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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== Composition of multiway data analysis ==

===Multiway data ===

Multiway data analysts use the term ''way'' to refer to athe dimensionnumber 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 data point with <math>I_0</math>-dimensions, <math>{\bf a}\in {\mathbb C}^{I_0}</math> is a [[Vector (mathematics and physics)|vector]], with a singleor data valuepoint forthat eachis discretestored orin continuousa value''one-way ofarray'' thedata single dimensionstructure.

* ''Two-way data:'' isA acollection [[matrixof (mathematics)|matrix]],<math>I_1</math> withdata points <math>{\bf a}\in single{\mathbb dataC}^{I_0}</math> valueis forstored eachin discretea or''two-way continuousarray'', value<math>{\bf ofA}\in two{\mathbb separateC}^{I_0\times dimensions;I_1}</math>. aA [[spreadsheet]] can be used to visualize such data in the case of discrete dimensions.

* ''Three-way data'': canA becollection viewedof asdata <math>{\bf a}\in stack{\mathbb C}^{I_0}</math> that has two modes of matricesvariation (oris similarly,stored asin a workbookthree-way ofarray, multiple<math>{\bf [[spreadsheet]]s),A}\in adding{\mathbb aC}^{I_0\times thirdI_1\times dimensionI_2}</math>. Such data might represent the temperature at different locations (two-way data) sampled over different times (the third dimension, 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.

In general, thea severalmultiway dimensionsdata represented is stored in thea datamultiway array setand may be measured at different times, or in different places, using different methodologies, and may contain inconsistencies such as missing data or discrepancies in data representation.

===Multiway model===

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}}</ref>

* [[Computer vision]] - TensorFaces<ref name=Vasilescu2002Tensorfaces>{{cite journal

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

|first2=D. |last2=Terzopoulos

|url=http://www.cs.toronto.edu/~maov/tensorfaces/Springer%20ECCV%202002_files/eccv02proceeding_23500447.pdf

|title=Multilinear Analysis of Image Ensembles: TensorFaces

|series=Lecture Notes in Computer Science 2350; (Presented at Proc. 7th European Conference on Computer Vision (ECCV'02), Copenhagen, Denmark)

|publisher=Springer, Berlin, Heidelberg

|doi=10.1007/3-540-47969-4_30

|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> and 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> analyzes facial 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

|first3=Xiao |last3=Zeng

|url=http://www.cs.toronto.edu/~maov/tensorfaces/Springer%20ECCV%202002_files/eccv02proceeding_23500447.pdf

|title="CausalX: Causal eXplanations and Block Multilinear Factor Analysis",

|conference=In the Proceedings of the 2020 25th International Conference on Pattern Recognition (ICPR 2020)

|___location=Milan, Italy

|pages=10736-10743

|year=2021

}}</ref>

* [[Electroanalytical chemistry]]

* [[Neuroscience]]

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|first2=Aitor|last2=Mimendia

|first3=Andrey|last3=Legin

|first4=Manel|last4=del Valle

|title=Multiway Processing of Data Generated with a Potentiometric Electronic Tongue in a SIA System

|year=2011

|journal=Electroanalysis

|volume=23

|doi=10.1002/elan.201000642

|issue=4

|pages=953–961

|doi=10.1002/elan.201000642

}}</ref>