Trilinear model

Depending on the data structure, different types of models are possible to be applied for data analysis. Thus, when data are ordered in one direction, linear univariant models can be applied (see (1)), and nonlinear models as well (see (2)). For data ordered in two directions, bilinear models can be applied (see (3)) or nonbilinear models. Finally, for data ordered in three directions, trilinear models can be applied (see (4)) or, failing that, nontrilinear models. 

In the case of data following a trilinear model structure [7], the profiles in the score matrices X in Xaug of (10) will present the same shape in the score profiles (or, in other words, the relative distribution of contamination patterns among samples in the different data matrices is the same) in the different individual data matrices D (obtained in the different sampling campaigns). In this case, the data set fulfills the trilinear model [7, 15, 21], which can be described by (11) (in this equation, superindex t in D k and X k matrices indicates that they conform to the trilinear model) ... 

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. 

MCR-ALS solutions can be additionally constrained to fulfill a trilinear model [82], When this trilinearity constraint is applied, the profiles in the three different modes (Useo, utemP, and VT) are directly recovered and can be compared with the profiles obtained using PARAFAC- or Tucker-based model decompositions. MCR-ALS results have already been compared with Tucker3-ALS and PARAFAC-ALS results in the resolution of different chemical systems [81],... 

Although the PARAFAC model is a trilinear model that assumes linear additivity of effects between species, the model can be successfully employed when there is a nonlinear dependence between analyte concentration and signal intensity. Provided that the spectral profiles in the X- and Y-ways are not concentration dependent, the resolved Z-way profiles will be a nonlinear function of analyte... 

In extending the PLS regression algorithm to three-way data, the only thing needed is to change the bilinear model of X to a trilinear model of X. For example, the first component in a bilinear two-way model... 

Depending on the purpose of the data analysis, several different models maybe appropriate. Choosing which model is better is therefore part of the overall validation procedure. For calibration problems, different models can be compared in terms of how well they predict the dependent variables. Sometimes a priori knowledge of the structure of the data is available (e.g., that fluorescence data can be well approximated by a trilinear model), but often this is not the case. If no external information is available on which comparison of different models can be based, other approaches have to be used. In the following, a discussion is given of how to assess the appropriateness of the model based on the mathematical properties of the data and the model. No exact rules will be given, but rather some guidelines that may be helpful for the data analyst. 

Harshman advocates the use of split-half analysis for determining the proper rank of models with unique axes, that is, models with no rotational freedom [Harshman 1984, Harshman De Sarbo 1984], In split-half analysis different subsets of the data are analyzed independently. Due to the uniqueness of the PARAFAC model, the same result - same loadings -will be obtained in the nonsplit modes from models of any suitable subset of the data, if the correct number of components is chosen. To judge whether two models are equal, the indeterminacy in trilinear models has to be remembered the order and scale of components may change if not fixed algorithmically. If too many or too few components are chosen, the model parameters will differ if a different data set is used for fitting the model. 

Figure 10.21. Bi- and trilinearity have at least three definitions in the underlying model (mathematical structure plus possible distributional assumptions), in the data and in the fitted model. It is perfectly possible to fit a (low-rank) bi- or trilinear model without having the data or the underlying model assumed (low-rank) bi- or trilinear. Two possible scenarios are shown where focus is primarily on the fitted model and on the underlying model respectively.

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. 

Some concentration nonlinearities can be treated by a simple extension of the trilinear model. If there are concentration nonlinearities, then the model estimates apparent concentrations in C. Making a simple nonlinear calibration curve between the values in C and the actual concentrations in the standards gives the solution since the apparent concentration of the analyte in the mixture can then be transformed to the actual concentration domain [Booksh et al. 1994],... 

Lee JK, Ross RT, Thampi S, Leurgans SE, Resolution of the properties of hydrogen-bonded tyrosine using a trilinear model of fluorescence, Journal of Physical Chemistry, 1992, 96, 9158-9162. 

Ross RT, Fee C, Davis CM, Ezzeddine BM, Fayyad EA, Leurgans SE, Resolution of the fluorescence spectra of plant pigment-complexes using trilinear models, Biochimica et Biophysica Acta, 1991, 1056,317-320. 

In excited-state spectroscopies, including fluorescence spectroscopy, spectroscopic intensity is usually linear in functions of each of three or more independent variables, so that a three-way array of data can be fit with a trilinear model. The presence of three or more linear relationships makes algebraic methods for resolving the spectra and other properties of individual components substantially more powerful than in the case of two linear relationships. The use of a general trilinear model is sometimes known as three-way factor analysis, three-mode factor analysis, or threemode principal component analysis. For a review of the mathematics and application to spectroscopy, see our survey article. ... 

In this chapter we focus on the application of trilinear models in fluorescence spectroscopy, because that is where we have most experience. However, these trilinear models are also applicable to other kinds of excited-state spectroscopy, such as transient absorption spectroscopy. Along the way, we also discuss bilinear and other models, including global analysis. ... 

We present here three types of trilinear models. We begin with a trilinear model which was first introduced by Carroll and Chang, who called it canonical decomposition (CANDECOMP), and by Harshman," who called it parallel factors (PARAFAC). We refer to it as PARAFAC. It can be written as... 

The second type of trilinear model, known as the Tucker2, or T2, model, named after Tucker, can be written as... 

Each type of trilinear model can be described as a special case of one of the others. For example, a T2 model with F, factors for the first way and Fj factors for the second way is a PARAFAC model with FjFj factors... 

The second method of obtaining an additional variable is to make time-resolved measurements. Trilinear models were first applied to the kinetics... 

In Section II we presented the standard general multilinear models, of which the bilinear and the PARAFAC and Tucker2 (T2) trilinear models are most important in spectroscopy. These models contain no information about the specimen except the linear dependence of spectral intensity on functions of each of the independent variables. However, some properties of the specimen are known, and a model that incorporates these known properties is preferred to one that does not. This is particularly true when the model is indeterminate without side conditions. In this section we discuss three settings for the application of knowledge about the specimen identifiable bilinear and T2 submodels, penalized general multilinear models, and submodels in which the dependence of the expected intensity from some components for some ways has a specific mathematical form. 

A decomposition algorithm is one that is inherently noniterative and that will yield the parameters of the model if no noise is present in the data. The singular value decomposition (SVD) is a well-known decomposition for bilinear models. There are decompositions for the PARAFAC, Tucker2, and Tucker3 trilinear models. 

The rate of convergence of unconstrained bilinear and T2 and T3 trilinear models is almost always quite fast. Convergence for PARAFAC and constrained models is much slower, even when we use a projection technique to accelerate convergence. 

The value of trilinear models is clearest in steady-state fluorescence measurements. They are also valuable in time-resolved fluorescence spectroscopy in situations where the appropriateness of a specific parametric equation for time decay, such as a sum of a few exponentials, is unclear. Although we are unaware of any work using trilinear models with other kinds of excited-state spectroscopy, trilinear models will be a valuable means of achieving component resolution whenever the absence of reliable parametric equations makes global analysis impossible. 

GRAM and PARAFAC are essentially trilinear models such that the data, R, is computationally modeled as the sum of the outer product of an integer number, N (the number of factors in the model), of three vectors, x, y, and z, as given in the following equation. 

In the second part of this chapter, an illustrative example of PARAFAC analysis for three-way data obtained in an actual laboratory experiment is presented to show how PARAFAC trilinear model can be constructed and analyzed to derive in-depth understanding of the system from the data. Thermal deformation of several types of poly lactic add (PLA) nanocomposites xmdergoing grass-to-rubber transition is probed by cross-polarization magic-angle (CP-MAS) NMR spectroscopy. Namely, sets of temperature-dependent NMR spectra are measured under varying clay content in the PLA nanocomposite samples. While temperature strongly affects molecular dynamics of PLA, the clay content in the samples also influences the molecular mobility. Thus, NMR spectra in this study become a three-way... 

The PARAFAC model can also be considered a trilinear model. Two sets of parameters are fixed, for example, the as and b% and the element is estimated as a linear function of the remaining parameter c. 

FIGURE 3 Scheme of MCR models (A) bilinear model for a row-wise augmented matrix, (B) bilinear model for a column-wise augmented matrix and (C) trilinear model for a cube. 

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