Analysis (MIA). This term is typically used when
the “images” are photographs for example, and the term covers a lot of
ground, such as finding particular objects within a photograph. While 2D
NMR spectra are typically plotted as contours, there is no reason why
they cannot be plotted as an image or heat plot, which is essentially a
photographic image. Classification using MIA methods involves a TUCKER1
analysis in which each image is the input, and the sample mode (only) is
reduced to a requested number of components.
ChemoSpec2Dcan carry out MIA. - PARAFAC. PARAFAC, or parallel factor analysis, is similar to a
method called TUCKER3 in which data reduction occurs on all three
dimensions and each dimension can have a different number of factors.
With PARAFAC, each dimension is reduced to the same number of
components.
ChemoSpec2Dcan carry out PARAFAC and it is discussed in greater detail below.
Keirs provides discussion and suggestions on terminology best practices (Kiers (2000)).
PARAFAC
Theory (Light)
PARAFAC is “parallel factor analysis.” This is a statistical technique that is somewhat analogous to PCA. Recall that PCA decomposes a 2D data set into scores and loadings, and is bilinear:
X(n × p) = C(n × R) × S(R × p) + E
Where X is the raw data, composed of n samples × p frequencies, C are the scores, and S are the loadings. R is the number of components selected. R is very much smaller than p, as noise and correlating variables have been reduced. Matrix C can be thought of as “concentrations” or weights. Matrix S is composed of “spectra” which serve as loadings. E consists of residuals/error. The goal of the PCA algorithm is to solve this equation and return C and S.
In comparison, PARAFAC decomposes a 3D data set into three matrices, and is trilinear. Because the data is 3D, standard matrix algebra cannot be applied. However, the math can be expressed as a summation:
$$ x_{ijk} = \sum_{r = 1}^{R} a_{ir}b_{jr}c_{kr} + \epsilon_{ijk} $$
Where xijk is an element of the 3D data array $\underline{\mathbf{X}}$. air is an element of the matrix A, and so forth for b/B and c/C. ϵ is the error term.
If $\underline{\mathbf{X}}$ is flattened by taking the K-dimension slices and concatenating them left-to-right to give a matrix X, then 2D matrix operations can provide a solution:
X = A(C ⊙ B)T + E
Here, ⊙ represents the Khatri-Rao product, a matrix multiplication variant needed in this situation. A, B and C are the component matrices as above (Bro and Smilde (2003), Smilde, Bro, and Geladi (2004) Appendix 4.A presents a number of alternative notations for PARAFAC). If you want to dig into the math, Kolda and Bader is the go-to source (Kolda and Bader (2009)).
Interpretation
Regardless of the mathematical representation or algorithmic
solution, the results provide A, B and C. Interpretation of the
component matrices depends upon how $\underline{\mathbf{X}}$ was constructed
(i.e. which dimension represents the samples). In the case of
ChemoSpec2D C contains values analogous
to scores in that they can be used to see how samples cluster (this is
because the samples are the third dimension of $\underline{\mathbf{X}}$). Standard matrix
multiplication of A × BT
for a particular column (component) gives a 2D loading plot (a
pseudo-spectrum) showing the contributions (loadings) of each peak to
the component. ChemoSpec2D uses the R package
multiway to carry out PARAFAC (Helwig (2019)).