Numerical experiments on unsupervised manifold learning applied to mechanical modeling of materials and structures

Comp es Rendus Mécanique Ruben Ibanez, Pie e Gilo mini, Elias Cue o and F ancisco Chines a Nume ical expe imen s on unsupe ised mani old lea ning applied o mechanical modeling o ma e ials and s uc u es Volume 348, issue 10-11 (2020), p. 937-958. <h ps://doi.o g/10.5802/c meca.53> Pa o he Thema ic Issue: Con ibu ions in mechanics o ma e ials Gues edi o s: Julie Diani, Oli ie Cas elnau and F ancisco Chines a © Académie des sciences, Pa is and he au ho s, 2020. Some igh s ese ed. This a icle is licensed unde he C ea i e Commons A ibu ion 4.0In e na ional License. h p://c ea i ecommons.o g/licenses/by/4.0/ Les Comp es Rendus. Mécanique son memb es du Cen e Me senne pou l’édi ion scien i ique ou e e www.cen e-me senne.o g Comp es Rendus Mécanique 2020, 348, n10-11, p. 937-958 h ps://doi.o g/10.5802/c meca.53 Con ibu ions in mechanics o ma e ials Nume ical expe imen s on unsupe ised mani old lea ning applied o mechanical modeling o ma e ials and s uc u es Ruben Ibaneza, Pie e Gilo minia, Elias Cue oband F ancisco Chines a∗,a aPIMM lab, A s e Me ie s Ins i u e o Technology, 151 Boule a d de Hôpi al, 75013 Pa is, F ance bA agon Ins i u e o Enginee ing Resea ch, Uni e sidad de Za agoza, Ma ia de Luna s/n, 50018 Za agoza, Spain E-mails: [email p o ec ed] (R. Ibanez), pie [email p o ec ed] (P. Gilo mini), ecue o@uniza .es (E. Cue o), ancisco[email p o ec ed] (F. Chines a) Abs ac . The p esen wo k aims a analyzing issues ela ed o he da a mani old dimensionali y. The in e es o he s udy is wo old: (i) i s , when oo many measu able a iables a e conside ed, mani old lea ning is expec ed o ex ac useless a iables; (ii) second, and mo e impo an , he same echnique, mani old lea ning, could be u ilized o iden i ying he necessi y o employing la en ex a a iables able o eco e single- alued ou pu s. Bo h aspec s a e discussed in he modeling o ma e ials and s uc u al sys ems by using unsupe ised mani old lea ning s a egies. Keywo ds. Nonsupe ised mani old lea ning, S a e a iables, Dimensionali y educ ion, k-PCA, S uc u al analysis, Ma e ial cons i u i e equa ions. Manusc ip ecei ed 21s June 2020, e ised 12 h July 2020, accep ed 7 h Oc obe 2020. 1. In oduc ion Recen ly, da a-d i en desc ip ion o ma e ials has been gaining popula i y. Many complex ma- e ial beha io s esis ing adi ional modeling p ocedu es, o ha a e oo complex om a mi- c os uc u al iewpoin , a e app oached by using da a-based desc ip ions. Di e en app oaches a e being conside ed. Among hem include hose based exclusi ely on measu ed da a, o he s ha ex ac he mani olds ela ed o da a, and o he s ha a emp o en o ce he modynamic and he momechanical consis ency. The in e es ed eade can e e o [1–7] and he nume ous e e ences he ein. This wo k ocuses on echniques based on he use o mani olds and hei as- socia ed mani old lea ning p ocedu es o ex ac ing hem om he a ailable da a. ∗Co esponding au ho . ISSN (elec onic) : 1873-7234 h ps://comp es- endus.academie-sciences. /mecanique/ 938 Ruben Ibanez e al. Figu e 1. Mul idimensional da a on one- (le ), wo- (cen e ), and h ee-dimensional ( igh ) mani olds embedded in RD. In gene al, da a in ol e many dimensions. Conside i s a sequence o h ee-dimensional (3D) ields de ined in a domain Ω⊂R3pa i ioned in o D oxels. Each o hese ields con ains many da a, one da um a each oxel. Each ield can be ep esen ed as a poin in a ec o space o dimension D( he numbe o oxels), whe e we can p esume each o he Dcoo dina e axes as epo ing he alue ha he ield o in e es akes in he associa ed oxel. Thus, each ield becomes a poin in ha high-dimensional space o dimension D,RD. I impo an co ela ions exis among he di e en ields, hese poin s a e expec ed o be dis ibu ed on a low-dimensional subspace embedded in he D-dimensional space. Techniques aiming a ex ac ing hese educed subspaces, he so-called slow mani olds, ske ched in Figu e 1, a e key ools o manipula ing da a and ex ac ing hei hidden in o ma ion. Thus, da a de ine in gene al slow mani olds embedded in e y la ge ec o spaces due o he signi ican hidden co ela ions among hem. The numbe o unco ela ed explica i e dimensions usually becomes much smalle han he a p io i assumed dimension o he space o accommo- da ing he da a. The ex ac ion o hese slow mani olds can be success ully accomplished by using linea and nonlinea dimensionali y educ ion echniques such as p incipal componen analysis (PCA) in he linea case and i s nonlinea coun e pa s (`PCA, ke nel-based PCA [k-PCA], LLE, SNE, e c.) [8–12]. These echniques can be applied o se e al physical sys ems. Mo eo e , when he slow mani- old is de e mined, he solu ion a any poin on i can be compu ed e y accu a ely om a sim- ple in e pola ion o neighbo ing da a on he mani old [13], enabling almos eal- ime p edic ions and he associa ed eal- ime decision-making. Howe e , ex ac ing knowledge om da a associa ed wi h an exis ing bu hidden model equi es he ollowing: •iden i ying he mani old in insic dimension, •disco e ing hidden pa ame e s, •disca ding useless pa ame e s, and •disco e ing he models o igina ing he da a. These ques ions a e add essed, illus a ed, and discussed in he p esen wo k in a pu ely me hodological manne , aiming a illus a ing he key concep s ha could open nume ous u u e possibili ies in he ield o mechanics o ma e ials, p ocesses, s uc u es, and sys ems. In ou p e ious wo ks, we add essed p oblems in ol ing housands o dimensions [13–16], p o ing ha despi e he appa en ichness in many cases, he embedding ega ds a ela i ely low-dimensional space. Howe e , in mos o he p oblems ha we ha e ea ed un il now, hei complexi y p e en ed ine analyses o hei solu ions. The p esen pape , which is pu ely me hodological, conside s simple p oblems de ined in low dimensions, wi h known solu ions C. R. Mécanique,2020, 348, n10-11, 937-958 Ruben Ibanez e al. 939 being easily isualizable, o acili a ing an analysis and discussion. O cou se, and as p o ed in he wo ks jus e e ed o, all he me hodologies apply o mul idimensional se ings. 2. Unsupe ised mani old lea ning Le us conside a ec o y∈RDcon aining expe imen al o syn he ic da a om measu emen s o nume ical simula ion. These esul s a e o en e e ed o as snapsho s. I hey a e ob ained by nume ical simula ion, hey consis o nodal alues o he essen ial a iable. The e o e, hese a iables will be somehow co ela ed and, no ably, he e will be a linea ans o ma ion W de ining he ec o ξ∈Rd, wi h d<D, which con ains he s ill unknown la en a iables such ha y=Wξ. (1) The D×d ans o ma ion ma ix W, which sa is ies he o hogonali y condi ion WTW=Id, is he main ing edien o he PCA and can be compu ed as de ailed in Appendix A om he co a iance ma ix associa ed wi h a numbe (M) o snapsho s y1,...,yM, which cons i u e he columns o ma ix Y. While PCA wo ks wi h he co a iance ma ix (i.e., YYT), mul idimensional scaling (MDS) wo ks wi h he G am ma ix con aining scala p oduc s (i.e., S=YTY) as desc ibed in Appendix A. On he o he hand, he k-PCA is based on he ac ha da a no linea ly sepa able in D dimensions could be linea ly sepa a ed i p e iously p ojec ed o a space in Q>Ddimensions. Howe e , he ue ad an age a ises om he ac ha i is no necessa y o w i e down he analy ical exp ession o ha mapping as desc ibed in Appendix A. 3. An illus a i e s uc u al mechanics case s udy Conside i s a hypo he ical mechanical sys em consis ing o a p isma ic beam whose h ee dimensions, heigh , wid h, and leng h, a e deno ed, espec i ely, by h,b, and L, all o hem being measu able quan i ies. In wha ollows, we conside a pa icula ou pu P ha cons i u es also a measu able quan i y (buckling c i ical load, e c.) assumed ela ed o hose pa ame e s om an exis ing bu ac ually hidden model e en i in wha ollows we will conside hypo he ical, and mos o he ime, unphysical models. Thus, we conside a se o da a composed o Mmeasu es yi={hi,bi,Li,Pi}, i=1,...,M, wi h hi,bi, and Libeing andomly chosen om a uni o m p obabili y dis ibu ion in hei espec i e in e als o exis ence, Ih,Ib, and IL, espec i ely, de ined om he ollowing:    Ih=[hmin,hmax] Ib=[bmin,bmax] IL=[Lmin,Lmax]. (2) Wi hou any o he p e-exis ing knowledge, one expec s he ou pu depending on he h ee ge- ome ical pa ame e s (i.e., P=P(h,b,L)). In wha ollows, we conside h ee di e en scena ios. 3.1. Ou pu depending on a single pa ame e In his sec ion, we assume a qui e simple model ha ela es he ou pu o a single pa ame e , P=αb,α∈R+, (3) whe e α=1000 in he nume ical es s ca ied ou . We pe o m M=1000 measu es, whe e Ih∈[0.1,0.2], Ib∈[0.15,0.2], and IL∈[1,1.5]. C. R. Mécanique,2020, 348, n10-11, 937-958 940 Ruben Ibanez e al. Figu e 2. Eigen alues λ∈[λ110−6,λ1]. The measu es cons i u e a se o M=1000 poin s in R4on which he k-PCA is applied by using he Gaussian ke nel κ(yi,yj)=exp− kyi−yjk2 2β2, (4) whe e β=10. Figu e 2 depic s he highes eigen alues among M esul ing om he k-PCA, hose lying be ween he highes alue λ1and 10−6λ1. The slow mani old associa ed wi h ξis ep esen ed by selec ing he i s h ee educed coo - dina es (ξ1,ξ2,ξ3) as shown in Figu e 3, whe e i s one-dimensional (1D) in insic dimension is no ed. This esul was expec ed om he conside ed model exp essed by (3). The poin s on he slow mani old a e colo ed depending on he alues o h,b,L, and P, e idencing ha bcons i u es he la en a iable and ha he ou pu Pscales ( isually) linea ly wi h i . The p ocess o colo ing he da a poin s in he embedded mani old dese es some addi ional commen s due o he ac ha his is used in all he analyses epo ed in he p esen pape . Man- i old lea ning echniques look o a low-dimensional mani old de ined by da a poin s. As soon as he slow mani old is ex ac ed, he di e en da a poin s can be mapped on i . This isualiza ion is only possible when he numbe o dimensions allows a simple g aphical ep esen a ion (as is he case o he p oblems add essed in he p esen pape ). Then, hese poin s can be colo ed de- pending on he alue o he di e en ini ial coo dina es, and one expec s ha i he e is a co e- la ion (di ec o in e se and linea o nonlinea ) be ween he ini ial and educed coo dina es, he colo s mus exhibi a ce ain g ading. E en i his analysis seems qui e dependen on he low dimensionali y o he embedding, in highe dimensional embeddings, he analysis can be pe o med by using local s a is ics. Thus, by conside ing a da a poin in he slow mani old and i s closes neighbo s, a local s a is ical analysis can be easily pe o med wi h he s anda d de ia ion indica ing he dispe sion o he da a (equi alen o he local dispe sion o colo s). One could be sligh ly su p ised by he nonlinea i y ha he mani old exhibi s despi e he linea i y o model (3). This nonlinea i y is an a i ac o he nonlinea ke nel (4) used. As he model is linea , one could expec he abili y o he PCA o add ess he p oblem a hand. Fo his pu pose, i su ices ans o ming he ke nel in o i s linea coun e pa , gi ing ise o he PCA κ(yi,yj)=yi·yj. (5) C. R. Mécanique,2020, 348, n10-11, 937-958 Ruben Ibanez e al. 941 Figu e 3. Slow mani old in he 3D space de ined by he i s educed coo dina es (ξ1,ξ2,ξ3). Each poin is colo ed acco ding o he alue o he coo dina e h( op le ), b( op igh ), L (bo om le ), o P(bo om igh ). Figu e 4. Slow mani old ξwhen conside ing he PCA linea dimensionali y educ ion. Each poin is colo ed acco ding o he alue o he a iable b. In his case, as expec ed, a single nonze o eigen alue esul s, and consequen ly he dimension o he educed space becomes one (i.e., ξ=ξ). The associa ed mani old is depic ed in Figu e 4. Now, we conside a sligh ly di e en model, again depending on a single a iable bu in a nonlinea manne , acco ding o P=αh2, (6) C. R. Mécanique,2020, 348, n10-11, 937-958 942 Ruben Ibanez e al. Figu e 5. Slow mani old in he 3D space de ined by he i s h ee educed coo dina es (ξ1,ξ2,ξ3). Each poin is colo ed acco ding o he alue o he coo dina e h(le ) o P( igh ). Figu e 6. Slow mani old in he 3D space de ined by he i s h ee educed coo dina es (ξ1,ξ2,ξ3). Each poin is colo ed acco ding o he alue o he coo dina e L(le ) o P( igh ). whe e again α=1000. Figu e 5 depic s he 1D slow mani old, whe e poin s a e colo ed acco ding o he alues o he a iables hand P. He e, e en i he di ec ela ion can be no ed, i s nonlinea - i y is much less e iden o isualize. Finally, we conside he model P=α L2, (7) whe e α=1000. Figu e 6 depic s he 1D slow mani old, whe e poin s a e colo ed acco ding o he alues o he a iables Land P o emphasize he in e se ela ion be ween hem. 3.2. Ou pu depending on wo pa ame e s In his sec ion, we conside a model in ol ing wo o he h ee a iables, in pa icula , P=αh3 L2, (8) whe e α=1000. Figu e 7 depic s he wo-dimensional (2D) slow mani old, whe e poin s a e colo ed acco ding o he alues o a iables h,b,L, and P. He e, he di ec and in e se e ec s o hand Lwi h espec o Pcan be no ed as well as he ac ha pa ame e bseems, and in ac is, useless. C. R. Mécanique,2020, 348, n10-11, 937-958 Ruben Ibanez e al. 943 Figu e 7. Slow mani old in he 3D space de ined by he i s h ee educed coo dina es (ξ1,ξ2,ξ3). Each poin is colo ed acco ding o he alue o he coo dina e h( op le ), b( op igh ), L(bo om le ), o P(bo om igh ). 3.3. Iden i ying hidden a iables The p e ious case s udies e ealed he abili y o ex ac he in insic dimensionali y o he slow mani old as well as he possibili y o iden i y useless pa ame e s. The p esen case add esses a e y di e en si ua ion in which he model in ol es he h ee a iables in he discussed case, h, b, and L. Howe e , only wo o hem we e measu ed, namely hand Lwi h he ou pu P, wi h b emaining inaccessible. Thus, we ha e P=αbh3 L2, (9) whe e α=1000. The e o e, he M=1000 collec ed da a yi,i=1,...,M, eads as yi={hi,Li,Pi}∈R3. Figu e 8 depic s he educed poin s (ξ1,ξ2,ξ3), which as can be seen a e dis ibu ed in a domain ω⊂R3. Howe e , no dimensionali y educ ion is no ed, and he embedding emains 3D. A di ec consequence is ha many alues o he ou pu Pexis o he same alues on he measu ed inpu s hand L, ela ed o he di e en alues o b, which a ec he ou pu P. Howe e , as bis no measu ed, i s alue is no conside ed in he da a poin s. Such a mul i alued ou pu does no ep esen any concep ual di icul y. I indica es ha e en i bo h a iables pa icipa e in he ou pu , he e may be o he s ha we e no conside ed o hose conside ed may be useless o explaining he ou pu . C. R. Mécanique,2020, 348, n10-11, 937-958 944 Ruben Ibanez e al. Figu e 8. Reduced ep esen a ion ξi∈R3o he da a yi∈R3, whe e he educed poin s a e colo ed acco ding o he alue o he ou pu P. Figu e 9. Reduced poin s colo ed acco ding o he alues o he coo dina es h(le ) and L ( igh ). To conclude abou he pe inence o hese a iables wi h espec o he conside ed ou pu , we conside colo ing he educed da a depending on he hand L alues ( e e o Figu e 9). We compa e hem wi h he one whe e he colo scales wi h he ou pu P epo ed in Figu e 8. Thus, one could conclude ha bo h a iables hand La e ele an o explaining he ou pu P. I we assume ha he ou pu Pshould be uni ocally explained om a small numbe o a iables, clea ly only wo a iables (he e hand L) a e no su icien . One ex a dimension su ices o eco e ing a single- alued ou pu , ha is, conside ing he educed poin s in ou dimensions R4. As isualizing hings in ou dimensions is qui e a di icul ask, o he sake o cla i y in he exposi ion, in wha ollows, we p opose add essing a simple model in ol ing lowe dimensional spaces. We conside he simple model P=αbh3, (10) whe e α=1000. Howe e , he M=1000 collec ed da a yi,i=1,...,M, only deal wi h hand P, (i.e., yi={hi,Pi}∈R2). Figu e 10 depic s he da ase yi={hi,Pi}, whe e i can be no ed ha many alues o he ou pu Pa e ound o he same alue o he a iable h. Fo uni ocally exp essing he ou pu P, we conside again he k-PCA. We compu e he educed da ase in a 3D space by conside ing he i s h ee coo dina es (ξ1,ξ2,ξ3), while colo ing hese C. R. Mécanique,2020, 348, n10-11, 937-958 Ruben Ibanez e al. 951 Figu e 22. Slow mani old ela ed o linea elas ic beha io and colo ed acco ding o he angen modulus (i.e., ∆σi/∆εi). Figu e 23. Slow mani old ela ed o nonlinea elas ic beha io and colo ed acco ding o he angen modulus (i.e., ∆σi/∆εi). ins an aneous angen modulus when loading and unloading. As Figu e 26 e eals, he mechanical mani old is now iche wi h poin s on he elas ic domain bounda y sepa a ed om he elas ic mani old. In he solu ion depic ed in Figu e 26, damage is no ac i a ed. In he p esence o damage, he educ ion in elas ic angen modulus scales wi h he plas- ic s ain. The slow mani old is colo ed acco ding o he angen modulus as shown in Figu e 27. •The las scena io adds an ex a ichness o he cons i u i e beha io . A he mechani- cal s a es wi hin he elas ic domain, he elas ic angen modulus is a ec ed by a la en C. R. Mécanique,2020, 348, n10-11, 937-958 952 Ruben Ibanez e al. Figu e 24. S ain–s ess poin s associa ed wi h s a es wi hin he elas ic domain wi h nonze o plas ic s ains (le ) and he associa ed slow mani old colo ed acco ding o he an- gen modulus ( igh ). Figu e 25. Slow mani old ela ed o damageable elas ic–plas ic beha io ope a ing wi hin he elas ic domain colo ed acco ding o he angen modulus. a iable ha is he p oduc o he plas ic s ain wi h ano he ex a a iable. The la e , which could ep esen s ain- a e sensi i i y, is assumed o ake a bi a y alues he e due o he ac ha we a e mo e in e es ed in me hodological aspec s han in physical con- side a ions. The conside ed mechanical s a es a e depic ed in Figu e 28. When apply- ing he k-PCA o he se o mechanical s a es yi={εi,σi,∆σi/∆εi}, he esul ing mani- old emains 3D. As expec ed, no dimensionali y educ ion is accomplished as Figu e 29 e eals, whe e many alues o he elas ic angen modulus can be ound o he same alues o he s ess and s ain. To in es iga e he na u e o his beha io , we depic in Fig- u e 30 he elas ic angen modulus e sus he plas ic s ain, whe e as expec ed i can be no ed ha he o me does no depend exclusi ely on he la e . By applying he k-PCA o he da a shown in Figu e 30, which a e nonsepa able in wo dimensions, one expec s C. R. Mécanique,2020, 348, n10-11, 937-958 Ruben Ibanez e al. 953 Figu e 26. Slow mani old ela ed o elas ic–plas ic beha io ope a ing wi hin he elas ic domain and on he elas ic domain bounda y colo ed acco ding o he angen modulus. Figu e 27. Slow mani old ela ed o elas ic–plas ic beha io ope a ing wi hin he elas ic domain and on he elas ic domain bounda y, colo ed acco ding o he angen modulus, when damage is ac i a ed. o sepa a e hem by embedding in a 3D space as p e iously discussed and as Figu e 31 p o es. Finally, Figu e 32 p esen s he slow mani old om Figu e 31 bu now colo ed wi h espec o he plas ic s ain o he ex a la en a iable. C. R. Mécanique,2020, 348, n10-11, 937-958 954 Ruben Ibanez e al. Figu e 28. Mechanical s a es wi hin he elas ic domain, colo ed acco ding o he angen modulus, he las depending on he p oduc o wo la en a iables, he plas ic s ain, and ano he a bi a ily chosen a iable. Figu e 29. Slow mani old ela ed o elas ic–plas ic beha io ope a ing wi hin he elas ic domain, colo ed acco ding o he angen modulus, wi h he las scaling wi h he p oduc o he plas ic s ain and an ex a la en a iable. Figu e 30. Elas ic angen modulus e sus he plas ic s ain. C. R. Mécanique,2020, 348, n10-11, 937-958 Ruben Ibanez e al. 955 Figu e 31. Slow mani old ela ed o he da a consis ing o he elas ic angen modulus and he plas ic s ain. Figu e 32. Slow mani old o he elas ic angen modulus colo ed acco ding o he plas ic s ain (le ) and he ex a la en a iable ( igh ). 6. Conclusion The p esen wo k in oduced, es ed, and discussed issues ela ed o mani old dimensionali y wi h wo majo pu poses: (i) i s , when oo many measu able a iables a e employed, mani old lea ning is able o disca d he useless a iables; (ii) second and mo e impo an , he same echnique can be employed o disco e ing he necessi y o employing and hen measu ing an ex a la en a iable ha is able o eco e and ensu e single- alued ou pu s. Bo h issues we e analyzed and discussed in wo case s udies, one wi h espec o s uc u al mechanics and he o he wi h espec o pa h-dependen ma e ial cons i u i e beha io s. Indeed, he physical in e p e a ion o he disco e ed la en a iable could imply he in oduc- ion o o he measu able a iables. This opic should be deeply analyzed. The main in e es in disco e ing hese mani olds is ha o a new accessible mechanical s a e, he ou pu can be in e ed by a simple in e pola ion om i s neighbo s on he mani old. The main aim o he p esen wo k is he cons uc ion and analysis o hese mani olds. Fu u e wo ks, cu en ly in p og ess, will ocus on hei use in pe o ming da a-d i en simula ions. Con lic s o in e es The au ho s decla e no compe ing inancial in e es s. C. R. Mécanique,2020, 348, n10-11, 937-958 956 Ruben Ibanez e al. Dedica ion This manusc ip was w i en wi h he help o he con ibu ions o all au ho s. All au ho s ha e gi en app o al o he inal e sion o he manusc ip . Acknowledgmen s The i s , hi d, and ou h au ho s a e suppo ed by hei espec i e ESI G oup esea ch chai s; hei suppo is g a e ully acknowledged. The i s au ho is suppo ed by CREATE-ID ESI-ENSAM esea ch chai . The hi d au ho is suppo ed by he ESI G oup Chai a he Uni e si y o Za agoza. The ou h au ho is suppo ed by CREATE-ID ESI-ENSAM esea ch chai . Appendix A. F om p incipal componen analysis o i s ke nel-based coun e pa A.1. P incipal componen analysis Le us conside a ec o y∈RDcon aining expe imen al esul s o syn he ic da a om a nume i- cal simula ion. These esul s a e o en e e ed o as snapsho s. I hey a e ob ained by nume ical simula ion, hey consis o nodal alues o he essen ial a iable. The e o e, hese a iables will be somehow co ela ed and, no ably, he e will be a linea ans o ma ion Wde ining he ec o ξ∈Rd, wi h d<D, which con ains he s ill unknown la en a iables, such ha y=Wξ. (11) The D×d ans o ma ion ma ix W, which sa is ies he o hogonali y condi ion WTW=Id, is he main ing edien o he PCA [8]. Assume ha he e exis Mdi e en snapsho s y1,...,yM, which we s o e in he columns o a D×M ma ix Y. The associa ed d×M educed ma ix Ξcon ains he associa ed ec o s ξi,i=1,...,M. The PCA usually wo ks wi h cen e ed a iables. In o he wo ds,          M X i=1 yi=0, M X i=1 ξi=0, (12) implying he necessi y o cen e ing da a be o e applying he PCA. The PCA p oceeds by gua an eeing maximal p ese ed a iance and minimal co ela ion in he la en a iable se ξ. The la en a iables in ξa e he e o e unco ela ed, and consequen ly he co a iance ma ix o ξ, Cξξ =E{ΞΞT}, (13) should be diagonal. To ex ac he dunco ela ed la en a iables, we p oceed om Cyy =E{YYT}=E{WΞΞTWT}=WE{ΞΞT}WT=WCξξWT. (14) P e- and pos -mul iplying by WTand W, espec i ely, and making use o he ac ha WTW=I, gi e us Cξξ =WTCy y W. (15) The co a iance ma ix Cy y can hen be ac o ized by applying he singula alue decomposi- ion, Cyy =VΛVT, (16) C. R. Mécanique,2020, 348, n10-11, 937-958 Ruben Ibanez e al. 957 whe e Vcon ains he o hono mal eigen ec o s; Λis a diagonal ma ix con aining he eigen al- ues so ed in descending o de . Subs i u ing (16) in o (15), we a i e a Cξξ =WTVΛVTW. (17) This equali y holds when he dcolumns o Wa e aken o be collinea wi h dcolumns o V. We hen p ese e he eigen ec o s associa ed wi h he dnonze o eigen alues, W=VID×d, (18) which gi es Cξξ =Id×DΛID×d. (19) We he e o e conclude ha he eigen alues in Λ ep esen he a iance o he la en a iables (diagonal en ies o Cξξ). A.2. Mul idimensional scaling The PCA wo ks wi h he co a iance ma ix o he expe imen al esul s, YYT. Howe e , he MDS wo ks wi h he G am ma ix con aining scala p oduc s (i.e., S=YTY) [8]. The MDS p ese es pai wise scala p oduc s: S=YTY=ΞTWTWΞ=ΞTΞ. (20) Compu ing he eigen alues o S, we a i e a S=UΛUT=(UΛ1/2)(Λ1/2UT)=(Λ1/2UT)T(Λ1/2UT), (21) which in u n gi es Ξ=Id×MΛ1/2UT. (22) A.3. Ke nel-based p incipal componen analysis The k-PCA is based on he ac ha da a no linea ly sepa able in Ddimensions could be linea ly sepa a ed i hey a e p e iously p ojec ed o a space in Q>Ddimensions. Howe e , he ue ad an age a ises om he ac ha i is no necessa y o w i e down he analy ical exp ession o ha mapping. The symme ic ma ix Φ=ZTZ, wi h Zcon aining he snapsho s zi∈RQ,i=1,...,M, associa ed wi h yi∈RD, has o be decomposed in o eigen alues and eigen ec o s. The p ocedu e o cen e ing da a ziis ca ied ou in an implici way. The eigen ec o decomposi ion eads as Φ=UΛUT, (23) gi ing ise o Ξ=Id×MΛ1/2UT. (24) The di icul ies o ope a ing in a high-dimensional space o dimension, in gene al, QÀD, and he mapping una ailabili y a e ci cum en ed by in oducing he ke nel unc ional κ(also known as he ke nel ick). This allows compu ing scala p oduc s in RQwhile ope a ing in RDby applying he Me ce heo em. This heo em es ablishes ha i κ(u, ) (whe e u∈RDand ∈RD) is con inuous, symme ic, and posi i e de ini e, hen i de ines an inne p oduc in he mapped space RQ. Many di e en ke nels exis ; some o hem a e epo ed in [8]. C. R. Mécanique,2020, 348, n10-11, 937-958 958 Ruben Ibanez e al. 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