Relating Principal Component Analysis on Merged Data Sets to a Regression Approach

Transcription

1 Relatng Prncpal Coponent Analyss on Merged Data Sets to a Regresson Approach Mchael Meyners & El Mostafa Qannar Fachberech Statstk, Unverstät Dortund, 44 Dortund, Gerany chael.eyners@udo.edu ENIIAA INRA, Laboratore de Statstque applquée à la caractérsaton des alents, 4407 Nantes, Cedex 03, France, qannar@entaa-nantes.fr Abstract A ethod for calculatng a consensus of several data atrces on the sae saples usng a PCA s based on a atheatcal background. We propose a odel to descrbe the data whch ght be obtaned e. g. by eans of a free choce proflng or a fxed vocabulary n a sensory proflng fraework. A regresson approach for ths odel leads to a Prncpal Coponent Analyss on Merged Data sets (PCAMD), whch provdes a sple ethod to calculate a consensus fro the data. Snce we use less restrctons on the varables under nvestgaton, the odel s claed to be ore general than the odel nduced by GPA respectvely SAIS, whch are wdely accepted ethods to analyse ths knd of data. Furtherore, the PCAMD provdes also addtonal opportuntes to copare and nterpret assessor perforances wth respect to the varables of the calculated consensus. An exaple fro a sensory proflng study of cder s provded to llustrate these possbltes. Keywords: prncpal coponent analyss, regresson analyss, erged data sets, sensory proflng, assessor perforance

2 Introducton For calculatng a consensus fro dfferent data atrces gven wth respect to the sae saples, two dfferent ethods are wdely accepted, naely Generalzed Procrustes Analyss (GPA; cf. Gower 975, ten Berge 977) and SAIS (whch abbrevates the French expresson "Structuraton des ableaux A ros Indces de la Statstque"; cf. Lavt et al. 994, Schlch 996). In partcular they are often used wthn sensory proflng applcatons, thus for nterpretaton purposes we confne ourselves to ths fraework. However, the generalsaton to other applcatons s obvous. We assue dfferent assessors whch took part n a sensory proflng sesson of several products. he calculated consensus should approprately represent the dfferences between the products under consderaton. Up to now, the superorty of nether GPA or SAIS has been shown n general, even though the results of a sulaton study as well as soe theoretcal consderatons have been gven by Meyners et al. (000). he probles arsng wth theoretcal consderatons due to the coplcated atheatcal background of both ethods. Furtherore ther algorths lead to teconsung prograng wth coputer software. A spler alternatve to calculate a consensus fro several data sets on the sae products has been gven by Kunert and Qannar (999). hey propose to defne a superatrx whch ncludes all ndvdual assessor atrces and to calculate a PCA of ths superatrx. he estated consensus of densonalty k s then gven by ts frst k prncpal coponents. In ths paper we relate ths ethod to a regresson odel whch wll be shown to be ore general than the odel nduced by GPA respectvely SAIS. A general relatonshp between PCA and Regresson Analyss has been gven by Jong and Kotz (999). Anyway, t wll be shown that n the partcular case consdered here the least squares approach splfes to a sple PCA on the Merged Data sets (PCAMD). hus the ethod of Kunert and Qannar

3 (999) wll have a stronger justfcaton. Furtherore, t wll be shown how dfferent behavour of the assessors s represented n the contrbuton of a partcular assessor to the dfferent densons of the consensus. hese values can also be easly calculated wthn ths ethod and provde several possbltes for nterpretaton. 3

4 Modellng sensory proflng data We rely on the notaton of Kunert and Qannar (999),. e. we assue a sensory proflng wth assessors on n products, whch gves us the data atrces X,, X. Meyners et al. (000) propose a reasonable odel for ths knd of data plctly gven by GPA respectvely SAIS. Under ther odel assupton the ndvdual assessor atrces arse fro an underlyng true consensus C (whch s obvously unknown) by ultplyng t wth an sotropc scalng factor α and an orthonoral rotaton atrx R. Furtherore a translaton atrx and soe ndependent errors wth ean zero, arranged n a atrx F, say, are added. hus we get X α C R + + F. It ght be wse to generalse ths odel to allow for dfferent scalng factors for the dfferent attrbutes of one assessor, even though nether GPA nor SAIS respect for these dfferences. hat s to say that even f there were no rando errors at all n ths ore general odel, usually none of the ethods would lead to the true consensus C. In a frst step t s necessary to enton that the data of a sensory proflng s usually pretreated at least n such a way that the atrces are centred colun-wse,. e. the su over the products s zero for each attrbute wthn each assessor. hs holds for GPA and SAIS as well as for the approach of Kunert and Qannar. It s clear fro the odel that the translaton does not affect the atrces anyore as soon as they are centred. hus wthout loss of generalty and for convenence of notaton we can neglect the translaton n the followng consderatons. he atrces R,,...,, account for possble confuson of the varables gven n C as well as they ght represent the use of dfferent lnear cobnatons of these varables by 4

5 assessor. However, the constrant of the beng rotaton atrces,. e. beng orthonoral and hence R R I where I s the dentty atrx, ples that the su of squares of each colun of R s equal to. Note that the k-th colun of R accounts for how uch the varables of C contrbute to the k-th attrbute of assessor. Hence fro the constrant t s clear that the overall squared contrbuton reans the sae for all attrbutes of ths assessor, whch usually ght not be fulflled n practce. We drop the orthonoralty assupton by allowng these atrces to be unconstraned and denote the B to dstngush the fro the rotaton atrces. However, on the opposte we constran the atrx C to have orthonoral coluns, whch s uch ore saller a constrant than the one we just dropped: We assue the product dfferences to be expressed n soe unnaed but orthogonal varables, whch leads us to the orthogonalty of the coluns. hs sees sensble, snce we are usually not nterested n correlated varables to descrbe the dfferences, and n partcular f we draw a graphcal representaton we usually take orthogonal axes. Under ths assupton the orthonoralty s no addtonal constrant, snce we can sply account for scale dfferences wthn the attrbutes of C by choosng the entres of the atrces B approprate,,...,. Hence t s also unnecessary to consder dfferent scalng factors for the attrbutes of an assessor, snce these can also be accounted for by B. However, t s well known that soe assessors tend to use sall ranges of scale, whereas others use a wde range of scale. Neglectng these dfferences would be sleadng snce wthn these ethods the latter ones would nfluence the consensus ore than the forer ones. hus we take dfferent sotropc scalng factors α nto account that represent the general behavour n usng the range of scale. hen the odel turns out to be X C α B + F. ( ) 5

6 o avod dfferent assessor nfluences due to dfferent behavour n usng the range of scale, the data atrces should be pre-scaled wth the nverse of an estator of α. he estates proposed by Kunert and Qannar (999) gve a reasonable value snce they lead to dentcal overall varabltes for each assessor. Anyway, ther denonator s just ntended to allow for coparsons wth the scalng factors of GPA and ght be replaced by any postve scalar wthout changng the relatve dstances of nterest between the saples wthn the consensus. Snce we are not nterested n coparng the results wth those of a GPA, we propose to replace ther value by. As t wll be shown later on, ths leads drectly to a straghtforward nterpretaton of the results whle coparng the assessor perforances. hus we propose to estate the scalng factors to be α ˆ trace( X X ) ( ) and to pre-scale the respectve atrx wth the nverse of ths value. After ths pre-scalng the su of squares of all entres of the data atrces s equal to for all assessors. Fro now on for convenence of notaton we assue the atrces to be pre-scaled accordng to (), thus denotng the pre-scaled atrces by () splfes to C B + E, ( 3 ) where E contans the rando errors after pre-scalng, whch obvously stll have ean zero. Wthn (3) the densons of the atrces are as follows: C s of order (n, q), where q s the nuber of densons n whch there are dfferences between the products and whch s unknown, B s a (q, p )-atrx, where p denotes the nuber of attrbutes used by assessor, and E s of order (n, p ). Furtherore we denote p p, the total nuber of attrbutes used. 6

7 Model (3) sees slar to the well-known lnear odel. However, unlke the usual lnear regresson odel we have several unknown paraeter atrces, naely C and B for,,. We are rather nterested n C, even though the estaton of the dfferent B can provde us wth useful nforaton upon the ndvdual assessor appearance as t wll be shown later on. It has been ponted out that ths odel ght be ore applcable than () snce there are no constrants about the ndvdual atrces whereas we have just a ore reasonable constrant for the atrx C. Note that no wthn the odel no densonalty constrants are gven at all,. e. n general the consensus C ght provde ore attrbutes than gven by each of the atrces B. In partcular, C ght theoretcally also nclude attrbutes that none of the assessors perceves, that s all entres of (the sae) partcular rows of B would be zero for each,,. In practce ths case ay occur f we have chosen non-representatve assessors fro a superpopulaton,. e. ther are non-neglgble dfferences between the products whch have not been detected by the panel. Of course we can only descrbe those product dfferences that have been perceved and accounted for by the assessors, thus we hope that we found a panel that accounts for all portant dfferences. Furtherore, fro a atheatcal pont of vew t s clear that the observed nuber k of densons wthn C s bounded by n as well as by p, snce we wll not need ore than n densons do descrbe all dfferences between n products, whereas on the opposte we cannot estate ore than p densons fro p attrbutes. In practcal applcatons we hope to fnd an estator for C wth a rather sall nuber of attrbutes that represent the product dfferences approprately, even though our odel ncludes also these rather unfalar and hopefully seldo cases. 7

8 Estaton of C and B It has been ponted out n the prevous secton that odel (3) contans several unknown data atrces. For estaton purposes we use the well-known least squares crteron,. e. wth the obvous notaton we want to nse the value of ˆ CB ˆ ˆ Eˆ, ( 4 ) where A trace( A A) ( 5 ) s the nor of a atrx A. Due to the fact that t s possble to estate those atrces all at once, an teratve algorth sees to be approprate. hus t ght be proposed to estate C n a frst step puttng n the observed and pre-treated data atrces for B, then to hold the estated C fxed and estate the atrces B. hs should be repeated terately untl convergence occurs. Anyway, we show that no teratons are necessary n practce, snce under the assuptons of odel (3) the estate for C s ndependent of the estate of B but reles only on the atrces,,,. Let us assue a fxed C that fulfls the constrant of orthonoral coluns. Mnsaton of (4) s then gven as the well-known Gauss-Markov-estator n a ultvarate lnear odel,. e. Bˆ ( C C) C, whch under the constrant C C I splfes to B ˆ C. ( 6 ) 8

9 9 hus we have CC ˆ and wth t ˆ ) trace( ) trace( ˆ CC CC CC +. Mnsaton of (4) s then equvalent to a axsaton of CC ˆ ( 7 ) wth respect to C. At ths stage t should be stated that ths s a two-densonal case of the three-way factor analyss odel gven by ucker (966) and dscussed by Brockhoff et al. (996). Anyway, we do not go nto detals here snce ths s beyond the scope of ths paper. Usng (5), (7) can be wrtten as CC CC CC ) trace( CC ) trace( C C ) trace( C C trace ( ) C C trace, ( 8 )

10 where ( ) contans the erged orgnal data atrces. It s one of the propertes of PCA that the axsaton of ths value wth respect to k densons wthn C s obtaned fro the frst k egenvectors of,. e. the frst k prncpal coponents of that atrx. Note that the axsaton does not depend on the values of the atrces B. hus we can estate C n the frst step and after that, f assessor dfferences are also to be consdered, the atrces B can be estated fro (6) replacng C by ts estate. Assessor perforance In practce t sees reasonable to look at the dfferent atrces B for the assessors only f we are nterested n the contrbuton of several varables of C to dfferent attrbutes wth respect to partcular saples. Usually t s of greater nterest to get an expresson of the contrbuton of the assessors to the dfferent prncpal coponents, even ore f these coponents have an useful nterpretaton. We assue Cˆ ( cˆ cˆ K cˆ ) to be the k-densonal approxaton of k C, where ĉ j, j,, k, are the frst k prncpal coponents of. It s also well-known fro PCA (cf. Jollffe 986) that the j-th egenvalue k of s gven by ˆ ˆ j c j c j. Slar to (8), ths value can be spltted nto factors whch leads to cˆ cˆ, j j j and the contrbuton of assessor to the k-th prncpal coponent s then gven by cˆ cˆ. j j j 0

11 hese values can be arranged n a table and provde, after dfferent transforatons, several nterpretatons of the assessor perforances. In a frst step the absolute values should be arranged as proposed n table. o splfy the nterpretatons, we assue that k s chosen equal to,. e. we consder all densons wth nonzero egenvalues. assessor PC PC K PC k Σ K k K k M M M M M K Σ K k k able : Contrbuton of the assessors to the total varaton accordng to the prncpal coponents. Fro table t can be seen how uch each assessor contrbutes to the varaton wth respect to each sngle denson. Snce the row-wse su s equal to,. e. all varaton of assessor s explaned wthn the prncpal coponents gven, the entres can be seen as the percentage of varaton of assessor explaned by the j-th prncpal coponent. Furtherore the colunwse su s equal to the respectve egenvalue as t s well-known fro PCA. As a atter of fact the egenvalues add up to when the pre-scalng has been done accordng to (). Note that usng the pre-scalng proposed by Kunert and Qannar (999) all values n table are dvded by,. e. to obtan the sae table wth these nterpretatons each entry has to be ultpled by.

12 assessor PC PC K PC k Σ K K k k k k M M M M M K k Σ K k k able : Colun-wse noralsed contrbuton of the assessors to the total varaton accordng to the prncpal coponents. For table the assessor-contrbutons have been noralsed colun-wse such that the coluns su up to,. e. the values account for the percentage the respectve prncpal coponent s nfluenced by assessor. Snce these values becoe qute sall as ncreases, t ght also be useful to ultply all entres wth so that they have ean. Furtherore ths ght splfy the nterpretaton when we are nterested rather n the perforance of each assessor than n the percentage of hs / her nfluence on the respectve denson: an assessor wth a value larger than for PC j contrbutes ore than the average assessors to ths coponent, whereas an assessor wth a value saller than contrbutes less. In partcular f a prncpal coponent s donated by one assessor, ths results n a very large value for the respectve assessor. As far as we can see t n ths representaton the row-wse sus have no useful eanng and are therefore otted. he dea to estate a consensus fro sensory proflng data by eans of a PCA has been called a "sple alternatve" by Kunert and Qannar (999). hus t should be entoned here

13 that our consderatons accordng to the assessor perforance respect for ths splcty, snce the calculatons are easly carred out. It deands only sple prograng n any software that provdes the possblty to calculate a PCA as well as covarances. Exaples We consder a sensory proflng of 0 saples of cder that have been assessed by 7 judges accordng to 0 varables wthn a fxed vocabulary proflng. We calculated a PCAMD on these data and confne ourselves on the assessor perforances, snce the nterpretaton of a product consensus s well-known and n the scope of ths paper of nor nterest. able 3 gves the values j for,, 7 and j,, 9. As t has been entoned n the prevous secton the rows su up to one whereas the coluns su up to the respectve egenvalues. Due to the roundng used the noralsed egenvalues cuulate not exactly to one. assessor PC PC PC 3 PC 4 PC 5 PC 6 PC 7 PC 8 PC 9 Σ Σ 3.5 (0.45).4 (0.6) 0.70 (0.0) 0.46 (0.07) 0.39 (0.06) 0.36 (0.05) 0.3 (0.05) 0.6 (0.04) 0. (0.03) able 3: Explaned varaton wthn the PC s for the dfferent assessors. 7 (.0) It can be seen that about half of the varaton of each assessor s explaned by the frst prncpal coponent, except for assessor 3 an 5 for who ths PC accounts for less than 40%. Assessor 3 has therefore a large value n PC, whle the reanng varaton of assessor 5 s 3

14 explaned by PC and 3. he cuulatve varaton wthn the frst two coponents s axsed by assessor 4,. e. hs saple space can be approxated pretty good wth the two-densonal representaton fro the prncpal coponents, whereas assessors and 5 are farly represented. assessor PC PC PC 3 PC 4 PC 5 PC 6 PC 7 PC 8 PC 9 Σ Σ able 4: Colun-wse noralsed varaton wthn the PC s for the dfferent assessors. Fro table 4 t can bee seen that there are only nor dfferences n the contrbuton of the assessors wthn the frst PC, even though assessors and 4 contrbute ore and 3 and 5 less than the average. Greater dfferences appear n PC, where assessor 3 (and also 4) contrbutes qute large, whereas assessor contrbutes less than the average. Even larger dfferences can be found n PC 4, 5 and 9, whle the saller egenvalues and wth t the saller eanng of these coponents have to be taken nto account,. e. we should avod any overnterpretaton of the values. o conclude, we ght assue assessor 4 to be represented qute well n the frst densons of the overall consensus and thus to agree well wth ths consensus, whle the assessors and 5 see to agree less wth t, whch leads to a sall representaton wthn the frst two PC s. 4

15 Dscusson he sple alternatve to GPA proposed by Kunert and Qannar (999) has been atheatcally justfed by fndng an approprate odel for sensory proflng data. Estatng the unknown paraeters wthn ths odel leads drectly to a very sple ethod for calculatng an overall consensus fro several data atrces. Furtherore t gves also a sple opportunty to copare the assessor perforances n the panel by calculatng the contrbuton of each assessor to the densons of the consensus. he exaple shows that ths ght lead to useful conclusons aong the assessor behavour. Acknowledgeent hs paper was wrtten whle the frst author vsted the ENIIAA Nantes. He wshes to thank the people at the ENIIAA for ther hosptalty and the Deutsche Forschungsgeenschaft (SFB 475, "Reducton of coplexty for ultvarate data structures") for the fnancal support of ths work. 5

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