Power transformations in correspondence analysis

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1 Powe tansfomations in oespondene analysis Mihael Geenae Depatment of Eonomis and Business Univesitat Pompeu Faba Ramon Tias Fagas, Baelona SPAIN Abstat: Powe tansfomations of positive data tables, pio to applying the oespondene analysis algoithm, ae shown to open up a family of methods with diet onnetions to the analysis of log-atios Two vaiations of this idea ae illustated The fist appoah is simply to powe the oiginal data and pefom a oespondene analysis this method is shown to onvege to unweighted log-atio analysis as the powe paamete tends to zeo The seond appoah is to apply the powe tansfomation to the ontingeny atios, that is the values in the table elative to expeted values based on the maginals this method onveges to weighted log-atio analysis, o the spetal map Two appliations ae desibed: fist, a matix of population geneti data whih is inheently two-dimensional, and seond, a lage oss-tabulation with highe dimensionality, fom a linguisti analysis of seveal books Keywods: Box-Cox tansfomation, hi-squae distane, ontingeny atio, oespondene analysis, log-atio analysis, powe tansfomation, atio data, singula value deomposition, spetal map This pape, pesented at the CARME007 onfeene in Rottedam in June 007, has been submitted to the speial issue on Coespondene Analysis and Related Methods of the jounal Computational Statistis & Data Analysis This eseah has been suppoted by the Fundaión BBVA, Madid, Spain Patial suppot of Ministy of Siene and Eduation gant BSM-06/357 is also gatefully aknowledged

2 Intodution Coespondene analysis (CA) has found aeptane and appliation by a wide vaiety of eseahes in diffeent disiplines, notably the soial and envionmental sienes (fo an up-to-date aount, see Geenae, 007a) The method has also appeaed in the majo statistial softwae pakages, fo example SPSS, Minitab, Stata, SAS, Statistia and XLSTAT, and seveal implementations in R (R Development Coe Team, 007) ae feely available (see, fo example, Nenadić and Geenae, 007, and the web site The method is outinely applied to a table of non-negative data to obtain a spatial map of the impotant dimensions in the data, whee poximities between points and othe geometi featues of the map indiate assoiations between ows, between olumns and between ows and olumns Methods based on log-atios have quite diffeent oigins in the physial sienes, notably hemisty and geology, (Aithison, 986) and lead to maps whee vetos between points depit the logaithms of the atios between data values in the oesponding pais of ows o olumns This methodology is patiulaly popula in the analysis of ompositional data (see, fo example, Aithison and Geenae, 00) Inteestingly, this log-atio analysis (LRA), with the slight but highly signifiant adaptation of weighting the ows and olumns of the table popotional to thei maginal totals (exatly as is done in CA), has been used extensively fo thee deades in the phamaeutial industy, oiginated by Lewi (976) In this ontext it has been alled the spetal map beause it depits the infomation fom biologial ativity speta The spetal map, whih we othewise all weighted LRA to distinguish it fom the unweighted fom, an also be used to analyse ontingeny tables (see Lewi, 998), in fat any atio-sale data, as long as thee ae no zeo values It is known that in spatial maps podued by LRA (unweighted o weighted), points that line up appoximately as staight lines suggest equilibium models in the ows o olumns oesponding to those points (fo example, see Aithison and Geenae, 00) CA does not have this popety, but has the advantage that it outinely handles data with zeo values, whih is one of the easons why it is so popula in eology and ahaeology, whee data tables ae often quite lage and spase So the pesent situation is one of two ompeting methods, eah with its patiula advantages, and no appaent diet theoetial link between them, apat fom the fat that both ae based on singulavalue deompositions It is known that CA and weighted LRA give vey simila esults if the vaiane in the table is low (this is a esult of the appoximation log(+x) x when x is small), but diffe when the vaiane is high see Geenae and Lewi (005) o Cuadas, Cuadas and Geenae (006) In the pesent pape we show that thee is a muh lose theoetial affinity between the two methods, in fat they both belong to the same family of methods defined by powe tansfomations

3 of eithe the oiginal data o etain atios alulated fom the data The powe tansfomation family, as embodied in the Box-Cox tansfomation (Box and Cox, 964), is usually used in statistis to symmetize the distibution of a esponse vaiable in a egession model to satisfy the egession assumptions (Hinkley, 975) In the analysis of fequeny data, assuming the ounts follow a Poisson distibution, the squae oot tansfomation is used to stabilize the vaiane (Batlett, 936) In eology abundane data is almost always highly ove-dispesed and a patiula shool of eologists outinely applies a fouth-oot tansfomation befoe poeeding with statistial analysis (Field, Clake & Wawik, 98) Hee we study the family of powe tansfomations in the ontext of oespondene analysis (CA) Some speial ases emege, notably the spetal map, whih is a limiting ase as the powe tansfomation paamete tends to zeo The main esult in this pape is thanks to the Box-Cox tansfomation f(x) = (/α) (x α ), whih onveges to log(x) as α tends to 0 Beause we ae making a ompaison with LRA, only stitly positive data will be onsideed In Setion we give two equivalent definitions of CA and show how powe tansfomations an geneate two espetive families of methods, the fist giving a diet link between CA and unweighted LRA, and the seond a diet link between CA and weighted LRA Popeties of these families ae illustated in Setion 3 using two examples, a data matix fom population genetis with high inheent vaiane, and a linguisti example with vey low vaiane Setions 4 and 5 teat elated topis and liteatue, and Setion 6 onludes with a disussion An Appendix is given with R ode that pemits viewing the smooth passage fom a CA map to a spetal map, as the powe paamete vaies fom to 0 (this ode is available fo download fom Powe families: fom oespondene analysis to log-atio analysis CA and LRA ae two of the many multivaiate methods based on the singula value deomposition (SVD) (see, fo example, Geenae 984: hapte 3) In the geometi intepetation of the SVD, the ows and/o olumns of the data matix define points in a multidimensional spae and the SVD identifies subspaes of low dimensionality whih aptue maximum sum-of-squaes in the data Diffeent weights fo the ows and olumns an be intodued into this sheme so that weighted sum-of-squaes is deomposed The weighting an be onsideed eithe as assigning diffeent weights to eah point, o as a hange of the Eulidean meti to a weighted one, o both of these at the same time, as is the ase in CA To establish notation, the following is a summay of standad theoy to obtain the pinipal oodinates of the ow and olumns points in a so-alled symmeti CA map (fo moe details, see Geenae, 007a), followed by the theoy fo the ompaable LRA map 3

4 Coespondene analysis Suppose that N is an I J table of non-negative data Fist divide N by its gand total n to obtain the so-alled oespondene matix P = (/n) N Let the ow and olumn maginal totals of P be the vetos and espetively these ae the weights, o masses, assoiated with the ows and olumns Let D and D be the diagonal maties of these masses The omputational algoithm to obtain oodinates of the ow and olumn pofiles with espet to pinipal axes, using the SVD, is as follows: Coespondene analysis Calulate the matix of standadized esiduals: S / T / = D ( P ) D () Calulate the SVD: S T = UDσ V whee U T U = V T V = I () / 3 Pinipal oodinates of ows: F = D UDσ (3) / 4 Pinipal oodinates of olumns: G = D VDσ (4) The ows of the oodinate maties in (3) and (4) efe to the ows o olumns, as the ase may be, of the oiginal table, while the olumns of these maties efe to the pinipal axes, o dimensions, of the solution The sum of squaes of the deomposed matix S is a quantity alled the total inetia, o simply inetia, of the data table: inetia = ( p ) I J I J ij i j ij φ = = i j (5) i= j= i j i= j= i j p The inetia is exatly Peason s mean-squae ontingeny oeffiient, that is the Peason hi-squae statisti fo the table divided by the gand total n of the table, and is used as a measue of total vaiane The squaed singula values σ k deompose the inetia, and the ow and olumn pinipal T T oodinates ae saled in suh a way that F D F G D G = = D σ, ie the weighted sum-of-squaes of the oodinates on the k-th dimension (o thei inetia in the dietion of this dimension) is equal to σ k, alled the pinipal inetia (o eigenvalue) on dimension k A two-dimensional solution, say, would use the fist two olumns of the oodinate maties, and the explained inetia aounted fo in the two-dimensional solution is the sum of the fist two tems σ +σ, usually expessed as peentages of the total inetia Standad oodinates ae defined as in (3) and (4) without saling on the ight by the singula values D σ, hene have weighted sum-of-squaes equal to 4

5 Notie in (5) how the inetia an be expessed in tems of eithe ontingeny diffeenes p ij i j o ontingeny atios p ij / i j The S matix in () an thus be witten equivalently as follows, in tems of the matix of ontingeny atios Q = D PD : S = D I D PD / T T T / ( )( )( ) I D (6) The pe- and post-multipliation of Q by the enting maties (I T ) and (I T ) T amounts to a weighted double-enting of the ontingeny atios This seond definition of CA is patiulaly useful fo ompaing with LRA Logatio analysis, weighted and unweighted A weighted LRA (ie, spetal map) is based on the logaithms of the elements of N: L = [log(n ij )]; hene we only onside stitly positive data hee Using the same masses and as in CA, the matix is then double-ented, and then a weighted SVD is pefomed, as summaized in the following omputational sheme: Weighted log-atio analysis (spetal map) Calulate the weighted, double-ented matix: S = D I L I D (7) / T T T / * ( ) ( ) Calulate the SVD: T S * = UDµ V whee U T U = V T V = I (8) / 3 Pinipal oodinates of ows: F = D UDµ (9) / 4 Pinipal oodinates of olumns: G = D VDµ (0) Notie that steps (8) (0) ae idential to () (4) of CA It is just the pe-poessing and fist step (7) that diffes The unweighted LRA is obtained simply by setting = (/I ) and = (/J ) in the above sheme, so that the initial matix S* is S* = ( IJ ) ( I (/ I ) ) L( I (/ J ) / T T ) () Sine the logaithm of the ontingeny atios is log(n ij ) log (n) log( i ) log( j ), and the doubleenting emoves the onstant log(n) and main effets log( i ) and log( j ), the only diffeene between the initial maties S and S* is that in (6) CA opeates on the ontingeny atios wheeas in (7) weighted LRA opeates on the log-tansfomed ontingeny atios 5

6 The total vaiane in weighted LRA (ie, the sum of squaes of matix S* in (7)) an be witten in tems of the logaithms of the double-atios : i i < j< j i n log n () ij i j i j j ni j nij Fo the unweighted LRA, again eplae the ow masses by (/I ) and the olumn masses by (/J ) 3 Powe families of analyses geneated by powe tansfomations The two foms of CA stating fom the oespondene matix in () o the ontingeny atios in (6) suggest two ways of intoduing a powe tansfomation Powe family : Pe-tansfom the matix P (o, equivalently N), by the powe tansfomation p ij (α) = p α ij Afte dividing out this matix by its total to obtain the new oespondene matix and ealulating the ow and olumn masses, poeed as in () to alulate the matix to be deomposed, denoted by S(α), and then ontinue as in () (4) above To standadize the analyses with diffeent values of the powe paamete α the singula values σ k ae divided by α, so the inetia is divided by α this is equivalent to dividing S(α) by α befoe applying the SVD Powe family : Pe-tansfom the matix Q of ontingeny atios by the powe tansfomation q ij (α) = q α ij Calulate S*(α) using the powe-tansfomed ontingeny atios, as in (6), followed by () (4) In this ase the masses i and j ae maintained onstant thoughout, equal to thei oiginal values iespetive of α Again, to standadize the analyses with diffeent values of the powe paamete α the singula values σ k ae divided by α, so the inetia is divided by α this is equivalent to dividing S*(α) by α befoe applying the SVD, o to dividing the powe-tansfomed ontingeny atios q ij (α) by α befoe double-enting and deomposing In powe family, whethe we double-ente (/α) q α α ij o (/α) (q ij ) makes no diffeene at all, beause the onstant tem will be emoved Hene, the analysis in this ase amounts to the Box-Cox tansfomation of the ontingeny atios: ( α ) q (3) α ij whih onveges to log(q ij ) as α 0 This shows that powe family onveges to weighted LRA as α 0 6

7 In powe family, we ae also analysing ontingeny atios of the fom (/α) q α α ij, o (/α) (q ij ), but then the atios as well as the weights and double-enting ae all with espet to ow and olumn masses that ae hanging with α At the limit as α 0, these masses tend to onstant values, ie /I fo the ows and /J fo the olumns; hene this shows that the limiting ase of powe family is the analysis of the logaithms with onstant masses, o unweighted LRA 3 Appliations 3 Two-dimensional example: the M-N system in population genetis If the data ae inheently two-dimensional then thee will be little diffeene in the unweighted and weighted LRA solutions, just a slight otation of the pinipal axes, so this seves as a good demonstation of the diffeene between the CA and LRA onfiguations This is the ase with the data set in Table fom population genetis, onening the estimated fequenies in 4 populations of thee goups in the M-N geneti system The two alleles, M and N, in this system ae odominant, so that the thee goups ae MN, M (denoting MM) and N (denoting NN) Figue shows the tansition in powe family, with fixed masses, fom CA (α = ) to weighted LRA (limit as α 0, ie log-tansfomation) in thee steps: α=075, α=050 and α=05 (in the Appendix the R ode is given to see a dynami smooth hange fom CA to LRA, using smalle steps α=099, 098,, 00, 00) This example is inteesting beause the CA solution shows the wellknown ah effet, with 864% inetia on the fist axis, and thus 56% on the seond As α desends the uve stats to staighten out until at the limit of the weighted LRA, the onfiguation is patially one-dimensional with 968% explained inetia on the fist pinipal axis (the inset boxes show the evolution of the total vaiane the uppe uve and the two pinipal inetias the two lowe uves as α desends fom to 0) The lineaity of M, MN and N in the final weighted LRA and the almost equal distane between the thee points imply a model fo the logatios: log(mn/m) = log(n/mn) + onstant, whih pefetly diagnoses the Hady-Weinbeg equilibium fo this geneti system: MN / M N = 4 (see, fo example, Geenae, 007b) The esult fo powe family, with hanging masses, is almost idential in this two-dimensional ase, the only notieable diffeene being the way the total inetia and the pats of inetia ae measued, sine the limiting ase as α 0 is the unweighted LRA, whee the peentage of inetia explained by the fist axis is slightly highe, 97% 7

8 3 Highe-dimensional example: the autho data The data set autho (see Geenae and Lewi, 005; Geenae, 007a: Chapte 0) onsists of ounts of the lettes a to z in samples of texts fom books (o haptes of books) by six famous English authos (Table ) This data set has an extemely low inetia, sine thee ae vey small diffeenes in the elative fequenies of the lettes, but the diffeenes between authos is still substantively meaningful Thee is one zeo value in this table (a ount of zeo ouenes of the lette q in the sample of text fom Faewell to Ams by Hemingway), whih we have eplaed by a, othewise LRA beaks down It is aleady known that CA and LRA will esemble one anothe when the inetia is low (Geenae and Lewi, 005; Cuadas et al, 006) Figue shows CA in the fist panel, weighted LRA in the last panel and the analysis of the powe-tansfomed ontingeny atios with α = 050 in the middle panel The diffeenes between the onfiguations of the books ae mino, as expeted, and the umulated peentage of inetia explained by the fist two axes is slightly lowe in the LRA map The benefit of the LRA appoah is that lettes that fom staight lines indiate linea models in the oesponding log-atios Fo example, as shown by Geenae and Lewi (005), the staight line fomed by k, y and x in the last panel of Figue indiates an equilibium elationship between these thee lettes whih amounts to: y x 0 k 08 In the CA map (fist panel of Figue ) suh elationships an not be diagnosed 4 Connetion with Hellinge analysis and distane Cuadas et al (006) have studied the onnetion between CA and an altenative to CA defined by Rao (995) based on Hellinge distanes This an also be thought of as a powe-tansfomed family if we stat fom the following equivalent fom of the matix S in () o (6) as follows, in tems of ow pofiles (the ows of P divided by thei ow sums, ie, the ows of D - P): S D D P D / T / = ( ) (4) Sine Rao s Hellinge analysis (HA) is based on the SVD of the matix: ~ / / T / S = D [( D P) ( ) ] (5) whih an be witten as: ~ S = D / [( D PD ) / T ] D / this suggests anothe family based on the powe tansfomation of the ontingeny atios D - P D - : 8

9 ~ T S( α ) = D [( D PD ] D (6) / α ) / (again, we would multiply this matix by /α befoe deomposing with the SVD) This family passes smoothly fom CA to HA as α hanges value fom to 05 (Cuadas, 007) HA does not seem to have any patial benefit ove CA o LRA, apat fom the advantage that is laimed that the meti between the ows does not depend on the olumn magins, as is the ase in CA Figue 3 shows the M-N example fo this family with α =, 075, 05 Thee is hadly any hange in the ow onfiguation and the peentage of inetia on the fist dimension, afte an initial inease, is less in HA This data set is two-dimensional in CA and LRA and in both powe families desibed in Setion 3, but is thee-dimensional in the ase of HA and the powe family desibed hee whih leads to it (apat fom the ase α =, whih is CA and thus two-dimensional) The intodution of a thid dimension ould be deemed a disadvantage of HA beause a size effet has now been mixed in with the analysis, wheeas CA and LRA onentate only on shape effets Bavaud (00, 004) looks at families of dissimilaity measues based on the ontingeny atios q ij, defined, fo example, between ows as: ( f q ) f ( q )) ( whee f ( q α ) = ( ) α q (7) j j ij i j fo whih α = gives the hi-squae distane, α = ½ gives the Hellinge distane (ie, the unweighted Eulidean distane between squae oot-tansfomed ow pofiles), and the limit as α tends to 0 gives a weighted distane based on the logaithms of the ow pofiles: ( ln( p / ) ln( p / )) j j ij i i j i Notie that this distane funtion is simila but not the same as the one impliit in weighted LRA, whih divides elements p ij in eah ow by thei espetive weighted geometi mean p p L p J i i ij, not by thei sum i 5 Powe vesions of elated methods The same idea an be applied to many elated methods, suh as multidimensional saling (MDS) and so-alled nonsymmeti oespondene analysis (NSCA) NSCA (see, fo example, Koonenbeg and Lombado 999) is a slight vaiation of CA theoy, also involving a genealized SVD of a matix The ows and olumns ae teated diffeently, depending on whethe the data ae onsideed as pediting the ows given the olumns, o the olumns given the ows Fo example, in the latte ase: 9

10 Nonsymmeti oespondene analysis fo pediting olumns, given ows Calulate the matix: ( S = D ( D P / T ( Calulate the SVD: S = UD σ V ) T (8) whee U T U = V T V = I (9) / 3 Pinipal oodinates of ows: F = D UDσ (0) 4 Pinipal oodinates of olumns: G = VDσ () Compae (8) with (4) the only diffeene is that the post-multipliation by D ½ is omitted Atually, to make a diet ompaison with CA, an equal weighting of /J should be intodued fo the olumns, ie, D ½ in the CA fomulation (4) should be eplaed by (/J) / = J / Vaious powe vesions an be onsideed, as befoe with hanging o fixed (ow) masses: (i) powe up the oiginal data α p ij, in whih ase the ow masses will hange aoding to α, also onveging to equal masses at the limit; (ii) powe up the pofiles α p / ) and aveage pofile ( ij i α j, keeping the ow masses equal to the oiginal ones fo all α ; (iii) α α powe up the ontingeny atios p ij /( i j ), keeping the ow and olumn masses equal to the oiginal ones fo all α Altenatively, we an use the idea of paametizing an analysis by showing the diffeene between CA and NSCA by inopoating a paamete, β say, whih allows a tansition fom one weighting system to anothe Fo example, let D w = β D + ( β )(/ J) I and eplae steps (8) and () above by: Matix to be deomposed ( S D D P D / T / = ( ) w () Pinipal oodinates of olumns: / G = D w VD (3) σ As β vaies fom to 0 the esulting maps will pass smoothly fom CA to NSCA espetively, whee the equal weighting of /J has been intodued into the NSCA definition Figue 4 shows thee snapshots of the tansition sine the ow masses ae appoximately equal thee is vey little hange in the onfiguations and peentages of inetia, only an inease in the inetias fo the nonsymmetial vesion 0

11 The same idea an be used to ompae CA with PCA in tems of thei espetive standadizations of the matix olumns, say, whee CA standadizes by the squae oot of the mean and PCA by the standad deviation This would make sense if the two methods wee analysing ompaable equalweighted ows, fo example if the ows add up to fo data that ae popotions (o peentages adding up to 00%) so that the pofiles wee the oiginal data and all ows eeived the same mass As befoe, the standadization ould be defined paametially as post-multipliation of the data matix by γ D ½ + ( γ ) D s, whee the olumns masses (means in this ase) ae in the diagonal of D and the olumn standad deviations ae in the diagonal of D s, so that as γ vaies fom to 0 the esulting maps pass smoothly fom CA to PCA In MDS we ae tying to math obseved distanes d ij with fitted distanes δ ij in a map To edue the influene of lage distanes in the fitting poess, a powe tansfomation an be intodued, fo example: d o ij α (( + d ) ) ( α) = ij α (4) This stats with the oiginal distanes when α = and onveges to a logaithmi tansfomation log(+ d ij ) as α 0 Caoll, Kumbasa and Romney (997) showed a diffeent onnetion between CA and MDS that is not govened by a powe tansfomation but is a limiting esult in the same spiit as those pesented hee Thei esult was that the CA of a suitably tansfomed distane matix has as a limiting ase lassial meti MDS (Householde-Togeson-Gowe saling, alled HoToGo saling by Willem Heise at CARME 007) We give Caoll et al s esult in ou pesent notation Suppose [d ii' ] is an I I squae matix of obseved distanes, and define a new table as follows: = (5) α n ii dii whee α > 0 and /α max{ d }, ie, squaed distanes ae subtated fom a numbe at least as ii lage as thei maximum so that the n ii' ae all nonnegative Then the CA of the matix N = [n ii' ] onveges to the lassial meti MDS solution as α 0 As in all ases above, a esaling needs to be intodued to make the solutions equivalent In the ase of CA, we pefom steps () and () on the oespondene matix P based on (3) and then the MDS oodinates ae: / / H = D UDσ (6) α

12 Hene H onsists of the standad oodinates D / U saled by the squae oots of the singula values (ie, the fouth oots of the inetias in the CA of N), then esaled by dividing by α The eigenvalues of the lassial MDS an be eoveed fom (I / α)σ k, emembeing that all these esults apply in the limit in patie, an α about one thousandth of the maximum of the 000 times this maximum, gives a solution vey lose to the MDS one d ii, ie /α about 6 Disussion and onlusion We have shown that CA and both unweighted and weighted LRA an be onneted by onsideing the powe tansfomation of the oiginal data matix o the matix of ontingeny atios espetively Fo the powe paamete α equal to we have simple CA in both ases, and as α tends to 0 we obtain unweighted o weighted ases espetively This shows that LRA is theoetially pat of the same family as CA, and not as diffeent as one might have thought The onnetion is espeially supising fo CA and the spetal map (weighted LRA) beause the two methods have been developed and applied extensively fo ove 30 yeas as ompletely sepaate methodologies The idea of linking methods by a paamete and espeially the dynami visualization of smooth hanges fom one method to anothe an be highly enlightening as to the popeties of these methods Vaious othe methods an be linked to CA in this way, as we have shown: CA to Hellinge analysis, CA to NSCA, in some ases CA to PCA and CA to vaious types of MDS Unfotunately, in these pages we an only show snapshots of some steps between the methods fo seleted values of the powe paamete, but we have povided one example of R ode in the appendix this ode an be used to get an idea of the dynami gaphis possibilities, and is easily adaptable to the othe ases desibed above The distintion between singula values, eigenvalues and inetias beomes a bit onfusing in this ase whee N is a squae matix The singula values of N ae atually eigenvalues (at least those oesponding to positive eigenvalues), and the inetias in the CA of N (often efeed to as eigenvalues) ae the squaes of the singula values of N

13 Refeenes Aithison, J (986) The Statistial Analysis of Compositional Data London: Chapman & Hall Aithison, J & Geenae, M J (00) Biplots of ompositional data Applied Statistis, 5, Batlett, M S (936) The squae oot tansfomation in analysis of vaiane Supplement to the Jounal of the Royal Statistial Soiety, 3, Bavaud, F (00) Quotient dissimilaities, Eulidean embeddability, and Huygens weak piniple In Data Analysis, Classifiation and Related Methods, eds H A L Kies et al, Heidelbeg: Spinge Bavaud, F (004) Genealized fato analyses fo ontingeny tables In Classifiation, Clusteing and Data Mining Appliations, eds D Banks et al, Heidelbeg: Spinge Box, G E P & Cox, D R (964) An analysis of tansfomations (with disussion) Jounal of Royal Statistial Soiety, Seies B, 35, Caoll, J D, Kumbasa, E and Romney, A K (997) An equivalene elation between oespondene analysis and lassial meti multidimensional saling fo the eovey of Eulidean distanesbitish Jounal of Mathematial and Statistial Psyhology, 50, 8 9 Cuadas, D (007) Contibuions a la genealitzaió de l anàlisi de omponents pinipals i de oespondènies Dotoal thesis, Univesity of Baelona Cuadas, C, Cuadas, D and Geenae, M J (005) A ompaison of methods fo analyzing ontingeny tables Communiations in Statistis Simulation and Computation, 35, Field, J G, Clake, K R &Wawik, R M (98) A patial stategy fo analysing multispeies distibution pattens Maine Eology Pogess Seies, 8, 37 5 Geenae, M J (007a) Coespondene Analysis in Patie Seond Edition London: Chapman & Hall / CRC Pess Geenae, M J (007b) Diagnosing models fom maps based on weighted logatio analysis Poeedings of Intenational Wokshop on Statistial Modelling, -6 July, Baelona URL Geenae, M J & Lewi, P J (005) Distibutional equivalene and subompositional oheene in the analysis of ontingeny tables, atio sale measuements and ompositional data Eonomis Woking Pape 908, Univesitat Pompeu Faba Unde evision fo publiation Oiginal unevised vesion available online at URL 3

14 Hinkley, D (975) On powe tansfomations to symmety Biometika, 6, 0 Koonenbeg, P M and Lombado, R (999) Nonsymmeti oespondene analysis: a tool fo analysing ontingeny tables with a dependene stutue Multivaiate Behavioal Reseah, 34, Lewi, P J (976) Spetal mapping, a tehnique fo lassifying biologial ativity pofiles of hemial ompounds Azneim Fosh (Dug Res), 6, Lewi, P J (998) Analysis of ontingeny tables In B G M Vandeginste, D L Massat, L M C Buydens, S de Jong, P J Lewi, J Smeyes-Vebeke (eds), Handbook of Chemometis and Qualimetis: Pat B, Chapte 3, pp 6 06 Amstedam: Elsevie Nenadić, O & Geenae, M J (007) Coespondene analysis in R, with two- and theedimensional gaphis: The a pakage Jounal of Statistial Softwae, to appea R Development Coe Team (005) R: A Language and Envionment fo Statistial Computing R Foundation fo Statistial Computing, Vienna, Austia URL Rao, C R (995) A eview of anonial oodinates and an altenative to oespondene analysis using Hellinge distane Qüestiió, 9,

15 Table Data set M-N : estimated popotions of thee geneti goups of the M-N system, with two o-dominant alleles M and N Population MN M N

16 Table Books fom whih text is sampled fo the autho data, and abbeviations used in Figue TD-Bu Thee Daughtes (Buk) FA-He Faewell to Ams (Hemingway) EW-Bu East Wind (Buk) Is-He Islands (Hemingway) D-Mi The Diftes (Mihene) SF6-Fa Sound and Fuy, h6 (Faulkne) As-Mi Asia (Mihene) SF7-Fa Sound and Fuy, h7 (Faulkne) LW-Cl Lost Wold (Clak) Pe-Ho Pendoi, h (Holt) PF-Cl Pofiles of the Futue (Clak) Pe3-Ho Pendoi, h3 (Holt) 6

17 Figue : Fom oespondene analysis (α = ) to weighted log-atio analysis (α 0), with thee intemediate steps, fo the M-N data, showing the symmeti maps (both ows and olumns in pinipal oodinates) The box shows a gaph of the values of the total inetia and two pinipal inetias as α deeases and the numeial value of α and the peentage of inetia explained on the fist dimension M MN N M MN N M MN 4 3 N M MN 4 3 N M MN 4 3 N

18 TD-Bu D-Mi LW-Cl EW-Bu FA-He SF7-Fa SF6-Fa PF-Cl Is-He Pd3-Ho As-Mi Pd-Ho a b d e f g h ij k l m n o p q s t u v w x y z TD-Bu D-Mi LW-Cl EW-Bu FA-He SF7-Fa SF6-Fa PF-Cl Is-He Pd3-Ho As-Mi Pd-Ho a b d e f g h i j k l m n o p q s t u v w x y z TD-Bu D-Mi LW-Cl EW-Bu FA-He SF7-Fa SF6-Fa PF-Cl Is-He Pd3-Ho As-Mi Pd-Ho a b d e f h g i j k l m n o p q s t u v w x y z Figue : Fom oespondene analysis (α = ) to weighted log-atio analysis (α 0), with one intemediate hybid analysis (α = ½) fo the autho data, showing the symmeti maps The box shows a gaph of the values of the total inetia and two pinipal inetias as α deeases and the numeial value of α and the peentage of inetia explained in the two-dimensional map

19 Figue 3: Fom oespondene analysis (α = ) to Hellinge analysis (α = 05) fo the M-N data The box shows a gaph of the values of the total inetia and two pinipal inetias as α deeases and the numeial value of α and the peentage of inetia explained on the fist dimension M MN N M MN N M MN N

20 TD-Bu D-Mi LW-Cl EW-Bu FA-He SF7-Fa SF6-Fa PF-Cl Is-He Pe3-Ho As-Mi Pe-Jo a b d e f g h ij k l m n o p q s t u v w x y z 6059 Figue 4: Fom oespondene analysis (β = ) to non-symmetial oespondene analysis (β = 0) fo the autho data, showing one intemediate hybid step (β = ½ ) The asymmeti map is shown with olumns in pinipal and ows in standad oodinates; the olumn (lette) pinipal oodinates have been multiplied by 4 fo bette legibility The box shows a gaph of the values of the total inetia and two pinipal inetias as β deeases and the numeial value of β and the peentage of inetia explained in the map TD-Bu D-Mi LW-Cl EW-Bu FA-He SF7-Fa SF6-Fa PF-Cl Is-He Pe3-Ho As-Mi Pe-Jo a b d e f g h i j k l m n o p q s t u v w x y z TD-Bu D-Mi LW-Cl EW-Bu FA-He SF7-Fa SF6-Fa PF-Cl Is-He Pe3-Ho As-Mi Pe-Jo a b d e f g h i j k l m n o p q s t u v w x y z

21 APPENDIX R ode to show dynami tansition fom CA to weighted LRA/spetal map (fo M-N data); lines stating with + indiate a ontinuation fom the pevious line # peliminaies nk<- xlim=(-5,5) ylim=(-05,05) sing<-matix(0,now=00,nol=nk) ine<-ep(0,00) # assume N is the oiginal data matix P <- N/sum(N) m <- apply(p,,sum) m <- apply(p,,sum) # analysis of ontingeny atios fo(i in 00:){ alpha<-i/00 Y <- diag(/(m^alpha)) %*% (P^alpha) %*% diag(/(m^alpha)) Y<-(/alpha)*Y m <- t(y) %*% asveto(m) Y <- Y - ep(,now(p)) %*% t(m) m <- Y %*% asveto(m) Y <- Y - m %*% t(ep(,nol(p))) Z <- diag(sqt(m)) %*% Y %*% diag(sqt(m)) inetia <- sum(z*z) ine[i] <- inetia svdz <- svd(z) sing[i,] <- svdz$d[:nk]^ FF <- diag(/sqt(m)) %*% svdz$u %*% diag(svdz$d) GG <- diag(/sqt(m)) %*% svdz$v %*% diag(svdz$d) if(gg[3,]<0){ FF[,]<- -FF[,] GG[,]<- -GG[,] } if(gg[3,]<0){ FF[,]<- -FF[,] GG[,]<- -GG[,] } plot((ff[,],gg[,]),(ff[,],gg[,]),xlim=xlim, ylim=ylim, + type="n",xlab="dim ",ylab="dim ") text(ff[,],ff[,],labels=ownames(n),ol="blue",font=) text(gg[,],gg[,],labels=olnames(n),ol="ed",font=) lines((xlim[]-05,xlim[],xlim[],xlim[]-05,xlim[]-05), + (ylim[],ylim[],ylim[]+05,ylim[]+05,ylim[])) fo(ii in 00:i){ text(xlim[]-05+05*ii/00,ylim[]+05*ine[ii], "", + ol="blak",font=) text(xlim[]-05+05*ii/00,ylim[]+05*sing[ii,],"",ol="blak") text(xlim[]-05+05*ii/00,ylim[]+05*sing[ii,],"",ol="blak") } text(xlim[]-0,ylim[]+045,ound(00*sing[ii,]/ine[ii],), + ol="blak",pos=4,ex=08,font=) text(xlim[]-05,ylim[]+045,ound(alpha,), ol="ed", pos=4, + ex=08,font=) if(i==00) Syssleep() Syssleep(0) }

22 # and now the spetal map fo alpha=0, ie weighted LRA Y <- asmatix(log(p)) m <- t(y) %*% asveto(m) Y <- Y - ep(,now(p)) %*% t(m) m <- Y %*% asveto(m) Y <- Y - m %*% t(ep(,nol(p))) Z <- diag(sqt(m)) %*% Y %*% diag(sqt(m)) inetia <- sum(z*z) svdz <- svd(z) FF <- diag(/sqt(m)) %*% svdz$u %*% diag(svdz$d) GG <- diag(/sqt(m)) %*% svdz$v %*% diag(svdz$d) if(gg[3,]<0){ FF[,]<- -FF[,] GG[,]<- -GG[,] } if(gg[3,]<0){ FF[,]<- -FF[,] GG[,]<- -GG[,] } plot((ff[,],gg[,]),(ff[,],gg[,]),xlim=xlim, ylim=ylim, + type="n",xlab="dim ",ylab="dim ") text(ff[,],ff[,],labels=ownames(n),ol="blue",font=) text(gg[,],gg[,],labels=olnames(n),ol="ed",font=) lines((xlim[]-05,xlim[],xlim[],xlim[]-05,xlim[]-05), + (ylim[],ylim[],ylim[]+05,ylim[]+05,ylim[])) fo(ii in 00:i){ text(xlim[]-05+05*ii/00,ylim[]+05*ine[ii], "", + ol="blak",font=) text(xlim[]-05+05*ii/00,ylim[]+05*sing[ii,],"",ol="blak") text(xlim[]-05+05*ii/00,ylim[]+05*sing[ii,],"",ol="blak") } text(xlim[]-0,ylim[]+045,ound(00*sing[ii,]/ine[ii],), + ol="blak",pos=4,ex=08,font=) text(xlim[]-05,ylim[]+045,ound(alpha,), ol="ed", pos=4, + ex=08,font=)