Analysis of multi-way (multi-mode) data

Abstract The concepts data matrix and multivariate data analysis are rapidly becoming popular and well-known words in chemistry. Many methods used in the laboratory can produce data arrays of a greater complexity than the data matrix. The broad picture easily gets lost here, not least because of the confusing nomenclature. There is a need for systematization and generalization. Methods available in psychometrics and methods used in chemical research are described and compared in this paper. The goal is to provide a systematic overview and a simple introduction to the subject. References are made to more detailed descriptions in the literature.

[1]  J. Lafon,et al.  Optimum computation of p bilinear forms , 1975 .

[2]  Paul Geladi,et al.  Can image analysis provide information useful in chemistry? , 1989 .

[3]  Paul Geladi,et al.  Data analysis of multivariate magnetic resonance images I. A principal component analysis approach , 1989 .

[4]  Paul Geladi,et al.  Principal Component Analysis , 1987, Comprehensive Chemometrics.

[5]  Forrest W. Young,et al.  Component models for three-way data: An alternating least squares algorithm with optimal scaling features , 1980 .

[6]  T. Howell,et al.  Global properties of tensor rank , 1978 .

[7]  Bruce R. Kowalski,et al.  Tensorial calibration: II. Second‐order calibration , 1988 .

[8]  J. Besag.  On the Statistical Analysis of Dirty Pictures , 1986 .

[9]  R. Cattell.  The three basic factor-analytic research designs-their interrelations and derivatives. , 1952, Psychological bulletin.

[10]  S. Wold,et al.  Multi‐way principal components‐and PLS‐analysis , 1987 .

[11]  Paul Geladi,et al.  Relationships between higher‐order data array configurations and problem formulations in multivariate data analysis , 1989 .

[12]  Thomas Lickteig.  Typical tensorial rank , 1985 .

[13]  J. Levin.  Three-mode factor analysis. , 1965, Psychological bulletin.

[14]  Avraham Lorber,et al.  Features of quantifying chemical composition from two-dimensional data array by the rank annihilation factor analysis method , 1985 .

[15]  Lohmöller Jb,et al.  [Trimodal factor analysis by Tucker: scaling, rotations, other models]. , 1979 .

[16]  S. Wold,et al.  Principal component analysis of multivariate images , 1989 .

[17]  J. Kruskal.  Three-way arrays: rank and uniqueness of trilinear decompositions, with application to arithmetic complexity and statistics , 1977 .

[18]  E. Davidson,et al.  Application of the method of rank annihilation to quantitative analyses of multicomponent fluorescence data from the video fluorometer , 1978 .

[19]  Bruce R. Kowalski,et al.  Generalized rank annihilation factor analysis , 1986 .

[20]  Paul Geladi,et al.  Image analysis and chemical information in images , 1986 .

[21]  T. Hirschfeld,et al.  The hy-phen-ated methods , 1980 .

[22]  J. C. van Houwelingen,et al.  Three-mode factor analysis of data on retention in normal-phase high-performance liquid chromatography , 1984 .

[23]  Bruce R. Kowalski,et al.  Generalized rank annihilation method , 1987 .

[24]  Bruce R. Kowalski,et al.  Generalized rank annihilation method : II. Analysis of bimodal chromatographic data , 1987 .

[25]  Lars Pallesen.  Statistical assessment of PCDD and PCDF emission data , 1987 .

[26]  John L. Lastovicka,et al.  The extension of component analysis to four-mode matrices , 1981 .

[27]  Richard R. Ernst,et al.  The accordion experiment, a simple approach to three-dimensional NMR spectroscopy , 1981 .

[28]  Gary D. Christian,et al.  Simultaneous multicomponent rank annihilation and applications to multicomponent fluorescent data acquired by the video fluorometer , 1981 .

[29]  C. Eckart,et al.  The approximation of one matrix by another of lower rank , 1936 .

[30]  K. Ramachandra,et al.  Vermeidung von Divisionen. , 1973 .

[31]  J. Chang,et al.  Analysis of individual differences in multidimensional scaling via an n-way generalization of “Eckart-Young” decomposition , 1970 .

[32]  Avraham Lorber,et al.  Quantifying chemical composition from two-dimensional data arrays , 1984 .

[33]  Gary D. Christian,et al.  Application of the method of rank annihilation to fluorescent multicomponent mixtures of polynuclear aromatic hydrocarbons , 1980 .

[34]  L. Tucker,et al.  Some mathematical notes on three-mode factor analysis , 1966, Psychometrika.

[35]  G. Bodenhausen,et al.  Principles of nuclear magnetic resonance in one and two dimensions , 1987 .

[36]  M. Atkinson,et al.  Bounds on the ranks of some 3-tensors , 1980 .

[37]  J. Gower.  Generalized procrustes analysis , 1975 .

[38]  C L Wilkins.  Linked gas chromatography infrared mass spectrometry. , 1987, Analytical chemistry.

[39]  J. Kruskal.  More factors than subjects, tests and treatments: An indeterminacy theorem for canonical decomposition and individual differences scaling , 1976 .

[40]  Ernest R. Davidson,et al.  Three-dimensional rank annihilation for multi-component determinations , 1983 .