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1 Avne Mhne Lernng Lernng Grh Moes Mu kehoo ernng o unree GM Er Xng Leure 5 Augus 009 Reng: Er Xng Er CMU Re: or Bs Assung he reers or eh CPD re goy neenen n noes re uy oserve hen he kehoo unon eooses no su o o ers one er noe: D D n π n π n n ML jk n j' k jk n j' k When soe vres re no oserve e z z z z Er Xng Er CMU
2 or unree grh oes For ree grh oes he kehoo eooses no su o ers one er y noe us rens. For unree grh oes he kehoo oes no eoose euse he norzon onsn s unon o he reers P K n C K C In gener we w nee o o nerene.e. rgnzon o ern reers or unree oes even n he uy oserve se. n Er Xng Er CMU Log Lkehoo or UGMs wh ur ue oens Suen sss: or UGM VE he nuer o es h onguron.e. X V s oserve n se D{ } n e reresene s oows: e e δ n o oun n ue oun n V \ In ers o he ouns he kehoo s gven y: D n D δ n D n n There s nsy n he kehoo δ n δ Er Xng Er CMU n
3 Er Xng Er CMU Dervve o Lkehoo Logkehoo: Frs er: Seon er: δ δ δ Se he vue o vres o Er Xng Er CMU Conons on Cue Mrgns Dervve o kehoo Hene or he u kehoo reers we know h: In oher wors he u kehoo seng o he reers or eh ue he oe rgns us e eu o he oserve rgns er ouns. Ths oesn e us how o ge he ML reers jus gves us onon h us e sse when we hve he. e *
4 or unree grh oes Is he grh eoose rngue? Are he ue oens ene on ues no suues? e.g. 4 no X X X X X X 4 X X 4 Are he ue oens u es or Gussns or reerze ore oy e.g. e k k? Er Xng Er CMU Proeres on o ue oens For eoose oes where oens re ene on ues he o ue oens eue o he er rgns or onons o he orresonng ue. Thus he n e sove y nseon!! I he grh s noneoose n or he oens re ene on non ues e.g. 4 we ou no eue o ues oens o er rgns or onons. Poen eresse s ur or: IPF Feurese oens: GIS Er Xng Er CMU
5 5 Er Xng Er CMU or eoose unree oes Deoose oes: G s eoose G s rngue G hs junon ree Poen se reresenon: Conser hn X X X. The ues re X X n X X he seror s X The er rgns us eu he oe rgns. Le us guess h We n very h suh guess sses he onons: n sry s s s ϕ Er Xng Er CMU or eoose unree oes on. Le us guess h To oue he ue oens jus eue he o he er rgns or onons.e. he seror us e ve no one o s neghors. Then. One ore ee: X X 4 X X
6 oneoose n/or wh non ue oens I he grh s noneoose n or he oens re ene on non ues e.g. 4 we ou no eue er rgns or onons o o ues oens. X X X X X 4 X 4 j j j s.. j { j } j / j j j / j Hoework! X X 4 Er Xng Er CMU or unree grh oes Is he grh eoose rngue? Are he ue oens ene on ues no suues? e.g. 4 no X X X X X X 4 X X 4 Are he ue oens u es or Gussns or reerze ore oy e.g. e k k? Deoose? M ue? Tur? Meho Dre IPF Gren GIS Er Xng Er CMU
7 Ierve Prooron Fng IPF Fro he ervve o he kehoo: we n erve noher reonsh: n whh ers y n he oe rgn. Ths s hereore eon euon or. Sovng n oseor s hr euse ers on oh ses o hs nonner euon. The e o IPF s o ho e on he rgh hn se oh n he nueror n enonor n sove or on he e hn se. We ye hrough ues hen ere: ee o o nerene here Er Xng Er CMU Proeres o IPF Ues IPF eres se o eon euons: However we n rove s so oorne sen gorh oornes reers o ue oens. Hene eh se w nrese he kehoo n w onverge o go u. Irojeon: nng sruon wh he orre rgns h hs he enroy Er Xng Er CMU
8 8 Er Xng Er CMU Dvergene Vew IPF n e seen s oorne sen n he kehoo usng he wy o eressng kehoos usng vergenes. Re h we hve shown zng he kehoo s euven o nzng he vergene ross enroy ro he oserve sruon o he oe sruon: Usng roery o vergene se on he onon hn rue: : n Er Xng Er CMU IPF nzes vergene Pung hngs ogeher we hve I n e shown h hngng he ue oen hs no ee on he onon sruon so he seon er n unee. To nze he rs er we se he rgn o he oserve rgn jus s n IPF. oe h hs s ony goo when he oe s eoose! We n nerre IPF ues s renng he o onon roes whe reng he o rgn roy wh he oserve rgn.
9 or unree grh oes Is he grh eoose rngue? Are he ue oens ene on ues no suues? e.g. 4 no X X X X X X 4 X X 4 Are he ue oens u es or Gussns or reerze ore oy e.g. e k k? Deoose? M ue? Tur? Meho Dre IPF Gren GIS Er Xng Er CMU Feurese Cue Poens So r we hve susse he os gener or o n unree grh oe n whh ues re reerze y gener ur oen unons. Bu or rge ues hese gener oens re eoneny osy or nerene n hve eonen nuers o reers h we us ern ro e. One souon: hnge he grh oe o ke ues ser. Bu hs hnges he eenenes n y ore us o ke ore neenene ssuons hn we wou ke. Anoher souon: kee he se grh oe u use ess gener reerzon o he ue oens. Ths s he e ehn eurese oes. Er Xng Er CMU
10 Feures Conser ue o rno vres n UGM e.g. hree onseuve hrers n srng o Engsh e. How wou we u oe o? I we use snge ue unon over he u jon ue oen wou e huge: 6 reers. However we oen know h soe rur jon sengs o he vres n ue re ue key or ue unkey. e.g. ng e on?e u? jk zzz... A eure s unon whh s vuous over jon sengs ee ew rur ones on whh s hgh or ow. For ee we gh hve ng whh s he srng s ng n 0 oherwse n sr eures or?e e. We n so ene eures when he nus re onnuous. Then he e o e on whh s ve sers u we gh s hve o reerzon o he eure. Er Xng Er CMU Feures s Mrooens By eonenng he eh eure unon n e e no rooen. We n uy hese rooens ogeher o ge ue oen. Ee: ue oen ou e eresse s: e ng ng?e?e e K K e kk k Ths s s oen over 6 osse sengs u ony uses K reers here re K eures. By hvng one nor unon er onon o we reover he snr ur oen. Er Xng Er CMU
11 Conng Feures Eh eure hs wegh k whh reresens he nuer srengh o he eure n wheher nreses or ereses he roy o he ue. The rgn over he ue s generze eonen y sruon uy GLIM: e ng u? ng?e?e u? zzzzzz L In gener he eures y e overng unonsrne nors or ny unon o ny suse o he ue vres: e e k k I How n we one eure no roy oe? Er Xng Er CMU Feure Bse Moe We n uy hese ue oens s usu: e kk I However n gener we n orge ou ssong eures wh ues n jus use se or: e Ths s jus our ren he eonen y oe wh he eures s suen sss! Lernng: re h n IPF we hve o ovous how o use hs rue o ue he weghs n eures nvuy!!! Er Xng Er CMU
12 Er Xng Er CMU o Feure Bse UGMs Se kehoo unon Inse o ozng hs ojeve rey we k s ower oun The rh hs ner uer oun Ths oun hos or µ n rur or Thus we hve n n D D / µ µ µ D Er Xng Er CMU Generze Ierve Sng GIS Lower oun o se kehoo Dene Re gn Assue Convey o eonen: We hve: D e e e e e D e e π π e e Λ D 0
13 Er Xng Er CMU GIS Lower oun o se kehoo Tke ervve: Se o zero where s he unnorze verson o Ue e e Λ D Λ e e e Re IPF: Er Xng Er CMU Sury IPF s gener gorh or nng o UGMs. eon euon or over snge ues oorne sen Irojeon n he ue rgn se Reures he oen o e uy reerze The ue esre y he oens o no hve o e ue For uy eoose oe reues o snge se eron GIS Ierve sng on gener UGM wh eurese oens IPF s se se o GIS whh he ue oen s u on eures ene s n nor unon o ue ongurons. IPF: GIS:
14 4 Er Xng Er CMU Where oes he eonen or oe ro? Revew: Mu Lkehoo or eonen y.e. A ML ese he eeons o he suen sss uner he oe us h er eure verge. D D Er Xng Er CMU Mu Enroy We n roh he oeng roe ro n enrey eren on o vew. Begn wh soe e eure eeons: Assung eeons re onssen here y es ny sruons whh ssy he. Whh one shou we see? The os unern or ee one.e. he one wh u enroy. Ths yes new ozon roe: α s.. H α Ths s Ths s vron vron enon o sruon! enon o sruon!
15 5 Er Xng Er CMU Souon o he MEn Proe To sove he MEn roe we use Lgrnge uers: So eure onsrns MEn eonen y. Proe s sry onve w.r.. so souon s unue. L µ α e e L e sne e e * * µ µ µ Er Xng Er CMU A ore gener MEn roe h h h s.. H n e α h e
16 Consrns ro D Where o he onsrns α oe ro? Jus s eore esure he er ouns on he rnng : α Ths so ensures onsseny uoy. Known s he eho o oens... w o rge nuers We hve seen se o onve uy: In one se we ssue eonen y n show h ML es oe eeons us h er eeons. In he oher se we ssue oe eeons us h er eure ouns n show h MEn es eonen y sruon. o uy g ye he se vue o he ojeve Er Xng Er CMU Geoer nerreon A eonen y sruon: E : h e A sruons ssyng oen onsrns M : Pyhgoren heore M M MEn : n s.. h M h h M M MLk : Er Xng Er CMU n s.. E M M 6
17 Sury Eonen y sruon n e vewe s he souon o n vron eresson he u enroy! The enroy rne o reerzon oers u erseve o he. Er Xng Er CMU
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