Rasmus Bro

PARAFAC. Tutorial and applications

Abstract This paper explains the multi-way decomposition method PARAFAC and its use in chemometrics. PARAFAC is a generalization of PCA to higher order arrays, but some of the characteristics of the method are quite different from the ordinary two-way case. There is no rotation problem in PARAFAC, and e.g., pure spectra can be recovered from multi-way spectral data. One cannot as in PCA estimate components successively as this will give a model with poorer fit, than if the simultaneous solution is estimated. Finally scaling and centering is not as straightforward in the multi-way case as in the two-way case. An important advantage of using multi-way methods instead of unfolding methods is that the estimated models are very simple in a mathematical sense, and therefore more robust and easier to interpret. All these aspects plus more are explained in this tutorial and an implementation in Matlab code is available, that contains most of the features explained in the text. Three examples show how PARAFAC can be used for specific problems. The applications include subjects as: Analysis of variance by PARAFAC, a five-way application of PARAFAC, PARAFAC with half the elements missing, PARAFAC constrained to positive solutions and PARAFAC for regression as in principal component regression.

Tracing dissolved organic matter in aquatic environments using a new approach to fluorescence spectroscopy

Abstract Dissolved organic matter (DOM) is a complex and poorly understood mixture of organic polymers that plays an influential role in aquatic ecosystems. In this study we have successfully characterised the fluorescent fraction of DOM in the catchment of a Danish estuary using fluorescence excitation–emission spectroscopy and parallel factor analysis (PARAFAC). PARAFAC aids the characterisation of fluorescent DOM by decomposing the fluorescence matrices into different independent fluorescent components. The results reveal that at least five different fluorescent DOM fractions present (in significant amounts) in the catchment and that the relative composition is dependent on the source (e.g. agricultural runoff, forest soil, aquatic production). Four different allochthonous fluorescent groups and one autochthonous fluorescent group were identified. The ability to trace the different fractions of the DOM pool using this relatively cheap and fast technique represents a significant advance within the fields of aquatic ecology and chemistry, and will prove to be useful for catchment management.

The N-way Toolbox for MATLAB

Abstract This communication describes a free toolbox for MATLAB® for analysis of multiway data. The toolbox is called “The N-way Toolbox for MATLAB” and is available on the internet at http://www.models.kvl.dk/source/ . This communication is by no means an attempt to summarize or review the extensive work done in multiway data analysis but is intended solely for informing the reader of the existence, functionality, and applicability of the N-way Toolbox for MATLAB.

A FAST NON-NEGATIVITY-CONSTRAINED LEAST SQUARES ALGORITHM

In this paper a modification of the standard algorithm for non‐negativity‐constrained linear least squares regression is proposed. The algorithm is specifically designed for use in multiway decomposition methods such as PARAFAC and N‐mode principal component analysis. In those methods the typical situation is that there is a high ratio between the numbers of objects and variables in the regression problems solved. Furthermore, very similar regression problems are solved many times during the iterative procedures used. The algorithm proposed is based on the de facto standard algorithm NNLS by Lawson and Hanson, but modified to take advantage of the special characteristics of iterative algorithms involving repeated use of non‐negativity constraints. The principle behind the NNLS algorithm is described in detail and a comparison is made between this standard algorithm and the new algorithm called FNNLS (fast NNLS).

Multiway calibration. Multilinear PLS

A new multiway regression method called N‐way partial least squares (N‐PLS) is presented. The emphasis is on the three‐way PLS version (tri‐PLS), but it is shown how to extend the algorithm to higher orders. The developed algorithm is superior to unfolding methods, primarily owing to a stabilization of the decomposition. This stabilization potentially gives increased interpretability and better predictions. The algorithm is fast compared with e.g. PARAFAC, because it consists of solving eigenvalue problems.

Blind PARAFAC receivers for DS-CDMA systems

This paper links the direct-sequence code-division multiple access (DS-CDMA) multiuser separation-equalization-detection problem to the parallel factor (PARAFAC) model, which is an analysis tool rooted in psychometrics and chemometrics. Exploiting this link, it derives a deterministic blind PARAFAC DS-CDMA receiver with performance close to non-blind minimum mean-squared error (MMSE). The proposed PARAFAC receiver capitalizes on code, spatial, and temporal diversity-combining, thereby supporting small sample sizes, more users than sensors, and/or less spreading than users. Interestingly, PARAFAC does not require knowledge of spreading codes, the specifics of multipath (interchip interference), DOA-calibration information, finite alphabet/constant modulus, or statistical independence/whiteness to recover the information-bearing signals. Instead, PARAFAC relies on a fundamental result regarding the uniqueness of low-rank three-way array decomposition due to Kruskal (1977, 1988) (and generalized herein to the complex-valued case) that guarantees identifiability of all relevant signals and propagation parameters. These and other issues are also demonstrated in pertinent simulation experiments.

Parallel factor analysis in sensor array processing

This paper links multiple invariance sensor array processing (MI-SAP) to parallel factor (PARAFAC) analysis, which is a tool rooted in psychometrics and chemometrics. PARAFAC is a common name for low-rank decomposition of three- and higher way arrays. This link facilitates the derivation of powerful identifiability results for MI-SAP, shows that the uniqueness of single- and multiple-invariance ESPRIT stems from uniqueness of low-rank decomposition of three-way arrays, and allows tapping on the available expertise for fitting the PARAFAC model. The results are applicable to both data-domain and subspace MI-SAP formulations. The paper also includes a constructive uniqueness proof for a special PARAFAC model.

Principal component analysis

Principal component analysis is one of the most important and powerful methods in chemometrics as well as in a wealth of other areas. This paper provides a description of how to understand, use, and interpret principal component analysis. The paper focuses on the use of principal component analysis in typical chemometric areas but the results are generally applicable.

Multi-Way Analysis with Applications in the Chemical Sciences: Smilde/Multi-Way Analysis with Applications in the Chemical Sciences

Foreword. Preface. Nomenclature and Conventions. 1 Introduction. 1.1 What is multi--way analysis? 1.2 Conceptual aspects of multi--way data analysis. 1.3 Hierarchy of multivariate data structures in chemistry. 1.4 Principal component analysis and PARAFAC. 1.5 Summary. 2 Array definitions and properties. 2.1 Introduction. 2.2 Rows, columns and tubes frontal, lateral and horizontal slices. 2.3 Elementary operations. 2.4 Linearity concepts. 2.5 Rank of two--way arrays. 2.6 Rank of three--way arrays. 2.7 Algebra of multi--way analysis. 2.8 Summary. Appendix 2.A. 3 Two--way component and regression models. 3.1 Models for two--way one--block data analysis: component models. 3.2 Models for two--way two--block data analysis: regression models. 3.3 Summary. Appendix 3.A: some PCA results. Appendix 3.B: PLS algorithms. 4 Three--way component and regression models. 4.1 Historical introduction to multi--way models. 4.2 Models for three--way one--block data: three--way component models. 4.3 Models for three--way two--block data: three--way regression models. 4.4 Summary. Appendix 4.A: alternative notation for the PARAFAC model. Appendix 4.B: alternative notations for the Tucker3 model. 5 Some properties of three--way component models. 5.1 Relationships between three--way component models. 5.2 Rotational freedom and uniqueness in three--way component models. 5.3 Properties of Tucker3 models. 5.4 Degeneracy problem in PARAFAC models. 5.5 Summary. 6 Algorithms. 6.1 Introduction. 6.2 Optimization techniques. 6.3 PARAFAC algorithms. 6.4 Tucker3 algorithms. 6.5 Tucker2 and Tucker1 algorithms. 6.6 Multi--linear partial least squares regression. 6.7 Multi--way covariates regression models. 6.8 Core rotation in Tucker3 models. 6.9 Handling missing data. 6.10 Imposing non--negativity. 6.11 Summary. Appendix 6.A: closed--form solution for the PARAFAC model. Appendix 6.B: proof that the weights in trilinear PLS1 can be obtained from a singular value decomposition. 7 Validation and diagnostics. 7.1 What is validation? 7.2 Test--set and cross--validation. 7.3 Selecting which model to use. 7.4 Selecting the number of components. 7.5 Residual and influence analysis. 7.6 Summary. 8 Visualization. 8.1 Introduction. 8.2 History of plotting in three--way analysis. 8.3 History of plotting in chemical three--way analysis. 8.4 Scree plots. 8.5 Line plots. 8.6 Scatter plots. 8.7 Problems with scatter plots. 8.8 Image analysis. 8.9 Dendrograms. 8.10 Visualizing the Tucker core array. 8.11 Joint plots. 8.12 Residual plots. 8.13 Leverage plots. 8.14 Visualization of large data sets. 8.15 Summary. 9 Preprocessing. 9.1 Background. 9.2 Two--way centering. 9.3 Two--way scaling. 9.4 Simultaneous two--way centering and scaling. 9.5 Three--way preprocessing. 9.6 Summary. Appendix 9.A: other types of preprocessing. Appendix 9.B: geometric view of centering. Appendix 9.C: fitting bilinear model plus offsets across one mode equals fitting a bilinear model to centered data. Appendix 9.D: rank reduction and centering. Appendix 9.E: centering data with missing values. Appendix 9.F: incorrect centering introduces artificial variation. Appendix 9.G: alternatives to centering. 10 Applications. 10.1 Introduction. 10.2 Curve resolution of fluorescence data. 10.3 Second--order calibration. 10.4 Multi--way regression. 10.5 Process chemometrics. 10.6 Exploratory analysis in chromatography. 10.7 Exploratory analysis in environmental sciences. 10.8 Exploratory analysis of designed data. 10.9 Analysis of variance of data with complex interactions. Appendix 10.A: an illustration of the generalized rank annihilation method. Appendix 10.B: other types of second--order calibration problems. Appendix 10.C: the multiple standards calibration model of the second--order calibration example. References. Index.

On the uniqueness of multilinear decomposition of N-way arrays

We generalize Kruskals fundamental result on the uniqueness of trilinear decomposition of three‐way arrays to the case of multilinear decomposition of four‐ and higher‐way arrays. The result is surprisingly general and simple and has several interesting ramifications. Copyright