Tucker models

P has the structure G(I B). Thus, the Tucker 1 model will always fit the data better than the Tucker2 model for equivalent numbers of components. However, if the structure of the Tucker2 model is appropriate, the Tucker2 model will provide a more accurate model with fewer parameters than the Tuckerl model. The Tucker3 model is a constrained version of the Tucker2 model and the PARAFAC model is a constrained Tucker3 model. Hence, there is a certain hierarchy between the different models as will be discussed in Section 5.1. 

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. 

The Tucker2 model can even be made less restrictive, e.g., by not compressing the second mode. This can be done by formulating the Tucker 1 model... 

The same type of nonuniqueness as is present in Tucker3 models is present in Tucker2 and Tucker 1 models. 

The Tucker 1 model is equivalent to a bilinear model of the data matricized to a two-way array (Section 4.2). Thus, standard two-way algorithms may be used for fitting the model. 

There are two aspects of interest in the models produced so far. These are the approximation of the data (e.g. TP ) and the residuals. For PCA, the matrix P represents the common variation in all batches and the matrix of scores T shows the behavior of the individual batches with respect to this common variation. The residuals show to what extent a batch fits the model. Hence, if the measurements of a new batch are available then two important sets of numbers can be calculated its scores and residuals. Suppose that the data of the new batch can be represented as Xnew (J x K), and in its matricized version as xnew (JK x 1). In order to find the scores and residuals of these data, it is assumed that they are preprocessed using the offsets and scales found during the model building. The scores and residuals of this new batch on the Tucker 1 model are obtained as... 

REIST Aerosol Science and Technology, Second Edition RHINE, TUCKER Modeling of Gas-Fired Furnaces and Boilers and Other Industrial Heating Processes ROSSITER Waste Minimization Through Process Design SAMDANI Safety and Risk Management Tools and Techniques in the CPI... 

By analogy with singular vector decomposition (SVD) and ordinary PCA, one can also define the TuckerS model in an extended matrix notation ... 

The extended matrix notation is represented here by the three dots that surround the core matrix. A graphical example of the TuckerS model is rendered in Fig. 31.19. 

The reaction manifold describing the automated determination of ammonia is shown in Fig. 6.1. Two alternative modes of sampling are shown discrete and continuous. Discrete 5 ml samples contained in ashed (450 °C) glass vials are sampled from an autosampler (Hook and Tucker model A40-11 1.5 min sam-ple/wash). For high-resolution work in the estuary, the continuous sampling mode is preferred. The indophenol blue complex was measured at 630 nm with a colorimeter and the absorbance recorded on a chart recorder. 

PREDICTING FIBER ORIENTATION - THE FOLGAR-TUCKER MODEL 443... 

To illustrate the effect of fiber orientation on material properties of the final part, Fig. 8.60 [5] shows how the fiber orientation distributions that correspond to 67 50 and 33% initial mold coverage affect the stiffness of the finished plates. The Folgar-Tucker model has been implemented into various, commercially available compression mold filling simulation programs and successfully tested with several realistic compression molding applications. 

FIGURE 12.2 Construction of a three-way array according to the unconstrained Tucker model. 

Smilde, A.K., Tauler, R., Henshaw, J.M., Burgess, L.W., and Kowalski, B.R., Multicomponent determination of chlorinated hydrocarbons using a reaction based sensor, 3 medium-rank second-order calibration with restricted Tucker models, Anal. Chem., 66, 3345-3351, 1994. 

Method Explicit Matrix Relations for Total Exchange Areas, Int.J. Heat Mass Transfer, 18, 261-269 (1975). Rhine, J. M., and R. J. Tucker, Modeling of Gas-Fired Furnaces and Boilers, British Gas Association with McGraw-Hill, 1991. Siegel, Robert, and John R. Howell, Thermal Radiative Heat Transfer, 4th ed., Taylor Francis, New York, 2001. Sparrow, E. M., and R. D. Cess, Radiation Heat Transfer, 3d ed., Taylor Francis, New York, 1988. Stultz, S. C., and J. B. Kitto, Steam Its Generation and Use, 40th ed., Babcock and Wilcox, Barkerton, Ohio, 1992. 

M. Rhine, R. J. Tucker, Modelling of Gas-Fired Furnaces and Boilers", McGraw-Hill in... 

In order to concisely describe multi-way models, the usual matrix product is not sufficient. Three other types of matrix products are introduced the Kronecker ( ), Hadamard ( ) and Khatri-Rao (O) product [McDonald 1980, Rao Mitra 1971, Schott 1997], The Kronecker product allows a very efficient way to write Tucker models (see Chapter 4). Likewise, the Khatri-Rao product provides means for an efficient way to write a PARAFAC model (see Chapter 4). The Hadamard product can, for instance, be used to formalize weighted regression (see Chapter 6). 

Ledyard Tucker was one of the pioneers in multi-way analysis. He proposed [Tucker 1964, Tucker 1966] a series of models nowadays called A-mode principal component analysis or Tucker models. An extensive treatment of Tucker models is given by Kroonenberg and de Leeuw [1980] and Kroonenberg [1983], In the following, three different Tucker models will be treated. 

Deriving a Tucker model from an assumed latent structure... 

There is a convenient interpretation of the Tucker model in terms of a new basis to express a three-way array. Consider the SVD of X X = USV. The matrix V is an orthonormal basis for the row-space of X. X can be expressed on this new basis by orthogonally projecting X on this basis the new coordinates are found by regressing X on V ... 

An important question is how the PARAFAC and Tucker3 models are related. PARAFAC models provide unique axes, while Tucker3 models do not. A Tucker model may be transformed (rotated) and simplified to look more like PARAFAC models. This can sometimes be done with little or no loss of fit. There is a hierarchy e.g. within the family of Tucker models, Tucker3, Tucker2 and Tuckerl, which is worth studying in more detail. PARAFAC models may be difficult or impossible to fit due to so-called degeneracies (Section 5.4), in which case a Tucker3 model is usually a better a choice. Further, the statistical properties of the data - noise and systematic errors - also play an important role in the choice of model. 

Consider a three-way array X (/ x J x K) and different Tucker models for this array. A Tucker3 model for this array is... 

Summarizing, a Tucker3 model compresses all three modes, a Tucker2 model compresses only two of the three modes, and a Tuckerl model compresses only one of the three modes. Hence the names Tuckerl, Tucker2 and Tucker3 (and, for example, a four-way Tucker model with compression in all modes would therefore be a Tucker4 model [Kiers 2000]). 

Constrained or restricted Tucker models have found use in analytical chemistry, specifically, in second-order calibration [Kiers Smilde 1998, Smilde el al. 1994a, Smilde el al. 1994b, Tauler et al. 1994] as well as in batch process modelling [Gurden el al. 2001, Gurden etal. 2002],... 

Going from the two-component PARAFAC model to the (2,10,20) Tuckerl model the number of parameters to estimate increases and hence also the model complexity increases. Obviously, the two-component PARAFAC model is the least complex of the models considered. If one of the modes (or two of the modes) is not compressed in Tucker models, this increases the number of parameters to be estimated considerably. 

The uniqueness properties of Tucker, constrained Tucker and PARAFAC models are discussed. A Tucker3 model finds unique subspaces, whereas a PARAFAC model finds unique axes. For constrained Tucker models, the situation is more complicated and no straightforward results are available. 

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