View source for Generalized Procrustes analysis

{{Short description|Method of statistical analysis}}
'''Generalized Procrustes analysis''' ('''GPA''') is a method of statistical analysis that can be used to compare the shapes of objects, or the results of surveys, interviews, or panels. It was developed for analysing the results of [[Social:Free-choice profiling|free-choice profiling]], a survey technique which allows respondents (such as [[Engineering:Sensory analysis|sensory panelists]]) to describe a range of products in their own words or language. GPA is one way to make sense of free-choice profiling data;<ref>{{Cite book | editor1-last = Meullenet | editor1-first = Jean-François | editor2-first = Rui | editor3-first = Christopher J | doi = 10.1002/9780470277539 | title = Multivariate and Probabilistic Analyses of Sensory Science Problems | year = 2007 | isbn = 9780470277539 | editor2-last = Xiong | editor3-last = Findlay }}</ref> other ways can be [[Multiple factor analysis|multiple factor analysis]] (MFA),<ref>{{Cite journal | last1 = Escofier | first1 = B. | last2 = Pagès | first2 = J. | doi = 10.1016/0167-9473(94)90135-X | title = Multiple factor analysis (AFMULT package) | journal = Computational Statistics & Data Analysis | volume = 18 | pages = 121–140 | year = 1994 }}</ref><ref> A comparison of GPA and MFA with sensory data is a chapter of the bookMFA is the core of a recent book: Pagès Jérôme (2014). ''Multiple Factor Analysis by Example Using R''. Chapman & Hall/CRC The R Series London 272 p</ref> or the STATIS method.<ref>{{Cite journal | last1 = Lavit | first1 = C. | last2 = Escoufier | first2 = Y. | last3 = Sabatier | first3 = R. | last4 = Traissac | first4 = P. | title = The ACT (STATIS method) | doi = 10.1016/0167-9473(94)90134-1 | journal = Computational Statistics & Data Analysis | volume = 18 | pages = 97–119 | year = 1994 }}</ref> The method was first published by J. C. Gower in 1975.<ref>{{Cite journal | last1 = Gower | first1 = J. C. | doi = 10.1007/BF02291478 | title = Generalized procrustes analysis | journal = Psychometrika | volume = 40 | pages = 33–51 | year = 1975 | hdl = 10.1007/BF02291478 | s2cid = 122244491 | hdl-access = free }}</ref>

Generalized Procrustes analysis estimates the scaling factor applied to respondent scale usage, generating a weighting factor that is used to compensate for individual scale usage differences. Unlike measures such as a [[Principal component analysis|principal component analysis]], GPA uses individual level data and a measure of variance is utilized in the analysis.

The Procrustes distance provides a metric to minimize in order to superimpose a pair of shape instances annotated by [[Landmark point|landmark point]]s. GPA applies the [[Procrustes analysis]] method to superimpose a population of shapes instead of only two shape instances. 

The algorithm outline is the following:
# arbitrarily choose a reference shape (typically by selecting it among the available instances)
# superimpose all instances to current reference shape
# compute the mean shape of the current set of superimposed shapes
# if the Procrustes distance between the mean shape and the reference is above a certain threshold, set the reference to mean shape and continue to step 2.

== See also ==
*[[Procrustes analysis]] 
*[[Orthogonal Procrustes problem]]

== References ==
{{Reflist}}
* {{cite book|author=I.L. Dryden and K.V. Mardia|title=Statistical Shape Analysis|publisher=John Wiley & Sons|year=1998|isbn=0-471-95816-6}}

[[Category:Multivariate statistics]]
[[Category:Euclidean symmetries]]
[[Category:Biometrics]]

{{Sourceattribution|Generalized Procrustes analysis}}