Multiway data analysis: Difference between revisions

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{{short description|Method of analyzing large data sets}}

'''Multiway data analysis''' is a method of analyzing large data sets by representing thea datacollection of observations as a [[multidimensionalmultiway array]]., <math> {\mathcal A}\in{\mathbb C}^{I_0\times I_1\times \dots I_c\times \dots I_C}</math>. The proper choice of arraydata dimensionsorganization into ''(C+1)''-way array, and analysis techniques can reveal patterns in the underlying data undetected by other methods.<ref name=Coppi1989>

{{cite book

|editor1-last=Coppi|editor1-first=R.

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==History==

The study of multiway data analysis was first formalized as the result of a conference held in 1988. The result of this conference was the first text specifically addressed to this field, Coppi and Bolasco's ''Multiway Data Analysis''.<ref name=Kroonenberg2008 Coppi1989>

{{cite book

|editor1-last=Coppi|editor1-first=R.

|page=xv

|editor2-last=Bolasco|editor2-first=S.

|title=Applied Multiway Data Analysis

|title=Multiway Data Analysis

|volume=702

|publisher=John Wiley & SonsNorth-Holland

|series=Wiley Series in Probability and Statistics

|___location=Amsterdam

|first=Pieter M.|last=Kroonenberg

|year=20081989

|publisher=John Wiley & Sons

|isbn=9780444874108

|year=2008

|isbn=9780470237991

}}</ref> At that time, the application areas for multiway analysis included [[statistics]], [[econometrics]] and [[psychometrics]]. In recent years, applications have expanded to include [[chemometrics]], [[agriculture]], [[social network analysis]] and the [[food industry]].<ref name=Bro1998>

{{cite thesis

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===Multiway data ===

Multiway data analysts use the term ''way'' to refer to athe "dimension"number ofsources theof data variation while reserving the word ''mode'' for the methods or models used to analyze the data.<ref name=Kroonenberg2008/>
{{rp|xviii}}.cite book

|page=xv

|title=Applied Multiway Data Analysis

|volume=702

|series=Wiley Series in Probability and Statistics

|first=Pieter M.|last=Kroonenberg

|publisher=John Wiley & Sons

|year=2008

|isbn=9780470237991

}}</ref>{{rp|xviii}}

In this sense, we can define the various ''ways'' of data to analyze:

* ''One- way data'': A isdata anpoint ''N''with <math>I_0</math>-dimensionaldimensions, <math>{\bf a}\in {\mathbb C}^{I_0}</math> is a [[Vector (mathematics and physics)|vector]] or an ''N''-dimensional data point that is stored in a ''one-way array'' data structure.

* ''Two-way data:'' isA a gridcollection of <math>I_1</math> data, apoints "data<math>{\bf [[matrixa}\in (mathematics)|matrix]]"{\mathbb thatC}^{I_0}</math> is stored in a ''two-way array'', <math>{\bf A}\in {\mathbb C}^{I_0\times I_1}</math>. A [[spreadsheet]] can be used to visualize such data in the case of discrete dimensions.

* ''Three-way data'': isA a cubecollection of data <math>{\bf a}\in {\mathbb C}^{I_0}</math> that has two modes of variation is stored in a threthree-way array, <math>{\bf A}\in {\mathbb C}^{I_0\times I_1\times I_2}</math>. Such data might represent the temperature at different locations (two-way data) sampled over different times (leading to three-way data)

* ''Four-way data'', using the same spreadsheet analogy, can be represented as a file folder full of separate workbooks.

* ''Five-way data'' and ''six-way data'' can be represented by similarly higher levels of data aggregation.

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|isbn=978-3-540-43745-1

|year=2002

}}</ref><ref name="MPCA-MICA2005">M.A.O. Vasilescu, D. Terzopoulos (2005) [http://www.media.mit.edu/~maov/mica/mica05.pdf "Multilinear Independent Component Analysis"], "Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR’05), San Diego, CA, June 2005, vol.1, 547–553."</ref> orand Human motion signatures<ref name="Vasilescu2002b">M.A.O. Vasilescu (2002) [http://www.media.mit.edu/~maov/motionsignatures/hms_icpr02_corrected.pdf "Human Motion Signatures: Analysis, Synthesis, Recognition," Proceedings of International Conference on Pattern Recognition (ICPR 2002), Vol. 3, Quebec City, Canada, Aug, 2002, 456–460.]</ref> computesanalyzes thefacial images and human joint angle data organizes in a multiway array. The multiway data analysis is employed to compute a set of causal factor representations.<ref name="Vasilescu2002tensorfaces">{{cite conference

|first=M.A.O. |last=Vasilescu

|first2=Eric |last2=Kim