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@article{Coissac:19:00,
	Abstract = {Motivation: Motivation: Molecular biology and ecology studies can produce high dimension data. Estimating correlations and shared variation between such data sets are an important step in disentangling the relationships between different elements of a biological system. Unfortunately, classical approaches are susceptible to producing falsely inferred correlations. Results: Here we propose a corrected version of the Procrustean correlation coefficient that is robust to high dimensional data. This allows for a correct estimation of the shared variation between two data sets and the partial correlation coefficients between a set of matrix data. Availability: The proposed corrected coefficients are implemented in the ProcMod R package available on https://git.metabarcoding.org/ lecasofts/ProcMod},
	Author = {Coissac, Eric and Gonindard-Melodelima, Christelle},
	Date-Added = {2019-11-15 09:19:38 +0100},
	Date-Modified = {2019-11-15 09:19:41 +0100},
	Doi = {10.1101/842070},
	Elocation-Id = {842070},
	Eprint = {https://www.biorxiv.org/content/early/2019/11/14/842070.full.pdf},
	Journal = {bioRxiv},
	Publisher = {Cold Spring Harbor Laboratory},
	Title = {Assessing the shared variation among high-dimensional data matrices: a modified version of the Procrustean correlation coefficient},
	Url = {https://www.biorxiv.org/content/early/2019/11/14/842070},
	Year = {2019},
	Bdsk-Url-1 = {https://www.biorxiv.org/content/early/2019/11/14/842070},
	Bdsk-Url-2 = {https://doi.org/10.1101/842070}}

@article{Chariton:10:00,
	Author = {Chariton, Anthony A and Roach, Anthony C and Simpson, Stuart L and Batley, Graeme E},
	Date-Added = {2019-11-13 19:17:47 +0200},
	Date-Modified = {2019-11-13 19:19:47 +0200},
	Journal = {Marine and Freshwater Research},
	Number = 10,
	Pages = {1109},
	Title = {Influence of the choice of physical and chemistry variables on interpreting patterns of sediment contaminants and their relationships with estuarine macrobenthic communities},
	Volume = 61,
	Year = 2010}

@article{Coissac-Eric:19:00,
	Author = {Coissac, Eric and Gonindard-Melodelima, Christelle},
	Date-Added = {2019-08-19 09:07:57 +0200},
	Date-Modified = {2019-08-19 09:12:35 +0200},
	Journal = {in prep},
	Title = {Assessing the shared variation among high-dimensional data matrices: a modified version of the Procrustean correlation coefficient},
	Year = {2019}}

@book{Theil:58:00,
	Address = {Amsterdam},
	Annote = {Le R2 ajuster de R

http://www.sudoc.abes.fr/xslt/DB=2.1//SRCH?IKT=12&TRM=005519756

},
	Author = {Theil, Henri, and Cramer, Jan Salomon, and Moerman, H. and Russchen, A.},
	Date-Added = {2019-07-11 15:10:21 +0200},
	Date-Modified = {2019-08-19 09:13:20 +0200},
	Keywords = {Economics, Mathematical; Forecasting; Business},
	Language = {en},
	Pages = {213},
	Publisher = {North-Holland Publishing Company},
	Title = {Economic forecasts and policy},
	Year = 1958}

@misc{CSoRgo:96:00,
	Author = {Cs{\"o}rg{\H o}, S{\'a}ndor and Faraway, Julian J},
	Date-Added = {2019-05-20 11:28:37 +0200},
	Date-Modified = {2019-05-20 11:32:25 +0200},
	Journal = {Journal of the Royal Statistical Society: Series B (Methodological)},
	Number = 1,
	Pages = {221--234},
	Title = {The Exact and Asymptotic Distributions of {Cram{\'e}r-Von} Mises Statistics},
	Volume = 58,
	Year = 1996}

@article{Robert:76:00,
	Abstract = {[Consider two data matrices on the same sample of n individuals,
               X(p $\times$ n), Y(q $\times$ n). From these matrices,
               geometrical representations of the sample are obtained as two
               configurations of n points, in Rp and Rq. It is shown that the
               RV-coefficient (Escoufier, 1970, 1973) can be used as a measure
               of similarity of the two configurations, taking into account the
               possibly distinct metrics to be used on them to measure the
               distances between points. The purpose of this paper is to show
               that most classical methods of linear multivariate statistical
               analysis can be interpreted as the search for optimal linear
               transformations or, equivalently, the search for optimal metrics
               to apply on two data matrices on the same sample; the optimality
               is defined in terms of the similarity of the corresponing
               configurations of points. which, in turn, calls for the
               maximization of the associated RV-coefficient. The methods
               studied are principal components, principal components of
               instrumental variables, multivariate regression, canonical
               variables, discriminant analysis; they are differentiated by the
               possible relationships existing between the two data matrices
               involved and by additional constraints under which the maximum
               of RV is to be obtained. It is also shown that the
               RV-coefficient can be used as a measure of goodness of a
               solution to the problem of discarding variables.]},
	Author = {Robert, P and Escoufier, Y},
	Date-Added = {2019-05-17 10:53:50 +0200},
	Date-Modified = {2019-05-17 10:54:04 +0200},
	Journal = {J. R. Stat. Soc. Ser. C Appl. Stat.},
	Number = 3,
	Pages = {257--265},
	Publisher = {[Wiley, Royal Statistical Society]},
	Title = {A Unifying Tool for Linear Multivariate Statistical Methods: The {RV-} Coefficient},
	Volume = 25,
	Year = 1976}

@article{Mayer:11:00,
	Abstract = {The integration of multiple high-dimensional data sets (omics
              data) has been a very active but challenging area of
              bioinformatics research in recent years. Various adaptations of
              non-standard multivariate statistical tools have been suggested
              that allow to analyze and visualize such data sets
              simultaneously. However, these methods typically can deal with
              two data sets only, whereas systems biology experiments often
              generate larger numbers of high-dimensional data sets. For this
              reason, we suggest an explorative analysis of similarity between
              data sets as an initial analysis steps. This analysis is based on
              the RV coefficient, a matrix correlation, that can be interpreted
              as a generalization of the squared correlation from two single
              variables to two sets of variables. It has been shown before
              however that the high-dimensionality of the data introduces
              substantial bias to the RV. We therefore introduce an alternative
              version, the adjusted RV, which is unbiased in the case of
              independent data sets. We can also show that in many situations,
              particularly for very high-dimensional data sets, the adjusted RV
              is a better estimator than previously RV versions in terms of the
              mean square error and the power of the independence test based on
              it. We demonstrate the usefulness of the adjusted RV by applying
              it to data set of 19 different multivariate data sets from a
              systems biology experiment. The pairwise RV values between the
              data sets define a similarity matrix that we can use as an input
              to a hierarchical clustering or a multidimensional scaling. We
              show that this reveals biological meaningful subgroups of data
              sets in our study.},
	Author = {Mayer, Claus-Dieter and Lorent, Julie and Horgan, Graham W},
	Date-Added = {2019-05-17 10:52:46 +0200},
	Date-Modified = {2019-05-17 10:52:51 +0200},
	Journal = {Stat. Appl. Genet. Mol. Biol.},
	Language = {en},
	Pages = {Article 14},
	Title = {Exploratory analysis of multiple omics datasets using the adjusted {RV} coefficient},
	Volume = 10,
	Year = 2011}

@article{El-Ghaziri:15:00,
	Abstract = {We review three measures of association between two datasets in
              view of their use in sensory data. The aim is threefold: (i) to
              show in which situations each measure of association is
              appropriate, (ii) to show their properties and how they can be
              applied efficiently to sensory data, (iii) to compare them. The
              three measures of association are multivariate correlation
              coefficient, RV coefficient and Procrustes similarity index. A
              particular emphasis is put on RV coefficient since it is very
              popular among sensory scientists. We stress the properties and
              shortcomings of this coefficient and propose an adjusted RV
              coefficient to be used instead of RV coefficient, particularly in
              situations where the number of samples is small or/and the number
              of variables is large.},
	Author = {El Ghaziri, Ang{\'e}lina and Qannari, El Mostafa},
	Date-Added = {2019-05-17 10:52:46 +0200},
	Date-Modified = {2019-05-17 10:52:51 +0200},
	Journal = {Food Qual. Prefer.},
	Keywords = {Multivariate correlation coefficient; Procrustes similarity index; RV coefficient; Permutation test; Adjusted RV coefficient},
	Month = mar,
	Pages = {116--124},
	Title = {Measures of association between two datasets; Application to sensory data},
	Volume = 40,
	Year = 2015}

@article{Jackson:95:00,
	Abstract = {Abstract:A multivariate measure of the concordance or
               association between matrices of species abundances and
               environmental variables was generally lacking in ecology until
               recently. Traditional statistical procedures comparing such
               relationships are often unsuitable because of non-linearity
               among species and/or environmental data. To address these
               problems, I propose a randomization test based on Procrustes
               analysis. One matrix is subject to reflection, rigid rotation,
               translation, and dilation to minimize the sum of the squared
               residual deviations between points for each observation and the
               identical observation in the target matrix. This is a classical
               Procrustes approach to matrix analysis. To assess the
               significance of this measure of matrix concordance, I use a
               randomization test to determine whether the sum of residual
               deviations is less than that expected by chance. The PROcrustean
               randomization TEST (PROTEST) may be used with either raw data
               matrices or with multivariate summaries of the original data
               (i.e. both direct or indirect gradient analysis). I provide
               examples of PROTEST analyses with benthic invertebrate
               communities, lake-water chemistry, lake morphology, and lake
               geographic position. Significant concordance between the benthic
               community and both lake-water chemistry and geographic position
               were found. PROTEST results differed from Mantel test results as
               the choice of distance measure with Mantel tests will influence
               the level of significance obtained.},
	Author = {Jackson, Donald A},
	Date-Added = {2019-05-14 18:22:51 +0200},
	Date-Modified = {2019-05-14 18:22:54 +0200},
	Journal = {{\'E}coscience},
	Month = jan,
	Number = 3,
	Pages = {297--303},
	Publisher = {Taylor \& Francis},
	Title = {{PROTEST}: A {PROcrustean} Randomization {TEST} of community environment concordance},
	Volume = 2,
	Year = 1995}

@article{Dray:07:00,
	Abstract = {Multivariate analyses are well known and widely used to identify
              and understand structures of ecological communities. The ade4
              package for the R statistical environment proposes a great number
              of multivariate methods. Its implementation follows the tradition
              of the French school of ``Analyse des Donnees'' and is based on
              the use of the duality diagram. We present the theory of the
              duality diagram and discuss its implementation in ade4. Classes
              and main functions are presented. An example is given to
              illustrate the ade4 philosophy.},
	Author = {Dray, St{\'e}phane and Dufour, Anne-B{\'e}atrice},
	Date-Added = {2019-05-13 19:18:23 +0200},
	Date-Modified = {2019-05-13 19:18:25 +0200},
	Journal = {Journal of Statistical Software, Articles},
	Number = 4,
	Pages = {1--20},
	Title = {The ade4 Package: Implementing the Duality Diagram for Ecologists},
	Volume = 22,
	Year = 2007}

@article{Dixon:03:00,
	Abstract = {Abstract. VEGAN adds vegetation analysis functions to the
              general-purpose statistical program R. Both R and VEGAN can be
              downloaded for free. VEGAN implements several ordination methods,
              including Canonical Correspondence Analysis and Non-metric
              Multidimensional Scaling, vector fitting of environmental
              variables, randomization tests, and various other analyses of
              vegetation data. It can be used for large data. Graphical output
              can be customized using the R language's extensive graphics
              capabilities. VEGAN is appropriate for routine and research use,
              if you are willing to learn some R.},
	Author = {Dixon, Philip},
	Date-Added = {2019-05-13 19:15:06 +0200},
	Date-Modified = {2019-05-13 19:15:09 +0200},
	Journal = {J. Veg. Sci.},
	Month = dec,
	Number = 6,
	Pages = {927--930},
	Title = {{VEGAN}, a package of {R} functions for community ecology},
	Volume = 14,
	Year = 2003}

@book{Bravais:44:00,
	Author = {Bravais, A},
	Date-Added = {2019-05-10 13:28:14 +0200},
	Date-Modified = {2019-05-10 13:28:16 +0200},
	Language = {fr},
	Publisher = {Impr. Royale},
	Title = {Analyse math{\'e}matique sur les probabilit{\'e}s des erreurs de situation d'un point},
	Year = 1844}

@article{Peres-Neto:01:00,
	Abstract = {The Mantel test provides a means to test the association between
              distance matrices and has been widely used in ecological and
              evolutionary studies. Recently, another permutation test based on
              a Procrustes statistic (PROTEST) was developed to compare
              multivariate data sets. Our study contrasts the effectiveness, in
              terms of power and type I error rates, of the Mantel test and
              PROTEST. We illustrate the application of Procrustes
              superimposition to visually examine the concordance of
              observations for each dimension separately and how to conduct
              hypothesis testing in which the association between two data sets
              is tested while controlling for the variation related to other
              sources of data. Our simulation results show that PROTEST is as
              powerful or more powerful than the Mantel test for detecting
              matrix association under a variety of possible scenarios. As a
              result of the increased power of PROTEST and the ability to
              assess the match for individual observations (not available with
              the Mantel test), biologists now have an additional and powerful
              analytical tool to study ecological and evolutionary
              relationships.},
	Author = {Peres-Neto, Pedro R and Jackson, Donald A},
	Date-Added = {2019-05-10 12:59:02 +0200},
	Date-Modified = {2019-05-10 12:59:05 +0200},
	Journal = {Oecologia},
	Keywords = {Mantel test; Matrix association; Multivariate analysis; Procrustes analysis; Randomization test},
	Language = {en},
	Month = oct,
	Number = 2,
	Pages = {169--178},
	Title = {How well do multivariate data sets match? The advantages of a Procrustean superimposition approach over the Mantel test},
	Volume = 129,
	Year = 2001}

@article{Gower:71:00,
	Author = {Gower, J C},
	Date-Added = {2019-04-19 12:11:20 +0200},
	Date-Modified = {2019-04-19 12:11:23 +0200},
	Journal = {Mathematics in the archaeological and historical sciences},
	Pages = {138--149},
	Publisher = {Edinburgh University Press Edinburgh},
	Title = {Statistical methods of comparing different multivariate analyses of the same data},
	Year = 1971}

@article{Lingoes:74:00,
	Abstract = {In connection with a least-squares solution for fitting one
              matrix,A, to another,B, under optimal choice of a rigid motion
              and a dilation, Sch{\"o}nemann and Carroll suggested two measures
              of fit: a raw measure,e, and a refined similarity measure,es,
              which is symmetric. Both measures share the weakness of depending
              upon the norm of the target matrix,B,e.g.,e(A,kB) $\neq$e(A,B)
              fork $\neq$ 1. Therefore, both measures are useless for answering
              questions of the type: ``DoesA fitB better thanA fitsC?''. In
              this note two new measures of fit are suggested which do not
              depend upon the norms ofA andB, which are (0, 1)-bounded, and
              which, therefore, provide meaningful answers for comparative
              analyses.},
	Author = {Lingoes, James C and Sch{\"o}nemann, Peter H},
	Date-Added = {2019-04-19 12:07:12 +0200},
	Date-Modified = {2019-04-19 12:07:17 +0200},
	Journal = {Psychometrika},
	Month = dec,
	Number = 4,
	Pages = {423--427},
	Title = {Alternative measures of fit for the Sch{\"o}nemann-carroll matrix fitting algorithm},
	Volume = 39,
	Year = 1974}

@article{SzeKely:13:00,
	Address = {Orlando, FL, USA},
	Author = {Sz{\'e}Kely, G{\'a}bor J and Rizzo, Maria L},
	Date-Added = {2019-04-19 00:38:28 +0200},
	Date-Modified = {2019-04-19 00:38:39 +0200},
	Journal = {J. Multivar. Anal.},
	Keywords = {Distance correlation, Distance covariance, High dimension, Multivariate independence, dCor, dCov, primary, secondary},
	Month = may,
	Pages = {193--213},
	Publisher = {Academic Press, Inc.},
	Title = {The Distance Correlation T-test of Independence in High Dimension},
	Volume = 117,
	Year = 2013}

@article{Szekely:07:00,
	Abstract = {Project Euclid - mathematics and statistics online},
	Author = {Sz{\'e}kely, G{\'a}bor J and Rizzo, Maria L and Bakirov, Nail K},
	Date-Added = {2019-04-19 00:25:31 +0200},
	Date-Modified = {2019-04-19 00:25:35 +0200},
	Journal = {Ann. Stat.},
	Keywords = {Distance correlation; distance covariance; multivariate independence},
	Language = {en},
	Month = dec,
	Number = 6,
	Pages = {2769--2794},
	Publisher = {Institute of Mathematical Statistics},
	Title = {Measuring and testing dependence by correlation of distances},
	Volume = 35,
	Year = 2007}

@article{Escoufier:73:00,
	Author = {Escoufier, Yves},
	Date-Added = {2019-04-19 00:11:19 +0200},
	Date-Modified = {2019-04-19 00:11:23 +0200},
	Journal = {Biometrics},
	Pages = {751--760},
	Publisher = {JSTOR},
	Title = {Le traitement des variables vectorielles},
	Year = 1973}

@article{Smilde:09:00,
	Abstract = {MOTIVATION: Modern functional genomics generates high-dimensional
              datasets. It is often convenient to have a single simple number
              characterizing the relationship between pairs of such
              high-dimensional datasets in a comprehensive way. Matrix
              correlations are such numbers and are appealing since they can be
              interpreted in the same way as Pearson's correlations familiar to
              biologists. The high-dimensionality of functional genomics data
              is, however, problematic for existing matrix correlations. The
              motivation of this article is 2-fold: (i) we introduce the idea
              of matrix correlations to the bioinformatics community and (ii)
              we give an improvement of the most promising matrix correlation
              coefficient (the RV-coefficient) circumventing the problems of
              high-dimensional data. RESULTS: The modified RV-coefficient can
              be used in high-dimensional data analysis studies as an easy
              measure of common information of two datasets. This is shown by
              theoretical arguments, simulations and applications to two
              real-life examples from functional genomics, i.e. a
              transcriptomics and metabolomics example. AVAILABILITY: The
              Matlab m-files of the methods presented can be downloaded from
              http://www.bdagroup.nl.},
	Author = {Smilde, A K and Kiers, H A L and Bijlsma, S and Rubingh, C M and van Erk, M J},
	Date-Added = {2019-04-19 00:08:07 +0200},
	Date-Modified = {2019-04-19 00:08:17 +0200},
	Journal = {Bioinformatics},
	Language = {en},
	Month = feb,
	Number = 3,
	Pages = {401--405},
	Title = {Matrix correlations for high-dimensional data: the modified {RV-coefficient}},
	Volume = 25,
	Year = 2009}

@article{Ramsay:84:00,
	Abstract = {A correlational measure for ann byp matrixX and ann byq matrixY
              assesses their relation without specifying either as a fixed
              target. This paper discusses a number of useful measures of
              correlation, with emphasis on measures which are invariant with
              respect to rotations or changes in singular values of either
              matrix. The maximization of matrix correlation with respect to
              transformationsXL andYM is discussed where one or both
              transformations are constrained to be orthogonal. Special
              attention is focussed on transformations which causeXL andYM to
              ben bys, wheres may be any number between 1 and min (p, q). An
              efficient algorithm is described for maximizing the correlation
              betweenXL andYM where analytic solutions do not exist. A factor
              analytic example is presented illustrating the advantages of
              various coefficients and of varying the number of columns of the
              transformed matrices.},
	Author = {Ramsay, J O and ten Berge, Jos and Styan, G P H},
	Date-Added = {2019-04-18 23:38:55 +0200},
	Date-Modified = {2019-04-18 23:39:01 +0200},
	Journal = {Psychometrika},
	Month = sep,
	Number = 3,
	Pages = {403--423},
	Title = {Matrix correlation},
	Volume = 49,
	Year = 1984}
