Modelling of spectroscopic batch process data using grey models to incorporate external information

Gurden, Stephen P.; Westerhuis, Johan A.; Bijlsma, Sabina; Smilde, Age K.

doi:10.1002/1099-128x(200102)15:2<101::aid-cem602>3.0.co;2-v

Cited by 44 publications

(31 citation statements)

References 41 publications

(51 reference statements)

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“…However, it also has direct applications in hybrid models. Ten Berge and Smilde (2002) discussed a 3PCA study of Gurden, Westerhuis, Bijlsma, and Smilde (2001), which involved a constrained 5 × 5 × 3 core array containing no more than five nonzero elements. Ten Berge and Smilde showed that this core, which has rank 5, is non-trivial, because the typical rank of 5 × 5 × 3 arrays is at least 6, which means that this core cannot be obtained almost surely from Tucker transformations of any arbitrary core.…”

Section: Discussionmentioning

confidence: 99%

Simplicity and Typical Rank Results for Three-Way Arrays

Berge

2011

Psychometrika

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Matrices can be diagonalized by singular vectors or, when they are symmetric, by eigenvectors. Pairs of square matrices often admit simultaneous diagonalization, and always admit block wise simultaneous diagonalization. Generalizing these possibilities to more than two (non-square) matrices leads to methods of simplifying three-way arrays by nonsingular transformations. Such transformations have direct applications in Tucker PCA for three-way arrays, where transforming the core array to simplicity is allowed without loss of fit. Simplifying arrays also facilitates the study of array rank. The typical rank of a three-way array is the smallest number of rank-one arrays that have the array as their sum, when the array is generated by random sampling from a continuous distribution. In some applications, the core array of Tucker PCA is constrained to have a vast majority of zero elements. Both simplicity and typical rank results can be applied to distinguish constrained Tucker PCA models from tautologies. An update of typical rank results over the real number field is given in the form of two tables.Key words: tensor decomposition, tensor rank, typical rank, sparse arrays, Candecomp, Parafac, Tucker component analysis.Three-way data arise when a complete set of measures is obtained for a set of individuals or objects on a set of variables, in a set of circumstances or conditions. For instance, when I subjects are evaluated on J tests on K different occasions, the data consist of an I × J × K three-way array. The array can be cut up into slices: there are K frontal slices of order I × J, J lateral slices of order I × K, and I horizontal slices of order J × K. Tucker (1966), Carroll andChang (1970), and Harshman (1970) have proposed generalizations of principal component analysis (PCA) for the analysis of three-way arrays. For a long time, these methods were well known in psychometrics, but did not receive much attention elsewhere. A few decades ago, however, chemometricians discovered that these methods are particularly useful for their own purposes of data analysis. Nowadays, three-way component analysis is at the heart of what is called tensor decomposition analysis, which has applications in various areas such as signal processing, numerical linear algebra, computer vision, numerical analysis, data mining, graph analysis, neuroscience, and others (Kolda & Brader, 2009). Although tensor decomposition analysis covers multiway arrays beyond the three-way level, only three-way component analysis will be considered in the present paper.Three-way component analysis poses various mathematical problems that have been studied extensively. Among these are the uniqueness properties of three-way decompositions and the transformational freedom that exists when they are absent, the possibilities of transforming arrays to simple forms (with as many vanishing elements as possible, that is), and the rank of three-way arrays. The present paper is focused on the latter two topics. The purpose is to explain simplifying transformations...

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Section: Discussionmentioning

confidence: 99%

Simplicity and Typical Rank Results for Three-Way Arrays

Berge

2011

Psychometrika

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“…Some tools to partly overcome this problem have been suggested, for example. plaid models [54], Quilt-PCA [55], Hierarchical PCA [56], and Gray Models [57,58]. It is not always easy to interpret the loading plots, due to the presence of very many variables.…”

Section: Discussionmentioning

confidence: 99%

Symbiosis of chemometrics and metabolomics: past, present, and future

Greef

Smilde

2005

Journal of Chemometrics

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120

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“…Tensor decompositions are in frequent use today in a variety of fields, including psychometrics, chemometrics, image analysis, graph analysis, and signal processing (Murakami & Kroonenberg, 2003;Vasilescu & Terzopoulos, 2002;Wang & Ahuja, 2003;Jia & Gong, 2005;Sun, Zeng, Liu, Lu, & Chen, 2005;Gurden, Westerhuis, Bijlsma, & Smilde, 2001;Nørgaard & Ridder, 1994;Smilde, Tauller, Saurina, & Bro, 1999;Smilde, Bro, & Geladi, 2004;Andersson & Bro, 1998). Tensors (i.e., X ∈ I1×I2×...×IN ), also called multiway arrays or multidimensional matrices, are generalizations of vectors (first-order tensors) and matrices (second-order tensors).…”

Section: Introductionmentioning

confidence: 99%

“…r Spectroscopy data (Smilde et al, 2004;Andersson & Bro, 1998)-for instance, X Batch number ×Time×Spectra Strength (Gurden et al, 2001;Nørgaard & Ridder, 1994;Smilde et al, 1999) r Web mining: X Users×Queries×Wep pages Click counts (Sun et al, 2005) r Image analysis: X People×Views×Illuminations×Expressions×Pixels Image intensity (Vasilescu & Terzopoulos, 2002;Wang & Ahuja, 2003;Jia & Gong, 2005) r Semantic differential data: X Judges×Music pieces×Scales Grade (Murakami & Kroonenberg, 2003) All of the above data sets are nonnegative, and the basis vectors and projections A (n) and interactions G can be assumed additive, that is, nonnegative. For the spectroscopy data, nonnegativity would yield batch groups containing, time, and spectra profiles additively combined by the nonnegative core, for the Web mining data giving groups of users, queries and Web pages interrelated with a strength given by the nonnegative core.…”

Section: Introductionmentioning

confidence: 99%

Algorithms for Sparse Nonnegative Tucker Decompositions

2008

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There is a increasing interest in analysis of large-scale multiway data. The concept of multiway data refers to arrays of data with more than two dimensions, that is, taking the form of tensors. To analyze such data, decomposition techniques are widely used. The two most common decompositions for tensors are the Tucker model and the more restricted PARAFAC model. Both models can be viewed as generalizations of the regular factor analysis to data of more than two modalities. Nonnegative matrix factorization (NMF), in conjunction with sparse coding, has recently been given much attention due to its part-based and easy interpretable representation. While NMF has been extended to the PARAFAC model, no such attempt has been done to extend NMF to the Tucker model. However, if the tensor data analyzed are nonnegative, it may well be relevant to consider purely additive (i.e., nonnegative) Tucker decompositions). To reduce ambiguities of this type of decomposition, we develop updates that can impose sparseness in any combination of modalities, hence, proposed algorithms for sparse nonnegative Tucker decompositions (SN-TUCKER). We demonstrate how the proposed algorithms are superior to existing algorithms for Tucker decompositions when the data and interactions can be considered nonnegative. We further illustrate how sparse coding can help identify what model (PARAFAC or Tucker) is more appropriate for the data as well as to select the number of components by turning off excess components. The algorithms for SN-TUCKER can be downloaded from Mørup (2007).

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Modelling of spectroscopic batch process data using grey models to incorporate external information

Cited by 44 publications

References 41 publications

Simplicity and Typical Rank Results for Three-Way Arrays

Simplicity and Typical Rank Results for Three-Way Arrays

Symbiosis of chemometrics and metabolomics: past, present, and future

Algorithms for Sparse Nonnegative Tucker Decompositions

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