Knowlg struturs (Doignon & Flmgn, 1985, 1999) Prmtr stimtion in probbilisti knowlg struturs with th pks pkg Florin Wiklmir n Jürgn Hllr Psyhoo 2012, Innsbruk, Fbrury 9 Gols Chrtrizing th strngths n wknsss in ll prts of knowlg omin Pris, non-numril hrtriztion of th stt of knowlg tht is omputtionlly trtbl Builing upon rsults from isrt mthmtis n xploiting th powr of urrnt omputrs Aptiv knowlg ssssmnt Effiintly intifying th urrnt stt of knowlg bs on sking miniml numbr of qustions Apting to th lry givn rsponss s xprin thrs o in n orl xmintion Prsonliztion in thnology-nhn lrning Automtilly slt ontnt tht prson is ry to lrn 2 A subomin of physis: Consrvtion of mttr (1) (Tgpr t l., 1997) ) Whn i mlts n prous wtr: (i) Th wtr wighs mor thn th i. (ii) Th i wighs mor thn th wtr. (iii) Th wtr n i wigh th sm. (iv) Th wight pns on th tmprtur. b) Aftr th nil rusts, its mss: (i) is grtr thn bfor. (ii) is lss thn bfor. (iii) is th sm s bfor. (iv) nnot b prit. ) Whn 10 grms of iron n 10 grms of oxygn ombin, th totl mount of mtril ftr iron oxi (rust) is form must wigh: (i) 10 grms. (ii) 19 grms. (iii) 20 grms. (iv) 21 grms. 3 A subomin of physis: Consrvtion of mttr (2) (Tgpr t l., 1997) ) Aftr 3 mtl nuts n 3 mtl bolts r join togthr: (i) Th totl mount of mtl is th sm. (ii) Thr is lss mtl thn bfor. (iii) Thr is mor mtl thn bfor. (iv) Th mount of mtl nnot b trmin. ) Photosynthsis n b srib s: WATER + CARBON DIOXIDE hlorophyll GLUCOSE sunlight Whih of th following sttmnts bout this rtion is NOT tru? (i) As mor wtr n mor rbon ioxi rt, mor gluos is prou. (ii) Th sm mount of gluos is prou no mttr how muh wtr n rbon ioxi is vilbl. (iii) Chlorophyll n sunlight r n for th rtion. (iv) Th sm toms mk up th GLUCOSE molul s wr prsnt in WATER n CARBON DIOXIDE. 4
Rspons pttrns (Tgpr t l., 1997) Stunts from grs four through twlv N = 1620 11111 01111 10111 11011 11101 11110 00111 01011 01101 01110 10011 10101 10110 11001 11010 11100 00011 00101 00110 01001 01010 01100 10001 10010 10100 11000 00001 00010 00100 01000 10000 00000 Dtrministi thory Dfinitions A knowlg omin is intifi with st of (ihotomous) itms. Th knowlg stt of prson is intifi with th subst K of problms in th omin th prson is pbl of solving. A knowlg strutur on th omin is olltion K of substs of tht ontins t lst th mpty st n th st. Th substs K K r th knowlg stts. 0 5050 200 250 Frquny 5 6 Consrvtion of mttr: Knowlg strutur (Tgpr t l., 1997) Probbilisti knowlg struturs Rtionl If thr r rspons rrors thn knowlg stts K n rspons pttrns R hv to b issoit. Dfinition A probbilisti knowlg strutur is fin by spifying knowlg strutur K on knowlg omin (i.., olltion K 2 with, K) mrginl istribution P K (K) on th knowlg stts K K th onitionl probbilitis P(R K) to obsrv rspons pttrn R givn knowlg stt K Th probbility of th rspons pttrn R R = 2 is prit by P R (R) = K K P(R K)P K (K) 7 8
Th bsi lol inpnn mol (BLIM) (Doignon & Flmgn, 1999) Assumption: Lol stohsti inpnn Givn th knowlg stt K of prson th rsponss r stohstilly inpnnt ovr problms th rspons to h problm q only pns on th probbilitis β q of rlss rror η q of luky guss Th probbility of th rspons pttrn R givn th knowlg stt K rs P(R K) = (1 β q ) (1 η q ). q K\R β q q K R q R\K η q q \(R K) Th pks pkg Provis funtionlity for prmtr stimtion in probbilisti knowlg struturs. Min funtions blim print, loglik plot, rsiuls simult s.pttrn,s.binmt Fitting n tsting bsi lol inpnn mols (BLIMs) Extrtor funtions gnrt rspons pttrns from givn BLIM onvrsion funtions Mximum liklihoo stimtion 9 Exmpl: Mximum liklihoo stimtion 10 EM lgorithm Formult th liklihoo s if w hv vilbl th bsolut frqunis M RK of subjts who r in stt K n prou pttrn R (omplt t) inst of th bsolut frqunis N R of th rspons pttrns R R (inomplt t). E(M RK ) = Expttion Comput N R P(K R, ˆβ (t), ˆη (t), ˆπ (t) ) Mximiztion Estimt ˆβ (t+1), ˆη (t+1), ˆπ (t+1) bs on m RK = E(M RK ) β β β β η η η η 11 12
Exmpl: Mximum liklihoo stimtion Mximum liklihoo stimtion blim(mttr97$k, mttr97$n.r, mtho="ml") Numbr of itrtions: 9474 Goonss of fit (2 log liklihoo rtio): G2(7) = 13.763, p = 0.055553 Minimum isrpny istribution (mn = 0.2858) Mn numbr of rrors (totl = 1.02435) rlss rror luky guss 0.3697625 0.6545893 Problms Goo fit (w.r.t. liklihoo rtio sttisti) not suffiint for mpiril vliity of knowlg strutur Fit my b obtin by inflting rlss rror rts β q n luky guss rts η q Wht w wnt: Goo fit with smll vlus of β q n η q How to pply onstrints on th rror probbilitis tht r motivt by th knowlg strutur? (inst of brut-for onstrints, Stfnutti & Robusto, 2009) How muh of th fit is u to inflting th rror probbilitis in ML stimtion? Minimum isrpny mtho Rtionl For rspons pttrn R n knowlg stt K onsir th istn (R, K) = (R \ K) (K \ R), whih is bs on th symmtri st-iffrn. It is th numbr of itms tht r lmnts of ithr, but not both sts R n K (numbr of rspons rrors). Exmpl (10001, 10100) = 2 13 15 Minimum isrpny mtho Rtionl For givn rspons pttrn R, onsir th minimum of th symmtri istns btwn R n ll th knowlg stts K K (R, K) = min (R, K). K K Th bsi i is tht ny rspons pttrn is ssum to b gnrt by los knowlg stt ls to xpliit (i.., non-itrtiv) stimtors of th rror probbilitis minimizs th numbr of rspons rrors n thus ountrts n infltion of rlss rror n luky guss probbilitis A prviously suggst implmnttion of this i by Shrpp (1999, 2001) os not llow for itm spifi stimts. 14 16
Minimum isrpny mtho Exmpl: Minimum isrpny stimtion Assumptions A knowlg stt K K is ssign to rspons pttrn R R only if th istn (R, K) is miniml Eh of th miniml isrpnt knowlg stts is ssign with th sm probbility with i RK = ˆP(K R) = i RK K K i RK { 1 (R, K) = (R, K) 0 othrwis β β β β η η η η Exmpl: Minimum isrpny stimtion 17 Minimum isrpny ML stimtion 18 blim(mttr97$k, mttr97$n.r, mtho="md") Numbr of itrtions: 1 Goonss of fit (2 log liklihoo rtio): G2(7) = 384.95, p = 0 Minimum isrpny istribution (mn = 0.2858) Mn numbr of rrors (totl = 0.2858) rlss rror luky guss 0.1269547 0.1588477 Moifi EM lgorithm Moify th E-stp in th EM lgorithm to implmnt th rstrition m RK = E(M RK N R, ˆβ (t), ˆη (t), ˆπ (t) ) = 0 whnvr (R, K) > (R, K). This ls to i RK P(K R, ˆβ (t), ˆη (t), ˆπ (t) ) m RK = N R K K i RK P(K R, ˆβ (t), ˆη (t), ˆπ (t) ) Th M-stp pros s usul. 19 20
Exmpl: Minimum isrpny ML stimtion Exmpl: Minimum isrpny ML stimtion β blim(mttr97$k, mttr97$n.r, mtho="mdml") β β β η η η η Numbr of itrtions: 133 Goonss of fit (2 log liklihoo rtio): G2(7) = 310.32, p = 0 Minimum isrpny istribution (mn = 0.2858) Mn numbr of rrors (totl = 0.2858) rlss rror luky guss 0.0481207 0.2376818 21 22 Outlook Th pks pkg fturs Fitting n tsting bsi lol inpnn mols (BLIMs) Rspons gnrtion from givn BLIM objt Mximum liklihoo, minimum isrpny, n MDML stimtion Work in progrss Smpling istributions for goonss of fit tsts Gnrliz MDML ritrion: troff btwn liklihoo mximiztion n rror minimiztion... Thnk you for your ttntion florin.wiklmir@uni-tubingn. http://cran.r-projt.org/pkg=pks http://r-forg.r-projt.org/projts/pks/ Florin Wiklmir [r, ut] Jürgn Hllr [ut] Psqul Anslmi [tb] 23 24
Rfrns Aitionl slis Rfrns Aitionl slis Rfrns Doignon, J.-P. & Flmgn, J.-C. (1985). Sps for th ssssmnt of knowlg. Intrntionl Journl of Mn-Mhin Stuis, 23, 175 196. Doignon, J.-P. & Flmgn, J.-C. (1999). Knowlg sps. Brlin: Springr. Shrpp, M. (1999). Extrting knowlg struturs from obsrv t. British Journl of Mthmtil n Sttistil Psyhology, 52, 213 224. Shrpp, M. (2001). A mtho for ompring knowlg struturs onrning thir quy. Journl of Mthmtil Psyhology, 45, 480 496. Stfnutti, L. & Robusto, E. (2009). Rovring probbilisti knowlg strutur by onstrining its prmtr sp. Psyhomtrik, 74, 83 96. Tgpr, M., Pottr, F., Millr, G. E., & Lkshminryn, K. (1997). Mpping stunts thinking pttrns by th us of th knowlg sp thory. Intrntionl Journl of Sin Eution, 19, 283 302. Exmpl: Gnrliz MDML stimtion blim(mttr97$k, mttr97$n.r, mtho="mdml", inrius=1) Numbr of knowlg stts: 15 Numbr of rspons pttrns: 32 Numbr of rsponnts: 1620 Mtho: Minimum isrpny mximum liklihoo Numbr of itrtions: 1679 Goonss of fit (2 log liklihoo rtio): G2(7) = 47.11, p = 5.3126-08 Minimum isrpny istribution (mn = 0.2858) Mn numbr of rrors (totl = 0.75031) rlss rror luky guss 0.5014542 0.2488576 Rfrns Exmpl: Gnrliz MDML stimtion β 25 Aitionl slis Error n gussing prmtrs bt t 0.00000 0.00000 b 0.65533 0.23622 0.00000 0.00000 0.32823 0.32550 0.00000 0.00000 26 β β β η η η η 27