Three-way data

Physically, there are three ways data is obtained. 

Multiway and particularly three-way analysis of data has become an important subject in chemometrics. This is the result of the development of hyphenated detection methods (such as in combined chromatography-spectrometry) and yields three-way data structures the ways of which are defined by samples, retention times and wavelengths. In multivariate process analysis, three-way data are obtained from various batches, quality measures and times of observation [55]. In image analysis, the three modes are formed by the horizontal and vertical coordinates of the pixels within a frame and the successive frames that have been recorded. In this rapidly developing field one already finds an extensive body of literature and only a brief outline can be given here. For a more comprehensive reading and a discussion of practical applications we refer to the reviews by Geladi [56], Smilde [57] and Henrion [58]. 

In the present terminology, a three-way data array is defined by n rows, p columns and q layers, with indices i,j and k, respectively. 

R. Tauler, A.K. Smilde and B.R. Kowalski, Selectivity, local rank, three-way data analysis and ambiguity in multivariate curve resolution. J. Chemom., 9 (1995) 31-58. 

Saurina, J. Hemandez-Cassou, S. Izquierdo-Ridorsa, A. Tauler, R., pH-gradient spectrophotometric data files from flow-injection and continuous flow systems for two- and three-way data analysis, Chem. Intell. Lab. Syst. 50, 263-271 (2000). 

Data obtained from environmental monitoring programs can be classified, according to their complexity, in data ordered in one direction (one-way data), two directions (two-way data), three directions (three-way data), and in multiple directions (multiway data) [9, 10]. Scalar numerical data (one variable measured in one sample) would correspond to data ordered in zero direction (zero-way), while vector data (for instance, different variables measured in one sample or one variable measured in different samples) are ordered in one direction. When different variables are measured in different samples, obtained data can be ordered in two directions, that is, in a data table or data matrix. Finally, the compilation of different... 

In this equation, whereas the same loading matrix (YT matrix) is common for the different individual data matrices Dt, k = 1, 2, 3, 4, four different score matrices Xjt, k = 1, 2, 3, 4 are considered to explain the variation in Daug. Since these four D. matrices have equal sizes (same number of rows or samples and of columns or variables) they can also be arranged in a three-way data cube, with the four data matrices in the different slabs of this cube. However, in the frame of the MCR-ALS method and of the general bilinear model in (10), it is preferable to consider them to be arranged in the column-wise augmented data matrix Daug. 

More complex than vectors or matrices (X, X andy, X and Y) are three-way data or multiway data (Smilde et al. 2004). Univariate data can be considered as one-way data (one measurement per sample, a vector of numbers) two-way data are obtained for instance by measuring a spectrum for each sample (matrix, two-dimensional array, classical multivariate data analysis) three-way data are obtained by measuring a spectrum under several conditions for each sample (a matrix for each sample, three-dimensional array). This concept can be generalized to multiway data. 

FIGURE 3.22 PARAFAC model for three-way data X. R components are used to approximate A by a trilinear model as defined in Equation 3.35. 

T. Kourti, Abnormal situation detection, three-way data and projection methods robust data archiving and modeling for industrial applications, Ann. Rev. Control, 27, 131-139 (2003). 

Centring can be complex for three-way data, and there is no inherent reason to do this, therefore, for simplicity, in this section no centring is used, so raw concentrations and chromatographic/spectroscopic measurements are employed. 

Olivieri, A.C. and Faber, N.M., Standard error of prediction in parallel factor analysis of three-way data, Chemom. Intell. Lab. Syst., 70, 75-82, 2004. 

Though there is a clear gain in the quality and quantity of information when going from two- to three-way data sets, the mathematical complexity associated with the treatment of three-way data sets can seem, at first sight, a drawback. To avoid this problem, most of the three-way data analysis methods transform the original cube of data into a stack of matrices, where simpler mathematical methods can be applied. This process is often known as unfolding (see Figure 11.10). 

FIGURE 11.10 Three-way data array (cube) unfolding ormatricization. Two-way datamatrix augmentation. 

Decompositions of three-way arrays into these two different models require different data analysis methods therefore, finding out if the internal structure of a three-way data set is trilinear or nontrilinear is essential to ensure the selection of a suitable chemometric method. In the previous paragraphs, the concept of trilinearity was tackled as an exclusively mathematical problem. However, the chemical information is often enough to determine whether a three-way data set presents this feature. How to link chemical knowledge with the mathematical structure of a three-way data set can be easily illustrated with a real example. 

Let us consider a three-way data set formed by several HPLC-DAD runs. If the data set is trilinear, X, Y, and Z will have as many profiles as chemical compounds in the original data set, and this number will be equal to the rank of the data set. 

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

The MCR-ALS decomposition method applied to three-way data can also deal with nontrilinear systems [81]. Whereas the spectrum of each compound of the columnwise augmented matrix is considered to be invariant for all of the matrices, the unfolded C matrix allows the profile of each compound in the concentration direction to be different for each appended data matrix. This freedom in the shape of the C profiles is appropriate for many problems with a nontrilinear structure. The least-squares problems solved by MCR-ALS, when applied to a three-way data set, are the same as those in Equation 11.11 and Equation 11.12 the only difference is that D and C are now augmented matrices. The operating procedure of the MCR-ALS method has already been described in Section 11.5.4, but some particulars regarding the treatment of three-way data sets deserve further comment. 

In the resolution of a columnwise augmented data matrix, the initial estimates can be either a single ST matrix or a columnwise augmented C matrix. The columnwise concentration matrix is built by placing the initial C-type estimates obtained for each data matrix in the three-way data set one on top of each other. The appended initial estimates must be sorted into the same order as the initial data matrices in D, and they must keep a correct correspondence of species, i.e., each column in the augmented C matrix must be formed by appended concentration profiles related to the same chemical compounds. When no prior information about the identity of the compounds in the different data matrices is available, the correct correspondence of species can be estimated from the two-way resolution results of each single matrix. 

The examples that are given in the following subsections show the power of multivariate curve resolution to resolve very diverse chemical problems. Different strategies adapted to the chemical and mathematical features of the data sets are chosen, and resolution of two-way or three-way data sets is carried out according to the information that has to be recovered. Because MCR-ALS has proved to be a very versatile resolution method, able to deal with two-way and three-way data sets, this is the method used in all of the following examples. 

The results presented below were obtained by multivariate curve resolution-alternating least squares (MCR-ALS). MCR-ALS was selected because of its flexibility in the application of constraints and its ability to handle either one data matrix (two-way data sets) or several data matrices together (three-way data sets). MCR-ALS has been applied to the folding process monitored using only one spectroscopic technique and to a row-wise augmented matrix, obtained by appending spectroscopic measurements from several different techniques. 

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... 

The information obtained in the FSIW-EFA exploratory analysis is used in the resolution step. The total number of compounds in the three-way data set has been found to be equal to three (active, excipient, and unknown). The iterative optimization process starts with a matrix ST containing the initial estimates found by... 

Navea, S., de Juan, A., and Tauler, R., Three-way data analysis applied to multispec-troscopic monitoring of protein folding, Anal. Chim. Acta, 446, 187-197, 2001. 

However, three-way data can also be formed with two object ways and one variable way and by one sample with three variable ways. Environmental data where several distinct locations are monitored at discrete time intervals for multiple analytes exemplifies three-way data with two object ways and one variable way. Excitation-emission-time decay fluorescence or gas chromatography with a tandem mass spectroscopic detector are instrumental methods that form three-way data with three variable ways. These data types are employed mostly for qualitative application. Herein, the desire of the analyst to elicit underlying factors that influence the ecosystem or to deconvolve highly overlapped spectral profiles to deduce the number, identity, or relaxation coefficients of constituents in a complex sample can be realized. The same procedures employed for quantitation lend themselves to the extraction of qualitative information. 

There are two competing and equivalent nomenclature systems encountered in the chemical literature. The description of data in terms of ways is derived from the statistical literature. Here a way is constituted by each independent, nontrivial factor that is manipulated with the data collection system. To continue with the example of excitation-emission matrix fluorescence spectra, the three-way data is constructed by manipulating the excitation-way, emission-way, and the sample-way for multiple samples. Implicit in this definition is a fully blocked experimental design where the collected data forms a cube with no missing values. Equivalently, hyphenated data is often referred to in terms of orders as derived from the mathematical literature. In tensor notation, a scalar is a zeroth-order tensor, a vector is first order, a matrix is second order, a cube is third order, etc. Hence, the collection of excitation-emission data discussed previously would form a third-order tensor. However, it should be mentioned that the way-based and order-based nomenclature are not directly interchangeable. By convention, order notation is based on the structure of the data collected from each sample. Analysis of collected excitation-emission fluorescence, forming a second-order tensor of data per sample, is referred to as second-order analysis, as compared with the three-way analysis just described. In this chapter, the way-based notation will be arbitrarily adopted to be consistent with previous work. 

There are six classes of three-way data, and four of these classes can be appropriately modeled with the basic trilinear, or PARAFAC (PARAllel FACtor), model, where the data cube is decomposed into A sets of triads, x, y, and z [16]. The trilinear... 

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