Scl nly of nvronmnl D - cdmc r 8-9 Prof. Frnndo Snò XRISS - PR 5 bl of onn Inroducon... xrc (D mprcl covrnc m)...7 xrc (D mprcl covrnc m)... xrc 3 (D mprcl covrnc m)... xrc 4 (D mprcl covrnc m)...3 xrc 5 (D mprcl covrnc m)...5 xrc 6 (D mprcl vrogrm m)...6 xrc 7 (D collocon)... xrc 8 (D collocon)...3 xrc 9 (D krgng)...4 xrc (D krgng)...7 Upd: 5/6/9 uhor: lbro Moln, L Prun, Mrko Rguzzon
Inroducon In h con h unknown gnl no modlld n drmnc wy (for xmpl lnr combnon of con nd n funcon or lnr combnon of pln funcon) bu n rm of ochc propr (for xmpl mn, covrnc funcon, c.). wo mhod r prnd: collocon nd krgng, h formr umng h h gnl h zro mn, h lr provdng n unbd m whn h gnl h n unknown mn. ollocon um h h obrvon r mpld from rndom proc n m nd hy r compod by gnl plu no: ( ) ( ) ( ),, wh h followng ochc fur: [ ( )] [ ( )] [ ( ) ( )] ( ), gnl covrnc funcon [ ( ) ( )] ( ) δ [ ( ) ( )],, no covrnc funcon, wh δ mnng h gnl nd no r uncorrld. W wn o m h gnl vlu n by ung lnr combnon of h vlbl obrvon: ( ). In ordr o drmn h combnon wgh, w nvok h Wnr-Kolmogorov prncpl: {[ ( ) ( )] } nmly w wn o mnmz h mn qur mon rror. h rul of h mnmzon : whr: (, ) (, ) K (, ) mn ( ) ( ) [x] [x] [x] (, ) (, ) K (, ) (, ) (, ) K (, ) M O (, )... (, ) M O M K K I wh h numbr of vlbl obrvon.
If n obrvon pon w lk bou flrng, bcu w wn o pr h gnl from no; f no n obrvon pon w lk bou prdcon. h mon rror h dffrnc bwn h ru gnl nd h md gnl, nmly: ( ) ( ) ( ). w do no know h ru gnl, w cn only m h rror vrnc: ( ) ( ) { } ( ) ( ),. L u conclud wh condron on h gnl covrnc mrx rucur. In gnrl, ymmrc mrx follow: ( ) ( ) ( ) ( ) ( ) O O K K 3 3,,,,, umng h h d r xrcd from onry proc n m: () [ ] con, h vrfd bcu ( ) [ ] ( ) ( ) [ ] ( ) ( ), h rulng gnl covrnc mrx : ( ) ( ) ( ) ( ) ( ) O K K 3 3 Fnlly, um h h d r rgulrly drbud n m (grd pon): w g h followng gnl covrnc mrx: 3 4 Δ ( ) ( ) ( ) ( ) ( ) ( ) Δ Δ Δ h o-clld oplz mrx. 3
Krgng ow um h h obrvon r mpld from rndom proc wh n unknown mn vlu h n gnrl dffrn from zro. h obrvon cn b modlld : whr: [ ( )] [ ( )] μ [ ( )] ( ) μ ( ) ( ) ( ) μ u( ) ( ) u unknown mn μ u u, [( )( ( ) )] [ ( ) ( )] u ( ) [ ( ) ( )] u (, ) μ [ ( ) ( )] ( ) [ ( ) ( )] [ u( ) ( )], gnl covrnc funcon, no covrnc funcon mnng h gnl nd no r uncorrld. Snc μ unknown, h drc m of h gnl covrnc funcon from h d qu complcd; hrfor h followng funcon dfnd n uch wy h do no dpnd on h mn vlu: [ ] [ ( u( ) u( ) ] ( ) (, ) ( ( ) ( ) u, h funcon clld vrogrm. W wn o m h gnl vlu n by ung lnr combnon of h vlbl obrvon: nd dmndng h h m unbd: h condon corrpond o forc h: ( ) [ ( )] μ.. whch n gnrl no fd n h c of collocon. In ordr o drmn h combnon wgh, h followng ym h o b olvd: ( Γ ) whr: α Γ M Γ (, ) (, ) M (, ) Γ M (, ) (, ) K (, ) (, ) (, ) K (, ) (, ) K K (, ) O M 4
I O wh h numbr of vlbl obrvon nd α h Lgrng mulplcor. h vrnc of h mon rror ( ) ( ) ( ) gvn by: ( ) [ ( )] α Γ. o h h vrnc of h mon rror of h krgng oluon lwy lrgr (or mo qul) hn h corrpondng vrnc of h collocon oluon. On h ohr hnd h unbn condon no gnrlly fd by collocon. ovrnc funcon mon h procdur con of hr p: mprcl covrnc funcon mon umng h h D proc onry n m (nvrn by rnlon n m) or, n gnrl, h h nd rndom fld homognou nd oropc (nvrn by roo-rnlon): mp ( ) covrnc funcon nrpolon ung pov dfn modl, lk for xmpl: ( ) ( ) ( ),, ( ) > whr, r prmr o b md. o h h vlu of h mprcl covrnc funcon n h orgn hould no b ud n h gnl covrnc funcon nrpolon bcu h um of h gnl nd no vrnc. no vrnc mon: ( ) ( ) ( ) ( ) mp If > mp for numrcl ron, hn h vrnc of h no h o b forcd qul o zro or up o n -pror vlu. 5
mp( ) Ĉ ( ) Vrogrm mon I lk h covrnc funcon mon, bu now h mprcl vrogrm vlud from h d. hr rlon bwn covrnc funcon nd vrogrm: ( ) ( ) ( ) ( ) h clld nugg ffc Howvr hr x vrogrm modl h do no hv h corrpondng covrnc modl, lk for xmpl: ( ) < ondr for xmpl, h covrnc modl no ccpbl bcu h condon no fd. ( ) ( ) ( ) ( ) < R ( ) ( ) ( ) ( ) no ccpbl 6
xrc ondr h followng obrvon: 3 4 5 6 7 8 9.6446.7845 -.646 4.84 5..7479 -.9-4.4.393-5.8349 m h mprcl covrnc funcon for 4, umng h h proc onry nd zro mn. Inrpol h mprcl covrnc funcon wh n xponnl modl: ( ). m h ndrd dvon of h wh no. Fr of ll, compu h mprcl covrnc funcon: mp mp mp ( ). 46 mp () 3 () ( ), 3. 47,. 6436 mp( ) mp 3 3 ( ) 4 4 4, 3 -.94, 4 -.479 In ordr o m h gnl vrnc, whch corrpond o h prmr of h covrnc nd w xrpol h vlu n : funcon, w connc wh rgh ln ( ) nd ( ) mp ( ) mp mp quon of h rgh ln png hrough wo pon: mp () ( ) mp mp mp ( ) ( ) ( ) mp 3.5858 7
.635 mp ( ) In ordr o g fr m of h prmr, w compu h corrlon lngh, h h m dnc for whch h covrnc funcon qul o hlf h vrnc:. 799 corrlon lngh c hn l u forc h h nrpold covrnc funcon qul o h mprcl covrnc funcon for : c [ ]. 54 ( ) mp log logmp ( ) Summrzng, fr m of h covrnc modl prmr 3.5858 nd. 54, whl h no ndrd dvon rul υ.635. In ordr o mprov h rough m, (lnrzd) l qur dumn cn b up. h fr p o lnrz h xpron of h covrnc modl wh rpc o h unknown prmr: whr: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) hn clcl l qur problm cn b olvd, condrng ll h vlbl obrvon (corrpondng n h c o h mprcl covrnc funcon), pr from : () ( ) () 3 ( 4) mp 3.47 mp.6436 obrvon mp.94.479 mp.8586-3.788.737-5.87 3 3 3 dgn mrx.633-5.8468 4 4 4.5435-7.7957 (don mx up wh h covrnc modl prmr ) mp ( ) 8
.9489.698.6436 3.788 4 3 known rm (don mx up wh h covrnc modl prmr ) I Q x δ δ δ norml Mrx 3.394-4.4796-4.4796.9769 ( ).753 3.33 x δ.54 3.5858 x.947 6.988 x x x δ ( ) 878. mp o h h l qur oluon hould b rd by condrng nw pproxmon vlu h m of h prvou p, unl h convrgnc rchd. 9
xrc ondr h m obrvon xrc. um h h d r mpld from onry proc wh n unknown mn. m h mn of h proc from h d. Rmov h md mn vlu from h d. m h mprcl covrnc funcon from h rdul d, for 4. Inrpol h covrnc funcon wh n xponnl modl: ( ). m h ndrd dvon of h wh no. Fr of ll, l u m h mn of h proc : m o h w cn obn h followng rducd obrvon:.368 3 4 5 6 7 8 9 m.478.5477 -.54 4.5673 4.8634.5 -.39-4.573.75-6.77 ompu h mprcl covrnc funcon for 4: mp( ) mp mp (). 3599 ( ). 9358,. 4677, mp mp () 3 3 ( ) 4 4 Drv fr m of h covrnc modl prmr, n xrc. quon of h rgh ln png hrough wo pon: mp () ( ) mp mp 3.439.6374 mp ( ) mp ( ) ( ) ( ) mp 3 -.4453 4, 3 -.46, 4
In ordr o g fr m of h prmr, w compu h corrlon lngh, h h m dnc for whch h covrnc funcon qul o hlf h vrnc: 7. corrlon lngh c hn l u forc h h nrpold covrnc funcon qul o h mprcl covrnc funcon for c : ( ) ( ) 68. log log mp mp Summrzng, fr m of h covrnc modl prmr 439 3. nd 68., whl h no ndrd dvon rul 6374. υ. Improv h m of h prmr by pplyng l-qur dumn, n xrc : () ( ) () ( ).46.4453.4677.9358 4 3 mp mp mp mp 7.56 -.556-5.367.673 4.9355 -.75.8983 -.855 4 3 4 4 3 3.789..4677.8983 4 3 I Q x δ δ δ.7747-3.99-3.99.978 ( ).849 3.54 x δ.68 3.439 x.7 6.943 x x x δ ( ) 8589. mp o h h l qur oluon hould b rd by condrng nw pproxmon vlu h m of h prvou p, unl h convrgnc rchd.
xrc 3 ondr h followng obrvon: 3 4 5 6 7 8 5.75.788.58 -.789 -.383 -.46-3.86 -.9 m h mprcl covrnc funcon lg, lg, lg, umng h h proc onry nd zro mn. m h gnl covrnc funcon uppong h blong o h fmly: ( ). m h ndrd dvon of h wh no. Fr of ll, compu h mprcl covrnc funcon: mp mp 4 mp, ( ) 8. ( ), 5. 867 () mp In ordr o m h covrnc modl prmr, l u wr h ym: mp () ( ) 3 mp mp ( ) ( ) 3. 5 ( ) ( ) 5.867 mp log log.938 3 mp 3 3.5 By ung h fr quon: () 5.867.938 mp mp 7.67 ( ) 8.4-7.67. 664
xrc 4 ondr h followng obrvon n h pln: P P P 3 P 4 P 5 P 6 P 7 x...3.5 4.3 3.9.4 x.3 4.7.5 3.5.8 3.. -.44 -.84.4.3 -.5.9.7 m h mprcl covrnc funcon umng h h rndom fld zro mn nd homognou nd oropc. ompu h mprcl covrnc funcon for,.5,. 5,. 5. Inrpol h mprcl covrnc funcon wh n xponnl modl: ρ ( ρ ) m h no vrnc. ( x x ) ( x x ) ρ. Fr of ll, l u rprn h obrvon pon: x hn, compu h mrx of h dnc bwn ch coupl of pon (ymmrc mrx wh zro dgonl): D ( ).64.846.969.447.387..3.96.4.8544.4.597.7464.86.5657 W cn now compu h mprcl covrnc funcon: mp.79 3.557.4698.58.63.66 ( ).56 7 3 4 5 6 7 3 4 5.34 5 (.5) ( ). 68 mp 3 4 4 5 4 6 5 6, d (,] x 3
9 34 36 47 57 5 6 7 5 6 (.5). 53 mp, (,] d 6 (.5) ( ). 98 mp 3 4 3 5 3 7 6 7 3, d (,3] W cn now compu fr m of h covrnc modl prmr by forcng h h modl p hrough (. 5) nd (. 5). W g h followng ym: mp (.5) (.5) mp (.5) (.5).5 mp.68 mp 4.9866.5.53 mp mp log 4.9866. 668 ondrng h fr quon, w obn:.668.5.68.68. 584.4478 h no vrnc rul:.56.584. mp ( ) 469 fr lnrzng h xpron of h covrnc modl: ( ρ ) ( ρ) ( ρ) ( ) ( ) ρ ( ) w cn mprov h m of h modl prmr by pplyng l-qur dumn:.5.5 (.5).68.5 mp.4478 -.34 ( ) ( ) mp.5.53.5.5.5.898 -.784.5.98.5.5 mp.5.8 -.6.5.68.5 δ.53 Q I δ x.5.5 δ.89 -.659 -.659.38 ( ) δ x.4.46.584 x.58 x x δ x.668.6 ( ).56.58. 476. 686 mp In h c h corrcon from l qur dumn vry mll. 4
xrc 5 ondr h followng obrvon n h pln: m h mprcl covrnc funcon umng h h rndom fld zro mn nd homognou nd oropc. ompu h mprcl covrnc funcon for,. 5,. 75. Inrpol h mprcl covrnc funcon wh n xponnl modl: ( ρ ) ρ ρ ( x ) ( ) x x x m h wh no vrnc. P P P 3 P 4 P 5 P 6 x.3.3.9.7. -. x.9..3.7.3.4-3.89 4.88 5.9.3 3.95.99 fr rprnng h obrvon pon: x compu h mrx of h dnc bwn ch coupl of pon: D ( ).7.8485.683.447.643.447.635.36.8.7 W cn now compu h mprcl covrnc funcon: mp.583.366.955.766.44 ( ) 4. 659 6 3 4 5 6 5 (.5) ( ) 6. 5 mp 4 5 6 3 4 5 6, (,.5] mp d.5.5.75 3 5 6 3 (.75). 563, 4 35 36 45 46 d (.5,] x 5
W cn now compu fr m of h covrnc modl prmr by forcng h h modl p hrough (. 5) nd (. 5). W g h followng ym: mp.5.75 6.5.563 mp.5 4. 899 log 4.899. 87 Ind of ubung h md n on of h wo quon nd hn drv h corrpondng, w wr mpl l qur problm: B.5.75.87.5.87.75 6.5 obrvon vcor.563 B B B 7. h no vrnc rul: ( ) 395 4.659 7.395 7.38.689.8386 dgn mrx.5 xrc 6 ondr h followng obrvon, rrgulrly drbud n m:.84.86.74 3.87 4.76 6.4 7. 8.3 9.35.4.89 5.53 6.8 7.85 8. 3.7 9.3 9.95..7 3.6 3.89 4.96 6.8 36.7.57 3.4 9.99.35 8.4 3.3 6.4 m h mprcl vrogrm for < 8. vrg h mprcl vrogrm ovr nrvl of Δ. Inrpol h mprcl vrogrm (h orgnl on nd h vrgd on) wh lnr modl:. m h wh no vrnc. ( ) b 6
h obrvon cn b modlld : whr: [ () ] [ () ] u [ () ( ' )] δ ' ( ) μ u( ) ( ) [( u() u( ' )) ] ( ' ) ( ) Fr of ll, compu h mrx of h dnc bwn ch coupl of pon: D..9.88 3.3..3 3.9 5.4.9 4.38. 3.5.89.37.48 6.6 5.4 4.6 3.3.4.76 7.9 6.7 5.9 4.6 3.7.79.3 8.48 7.46 6.58 5.45 4.56 3.8.3.9 9. 8.9 7. 6.8 5.9 3.7.95.9.63.37 9.35 8.47 7.34 6.45 4.97 4. 3.8.89.6.88 9.86 8.98 7.85 6.96 5.48 4.7 3.69.4.77.5.3.3.4 9.9 8.4 6.9 6.6 5.3 3.84 3..95.44 3.5.3.5. 9.3 7.65 6.89 5.86 4.57 3.94.68.7.73 4. 3...9. 8.7 7.96 6.93 5.64 5. 3.75 3.4.8.7 5.34 4.3 3.44.3.4 9.94 9.8 8.5 6.86 6.3 4.97 4.46 3..9. whr d. For ch dnc vlu, w cn compu h corrpondng mprcl vrogrm vlu, obnng h followng mrx: Γ ( mp) whr ( mp) 59.96 66.5858.3 3.896 3.99.7648 [ ] ( ) ( ( ) ( ) 5.64.648 7.584.83 mp nd. 7.5 8.956 4.8.69.535 O L K K K K K 7
cr plo gnrlly ud o rprn h cloud of h mprcl vrogrm vlu: ( mp ) () h mprcl vrogrm h o b nrpold wh propr modl; o h m uful o vrg, o o dnfy h hp of h vrogrm modl. L u condr Δ, w g: ( mp ) ( ) ( mp ) ( mp) () [ 59.96 66.5858.... ] 9. 494 < d 6 ( mp ) ( mp) () 3 [ 3.896 5.64 3.99... ] 35. 364 3 d 4 < ( mp ) ( mp) () 56.63 ( ) 5 5 5 4< d 6 ( mp ) ( mp) ( ) 7.5649 ( 9) 7 7 7 6< d 8 Rprnng h vrgd mprcl vrogrm, y o rcognz lnr rnd: (mp) ( ) 4 6 8 ow w cn compu h prmr of h vrogrm modl: ( ) b > 3 5 7 by pplyng l qur dumn o h orgnl (no vrgd) mprcl vrogrm, nglcng h vlu n h orgn whch lwy qul o zro: 8
59.96 66.5858 3.896 5.64 M..9 3.3 3.9 M M Q I 687.4 336. 336. 87 numbr of vlu of h mprcl vrogrm for < 8.57.994.994.4988 9.58.55 x 3856.37 b 3.563 ( ) mp ( ) h no vrnc rul: b 3.563. b 4 6 8 h modl nrpolon cn b lo compud from h vrgd vrogrm, hu rducng h compuonl burdn, bu nroducng om pproxmon du o h dffrn ccurcy of h md mprcl vrogrm vlu (h hould b modlld n h Q mrx of h l-qur problm). 9.494 35.364 56.63 7.5649 3 5 7 Q I 84 6 6 4.5...5 96.35 9.34 x 83.963 b 9.86 9
(mp) () h no vrnc rul: b.897. b 3 5 7
xrc 7 ondr wo obrvon ( fgur) h r xrcd from D onry proc wh zro mn nd known covrnc funcon ( ). h obrvon r ffcd by wh no wh known vrnc.5. 3.837.35 - m h gnl vlu n h mn pon bwn h wo obrvd vlu ( ), ccordng o h Wnr Kolmogorov prncpl. ompu h vrnc of h mon rror. h obrvon cn b modlld : whr [] [ ] [ ] (gnl nd no r uncorrld) In ordr o olv h collocon problm, w compu h gnl covrnc mrx n h obrvon pon: ( ) ( ) ( ) ( ) h no covrnc mrx n h obrvon pon: I nd h gnl covrnc vcor bwn h obrvon pon nd h prdcon pon: () () h collocon m of h gnl n rul: ( ) ( )
whr:.353.5.353.5.5.5.353.353.5.353.353.5.353.895.5.876.895.876.876.876.895 [.3679.3679] [.656. 656].35 3.837.895 ( ) [.656.656]. 34 o h bcu h wo obrvon hv h m no vrnc nd hv h m dnc o h prdcon pon. o lo h, hppn nd wh krgng. h vrnc of h mon rror rul: ( ) ( ) [.3679.3679] [.656.656].954. 846.895.876.3679.3679.876.3679.895.3679 If w do no um h h proc zro mn, fr w hv o m h mn of h proc by ung h covrnc nformon o dfn h Q mrx of h l qur problm: whr: μ.35 3.837 ( Q ) Q [.5.5].544 m μ Q.35 3.837 hn w cn pply collocon o h rdul d (fr ubrcng h md mn) nd, n h nd, w cn ror h rmovd mn vlu o h gnl m:.893 μ.893 ( ) ( μ) [.656.656].544.544.544 mo h procdur h o b ppld only whn h proc no umd o b zro mn. If h covrnc funcon known, h procdur ld o h m rul of krgng.
xrc 8 ondr h followng obrvon n h pln: x x x P.6 P -.883 P -.9378 P P P P x m h gnl vlu n (,) P umng h h obrvon r mpld from homognou nd oropc rndom fld wh zro mn nd covrnc funcon ρ 4 ( ρ) ρ (wh ρ h ucldn dnc bwn coupl of pon). h obrvon r corrupd by wh no wh vrnc. 5 ompu h vrnc of h mon rror. ( ρ ). ρ h obrvon cn b modlld : whr: (,) ( ), ( ), ( ) ( ) ( ) ( ).3679 ( ) ( ) ( ) ( ) ( ) 3.3679.46 3.3679.46.3679.5.5.5 h opml combnon wgh ccordng o h Wnr Kolmogorov prncpl r: ( ) [. 78.7767.7767 ] 3
whr: ( ).46 ( ).3679 ( ).3679 ( ).76.469.469.469.988.4949.469.4949.988 h collocon m n P (, ) rul: ( P ) ( ). 776 wh h followng mon rror vrnc: ( P ) ( ) ( ) ( ).889. xrc 9 ondr h m obrvon xrc 7, now umng h h proc h n unknown conn mn. m h gnl vlu n nd h corrpondng mon rror vrnc by pplyng krgng. 3.837 ( ) (gnl covrnc).35.5 (wh no vrnc) - h obrvon cn b modlld : whr [] μ [] [ ] (gnl nd no r uncorrld) I 4
W wn o m h gnl n lnr combnon of h vlbl obrvon: ( ) whr h coffcn hv o b drmnd by krgng. Fr of ll, drv h vrogrm from h covrnc funcon: ( ) ( ) ( ) ( ) () () ( ) ( ) ( ) hn w compu h vrogrm mrx n h obrvon pon: ( ) Γ nd h vrogrm vcor bwn h obrvon pon nd h prdcon pon: ( ) whr: 5
h krgng ym cn b wrn : ( ) Γ α whr: Γ By xpndng, w g ym of hr quon wh hr unknown (h numbr of unknown of krgng oluon lwy qul o h numbr of obrvon plu on). ( ) ( ) α α h oluon of h ym : M 444 4 3 4 44 4 α ( ) ( ) 93. d M ( ) ( ).373.5.5.5.4486.4486.5.4486.4486 d M M.348.5.5.63.63 M α hrfor, h krgng m of h gnl n rul: ( ) 544. 3.837.35 h vrnc of h mon rror gvn by: ( ) ( ).9569.63.348.63.63.348 Γ α o h h krgng m corrpond o h mn vlu of h wo vlbl obrvon. o lo h h mon rror vrnc of h krgng oluon hghr hn h corrpondng vrnc of h collocon oluon, xpcd. 6
xrc ondr h m obrvon xrc 8, now umng h h proc h n unknown conn mn. m h gnl vlu n P (,) nd h corrpondng mon rror vrnc by pplyng krgng. x x P.6 P -.883 P -.9378 P x P P P x h obrvon cn b modlld : whr now [] μ. h vrogrm drvd from h gvn covrnc funcon ρ 4 ( ρ) ρ ( ) ( ρ) ρ 4 Γ ρ In ordr o olv h krgng problm w compu h followng mrc: 3 3.5.5.5 3 h krgng ym cn b wrn : whr: Γ α M Γ.5.3679.3679.3679.5.46.3679.46.5 7
.46.3679 b.3679 h oluon of h ym : M 3 α.3.3439 b.3439.5433 o h h krgng m of h gnl rul: ( P ).3.6.3439 (.883).3439 (.9378). 639 h mon rror vrnc gvn by: ( P ) α.5433.3758. 93 whch hghr hn h on obnd by collocon. 8