Statistical Analysis of Environmental Data - Academic Year Prof. Fernando Sansò

Transcription

1 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

2 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.

3 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

4 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

5 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

6 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

7 xrc ondr h followng obrvon: 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, mp( ) mp 3 3 ( ) 4 4 4, , 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

8 .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 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 mp dgn mrx (don mx up wh h covrnc modl prmr ) mp ( ) 8

9 known rm (don mx up wh h covrnc modl prmr ) I Q x δ δ δ norml Mrx ( ) x δ x 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

10 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: m ompu h mprcl covrnc funcon for 4: mp( ) mp mp () ( ). 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 mp ( ) mp ( ) ( ) ( ) mp , , 4

11 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 nd 68., whl h no ndrd dvon rul υ. Improv h m of h prmr by pplyng l-qur dumn, n xrc : () ( ) () ( ) mp mp mp mp I Q x δ δ δ ( ) x δ x x x x δ ( ) 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.

12 xrc 3 ondr h followng obrvon: 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. ( ), () mp In ordr o m h covrnc modl prmr, l u wr h ym: mp () ( ) 3 mp mp ( ) ( ) 3. 5 ( ) ( ) mp log log mp By ung h fr quon: () mp mp 7.67 ( )

13 xrc 4 ondr h followng obrvon n h pln: P P P 3 P 4 P 5 P 6 P 7 x x 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 ( ) W cn now compu h mprcl covrnc funcon: mp ( ) (.5) ( ). 68 mp , d (,] x 3

14 (.5). 53 mp, (,] d 6 (.5) ( ). 98 mp , 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 mp mp log ondrng h fr quon, w obn: h no vrnc rul: 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 ( ) ( ) mp mp δ.53 Q I δ x.5.5 δ ( ) δ x x.58 x x δ x ( ) mp In h c h corrcon from l qur dumn vry mll. 4

15 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 x fr rprnng h obrvon pon: x compu h mrx of h dnc bwn ch coupl of pon: D ( ) W cn now compu h mprcl covrnc funcon: mp ( ) (.5) ( ) 6. 5 mp , (,.5] mp d (.75). 563, d (.5,] x 5

16 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 mp log Ind of ubung h md n on of h wo quon nd hn drv h corrpondng, w wr mpl l qur problm: B obrvon vcor.563 B B B 7. h no vrnc rul: ( ) dgn mrx.5 xrc 6 ondr h followng obrvon, rrgulrly drbud n m: 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

17 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 whr d. For ch dnc vlu, w cn compu h corrpondng mprcl vrogrm vlu, obnng h followng mrx: Γ ( mp) whr ( mp) [ ] ( ) ( ( ) ( ) mp nd O L K K K K K 7

18 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) () [ ] < d 6 ( mp ) ( mp) () 3 [ ] d 4 < ( mp ) ( mp) () ( ) < d 6 ( mp ) ( mp) ( ) ( 9) < d 8 Rprnng h vrgd mprcl vrogrm, y o rcognz lnr rnd: (mp) ( ) ow w cn compu h prmr of h vrogrm modl: ( ) b > by pplyng l qur dumn o h orgnl (no vrgd) mprcl vrogrm, nglcng h vlu n h orgn whch lwy qul o zro: 8

19 M M M Q I numbr of vlu of h mprcl vrogrm for < x b ( ) mp ( ) h no vrnc rul: b b 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) Q I x b

20 (mp) () h no vrnc rul: b.897. b 3 5 7

21 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 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: ( ) ( )

22 whr: [ ] [ ] ( ) [ ]. 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: ( ) ( ) [ ] [ ] 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 ( Q ) Q [.5.5].544 m μ Q 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) [ ] 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.

23 xrc 8 ondr h followng obrvon n h pln: x x x P.6 P P 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 ( ) ( ) ( ) ( ) ( ) h opml combnon wgh ccordng o h Wnr Kolmogorov prncpl r: ( ) [ ] 3

24 whr: ( ).46 ( ).3679 ( ).3679 ( ) 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 ( ) (gnl covrnc).35.5 (wh no vrnc) - h obrvon cn b modlld : whr [] μ [] [ ] (gnl nd no r uncorrld) I 4

25 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

26 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 α ( ) ( ) 93. d M ( ) ( ) d M M M α hrfor, h krgng m of h gnl n rul: ( ) h vrnc of h mon rror gvn by: ( ) ( ) 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

27 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 P 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: h krgng ym cn b wrn : whr: Γ α M Γ

28 b.3679 h oluon of h ym : M 3 α b o h h krgng m of h gnl rul: ( P ) (.883).3439 (.9378). 639 h mon rror vrnc gvn by: ( P ) α whch hghr hn h on obnd by collocon. 8