View source for Multiway data analysis

{{short description|Method of analyzing large data sets}}
'''Multiway data analysis''' is a method of analyzing large data sets by representing a collection of observations as a multiway 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 data organization 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.
  |editor2-last=Bolasco|editor2-first=S.
  |title=Multiway Data Analysis 
  |publisher=North-Holland
  |location=Amsterdam
  |year=1989
  |isbn=9780444874108
}}</ref>

==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= Coppi1989>
{{cite book
  |editor1-last=Coppi|editor1-first=R.
  |editor2-last=Bolasco|editor2-first=S.
  |title=Multiway Data Analysis 
  |publisher=North-Holland
  |location=Amsterdam
  |year=1989
  |isbn=9780444874108
}}</ref>  At that time, the application areas for multiway analysis included [[Statistics|statistics]], [[Econometrics|econometrics]] and [[Psychometrics|psychometrics]]. In recent years, applications have expanded to include [[Chemistry:Chemometrics|chemometrics]], [[Earth:Agriculture|agriculture]], [[Philosophy:Social network analysis|social network analysis]] and the food industry.<ref name=Bro1998>
{{cite thesis
  |url=http://curis.ku.dk/ws/files/13035961/rasmus_bro.pdf
  |title=Multi-way Analysis in the Food Industry: Models, Algorithms, and Applications
  |first=Rasmus|last=Bro
  |degree=Ph.D.
  |publisher=[[Organization:University of Amsterdam|University of Amsterdam]]
  |date=20 November 1998
}}</ref>

== Composition of multiway data analysis ==

===Multiway data ===
Multiway data analysts use the term ''way'' to refer to the number sources of data variation while reserving the word ''mode'' for the methods or models used to analyze the data.<ref name=Kroonenberg2008>
{{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]] or data point that is stored in a ''one-way array'' data structure. 
* ''Two-way data:'' A collection of <math>I_1</math> data points <math>{\bf a}\in {\mathbb C}^{I_0}</math> is stored in a ''two-way array'', <math>{\bf A}\in {\mathbb C}^{I_0\times I_1}</math>. A [[Engineering:Spreadsheet|spreadsheet]] can be used to visualize such data in the case of discrete dimensions.
* ''Three-way data'': A collection of data <math>{\bf a}\in {\mathbb C}^{I_0}</math> that has two modes of variation is stored in a three-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.

In general, a multiway data  is stored in a multiway array and 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===

===Multiway application===
Multiway data analysis can be employed in various multiway applications so as to address the problem of finding hidden multilinear structure in multiway datasets. Following are examples of applications in different fields:<ref>
{{cite thesis
  |url=http://www.cs.rpi.edu/research/pdf/07-06.pdf
  |title=Unsupervised Multiway Data Analysis: A Literature Survey
  |first1=Evrim|last1=Acar 
  |first2=Bulent|last2=Yener
  |publisher=[[Organization:Rensselaer Polytechnic Institute|Rensselaer Polytechnic Institute]]
}}</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
* [[Biology:Neuroscience|Neuroscience]]
* [[Process analysis]]
* [[Philosophy:Social network analysis|Social network analysis]]/web-mining

=== Multiway processing ===
Multiway processing is the execution of designed and determined multiway model(s) transforming multiway data to the desirable level by addressing the specific need of particular multiway application. A typical example of data generated with a potentiometric electronic tongue illustrates relevant multiway processing.<ref>
{{cite journal
  |first1=Raul|last1=Cartas
  |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
 |issue=4
 |pages=953–961
 |doi=10.1002/elan.201000642
}}</ref>

== See also ==
*[[Multilinear subspace learning]]

==References==
{{reflist}}

[[Category:Data analysis]]

{{Sourceattribution|Multiway data analysis}}