13(3):369{396. in the Social Sciences, Street, Houston, TX :539{565. Sciences, Inc., Seattle.

Transcription

1 p t 1 X Votng :7557 :780 3 :8317 :7339 Colos :478 :494 3 :5440 :4540 Natons :843 : :3897 :3969 Colas :6985 : :7878 :7570 Table : Relatve eos (RE) fo two ntal conguatons usng the stess ( 1) cteon. The numecal esults suggest that both X and t 1 mpove on n pactce. In patcula, the elatve eos of the easly computed optmal dlatons of the classcal solutons anged fom :843 to :8317. The elatve eos ae somewhat lage fo the stess cteon than fo the sstess cteon. Fo both ctea, howeve, optmal dlaton of the classcal soluton seems well woth the modest computatonal expense equed to obtan t. Refeences Ctchley, F. (1988). On cetan lnea mappngs between nne-poduct and squaed-dstance matces. Lnea Algeba and Its Applcatons, 105:91{ 107. Ekman, G. (1954). Dmensons of colo vson. Jounal of Psychology, 38:467{474. Glunt, W., Hayden, T. L., and Lu, W.-M. (1991). The embeddng poblem fo pedstance matces. Bulletn of Mathematcal Bology, 53:769{ 796. Geen, P. E., Camone, J., F. J., and Smth, S. M. (1989). Multdmensonal Scalng: Concepts and Applcatons. Allyn and Bacon, Boston. Goenen, P. J. F. (1993). The Majozaton Appoach to Multdmensonal Scalng. DSWO Pess, Leden, The Nethelands. Hand, D. J., Daly, F., Lunn, A. D., McConway, K. J., and Ostowsk, E. (1994). A Handbook of Small Data Sets. Chapman & Hall, New Yok. usng Newton's method. Computatonal Statstcs, 13(3):369{396. Kuskal, J. B. and Wsh, M. (1978). Multdmensonal Scalng. Sage Publcatons, Bevely Hlls. Sage Unvesty Pape sees on Quanttatve Applcatons n the Socal Scences, Malone, S. W. and Tosset, M. W. (000). A study of the ctcal ponts of the SSTRESS cteon n metc multdmensonal scalng. Techncal Repot 00-06, Depatment of Computatonal & Appled Mathematcs, Rce Unvesty, 6100 Man Steet, Houston, TX Meulman, J. J. (199). The ntegaton of multdmensonal scalng and multvaate analyss wth optmal tansfomatons. Psychometka, 57:539{565. Romesbug, H. C. (1984). Cluste Analyss fo Reseaches. Lfetme Leanng Publcatons, Belmont, CA. SAS Insttute (1997). SAS/STAT Softwae: Changes and Enhancements though Release 6.1. SAS Insttute, Inc., Cay, NC. Statstcal Scences (1993). S-PLUS fo Wndows Refeence Manual, Veson 3.1, volume. Statstcal Scences, Inc., Seattle. Taazaga, P. and Tosset, M. W. (1998). An appoxmate soluton to the metc SSTRESS poblem n multdmensonal scalng. Computng Scence and Statstcs, 30(1):9{95. Togeson, W. S. (195). Multdmensonal scalng: I. Theoy and method. Psychometka, 17:401{ 419. Wsh, M. (1971). Indvdual deences n peceptons and pefeences among natons. In Kng, C. W. and Tget, D., edtos, Atttude Reseach Reaches New Heghts, pages 31{38. Amecan Maketng Assocaton, Chcago. Wsh, M., Deutsch, M., and Bene, L. (197). Deences n peceved smlaty of natons. In Romney, A. K., Shepad, R. N., and Nelove, S., edtos, Multdmensonal Scalng: Theoy and Applcatons n the Behavoal Scences, volume, pages 89{313. Semna Pess, New Yok. Keasley, A. J., Tapa, R. A., and Tosset, M. W. (1998). The soluton of the metc STRESS and SSTRESS poblems n multdmensonal scalng

2 of n 15 New Jesey congessmen on 19 envonmental blls. Enty j s the numbe of tmes that congessmen and j voted deently. Ths s Data Set 94 n Hand et al. (1994), ognally analyzed by Romesbug (1984).. Colos. Ths dssmlaty matx was constucted fom Ekman's (1954) smlaty data on colo pecepton. In Ekman's expement, each of 31 students ated the smlaty of 91 pas of n 14 colo stmul on a scale of 0 (\no smlaty at all") to 4 (\dentty"). We dened coespondng dssmlates by scalng the mean smlaty atngs to ange fom 0 to 10, then subtactng each of them fom Natons. Ths dssmlaty matx was constucted fom Wsh's smlaty data on peceptons of natons (Wsh, 1971; Wsh et al., 197; Kuskal and Wsh, 1978). In Wsh's expement, each of 18 psychology students ated the smlaty of 66 pas of n 1 natons on a scale of 1 (\vey deent") to 9 (\vey smla"). The mean smlaty atngs ae epoted n Fgue 7 of Kuskal and Wsh (1978). We dened coespondng dssmlates by subtactng each of the mean smlaty atngs fom Colas. Ths dssmlaty matx was epoted by Geen, Camone, and Smth (1989); t also appeas as Table 7.1 n Goenen (1993). Each of 38 students ated the smlaty of 45 pas of n 10 colas on a 9-pont scale and the coespondng dssmlates wee then summed ove the 38 students. Fo each dssmlaty matx and each taget dmenson p ; 3, we calculated the followng con- guatons: 1. The classcal soluton speced n Theoem. Ths soluton s denoted by.. The appoxmate soluton poposed by Taazaga and Tosset (1998). To obtan ths soluton, we st used the S-PLUS (1993) functon nlmnb to solve Poblem (), obtanng 1 ; : : :; p. We then poceeded as though computng the classcal soluton, usng nstead of +. Ths soluton s denoted by X. 3. The optmal dlatons of the classcal soluton fo the sstess ( ) and stess ( 1) ctea. These solutons ae denoted by t. 4. Nomnal mnmzes of the sstess ( ) and stess ( 1) ctea. These wee computed usng PROC MDS n SAS (1997) wth optons ft p t X Votng :4661 : :6537 :4714 Colos :1816 :160 3 :035 :163 Natons :4117 :403 3 :3771 :303 Colas :3011 : :343 :04 Table 1: Relatve eos (RE) fo two ntal conguatons usng the sstess ( ) cteon. (fo ) o ft1 (fo 1), fomula0, and levelabsolute. These mnmzes ae denoted by SAS. Instead of (X), SAS epots p (X) kd (X)? k p : If SAS succeeds n ndng a global mnmze of, then necessaly p (SAS ) t : To quantfy the eo of t elatve to the eo of, we computed RE t t? p (SAS )? p (SAS ) the nomnal eo of the optmally dlated classcal soluton dvded by the nomnal eo of the classcal soluton. Fo compason, we also calculated the same quantty usng X nstead of t. Results fo the sstess ( ) cteon ae epoted n Table 1. Fo, theoetcal esults n Sectons and 3 guaantee that 0 RE (X ) RE t 1: The numecal esults suggest that both X and t substantally mpove on n pactce. In patcula, the elatve eos of the easly computed optmal dlatons of the classcal solutons anged fom :1816 to :6537. Results fo the stess ( 1) cteon ae epoted n Table. Fo 1, theoetcal esults n Secton 3 guaantee only that 0 RE t 1 1: ;

3 Theoem 3 (Genealzed Ctcal Pont Theoem) If ( X) 0, then () hd ( X); kd ( X)k ; t ^B?? D t () kd ( X)k + kd ( X)? k k k ; () If ( Y ) 0, then ( X) < ( Y ) f and only f kd ( Y )k < kd ( X)k; (v) kd ( X)? k k k. Pat (v) of Theoem 3 states that, fo any xed dssmlaty matx, the ntepont dstance matces of all statonay conguatons le on the same sphee of adus kk, centeed at. Theoem 3 does not addess the exstence of nonglobal mnmzes, but pat () states that global mnmzes ae those statonay conguatons of maxmal nom. Ths emakable fact would seem to have pofound mplcatons fo global optmzaton of. We beleve that what follows s the st successful attempt to explot the geomety of the ctcal ponts of fo the pupose of mnmzng. Suppose that a conguaton matx X has been poposed as a possble mnmze of. (Moe speccally, suppose that X has been poposed as the ntal conguaton fom whch an teatve optmzaton method wll stat.) If () n Theoem 3 s not satsed, then X cannot be a mnmze of. The followng esult of Malone and Tosset (000) states that, by dlatng X so that () s satsed, we necessaly decease. Theoem 4 (Dlaton Theoem) Let be a xed dssmlaty matx, let X be a xed conguaton matx fo whch kd(x)k > 0, and consde the functon : <! < dened by (t) (tx). Let Then! 1 t hd (X); kd (X)k : () D (t X) les on the sphee descbed by (v) n Theoem 3; and () t s a global mmmze of. Agan we specalze to the case and evst the methods descbed n Secton. Poblem () was deved by eplacng each + n Theoem wth a feely vayng. Suppose, nstead, that we wte t + and allow t < to vay. Ths esults n the objectve functon g (t) 1 t (v v 0 )? and the optmzaton poblem mnmze g (t) subject to t 0: (3) Recognzng that Poblem (3) s the poblem of optmally dlatng the classcal soluton, we can apply Theoem 4 to compute ts soluton explctly, wthout ecouse to numecal optmzaton. Because Poblem (3) s a specal case of Poblem () wth fewe degees of feedom, we conclude that optmally dlatng the classcal soluton fo s necessaly nfeo to solvng Poblem (). Howeve, t s much ease to compute t n Theoem 4 than t s to solve Poblem (). Futhemoe, the numecal esults epoted n Secton 4 suggest that the optmal dlaton of the classcal soluton s often compettve wth the soluton of Poblem (). Fnally, suppose that ( j ) s actually a dstance matx,.e. that D n (n? 1), and wte D ^D ^dj : Then t s well-known see Meulman (199) fo dscusson that ^d j. Because j g (t) t ^d? j j ; t ^D? X we have the followng esult: Theoem 5 Let be a xed dssmlaty matx. Gven p, let denote the conguaton matx de- ned n Theoem. Gven, let t denote the optmal dlaton of. If Dn (n? 1), then t 1, wth equalty f and only f D n (p). 4 Numecal Results We now epot the esults of some numecal expements that nvestgated the extent to whch optmal dlaton mpoves classcal solutons wth espect to the sstess ( ) and stess ( 1) ctea. We also compae optmal dlaton of classcal solutons wth solutons to Poblem (). These esults extend esults epoted by Taazaga and Tosset (1998). We studed the followng dssmlaty matces: 1. Votng. Ths dssmlaty matx s a dsageement matx constucted fom the votng ecods j

4 Classcal MDS Gven n, let e denote the n-vecto (1; : : : ; 1) 0. Gven an nn matx B, let b denote the n-vecto dag(b). Let denote the lnea tansfomaton dened by (B) be 0 + eb 0? B: Gven p n, let D n (p) denote the set of matces that can be ealzed as the ntepont dstances of some x 1 ; : : : ; x n < p and let n (p) denote the set of symmetc postve semdente n n matces of ank no geate than p. Then the followng esult s well-known: Theoem 1 (Embeddng Theoem) D D n (p) f and only f thee exsts B n (p) such that (B) D. The lnea tansfomaton does not have a unque nvese. In fact, fo any s < p such that s 0 e 1, the lnea tansfomaton s dened by s (D)? 1 (I? es0 ) D (I? es 0 ) s an nvese of, whee I denotes the n n dentty matx. We ae nteested n the nvese 1 obtaned by settng s en. See Ctchley (1988) fo a detaled study of the popetes of and 1. Let kk denote the Fobenus nom. Classcal MDS can be dened by the optmzaton poblem mnmze kb? 1 ( )k subject to B n (p); (1) whch s mplct n Togeson (195). The objectve functon was late dubbed the stan cteon. The followng explct soluton to Poblem (1) s also well-known: Theoem (Classcal MDS) Gven, let 1 n denote the egenvalues of B 1 ( ) and let v 1 ; : : :; v n denote the coespondng egenvectos. Gven p n, let + max( ; 0) fo 1; : : :; p. Then ^B 1 + v v 0 s a global mnmze of Poblem 1. Futhemoe, f s the n p conguaton matx whose th column s ( + )1 v, then ^B D : Because the classcal soluton,, can be computed explctly, t s often used as the ntal conguaton fom whch optmzaton of commences. Now we specalze to the case of and wte ^B? + (v v) 0? 1 Taazaga and Tosset (1998) obseved that a conguaton matx wth a smalle sstess value than ( ) can be obtaned by eplacng the + wth fee vaables < p, esultng n the objectve functon f() (v v) 0? 1 and solvng the p-vaate optmzaton poblem: mnmze f() subject to 0; ; : () Ths s a faly easy poblem to solve numecally and numecal esults suggest that the esultng mpovement on classcal MDS s often substantal; howeve, mpovement s only guaanteed when and soluton eques an teatve algothm fo boundconstaned optmzaton. 3 Optmal Dlatons Glunt, Hayden, and Lu (1991) establshed the emakable esult that, fo any xed dssmlaty matx, all of the ntepont squaed dstance matces geneated by statonay conguatons of the sstess cteon wth unt weghts le on the suface of a sphee. Malone and Tosset (000) extended ths esult fom the specal case of to the geneal case of > 0. Let ( j ) denote a xed dssmlaty matx. Gven a conguaton matx X, let D(X) (d j ) denote the coespondng matx of ntepont Eucldean dstances. Fo > 0, let ( ) and j let D (X) (d ). We ae nteested n statonay j conguatons fo the eo cteon (X) kd (X)? k ;.e. n conguaton matces X fo whch ( X) 0. Gven squae matces A (a j ) and B (b j ), let X ha; B a j b j j denote the Fobenus nne poduct of A and B. Let k k denote the coespondng Fobenus nom.

5 Optmal Dlatons fo Metc Multdmensonal Scalng 1 Samuel W. Malone, Depatment of Mathematcs, Duke Unvesty, Duham, NC swm3@duke.edu E-mal: Mchael W. Tosset, Depatment of Mathematcs, College of Wllam & May, P.O. Box 8795, Wllamsbug, VA E-mal: tosset@math.wm.edu Abstact We deve a smple fomula that optmally dlates a conguaton of ponts wth espect to a speced membe of a paametzed famly of eo ctea. Ths famly ncludes the popula stess and sstess ctea. We pesent examples that demonstate that Togeson's classcal soluton, often used as an ntal conguaton fo teatvely mnmzng s/stess, s not optmally dlated. Optmally dlatng the classcal soluton befoe commencng optmzaton s a smple tactc that may have pactcal mplcatons fo global optmzaton stateges. 1 Intoducton Multdmensonal scalng (MDS) s a geneal tem fo technques that constuct conguatons of ponts n a taget metc space fom nfomaton about ntepont dstances. In the case of two-way MDS, the nfomaton s speced n the fom of a dssmlaty matx,.e. a matx ( j ) such that j 0, 0, and j j. Typcally, the taget metc space s p-dmensonal Eucldean space. In most applcatons, such as the econstucton of molecula conguatons fom nfomaton about nteatomc dstances (p 3), the taget dmenson s small. Fo a conguaton of ponts x 1 ; : : : ; x n < p, the n p conguaton matx X s the matx whose ows ae the x 0. Fom X t s easy to compute the Eucldean ntepont dstance matx D(X) (d j ). The goal of metc two-way MDS s to constuct a conguaton matx fo whch the ntepont dstances d j appoxmate the dssmlates j. Two popula ways of measung the dscepancy between a dstance matx and a dssmlaty matx ae the stess and sstess ctea. The fome s based on the squaed eos between the dstances and dssmlates; the latte s based on the squaed 1 Ths eseach was suppoted by gants DMS and DMS fom the Natonal Scence Foundaton. eos between the squaed dstances and squaed dssmlates. Let (D; ) X j w j [(d j )? ( j ) ] ; whee the w j ae nonnegatve weghts. Wte D D(X) and let (X) (D(X)); then 1 s the metc stess cteon and s the slghtly less popula metc sstess cton. In pactce, one often sets each w j 1. Ths s the only case consdeed n ths epot. Howeve, one can use the weghts ethe to accommodate mssng data (by settng the appopate w j 0) o to weght moe elably measued dssmlates moe heavly. Havng selected an eo cteon, metc MDS attempts to nd a conguaton matx X that mnmzes. Unfotunately, mnmzes must be computed by an teatve algothm fo numecal optmzaton. A suvey of some of the moe ecent algothms fo mnmzng 1 and was made by Keasley, Tapa, and Tosset (1998). These algothms nd local mnmzes. To mpove the chance that the algothm wll convege to a global mnmze, the use s advsed to choose a good ntal conguaton. The conventonal choce the default ntal conguaton used n many mplementatons s the conguaton constucted by classcal MDS. Classcal MDS nvolves nothng moe complcated than a sngle spectal decomposton of a symmetc matx and usually povdes an excellent conguaton fom whch to begn mnmzng. In Secton we bey descbe classcal MDS and a closely elated pocedue that necessaly mpoves upon t when. In Secton 3, we ntoduce a smple devce fo mpovng any ntal conguaton wth espect to any choce of. In Secton 4 we pesent numecal esults that suggest that ths devce often mpoves the classcal soluton substantally.