Optimal Groups using the Akaike Information Criterion

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Dominick Osborne
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1 Optmal Goups usg the Akake Ifomato Cteo Pepaed by Geoge Pedes ad Chul Pak Peseted to the Isttute of Atuaes of Austala XVth Geeal Isuae Sema 6-9 Otobe 5 Ths pape has bee pepaed fo the Isttute of Atuaes of Austala s Isttute XVth Geeal Isuae Sema 5. The Isttute Coul wshes t to be udestood that opos put fowad hee ae ot eessaly those of the Isttute ad the Coul s ot esposble fo those opos. The Isttute wll esue that all epodutos of the pape akowledge the Autho/s as the autho/s, ad lude the above opyght statemet: The Isttute of Atuaes of Austala Level 7 Challs House 4 Mat Plae Sydey NSW Austala Telephoe: Fasmle: Emal: atuaes@atuaes.as.au Webste:

2 Optmal Goups usg the Akake Ifomato Cteo. Abstat I ay tal data aalyss, t s eessay to ga a udestadg of the data at a bas level. What s thee, what s t, what dosyases ase fom the sue s poessg of the data, s the data a appopate fom fo modelg? A good statg pot s to ostut hstogams of the data. The hstogams ae ostuted fom aw data. The most ommo questo asked whe ostutg hstogams s How may bs should I use. Though the use of the Akake Ifomato Cteo we wll show how a optmal umbe of bs s alulated ad tu use ths appoah to estmate a optmal umbe of homogeous goupgs attempts to lassfy atg vaable ategoes a pg exese. The bas dea fo the method s obtaed fom Sakamoto985 & late by Taylo 987. Keywods: bs, optmal goupgs, hstogam, pg, Akake Ifomato Cteo. Itoduto Followg the wok of Sakamoto985, the Akake Ifomato CteoAIC s a bass of ompaso ad seleto amog seveal models. The AIC was todued by Akake 973. I hs pape Akake showed the mpotae of the Kullbak-Leble 95 fomato quatty statsts ad deved AIC as ts estmato. The AIC s a bass of ompaso ad seleto amog seveal statstal models. As we all kow the goodess of ft of paametes of a model a be alulated by the expeted log lkelhood, amely, the lage the expeted log lkelhood the bette the explaato. The log lkelhood a be egaded as a estmato of the expeted log lkelhood. The mea expeted log lkelhood s the quatty defed as the mea, wth espet to the data x, of the expeted log lkelhood of the maxmum lkelhood model. That s, the lage the mea expeted log lkelhood the bette the ft to the model. At fst sght, t would seem that the mea expeted log lkelhood a be estmated by the maxmum log lkelhood. The maxmum log lkelhood, howeve, s show to be a based estmato of the mea expeted log lkelhood. The maxmum log lkelhood has a geeal tedey to ove estmate the tue value of the mea expeted log lkelhood. Ths tedey s moe pomet fo models wth lage umbe of fee paametes. Ths meas that f we hoose the model wth the lagest maxmum log lkelhood, a model wth a ueessaly lage umbe of fee paametes s lkely to be hose. I lookg at the elatoshp betwee the bas ad the umbe of fee paametes of a model, t s foud that maxmum log lkelhood of a model umbe of fee paametes of the model

3 s a asymptotally ubased estmato of the mea expeted log lkelhood. As defed by Akake 987 hs AIC estmato of Kullbak Leble fomato s AIC - x maxmum log lkelhood of the model xumbe of fee paametes of the model O deoted as AICk - λθ k K s Akake s poposed teo fo model seleto. A model whh mmses the AIC deoted by MAICE s the most sgfat model osdeed appopate. Whe thee ae seveal models whose values of the maxmum lkelhood ae about the same level, we would hoose the model wth the smallest umbe of fee paametes. I ths sese AIC ealses, the pple of pasmoy. 3. Costutg a optmal hstogam? As Sakamoto985 defed, let x,, x be a sample of measuemets fom a eta populato ad let x ad x be the smallest ad lagest values the sample. A teval [x -.5p, x.5p] s dvded to g lasses, whee p s the peso of eah soe. Oe way to deteme the ategoy sze g s to take [ / -], whee the opeato [] epesets the tege pat as ths s used as the fst set of bs. Usg the tal set of bs a hstogam s ostuted. The AIC s alulated to hek the goodess of ft of vaous models to the tal fequey table. Usg ths appoah wll allow the devato of a optmal fequey hstogam. So osde whee ad ae the umbe of bs to be gouped the two eds of the hstogam. Also let be the umbe of bs that ae gouped the mddle seto. Sakamoto985 defed a model as beg Model,,. The espetve model s defed as p p p3... p ϑ p... p ϑ p... p ϑ usg the fat that p ϑ ϑ ϑ alulatg the maxmum log lkelhood wll gve

4 log log log,, / AIC Let be the fequees the th teval ad be the umbe of bs the hstogam as a esult of goupg tal lasses esultg log log log,, AIC We wll use the h-squaed statst to test the applato of the AIC seletg the optmal umbe of bs. Gve ths we wll eed to defe the h-squaed statst, as t s the best kow goodess of ft statst. It a be used fo both otuous ad dsete sample data. To alulate the h-squaed statst, you must fst beak up the x-axs doma to seveal bs. The h-squaed statst s the defed as: E E N χ Whee N the obseved umbe of sample the th b. E I the expeted umbe of samples the th b. N The umbe of bs We kow that a weakess of the h-squaed statst s that thee ae o lea gudeles fo the seletg the umbe ad loato of the bs. I some stuatos, you a eah dffeet olusos fom the same data depedg o how you spefed the bs. We eogse the exstee of othe o-paamet statsts that ould be used but fo the puposes of ths pape we wll oly use h-squaed.

5 The followg example llustates whe the optmal umbe of bs ae alulated by usg the AIC the most appopate dstbuto s seleted. I the example below a sample of obsevatos have bee smulated fom the omal dstbuto. Below s the summay statsts of the sample of obsevatos. Desptve Statsts of sample Mea Stadad Eo Meda Stadad Devato Sample Vaae 9.48 Kutoss Skewess.897 Rage Mmum 7. Maxmum Sum Cout The gaph o the left s the hstogam based o equal b legths, the gaph o the ght s based o the AIC optmal umbe of bs. Hstogam wth Equal bs Optmal Hstogam We used two othe methods alulatg the bs whe attemptg to ft the dstbuto of the smulated set of values. The methods used wee the equal pobabltes & equal tevals.

6 I all, thee methods wee used, ludg the AIC opto to defe a optmal umbe bs. The gaphs below show how the ftted dstbuto alteed based o the appoah seleted ad that the AIC bs povded the best ft to the sampled data. Gaph : The IvGauss dstbuto was seleted whe attemptg to ft the smulated omal dstbuted data. The equal pobablty appoah was used to selet the umbe of bs. 8 IvGauss.4, Shft Gaph : The Webull dstbuto was seleted whe attemptg to ft the smulated omal dstbuted data. The equal b appoah was used. 8 Webull3.44, 7.46 Shft Values x ^ Values x ^ < 5.% 9.% 5.% > Gaph 3: Based o usg the AIC optmal bs. The omal dstbuto was seleted whe attemptg to ft the smulated omal dstbuted data. < 5.% 9.% 5.% > Nomal9.885, Values x ^ < 5.% 9.% 5.% > Usg the AIC to obta a optmal umbe of homogeous goupgs.

7 Smla to the applato of the AIC fo seleto of optmal umbe of bs, we have futhe looked at how we ould use suh a appoah to detfy homogeous goups. Ths, fo example, s of use whe tyg to goup postodes o seletg sum sued bads a pg evew. Ofte you would use statstal tehques suh as luste aalyss whh eques spefato of the umbe of lustes equed. It s ofte dffult dedg how may goupgs oe should use. A optmal umbe of lustes/bads o goupgs a be establshed by applyg a smla appoah as to fdg the optmal umbe of bs usg the AIC. Dayto 998 poposed the use of AIC to detfy optmal subsets of meas o popotos based o depedet goups. Ths s smla to odutg a aalyss of vaae ad testg the hypothess of equalty of the meas. Whe testg a lage umbe of goups ad fo o-paamet alteatves t would volve testg a lage umbe of pas. A example of aothe patal use of AIC appoah s to goup dvdual polyholde age to homogeeous age bads the otext of pvate moto pg. The eommeded appoah to podue elatvty fo polyholde age s fttg a otuous polyomal fo the most of ages, exept whee oe should allow fo age spef haatests. The most ommo appoah modellg sk pemums s to ft dvdual fequey ad sze models fo eah lam type the ombe to fom a sk pemum model. If the dvdual fequey ad sze model adopted a polyomal futo fo age elatvty, the the fal sk pemum elatvtes fo age s also smooth wthout ay futhe fttg poess. Ths fal smoothed elatvty fo age s deal fo atg puposes. If the uet atg system that a sue has a allow elatvtes fo dvdual age, the the fal smoothed elatvty wthout ay goupg should be used. Howeve, t s ofte equed pate that the otuous elatvtes to be gouped to age bads mostly due to the flexblty of atg systems. We have used AIC appoah to goup otuous elatvtes to optmal umbe of goups. The followg summases the steps we have take ad pesets ou fdgs. Fom ages 6 to, a smoothed fal sk pemum elatvty fo age s smulated ad soted a ode. Bas paametes fo AIC method ae alulated. eg. Iteval wdth, Ital umbe of tevals, mmum/maxmum pots ad tal hstogam The usg a teatve poess o dffeet ombato of goups, the optmal ombato of goups wth the lowest AIC was foud. Compae the elatvtes suggested by AIC goupgs to that of smoothed elatvty. AIC values fo some models of the Polyholde Age example.

8 Rak Model AIC AIC-mAIC 4,, ,, ,, ,5, ,, ,, ,6, ,6, ,, ,6, ,7, ,, ,, ,5, ,4, ,3, ,6, ,3, ,5, The ext gaph ompaes two elatvtes; Relatvty fo Polyholde Age Smoothed Relatvty Relatvty AIC Optmal Relatvty Polyholde Age Fgue Shows the optmal umbe of goupgs detfed by the AIC. I the exese, the optmal AIC was foud to have fve dsttve goups. 5. Coluso

9 I ths pape, we have desbed the use of the Akake Ifomato Cteo to selet the optmal umbe of bs whe ostutg a hstogam. Though smulato of data t was also show that by usg the bs fom the AIC appoah mpoved the auay of estmatg the appopate dstbuto fo a patula data set. A example was also povded as to how the AIC ould be used to povde optmal goupgs fo patula atg vaables a pg aalyss. Refeees:. Dayto, C 998 Best Subsets usg the Ifomato teo, Uvesty of Maylad.. Kullbak, S & Leble, R.A. 95. O fomato ad suffey. Auals of Mathematal Statsts. 3. Sakamoto, Y Categoal Data Aalyss by AIC. Kluwe Aadem Publshes, Dodeht, Hollad, Lodo, Bosto, 985, 4. Taylo, C.C., 987. Akake s Ifomato Cteo ad the Hstogam. Bometka, 74, ,

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