Three-way PCA

Three-way PCA allows a much easier interpretation of the information contained in the data set, since it directly takes into account its three-way structure. If the TuckerS model is applied, the hnal result is given by three sets of loadings together with a core array describing the relationship among them. If the number of components is the same for each way, the core array is a cube. Each of the three sets of loadings can be displayed and interpreted in the same way as a score plot of standard PCA. 

An example of application of three-way PCA is a data set from the Venice lagoon. In it, 11 chemical variables (chlorophyll-a, total suspended matter, water transparency, fluorescence, turbidity, suspended solids, NH4, NOj", P, COD, and BOD5) have been measured monthly in 13 sampling sites (see Figure 5.5) during the period May 1987 to December 1990, for a total of 44 months. 

Table 5.7 shows some types of data sets on which three-way PCA can be successfully apphed. 

FIGURE 5.6 The results of the three-way PCA applied to the Venice data set. (a) Plot of the loadings of the variables (b) plot of the loadings of the sampling sites (c) plot of the loadings of the months (d) plot of the loadings of the month on axis 1 versus time. 

The above examples of PCA applications deal with a two-way data analysis (subjects and variables). A three-way PCA (subjects, variables, and conditions) has shown that trace metal concentrations can be used to classify healthy and diseased blue crabs [72]. 

The results of the three-way PCA applied to the Venice data set. a) plot of the loadings of the variables. 

The simplest versions are unfolding methods where the multiway array is unfolded to a matrix. In above example the objects measured at different time points can be arranged as blocks of rows in a single matrix. Usual decomposition methods like PCA can then be applied. Of course, in this case the three-way nature of the data is ignored. 

Figure 6.10. Illustration of an unfolding of a three-way table, X, into a two-way table. In the amphetamine example, the three- way table has 10 rows, 4 columns, and 6 sheets of columns, which gives a 10 rows and 24 columns large two-way table after unfolding. The weight table, W, is also unfolded to form the usual row vector, w. After applying the PLS or PCA algorithm to the data, both X and w...

FIGURE 11.11 Bilinear models for three-way data, unfolded PCA and unfolded MCR. 

Both PCA and MCR-ALS can be easily extended to complex data arrays ordered in more than two ways or modes, giving three-way data arrays (data cubes or parallelepipeds) or multiway data arrays. In PCA and MCR-ALS, the multiway data set is unfolded prior to data analysis to give an augmented two-way data matrix. After analysis is complete, the resolved two-way profiles can be regrouped to recover the profiles in the three modes. The current state of the art in multiway data analysis includes, however, other methods where the structure of the multiway data array is explicitly built into the model and fixed during the resolution process. Among these... 

Barbieri, P., Andersson, C.A., Massart, D.L., Predozani, S., Adami, G., and Reisen-hofer, E., Modeling bio-geochemical interactions in the surface waters of the Gulf of Trieste by three-way principal component analysis (PCA), Anal. Chim. Acta, 398, 227-235, 1999. 

Visual inspection of the estimated factors is not to be trusted in the presence of degenerate factors, which occur when two or more factors are collinear in one or more of the three ways. When this is the case, in the concentration way or Z-way, the PARAFAC model is still valid, but the rotational uniqueness of the X-way and Y-way protiles of the degenerate factors is lost. This often results in estimated protiles that are hard to interpret. If the collinearity occurs in the X-way or Y-way, the PARAFAC model may not be appropriate, and the constrained Tucker3 model should be used instead. Collinearity in the X-way or Y-way can be checked by successively performing PCA on data unfolded to an / x (J K) matrix, and then to a. / x (l K) matrix. If there are no collinearities in the X- or Y-ways, the optimal number of factors determined by both unfoldings will be the same. 

Different schemes have been developed to allow the validation of three-way models. One method used for PARAFAC models is split-half analysis, where the data set is split into two parts and individual modelling is performed on the two halves. If the two models show similar spectral loadings, the variation expressed in the two halves is comparable and it can be assumed that an appropriate number of components have been chosen. As for PCA, visual interpretability is also important and usually substantially easier for PARAFAC, because the components directly represent chemically meaningful phenomena. Other tools that may aid in deciding the correct number of components can be found in the literature.15... 

In principal component analysis (PCA), a matrix is decomposed as a sum of vector products, as shown in Figure 1.6. The vertical vectors (following the object way) are called scores and the horizontal vectors (following the variable way) are called loadings. A similar decomposition is given for three-way arrays. Here, the array is decomposed as a sum of triple products of vectors as in Figure 1.7. This is the PARAFAC model. The vectors, of which there are three different types, are called loadings. 

Finally, two important decomposition methods for two-way analysis (PCA) and three-way analysis (PARAFAC) are introduced briefly, because these methods are needed in the following chapter. 

In this section the use of different types of notation in two-way analysis is illustrated with the PCA model or the SVD of a two-way array X. This will facilitate understanding the notation for three-way component models (Chapter 4). 

Equation (3.11) shows that the elements of T and P give a model of X. This is one way of writing a PCA model of X and this type of notation can also be extended easily to the three-way case, as will be shown in Chapter 4. 

The similarity between a Tucker 1 model of a three-way array X and a PCA model of a two-way array X has already been mentioned in Section 4.1. This similarity is the basis for the most popular three-way regression model, the Tucker 1-PLS model. In the literature this regression method has also been called three-way partial least squares regression [Wold et al. 1987]. This name should be avoided for the Tucker 1 version specifically, because it gives rise to much confusion as to what is multi-way methodology. Even though the Tuckerl version of three-way partial least squares regression is indeed one way to implement a three-way method, it is actually the model that assumes the least three-way structure in the data. 

Important multi-way component and regression models have been described in this chapter. PARAFAC and Tucker3 are the best-known methods which can both be viewed as extensions of ordinary two-way PCA. PARAFAC is an extension in the sense that it provides the bestfitting rank R component model of a three-way data set, and Tucker3 is an extension of PCA... 

Two-dimensional plots (score and loading plots in PCA and for three-way analysis)... 

With a stack of low-rank bilinear data matrices such as the one above, a three-way array can be built. The underlying model of such an array is not necessarily low-rank trilinear if for example there are retention time shifts from sample to sample. However, the data can be still be fitted by a low-rank trilinear model. If the deviations from the ideal trilinearity (low-rank - one component per analyte) are small, directly interpretable components can be found. If not, components are still found in the same sense as in PCA, where the components together describe the data, although no single component can necessarily be related to a single analyte. 

Xie et al. [1999b] use the multilinear engine to find a model with five two-way factors and two three-way factors for the ALERT data. Because of the use of five PCA factors, the model explains a high percentage of the total sum of squares. The seven factors need rotation to become interpretable. For the two-way factors, the variation over the 11 years and the seasonality over the weeks are interpreted. The authors also take up rescaling of the factors. The interpretation of the element factors needs a backscaling to ng nr3 to make the interpretation more meaningful. 

To apply standard PCA, these three-way data arrays X have to be matricized to obtain a two-way data table. This can be done in different ways, according to what one is interested in focusing on. 

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