Bilinear estimation of pollution source profiles in receptor models. Clifford H Spiegelman Ronald C. Henry NRCSE

Transcription

1 Blear esmao of olluo source rofles receor models Eu Sug Park Clfford H Segelma Roald C Hery NRCSE T e c h c a l R e o r S e r e s NRCSE-TRS No 9

2 Blear esmao of olluo source rofles receor models Eu Sug Park, Clfford H Segelma, ad Roald C Hery 3 Uversy of Washgo Naoal Research Ceer for Sascs ad he Evrome Seale, WA Texas A&M Uversy Dearme of Sascs College Sao, TX Uversy of Souher Calfora Dearme of Cvl & Evromeal Egeerg 36 Sourh Vermo Ave Los Ageles, CA Address for corresodece: NRCSE, Uversy of Washgo, Box 357, Seale, WA E-mal: eark@sawashgoedu

3 Summary Receor models am o defy he olluo sources based o ar olluo daa Ths arcle s cocered wh esmao of he source rofles (olluo reces) ad her corbuos (amous of olluo) We ake a cosraed olear leas squares aroach To avod havg fely may soluos, we rese ew ses of model defably codos, whch are ofe reasoable racce The resulg esmaors are show o be cosse ad asymocally ormal uder arorae defably codos Smulaos ad a alcao o real ar olluo daa llusrae he resuls Key words: Receor model; Model defably; Cosraed olear leas squares; Cossecy; Asymoc ormaly; VERTEX

4 Iroduco Receor modelg s a colleco of mehods used o model ar olluo daa Ar qualy daa ycally cosss of coceraos o ffy or sxy comouds of arbore gases or arcles measured over me The basc assumos receor modelg s coservao of mass ad chemcal mass balace (see, eg, Hoke, 985, 99) If here are q olluo sources, he h measureme from he receor, y = ( y, y,,y ), ca be rereseed as q y = αkpk + ε, =, L, () k = where P k = ( k, k,, k ) s he k h source rofle whch cosss of he fracoal amou of each seces he emssos from he k h source, α k s he corbuo from he k h source o he h day, ad ε = ( ε, ε,, ε ) s he measureme error o he h observao For examle, a rofle for a refery mgh look lke Proae, %; - Buae, 8%; -Peae, 7%; -Peae, 7%; -Mehyleae, 7%; oher chemcal seces, 3% The obecves receor modelg are o defy olluo sources ad assess he corbuo of each source based o hs daa There have bee wo radoal aroaches o receor modelg, whch are he chemcal mass balace (CMB) receor model ad mulvarae receor model (Hoke, 99) I CMB, he umber of sources, q, ad he source rofles, P k s, are assumed kow, ad he ma obecve s o esmae he source corbuos, α k s I ha case, he roblem reduces o he ordary lear leas squares regresso Several examles of CMB mehods such as racer eleme mehod, lear rogrammg mehod, ordary lear leas squares mehod, effce varace leassquares mehod, rcal comoe regresso mehod, ad rdge regresso mehod ca be foud Hery e al (984) ad Hoke (985) The CMB mehods are erformed o oe observao a a me The CMB assumos, however, o he kow umber of sources ad he kow rofles are ofe o useful racce Such lmaos of CMB

5 3 leads o he use of mulvarae aalyss receor modelg I marx erms, he model () ca be wre as Y = AP + E () where A: q source corbuo marx P: q source comoso marx E: error marx Examles of hose mulvarae models clude rcal comoe aalyss, facor aalyss, arge rasformao facor aalyss, self-modelg curve resoluo, ad so o (see, eg, Hery 99) The advaage of mulvarae receor modelg s ha does o requre a ror kowledge of he source characerscs Mulvarae receor modelg res o ge he esmaes for he umber of sources, q, her rofles, P, ad corbuos, A, all ogeher from daa However, hs goal cao be easly acheved sce here could be fely may soluos for A ad P eve wh he kow umber of sources There have bee some aems o avod hs roblem by lacg he cosras o he arameers (see Hery ad Km 99; Yag 994) Those cosras ca be obaed from ror kowledge of he roblem uder sudy or from he daa self Ths ssue wll be addressed erms of model defably more geeral Seco The frs mehod develoed by a sasca receor modelg feld was Source Aorome wh oe Source Ukow (SASU) by Badee-Roche ad Ruer (99) They suosed ha q = ad oe source rofle s kow ad oe s ukow They reaed he source corbuos as radom quaes havg a Drchle (Bea he case of q = ) dsrbuo, ad red o esmae he ukow source rofle ad he arameers of he dsrbuo of source corbuos by assumg ha he lm he ukow source s observed Segelma ad Daer (993) red a relaed esmae They wroe each P k = (s k,, s k ) as wo dmesoal robably mass fucos, (s k,

6 4 -s k ), =,, The he rao of robably masses gve o a seces o wo dffere days s calculaed If ha rao s exreme (eher bg or small) he a caddae for a wo dmesoal source rofle s foud Eher he mehods foud Badee-Roache ad Ruer (99) or hose foud Segelma ad Daer (993) are examles of racer mehods lookg for sgle seces ha s dcave of a sgle olluo source The assumo of havg racer eleme for each source makes ay ossble source roao or rasformao mossble, ad defably of model arameers s auomacally acheved Uforuaely, hs assumo could be urealsc racce sce bg ces have a umber of olluo seces ha do o occur by hemselves Yag (994) red cofrmaory facor aalyss model (see, eg, Aderso 984, sec 4) uder he assumos ha he umber ad yes of corbug sources are kow a ror He reaed he source corbuos as he radom vecors havg a dsrbuo wh some ukow mea vecor γ ad covarace marx Φ, ad showed he esmaors obaed by maxmzg a obecve fuco are cosse ad asymocally ormal Hs obecve fuco s acually he log-lkelhood fuco of he observaos whe hey follow a mulvarae ormal dsrbuo alhough he makes o ormaly assumos abou he observaos As a maer of fac, may evromeal egeers wa o vew he source corbuos as fxed arameers o radom varables, ad he assumo of ror kowledge of he yes of all sources he model s o a comforable assumo Ideffably of he model arameers The umber of sources, q, eeds o be deermed We are cocered wh he umber of maor olluo sources o he umber of all olluo sources sce here could be mllos of sources aure, ad would be mossble or meagless f we ry o defy all of hose sources Therefore, q meas he umber of maor olluo sources hereafer May ar olluo daases ycally coss of he measuremes o ffy or

7 5 sxy varables (VOC chemcal seces) The daa se s ofe oo large o hadle all a oce Furhermore, o all of he seces are helful fdg he maor olluo roducs I may evromeal alcaos some seces have a few commo maor sources ad some have may more muscule sources If he seces used esmag q come from dffere ses of sources each wh dffere umber of sources he esmaed umber of sources s o lkely o be erreable I s crucal o selec a arorae se of seces o esmae he umber of maor sources, q I could be doe by evromeal exer s udgemes or he lack of such source, by seces seleco algorhms such as SPECIESA or SPECIESB (see Park 997) I hs seco we assume ha a arorae se of seces s seleced ad he umber of maor olluo sources, q, s correcly esmaed We also assume ha model () each row of marx E has mea vecor ad varace-covarace marx Σ, ad A ad P are ukow cosa marces We lace hyscal cosras o A ad P The elemes of A ad he elemes of P are oegave, ad he row sum of P s Tha s, α k k k =,, =, (3) where =, L,, k =, L, q, =, L, The cosra k = = dcaes ha oly he relave amou of each seces a source s of our eres Our sources have fxed raos of he chemcal seces As log as he relave amous of seces are gve, we cosder he source defed We frs eed o roduce he defo of he model defcao Defo Le Y be a marx of he observable radom varables, θ be a marx of he arameers of eres, ad le F Y (C;θ) be he dsrbuo fuco of Y for arameer θ evaluaed a Y = C The arameer θ s defed f, for ay θ ad θ he arameer sace,

8 6 F Y (C;θ ) = F Y (C;θ ) for all C mles ha θ = θ If he he arameer θ s defed, we also say ha he model s defed Uder he model (), he dsrbuo of Y s deermed by AP ad Σ ( he ormal error case) Tha s, F Y (C; A P,Σ ) = F Y (C; A P,Σ ) mles ha A P = A P ad Σ = Σ ad vce versa I does o, however, mly ha (P, A ) = (P, A ) whch are he arameers of our eres Thus, our case, he defo ca be reduced o he followg: Defo The arameer (P, A) s defed f, for ay (P, A ) ad (P, A ) he arameer sace, A P = A P mles ha P = P ad A = A We also defe ear defably of he model arameers Defo 3 The arameer (P, A) s early defed f, for ay (P, A ) ad (P, A ) he arameer sace, A P = A P mles ha P P ad A A Prooso Assume rak(a) = q ad rak(p) = q The A * P * = AP mles ha A * = AR ad P * = R - P for a osgular marx R = (A A) - A A *

9 7 Sce boh A ad P are ukow, our model () suffers from odefably of model arameers eve whou he error marx, e, Y = AP = ARR - P for ay osgular marx R Eve he reasoable cosras ha we u, (3), do o remove hs odefably Ths ye of odefably s ofe referred o as facor deermacy he coex of facor aalyss Sce here are q elemes he marx R, we eed o u q deede codos o P or A o rule ou hs deermacy Preassgg zeros secfed osos of P s usually doe he cofrmaory facor aalyss Bu, requres some ror kowledge abou he source rofles o be esmaed If formao abou he yes of all he sources s avalable (as assumed Yag 994) oe ca ge he dea of where o assg s he marx P ad hs deermacy would be ake ou Sce our case he source rofles are ormalzed o sum o, hs us q deede codos o P Thus he umber of free arameers R reduces o q(q-), ad so we eed oly q(q-) more deede codos Oe se of such codos are C here are a leas q- zero elemes each row of P, C he rak of P (k) s q-, where P (k) s he marx comosed of he colums coag he assged s he kh row wh hose assged s deleed These codos ca be easly foud usual mulvarae aalyss exbook (see, eg Aderso 984) Noe ha C ad C are auomacally sasfed f we have racer eleme for each source A smlar se of codos ca also be aled o he source corbuo marx A D There are a leas q- zero elemes each colum of A D The rak of A (k) s q-, where A (k) s he marx comosed of he rows coag he assged s he h colum wh hose assged s deleed These codos are closely relaed o Hery s assumo ha he daa coas some os such ha each source s mssg (Hery 997) He argued ha f here are a leas (q-) edge os (os ha have oe source mssg) for each source ad he edge os do o have ay mulcolleares of dmeso less ha q- he he soluo o he

10 8 geeral mxure roblem s uque I o error case hese codos ca be covered o model defably codos The codo D mles ha o wo sources have he same se of q- edge os ad he edge os (a leas q- of hem) are learly deede To hel solve facor deermacy roblem, here, we also rese wo ew ses of assumos for defably or ear defably of A ad P by modfyg Hery s edge o assumo We eed oly oe se of assumos o hold for A ad P o be defable The frs se of our basc assumos are: A Each source s mssg o some days ad we kow whe a source s mssg A The average corbuo of h source whe k h (k ) source s mssg s equal o he average corbuo of h source for all days A3 The source corbuo marx A s of full colum rak ad he source comoso marx P s of full row rak, e, rak(a) = q ad rak(p) = q The secod se of our basc assumos are: B Each source s mssg o some days ad we kow whe a source s mssg B The dfferece bewee he average corbuo of h source whe k h (k ) source s mssg ad he average corbuo of h source for all days s small B3 The source corbuo marx A s of full colum rak ad he source comoso marx P s of full row rak, e, rak(a) = q ad rak(p) = q Remark The assumo A (or B) s equvale o reassgg zeros a secfed oso of he source corbuo marx Ths usually requres less ror formao ha he codos based o he source comoso marx Alhough Hery

11 9 (997) assumed he exsece of a leas q- edge os for each of q sources, here, A (or B) allows havg less ha q- edge os as log as he oher assumos are sasfed ( k Defg α ) as he average corbuo of h source whe he k h source s mssg ad α as he average corbuo of h source for all days, we ca reexress he above assumos as follows Of course we requre ha k For A-A3, A α k = whe I k, k =,, q Here I k s defed o be a subse of {,,, } for whch he k h source s mssg ( k A α ) =α, =,, q, k A3 rak(a) = q, rak(p) = q For B-B3, B α k = whe I k, k =,, q Here I k s defed o be a subse of {,,, } for whch he k h source s mssg ( k B α -α ) ε, =,, q B3 rak(a) = q, rak(p) = q The followg resuls show ha uder each se of assumos, A-A3 ad B-B3, odefably of he model arameers ca be removed Tha s, A * = A ad P * =P (or A * A ad P* P) The roofs are foud Aedx B Resul Le Assumos A-A3 hold The R = I

12 where I s he q q Idey marx ad R s ay osgular marx sasfyg A * = AR ad P * = R - P Resul ε Le Assumos B-B3 hold Defe B = + α k q ( q ) λ q where λ q s he smalles egevalues of PP If B s small eough he he dagoal elemes of R are close o, ad he off-dagoal elemes of R are close o Remark We emhasze ha all he codos ced hs aer are suffce codos bu o ecessary codos for model defably O closer seco f we kew ha eres of he A marx sasfy a = a we could dfferece he corresodg observaos ad creae zeros Thus by dog ycal me seres dfferecg we may creae daa ha sasfes he defably codos whe he orgal daa does o 3 Esmao of source rofles ad corbuos The umber of arameers model () creases o fy as he samle sze creases Kefer ad Wolfowz (956) addressed he ssue of esmag he srucural arameer cossely whe here are fely may cdeal arameers They assumed ha he cdeal arameers were deedely dsrbued chace varables wh a commo ukow dsrbuo fuco Ths assumo was made Badee-Roche ad Ruer (99) ad Yag (994) We do o make such assumo for our cdeal arameers, he rows of A Isead of reag hem as chace varables, we us leave hem as ukow arameers, whch s he way ha may scess ad hs alcao mos evromeal egeers wa o vew hem To acheve a cosse sequece of esmaors we eed o furher resrc a arameer sace for A, as well as ulzg he defably codos he fg rocedure Two models, Quas Radom Fucoal Model (whch s a geeralzao of he model used Kefer ad Wolfowz (956)) ad

13 Relcaed Fucoal Model, are cosdered, ad a se of algorhms, VERTEX, o fd he leas squares soluo s roduced Each of hese algorhms ca be selecvely mlemeed accordg o he ses of defably codos used The resulg esmaors are show o be cosse, ad also he uceraes assocaed wh hem are rovded 3 Quas radom fucoal model To overcome he dffculy of havg fely may arameers we frs resrc he arameer sace of A by assumg ha he frs ad he secod samle momes of he rows of A coverge o some fxed vecor ad marx, resecvely Ths model s referred o as quas-radom fucoal model Gleser (983) We assume y = α P + ε, where he ε are deede decally dsrbued -dmesoal radom row vecors wh zero mea vecor, osve defe covarace marx Σ, ad {α } s a fxed sequece sasfyg ad α = α α = K = ( α α ) ( α α ) K = where α s a q-dmesoal vecor ad K s a q q osve defe marx We choose he esmaors of A ad P so as o mmze he sum of leas squares, Q (P, A) = r[(y - AP) (Y-AP)] = y α P subec o he cosras, (3), ad defably codos = q k k k = (4)

14 VERTEX Sce boh of A ad P are ukow arameers, our esmao rocedure, VERTEX, cosss of wo ses: ) Gve P, A ca be esmaed by à =YP (PP ) - ) Fd ˆP whch mmzes Q ( P, A ) = r ( Y AP )( Y AP ) [ ] [ ] [( ( ) ) ( ( ) )] [ ( ( ) )] [( )( ( ) P) ] = r ( Y YP ( PP ) P) ( Y YP ( PP ) P) = r I P PP P Y Y I P PP P = r Y Y I P PP P = r S + y y I P PP where he vecor y = Y = y ad he marx = S = ( Y y) ( Y y) = ( y y) ( y y) = where s a -dmesoal colum vecor cossg of s over he feasble se Ω for whch he cosras o P ad he defably codos C ad C are sasfed The cossecy ad he asymoc ormaly of ˆP ca be rove by adag he roeres of leas squares esmaors Fuller (987) We sae he asymoc resuls for ˆP Theorem ad Theorem The roofs of all our heorems are foud he aedx B Theorem (Cossecy of ˆP) Le A, P ad Σ be he rue values of A, P, ad Σ resecvely Assume α = α α = where α s he h row of A ad α s a q-dmesoal row vecor ad K = ( α α ) ( α α ) K where K s a =

15 3 full rak marx Le he defably codos C-C hold Le W be he subse of +(+)/-dmesoal Eucldea sace ad ω ω s he eror of W ad Σ =σ I The, whe, = ( α P, vech( Σ + P K P ) ) Assume vec( P ˆ) vec( P ) Remark 3 Noe ha gw ( ;ˆ) θ q s a cosse esmaor of σ, ad ha AP ( ˆ) YP ˆ ( PP ˆˆ = ) coverges o usual leas squares esmae of A whe P s kow Remark 4 The samle mea of he esmaed daly source corbuos, where ˆα s he h row of AP ( ˆ) YP ˆ ( PP ˆˆ = ), ca be show o be cosse; = ˆα α = ˆα, Theorem (Asymoc Normaly of ˆP) Le he assumos for Theorem hold Le r be he umber of free arameers P ad θ be he r-dmesoal vecor cossg of hose free arameers Assume ha he rue arameer value θ (Θ), where Θ, he arameer sace for θ, s a covex comac subse of r-dmesoal Eucldea sace Assume he error covarace marx Σ = σ I ad he errors have fe fourh momes Le ˆθ be he value of θ ha mmzes The [ ] g( y, vechs; θ ) = r ( S + y y)( I P ( PP ) P) d ( ˆ θ θ) N(, H BGB H ), where

16 4 G s he lmg covarace marx of y vechs [ { } [,( ) ], B = (B, B ), { }] B = L ( I P( PP ) P) ( PP ) P + M ( PP ) P ( I P( PP ) P ) Φ, B = [( ) ] L I P( PP ) P α, H = L( ( I P( PP ) P) K ) L+ L ( I P( PP ) P) αα L, L [ ] vecp = s he marx of aral dervaves of P wh resec o θ evaluaed a θ = θ, M vecp = s he marx of aral dervaves of P wh resec o θ evaluaed a θ = θ, ad Φ s he marx A ( + ) marx such ha veca =Φ vecha for ay symmerc Remark 5 Noe σ = for uder our assumo ha Σ = σ I Remark 6 If θ s o he boudary of Θ, e, some of he elemes of θ s zero, he he lmg dsrbuo of ˆθ would o be a ormal dsrbuo I would be a mxure of o mass a zero ad a ormal dsrbuo 3 Relcaed fucoal model Cossder he model Y = UAP + E, (5) where U m L m L =, M M O M m = N, m s a m -dmesoal colum vecor = L m cossg of s, A s he q source corbuo marx, P s he q source comoso

17 5 marx, ad E s a N error marx The h observao he h relcao, y wh double scr oao, s rereseed by y = α P+ ε, =, L,, =, L, m where α s he q-dmesoal row vecor corresodg o he h source corbuo, ad ε s a radom error corresodg o he h observao he h relcao We assume ε s are deede ad decally dsrbued wh mea vecor ad varace-covarace marx Σ, ad α s ad P are ukow aramers Ths model s recogzable as a relcaed fucoal model (see, eg, Gleser 983) We have ( ) = E Y UAP ad Var( Y) = I Σ N Noe ha U s a kow N marx Uder he defably codos, A-A3 or B- B3, descrbed seco, hs model s defed (or early defed) Tha s, UA P = UA P mles ha A = A ad P = P ( or A A ad P P ) Le m = N Here, N s he oal umber of observaos he daa = The leas squares esmaors of A ad P are obaed by mmzg he sum of squares, [ ] Q N (P, A) = N r ( Y UAP) ( Y UAP) (6)

18 6 over he feasble se Θ, where [ ] [ ] = N r ( Y UY + UY UAP) ( Y UY + UY UAP) = N r[ ( Y UY ) ( Y UY )]+ N r ( UY UAP) ( UY UAP) = N r[ ( Y UY ) ( Y UY )]+ N r[ ( Y AP) U U( Y AP) ] = N r ( Y UY ) [ ( Y UY )]+ N r[ M( Y AP)( Y AP) ] Θ PA, αk, k, k,, L,, k, L, q,, L,, = = ( ) = = = = Ι Α where I A (= A-A3 or B-B3) s a se of he defably codos defed seco [ ] Sce N r ( Y UY ) ( Y UY ) does o deed o A or P, mmzg Q N (P, A) s [ ] equavele o mmzg Q * N (P, A) = N r M( Y AP)( Y AP) wr A ad P A fg algorhm for esmag A ad P uder hs model s gve below: VERTEX ) Gve A, P ca be esmaed by [ ] ( ) = ( ) P UA UA UA UY A MA A MY = ( ) ( ) m where M = m O m ad Y = m m m m = m = M m = y y y ) Fd  whch mmzes [ ] Q * N (P, A) = N r M( Y AP )( Y AP )

19 7 ( )( ( ) ) ( ) ( ( ) ) ( ) ( I A( A MA) A M) ( ) = ( ) N r M Y A A MA A MY Y A A MA A MY N r M I AAMA AMYY I AAMA AM = ( ) = ( ) over he feasble se Ω A where N r M I A A MA A M YY N r Y M I A A MA A M Y [ ] = ( ) { k } Ω A = Aα, =, L,, k =, L, q, Ι Α where I A (= A-A3 or B-B3) s a se of he defably codos defed seco Defo 4 Le A be a sequece of radom marces ad A be a cosa marx The A A meas P( A A > ε ) as for each ε >, where A = a The asymoc resuls for ˆP are saed Theorem 3 ad Theorem 4, whch are esablshed based o he asymoc resuls for  I he followg roosos we frs sae he asymocs for  Prooso (Cossecy of Â) Le A, P ad Σ be he rue values of A, P, ad Σ resecvely Le he arameer sace for A, Ω A be a comac subse of q-dmesoal Eucldea sace coag A Assume he defably codos A-A3 Seco are sasfed Also assume A P s he eror of a subse of -dmesoal Eucldea sace The, whe m, lm m N = c >, =,,,  A

20 8 Remark 7 Uder he defably codos B-B3 Seco, we ge he aroxmae cossecy of Â, e, P( A A > ε ) as for raccally small (o arbrary) ε > Le ˆ ˆ P A MA ˆ A ˆ MY = ( ) As a mmedae cosequece of Prooso ad he fac ha Y A P, ad by usg he couous mag heorem, we oba he cossecy of ˆP as follows Theorem 3 (Cossecy of ˆP) Uder he defos ad he assumos of Prooso, whe m, lm m N = c >, =,,, ˆP P We ow esablsh he asymoc ormaly of  Prooso 3 (Asymoc Normaly of Â) Le he assumos for Prooso hold Le r be he umber of free arameers A ad θ be he r-dmesoal vecor cossg of hose free arameers Assume ha he rue arameer value θ (Θ), where Θ, he arameer sace for θ, s a covex comac subse of r-dmesoal Eucldea sace Assume he errors have fe fourh momes Le ˆθ be he value of θ ha mmzes The gy (, N MA ; ) [ I ] = r Y N M( A( A N MA) A N M) Y ( ) d m ( ˆ θ θ ) N, H B( C Σ )B H, where m = m{ m}, Σ s error covarace marx,

21 9 [ ( ) ] [ ( ) ] B = Q C I A ACA ( ) AC P, H = Q C I A A CA ( ) A C PP Q, c C = c O where c c = lm m m, C N c c = c c O where c c c = lm m m, ad N veca Q = s he marx of aral dervaves of A wh resec o θ evaluaed a θ = θ Remark 8 The same comme as Remark 5 ca be made here f θ s o he boudary of Θ Remark 9 The asymoc ormaly of ˆ ˆ P A MA ˆ A ˆ MY = ( ) ca also be esablshed by sadard argumes for olear leas squares esmaors However, he resulg dsrubuo has a exremely comlcaed covarace marx Thus, we emloy Boosra mehod o oba he aroxmae covarace marx The esmaes are aymocally ubased by heorem 3 Uder he relcaed fucoal model (5), resamlg ca be doe wo ways, Case resamlg ad Model based resamlg We ada he algorhms Davso ad Hkley (997) for our model: For h =, L, H (H: boosra sze), A) Case resamlg For each source corbuo α ( =, L, ), choose y, L, y by radomly samlg wh relaceme from y L y,, m Combg y, L, y, =, L,, leads o a boosra samle, Y m m

22 B) Model based resamlg Fd  ad ˆP based o he orgal daa Comue resduals by R = I DH Y UAP ˆ ˆ where DH s a dagoal marx ( ) ( ) = ( ) ( ) ( ) ( ) cossg of he dagoal elemes of H UAˆ UAˆ UAˆ UAˆ 3 Radomly samle ε from r r, L, r N r where r ( =, L, N) s he h row of he N N marx R ad r s he resdual mea vecor, e, r = N r, L, r 4 Se Y = UAP ˆ ˆ + E where E ε, L, ε N = ( ) = = Boosra esmaors ˆP are obaed for H boosra samles, ad he samle covarace marx of hose ˆP s used as aroxmae covarace marx of ˆP Remark I has bee observed from he smulao sudy ha case resamlg leads o more sable boosra esmaors ha model based resamlg Accuracy of boosra esmaors ˆP from model based resamlg deeds heavly o how good he esmae  from he orgal daa s sce  s reaed as A model based resamlg 4 Smulaos I hs seco we cosder smulaed examles o llusrae he roosed mehods For VERTEX, he daa s geeraed by he model () where =, = 9, ad q = 3 The errors are deedely geeraed from he ceered logormal dsrbuo so ha hey have mea ad varace-covarace marx σ I The source comoso marx P (acually, P ) s gve Table The value of σ was chose so ha he rooros of he

23 error sadard devaos o he model sadard devaos are mosly bewee ~3% The elemes of source corbuo marx A are ake from he uform radom umbers (uform(,)) The rue mea source corbuos are gve Table The resulg daa marx Y cosss of oegave umbers Table True source comoso rofles ( P ) Seces Source Source Source Table True mea source corbuos Source Source Source 3 α To aly VERTEX, we assume ha s kow beforehad ha seces ad 4 are mssg source, seces 5 ad 7 are mssg source, ad seces ad 9 are mssg source 3, e, some of he elemes of he source comoso marx are resecfed Table 3 shows he resulg esmaes of he source comosos from VERTEX, ad Table 4 shows he asymoc sadard devaos of he esmaors Table 3 Esmaed source comoso rofles ( ˆP VERTEX Seces Source Source Source

24 Table 4 Asymoc sadard devaos of ˆP VERTEX Seces Source Source Source The VERTEX rovdes he esmaes for all of source corbuos, bu oly he samle mea of hose esmaed source corbuos s reored here The esmaed mea source corbuos s gve Table 5 Table 5 Esmaed mea source corbuos Source Source Source 3 ˆα VERTEX Noe ha he umbers Table 5 are he esmaes of he absolue source corbuos hs case sce he rue source rofles are geeraed so ha he sum of seces each rofle s I real suaos, he sum could be ay osve umber, say c k, k=,, 3 I ha case, αˆ, αˆ, αˆ ( 3) wll be he esmae of cα, cα, c3α3 ( ), =, L,, ad ( α ˆ, α ˆ, α ˆ 3) wll be he esmae of ( cα, cα, c3α3) Fg shows he rcal comoe lo of he daa, he esmaed source rofles, ad he rue source rofles I ca be see ha he source rofles obaed from VERTEX gve very good aroxmao o he rue source rofles

25 3 z z 45 z z 45 z3 - - z Fg Prcal comoe los of he daa (o), he rue sources (+), ad he fed sources by VERTEX (*) To llusrae VERTEX, he daa s geeraed based o he model (5) where N = 7, = 4 (assumg he source corbuos are reeaed every 4 hour), = 7, ad q = 3 Alhough he umber of relcaos m eed o be equal, for he sake of brevy, he same umber of relcaos are used for he source corbuos Thus, m = m = = m 4 = 3 The source rofles ormalzed o sum o are gve Table 6 The source corbuo marx A s geeraed o sasfy he codos A-A3 seco I s assumed ha source s mssg o 8h hour, source s mssg o 7h hour, ad source 3 s mssg o 6h hour, ad whe each source s mssg, he average source corbuos of he oher sources say he same The errors assocaed wh N observaos are deedely geeraed from he ceered logormal dsrbuo so ha he rooros of he error sadard devaos o he model sadard devaos (whch ca be defed as he squarerooed dagoal elemes of P K P hs case) are abou ~3%

26 4 Table 6 True source comoso rofles ( P ) Seces Source Source Source Noe ha, VERTEX, he cosraed mmzao s doe wh A Oce we ge he esmaed source corbuos, ÂVERTEX, he source comosos are esmaed by ordary leas squares, e, ˆ ˆ P A A ˆ A ˆ Y VERTEX VERTEX VERTEX = ( ) Table 7 shows he VERTEX esmaed source rofles ormalzed o sum o Alhough he oegavy cosras for he comosos were o used, he esmaes of source rofles are all oegave I s observed from he smulao ha oly whe he rue source comoso marx coas zeros, he corresodg esmaes (of zeros) are egave I ha case, would be a aural choce o relace he egave esmaes wh ad reormalz each source rofle Fg shows he rcal comoe lo of he daa, he rue source rofles, ad ˆP VERTEX I ca be see ha ˆP VERTEX comoso marx gves a very good aroxmao o he rue source Table 7 Esmaed source comoso rofles ( ˆP VERTEX Seces Source Source Source

27 5 z z z z z z Fg Prcal comoe los of he daa (o), he rue sources (+), ad he fed sources by VERTEX (*) Table 8 ad Table 9 show he boosra sadard devaos of ˆP VERTEX based o boosra samles from Case resamlg ad Model based resamlg, resecvely Table 8 Esmaed sadard devaos of ˆP VERTEX from Case resamlg Seces Source Source Source

28 6 Table 9 Esmaed sadard devaos of ˆP VERTEX from Model based resamlg Seces Source Source Source Alcaos I hs seco we rese some alcaos of our mehods o real ar olluo daases 5 Examle (Ar olluo comoso) As ar of a large ar qualy sudy, hourly coceraos of hydrocarbo gases were deermed by auomaed gas chromaograhy a wo ses Houso Texas from Jue o November 993 The orgal daa cosss of,54 hourly observaos (afer al screeg of he oulers) o 54 volale orgac comouds (VOC) ad oal omehae orgac carbo (TNMOC) The wd daa cossg of hourly average wd dreco, sadard devao of he wd dreco, ad resula wd dreco ad seed were also rovded These daa were used Hery, Seglema, Colls, ad Park (997) The mora seces were seleced by examao of he scaerlos, he correlao marx, ad evromeal egeer s udgeme for furher aalyss Hery e al (997) We use he same se of seces Accordg o Hery e al (997) Idusral ad Idusral 3 show esecally hgh emssos for he wd dreco 8 o o 9 o To he daase cossg of 83 observaos wh he 8-9 wd dreco, VERTEX s aled aga wh ˆq = 3 Here, formao eeded for resecfcao of zero eleme s obaed from he SAFER resul Hery e al (997) Table shows he esmaed source rofles The fed sources are very close her comosos o Idusral,

29 7 Idusral 3, ad Idusral 5 from SAFER The R (r ) values are gve Table Fg 3 shows he rcal comoe lo of he daa ad he fed sources by VERTEX Table Esmaed source comoso rofles ( ˆP VERTEX ) Seces Source Source Source 3 Ehae N_ro Aceyl Tee Bu3dm Peam Pe3m Hexa3m Ebz M_xyl Bz35m Bz4m Table R values bewee ˆP VERTEX ad ˆP SAFER Source Source Source 3 Idusral Idusral Idusral z z z z z z Fg 3 Prcal comoe los of he daa (o) ad he fed sources by VERTEX (*)

30 8 5 Examle (Ar olluo saal) As he secod examle we cosder measuremes o sulfur doxde gas colleced from receor ses he earby Grad Cayo Naoal Park (Hery, 99) The resulg daa se cosss of 53 observaos o varables (here receor ses) The umber of sources, q, s esmaed o be 3 by he NUMFACT algorhm (Hery, Park, ad Segelma, 997) Physcally, here are hree kow source regos of sulfur doxde gases he rego These sources are beleved o corresod o olluo sources souher Calfora, coer smelers souher Arzoa ad orher Mexco, ad elecrc ower las No all he source rofles have he requred umber of zeros o aly VERTEX, so we aly VERTEX wh defably codos A-A3 Source s assumed o be mssg o Day, Source s mssg o Day, ad Source 3 s assumed o be mssg o Day 44 These edge os (obs,, ad 44) are ake by he rcal comoe los of he daa ad he SAFER f resuls (Hery ad Km, 99) The esmaed source rofles ormalzed o sum o ad sadard errors based o 8 boosra samles from Model based resamlg aear Table Fg 4 shows he rcal comoe lo of he daa ad he fed sources From he lo ca be see ha he esmaed source rofles gve a reasoable f o he daa Table Esmaed source comoso rofles ( ˆP VERTEX )+ Varables Source Source Source (3) 5 (4) 35 (3) 886 (64) 635 (6) 7 (4) 586 (37) 78 (44) 857 (35) 346 (65) 9 () 77 (44) 38 (5) 684 (4) 476 (86) (9) 686 (4) 64 (56) 7 (43) (8) 994 (7) 87 (67) 686 (46) 99 (9) 78 (4) 75 (78) 78 (98) 686 (59) 47 (46) 37 (9) 67 (77) 73 (38) 55 (6) + Sadard errors are gve areheses

31 9 z z z z 5 z3-5 - z Fg 4 Prcal comoe los of he daa (o) ad he fed sources by VERTEX (*) 6 Coclusos Ths arcle has bee cocered wh cosse esmao of source rofles ad uceray esmao To elmae model odefably roblem, ew ses of defably codos based o he source corbuo marx were roosed addo o a se of radoal defcao codos ha use reassged s he source comoso marx These ew codos usually requre less ror formao ha he codos based o he source comoso marx As a mehod of esmag he source comosos ad he corbuos, smulaeously, he cosraed olear leas squares aroach was suggesed Two algorhms o fd he leas squares soluo, VERTEX ad VERTEX were reseed Each of hese ca selecvely be mlemeed accordg o he defably codos ha ca be acheved he roblem uder sudy The esmaors from VERTEX were show o be cosse ad asymocally ormal uder arorae defably codos

32 3 Ackowledgemes Ths Research was arally suored by he Texas A&M Ceer for Evromeal ad Rural Healh hrough Gra RF9768 from he Naoal Isue of Evromeal Healh Sceces ad by Gra DMS from he Chemsry ad Sascs ad Probably rograms a he Naoal Scece Foudao (CH Segelma ad ES Park)

33 3 Refereces Aderso, TW (984) A Iroduco o Mulvarae Sascal Aalyss (d ed), New York: Wley Badee-Roche, K, ad Ruer, D (99) Source aorome wh oe source ukow, Chemomercs ad Iellge Laboraory Sysems,, Davso, AC, ad Hkely, DV (997) Boosra Mehods ad her Alcao, New York, Cambrdge Uversy Press Fuller, WA (987) Measureme Error Models, New York: Wley Gleser, LJ (983) Fucoal, srucural ad ulrasrucural errors--varable models, roceedgs of he busess ad ecoomcs seco, Amerca Sascal Assocao, Washgo, DC, Hery, RC (99) Mulvarae receor models I Receor Modelg for Ar Qualy Maageme (ed P Hoke), 7-47 Amserdam: Elsever (99) Persoal commucao (997) Hsory ad fudameals of mulvarae ar qualy receor models, Chemomercs ad Iellge Laboraory Sysems, 37, 37-4 Hery, RC, Park, ES, ad Segelma, CH (997) Esmag he umber of facors o clude a mulvarae mxure model, Techcal Reor No79 Hery, RC, ad Km, BM (99) Exeso of self-modelg curve resoluo o mxures of more ha hree comoes ar Fdg he basc feasble rego, Chemomercs ad Iellge Laboraory Sysems, 8, 5-6 Hery, RC, Lews, CW, ad Hoke, PK (984) Revew of receor model fudameals, Amosherc Evrome, 8, Hery, RC, Lews, CW, ad Colls, JF (994) Vehcle-relaed hydrocarbo source comosos from ambe daa: he grace/safer mehod, Evromeal Scece ad Techology, 8, 83-83

34 3 Hery, RC, Segelma, CH, Colls, JF, ad Park, ES (997) Reored emssos of orgac gases are o cosse wh observaos, Proceedgs of Naoal Academy of Scece, 94, Hoke, PK (985) Receor Modelg Evromeal Chemsry, New York: Wley (99) A roduco o receor modelg, Chemomercs ad Iellge Laboraory Sysems,, -43 Kefer, J ad Wolfowz, J (956) Cossecy of he maxmum lkelhood esmaor he resece of fely may cdeal arameers, Aals of Mahemacal Sascs, 7, Park, E S (997) Mulvarae receor modelg from a sascal scece vewo, uublshed PhD dsserao, Texas A&M Uversy, De of Sascs Rao, C R (973) Lear Sascal Iferece ad s Alcaos (d ed), New York: Wley Searle, S R (98) Marx Algebra Useful for Sascs, New York: Wley Segelma, DH, ad Daer, S (993) Mulvarae chemomercs, a case sudy: alyg ad develog receor models for he 99 El Paso wer PM receor modelg scog sudy I Mulvarae Evromeal Sascs (ed CR Rao), Amserdam: Elsever Yag, H (994) Cofrmaory facor aalyss ad s alcao o receor modelg, uublshed PhD dsserao, Uversy of Psburgh, De of Mahemacs ad Sascs

35 33 A : q source corbuo marx Aedx A: Noaos P : q source comoso marx E : N error marx m : # of relcaos for h source corbuo m = = N: oal umber of observaos m : m -dmesoal colum vecor cossg of s U m L m m L =, M = M M O M L m m O m m, Y = m m m = m = M m = y y y c m lm, m m m N = m = { }, c = lm m m N c C = c O, C c c c = c c O c c Σ : varace-covarace marx of error vecors, e, Var(ε ) = Σ X = AP, W vecx =, F vecx = veca Q = veca, R = vecp, T = vecp, U =, α = c α, K = c c c α α α α = = = =

36 34 Aedx B: Proofs B Proof of rooso Le s call a ew source corbuo marx ad a ew source comoso marx obaed by a lear rasformao A * ad P *, resecvely Ad, assume A * P * = AP By osmullyg A * P * = AP by P, we ge A * P * P = APP A * P * P (PP ) - = A sce PP s of full rak by he assumo Leg S = P * P (PP ) -, A = A * S (A) Smlarly, remullyg A * P * = AP by A, we ge A A * P * = A AP (A A) - A A * P * = P sce A A s of full rak by he assumo Leg R = (A A) - A A *, P = RP * (A) By (A), (A), ad he assumo ha A * P * = AP, we have AP = A * S RP * = A * P * Sce A * s of full colum rak ad P * s of full row rak, we ge from (A), (A * A * ) - A * A * S RP * P * (P * P * ) -* = (A * A * ) - A * A * P * P * (P * P * ) - ad hece SR = I Noe ha boh of S ad R are q q full rak marces Hece, S = R - Usg hs ad (A), we ge A * = AR, ad from (A), P * = R - P Thus, f A * P * = AP, he A * = AR ad P * = R - P for a osgular marx

37 35 R = (A A) - A A * Noe ha R ca have dffere exressos B Proof of resul We eed he followg lemmas o rove resul ad resul Assume he hyscal cosras for he source comoso marx ad he source corbuo marx, (3), hold hroughou The Lemma A Le r k deoe he (k,) h eleme of R where k =, L, q, =, L, q q r k = =, k =, L, q Proof We have P = R P* from P * = R - P Thus, (k,) h eleme of he marx P ca be exressed as q r k = = k Due o he cosra ha row sum of P s, ad by erchagg he summaos, q r k = = = q = =, k rk = = = I follows from he cosra = = ha q r k = = Lemma A Uder he assumos A-A3, r k =, k =,L q, =, L, q, k Proof From A * = AR, he (,) h eleme of he marx A * ca be exressed as

38 36 α r + α r + L + α r = α, =, L,, =, L, q q q ad hece α r + α r + L + α r = α (A3) q q Say kh ( k =, L, q ) source s mssg o some days The ( k) ( k) ( k) ( k) ( k α r + L + α r + α r + L + α r q =α ), (A4) k k, k+ k+, q ( k =, L, q sce α ) ( k k = Here α ) ( k s defed he smlar way as α ) Subsracg (A4) from (A3), ( k) ( k) ( k) ( k) ( α α ) r + L + ( α α ) r + α r + ( α α ) r + L + ( α α ) r q k k k, k k k+ k+ k+, q ( k =α α ), (A5) By alyg A, we have for k α k r k = Ths mles r k = sce α k by A3 Proof of resul From lemma A, Usg hs ad lemma A ogeher, we ge Ths comlees he roof r k =, k =,L q, =, L, q, k q rk = rkk = =, k =,L q B3 Proof of resul We eed he followg lemmas addoaly o rove resul Lemma A3 Uder he assumos B ad B3, for k =, L, q, =, L, q, r k q λ q

39 37 where λ q s he smalles egevalue of PP Proof Recall ha P = RP Posmullyg hs by P ( P P ), we ge Le C = R = PP ( P P ) PP ad D P P The (k,) h eleme of C ca be exressed by = ( ) c k = k = Each summad,, he above equao s bouded by due o he cosras ha k ad Hece, k =,, q k L, =, L, q c =, (A6) k k = Noe ha he marx PP s symmerc By he dagoably of symmerc marces (see, eg, Searle 98, sec 6b), we have he exresso ha PP = UΛU where Λ s he dagoal marx of he egevalues of PP ad U s he orhogoal marx cossg of he egevecors of PP, ad hece D = ( P P ) = ( UΛU ) = UΛ U (A7) Usg (A7), he (,) h eleme of D ca be exressed by d q = usus, =, L, q, =, L, q s= λs Noe ha u s, u s sce U s a orhogoal marx Leg λ q be he smalles egevalue of PP, we oba for k =, L, q, =, L, q, q q q d = usus usus λ λ λ s= s s= s q (A8)

40 38 by he ragle equaly ad he oegavy of he egevalues of a oegave defe marx Noe ha λ q > due o he assumo B3 The (k,) h eleme of he marx R = CD s wre by By he ragle equaly, (A6), ad (A8), k =, L, q, =, L, q r = q c d k k = q q q q q rk = ckd ck d ck λ λ, = = q = Lemma A4 Uder he assumos B - B3, for k =, L, q, =, L, q, k, q r k B ε where B s defed by B = + αk q ( q ) λq Proof From (A5) he roof of lemma A, we ge ( k) ( k) ( k) ( k) ( α α ) r + L + ( α α ) r + α r + ( α α ) r + L + ( α α ) r q k k k, k k k+ k+ k+, q =α α Tha s, ( k ), ( k) ( k) ( k) ( k) α r = ( α α ) ( α α ) r L ( α α ) r ( α α ) r k k ad hece for k k k k, k+ k+ k+, =, L, q, =, L, q, k, ( k) L ( α α ) r ( k) ( k) ( k) ( k) α r = ( α α ) ( α α ) r L ( α α ) r ( α α ) r k k k k k, k+ k+ k+, q q q L ( k) ( α α ) r q q q ( k) ( k) ( k) ( k) α α + α α r + L + α α r + α α r + k k k, k+ k+ k+, L + α α q ( k) q r q

41 = + ε ε q ε q ε ε q L ( q ) λ λ λ q q q by he ragle equaly, B, ad lemma A3 Thus α r k k ε + q ( q ) λq I follows from B3 ha r k ε + αk q ( q ) B λq =, where k =, L, q, =, L, q, k Proof of resul From lemma A4, rk B, k =,L q, =, L, q, k Usg hs ad lemma A ad by he assumo ha B s small eough, we ge Ths comlees he roof q = rk r, k kk = =,L q B4 Proof of heorem We eed he followg lemmas o rove he heorem Lemma A5 Le y = α P + ε, where he ε are deede decally dsrbued - dmesoal radom row vecors wh zero mea vecor, osve defe covarace marx Σ Le{α } be a fxed sequece sasfyg ad α = α α = K = ( α α ) ( α α ) K =

42 4 The, whe, y = y α P, = ad S = ( y y) ( y y) Σ + P K P = Proof Ths resul follows from WLLN A dealed roof ca be foud Park (997) Lemma A6 Le g(x,y) be a couous real valued fuco defed o he Caresa roduc A B, where A s a subse of -dmesoal Eucldea sace ad B s a comac subse of q-dmesoal Eucldea sace Le x be a eror o of A Assume ha he o y s he uque o for whch My Bg( x, y) s aaed Le y m (x) be a o B such ha gxy (, ( x)) = M gxy (, ) The y m (x) s a couous fuco of x a x = x Proof Aedx 4B of Fuller (987) Proof of heorem Le w m y B = ( y, vech( S) ), ad θ = vec( P) Le Θ be he comac subse [ { }] of q-dmesoal Eucldea sace Defe gw ( ; θ ) =r ( S + y y) I P ( PP ) P Noe ha gw ( ; θ ) s a couous real valued fuco defed o W Θ By lemma A5, as, We show ha w ω Mg( ω ; θ) = Mr ( Σ + P P + P ααp ) I P ( PP ) K P s θ Θ θ Θ [ { }] uquely aaed a θ = θ f Σ = σ I Assumg Σ = σ I, [ { }] g( ωθ, ) = r ( σ I + P P + P ααp ) I P ( PP ) K P

43 4 [ ] [ α { ( ) } α] [ K { } ]+ [ α { PP } ] α = σ r[ I P ( PP ) P]+ r KP{ I P ( PP ) P} P + r P I P PP P P = σ ( q) + r P I P ( PP ) P P r P I P ( PP ) Noe ha I P ( PP ) P s a roeco marx ad so colums of I P ( PP ) P are orhogoal o ad learly deede of colums of P (rows of P) Sce K s of full rak, K P also sa he row sace of P Sce he roeco marx s uque (see, eg, [ { } ] Rao 973, sec c4), r K P I P ( PP ) P P has a uque mmum whe P ( PP ) P = P( PP ) P, whch s rue for P =RP for ay q q osgular marx R By he defably codos o P, C-C, R should be a dey marx, ad hece, [ { } ] r K P I P ( PP ) P P K has a uque mmum a P = P Thus, Mg( ω ; θ) θ Θ =σ ( q) + Mr{ K P ( I P ( PP ) P) P}+ r αp( I P ( PP ) P) Pα θ Θ { } =σ ( q) + r[ K P { I P ( P P ) P} P]+ r αp{ I P( PP ) P} Pα [ ] =σ ( q) s uquely aaed a P = P, e, θ = θ By lemma A6, vec( P ˆ) whch s he value of vec(p) such ha g( w ; vec( P ˆ)) = Mg( w ; vec( P)) s a couous fuco of w ad he θ Θ resul follows from he couous mag heorem B5 Proof of heorem We wll eed he followg lemmas he sequel o rove heorem Lemma A7 Le y = α P + ε, where he ε are deede decally dsrbued - dmesoal radom row vecors wh zero mea vecor, osve defe covarace marx Σ, ad fe fourh momes Le {α } be a fxed sequece sasfyg

44 4 α = α α = ad [ ] Le ˆ y, vechs γ = ( ) K ad γ = ( α α ) ( α α ) K = [ ( ) ] = α P, vechm + vechσ where m = PKP The d G ( ˆ γ γ ) N(,I), where he elemes of G are he covaraces of he elemes of ˆ γ, σ ( ) = Cov y, y σ, Cov( y, Sk ) = τ k, CS (, Skl ) = ( mkσl + mlσk + mkσl + mlσk + κ, kl )+ O( ), κ = E εε σ ε ε σ = E( εε ), ad τ = E( εε ε ) {( )( )}, kl k l kl k k Proof I s a drec adaao of heorem C of Fuller (987) Lemma A8 Uder he defos ad he assumos of lemma A7, d ( ˆ γ γ ) N(,G ), where he elemes of G are he lmg values of he elemes of G, lm Cov y, y σ, lm Cov y, S lm Cov S, S m σ m σ m σ m σ κ, ad m s are he elemes of m = P K P ( ) = ( k ) = τ k ( kl ) = ( k l + l k + k l + l k +, kl ) Proof I follows from he fac ha G G robably ad Slusky heorem [ { }] Lemma A9 Le g ( vechs, θ ) = r S I P ( PP ) P The

45 43 (a) [ { } g( vechs, θ)= L ( I P( PP ) P) ( PP ) P { ( ) }] ( + ) [ ] + M ( P P ) P I P ( P P ) P Φ vech S Σ P K P vecp vecp where L = = P= P vecp vecp, M = = P= P, ad Φ s he ( + ) marx such ha vec S I + PKP = Φ vech S I + P K P ( ( )) ( ( )) (b) [ ] + g( vechs, θ )= { L I P( PP ) P } K L o( ) vecp vecp where L = = P= P Proof A dealed roof ca be foud Park (997) Lemma A Le g ( y, θ ) = y( I P ( PP ) P) y The, [ ] (a) g θ ( y, ) =L { I P( PP ) P} α ( y αp)+ o( ) [ ] + (b) g θ ( y, ) = L I P PP P αα L O { ( ) } ( ) Proof A dealed roof ca be foud Park (997) Proof of heorem Le Θ = { θ, =,,r} Frs, we show wh robably aroachg oe ˆθ s he eror of he arameer sace Tha s, we eed o show ha here s a oe r-ball wh ceer ˆθ, all of whose os belog o Θ Le s deoe he se of all os θ R r such ha θ a < d, whch s a oe r-ball of radus d ad ceer a, by Bad ( ; ) By he assumo, θ I (Θ), ad hece here exss a δ > such ha B(θ ; δ ) Θ By heorem ˆθ s cosse for θ Therefore, wh robably aroachg oe as creases, ˆθ -θ < ε Seg ε = δ /, we have

46 44 B( ˆθ ; δ /) B(θ ; δ ) Θ wh robably aroachg oe as creases Thus ˆθ I(Θ) wh robably aroachg oe as creases The res of roof s based o Taylor s heorem ad he asymoc ormaly of y ad S The same argume he roof of heorem 4B Fuller (987) ca be used for our case oo We oly eed o calculae he frs ad secod dervaves of he obecve fuco g( ys, ; θ ) wh resec o θ sce he obecve fuco whch we mmze s dffere from ha of Fuller From (4B9) Fuller (987), we oba wh robably aroachg oe as, ˆ (, ; θ ) (, ; θ ) θ θ = g y vechs g y vechs (A9) where he elemes of θ * are evaluaed a os o he le segme ogθ ad ˆθ Noe ha our obecve fuco g( y, vechs; θ ) ca be reexressed as follows: [ { }] g( y, vechs; θ ) = r ( S + y y) I P ( PP ) P [ { }] Le g ( vechs, θ)= r S I P ( PP ) P = r[ S{ I P ( PP ) P} ]+ y{ I P ( PP ) P} y { } ad g ( y, θ)= y I P ( PP ) P y The, g( y, vechs; θ) = g( vechs, θ) + g( y, θ) { } = g( vechs, θ) + g( y, θ) (A) By lemma A9 (a), [ { } g( vechs, θ)= L ( I P( PP ) P) ( PP ) P { ( ) }] + [ ( )] + M ( P P ) P I P ( P P ) P Φ vech S Σ P K P

47 45 By lemma A (a), g( y, θ) =L[ { I P( PP ) P} α] ( y αp)+ o( ) Le ad [ { } { ( )}] B = L ( I P( PP ) P) ( PP ) P + M ( PP ) P I P( PP ) P [ ] B = L { I P( PP ) P } α Φ The follows from (A) ha gys (, ; θ ) = Bvech[ S ( Σ + PK P) ] B( y α P) + o( ) (A) Noe ha g( y, vechs; θ) = g( vechs, θ) + g( y, θ) (A) By lemma A9 (b), By lemma A (b), [ ] + g( vechs, θ )= { L I P( PP ) P } K L o( ) [ ] + g( y, θ) = { L I P( PP ) P} αα L O( ) [ ] [ ] Le H = L { I P PP ( ) P } K L ad H = L { I P( PP ) P} αα L The follows from (A) ha By heorem, g( vechs, θ) = H + H + o () (A3) ˆθ θ as, ad so θ also coverges o θ sce θ s bewee ˆθ ad θ Because he aral dervaves are couous,

48 46 g( y, vechs; θ ) g( y, vechs; θ) = + o () = H + H +o () (A4) where we have used (A3) Leg H = H + H ad B = ( B, B ), we oba ha [ { ( )} ] + ˆ θ θ = H B y α P, vechs vech Σ + P K P o ( ), (A5) where we have used (A9), (A), ad (A4) Le B = [{ } ] L I P ( PP ) P α, B = (B, B ) ad [ ] + [ ] H =L { I P( PP ) P} K L L { I P( PP ) P} αα L The, by he assumos α α, K K, ad hece by he couous mag heorem, B B, H H (A6) I follows from (A5), (A6), lemma A8, ad he couous mag heorem ha where G s defed lemma A8 d ( ˆ θ θ) N,H BGB H ( ) B6 Proof of rooso By he WLLN, as m, =,,, Y A P Le gy (, N MA ; ) heorem, [ { } ] = r Y M A A MA I ( ) A M Y The by he couous mag N gy (, N MA ; ) gap (, CA ; )

49 47 c where C = c O ad c c m lm > m N = Noe ha M g( AP, C; A) = M r P A C I A( A CA) A C A P θ Θ θ Θ [ { ( ) } ] =M r P A C I C A( A CA) A C C A P θ Θ [ { ( ) } ] s uquely aaed whe CAACA ( ) AC = CA( ACA) AC sce he roeco marx s uque By he defably of he model arameers dscussed seco, hs mles ha A = A Thus, M g( AP, C; A) s uquely aaed a A = A By lemma A6, θ Θ Â whch s he value of A such ha gy (, N MA ; ˆ) = MgY (, N MA ; ) ( ) fuco of Y, N M ad he resul follows from he couous mag heorem A s a couous B7 Proof of rooso 3 We eed he followg lemmas he sequel o rove he rooso m Lemma A Le M = m O ad m m = m{ m} The ( ) (a) vec M ( Y A P ) d N, I ( ) ( Σ ) ( ) d (b) vec m ( Y A P ) N, C Σ where C c c = c c O, c c c = lm m m, ad c N m lm m N =

50 48 ( ) m y a P m y a P Proof (a) ( ) M ( Y AP) = M m( y ap) ( ) By he mulvarae CLT, m y d ap N,Σ, =,, L The cocluso follows because y, L, are deede y ( ) (b) The cocluso s mmedae because m ( Y AP) = m M M ( Y AP) ad m M m m = m m O m m c c m, =, L, c c O c c = C Lemma A Le M be a dagoal marx ad A be a q marx, ad θ = veca The (a) M I AAMA AM A M I AAMA AM A { ( ) } = { ( ) } A (b) AM I AAMA AM M I A A MA A { ( ) }= ( ) M (c) AM { I AAMA ( ) AM} A= q q (d) AM { I AAMA ( ) AM} A { } = A M I A A MA A A A M M I A A A MA A { ( ) } + { ( ) M } Proof (a) For a eleme θ of θ, { I A( A MA) A M}

51 49 ( ) ( ) = A A MA ( AMA ) AM A A M A A A MA M = A ( ) A MA A M A ( A MA) ( ) ( ) AMA AMA AM AA A MA ( ) M = ( ) ( ) + ( ) A AMA AM AAMA A MA A A M AMA A AM AAMA ( ) M = A ( AMA) AM ( ) A MA A AMA θ ( AMA) AM ( ) AM A A A MA AMA ( ) AM ( ) AA A MA M ={ I A A MA A M} A A AMA AM AAMA M I A A MA A ( ) ( ) ( ) ( ) M { } (A7) Posmullyg boh sdes of he above equao by A, we ge M { I A( A MA) A M} A =M{ I A A MA A M} A A A MA A MA MA A MA M { I A A MA A ( ) ( ) ( ) ( ) M } A =M I A A MA A M A ( ) MA( A MA) A MA MA ( ) { } =M{ I A A MA A M} A ( ) (b) Premullyg boh sdes of he equao (A7) by A, we ge AM I AAMA ( ) AM { } =AM{ I AAMA AM} A A AMA AM AMAAMA M I A A MA A ( ) ( ) ( ) ( ) M { }

52 5 =( AMAM) A AMA AM A ( ) M I A( A MA) A M A = M { I A A MA A ( ) M } (c) Posmullyg boh sdes of (b) by A { } A A AM I AAMA AM A M I A A MA A { ( ) } = { ( ) M } A = ( MA MA )= q q (d) Dffereag boh sdes of (a) by θ, we ge M I AAMA AM A M I AAMA AM A { ( ) } + { ( ) } = M { I A A MA A M} A ( ) M I A( A MA) A M { } A Premullyg boh sdes of he above equao by A, we ge AM I AAMA AM A AM I AAMA AM A { ( ) } + { ( ) } = AM { I AAMA AM} A ( ) AM I AAMA ( ) AM I follows ha AM { I AAMA ( ) AM} A { } = A M I A A MA A A A M M I A A A MA A { ( ) } + { ( ) M } where we have used (b) A [ { } ] Lemma A3 Le gy (, N MA ; ) = r Y N M A A N MA I A N ( ) M Y ad veca Q = Assume ha m N c = o( m ), =, L, The [ ] (a) gy (, N M, θ ) Q C I A ( A CA ) A C P vec Y A P o( m = { } ( ) + )

53 5 [ ] (b) gy (, N M, θ ) = Q C I A A CA A C PP Q { ( ) } + o () Proof (a) For a eleme θ of θ, gy (, N M, θ) = N r( Y AP) M { I A( A MA) A M} ( Y AP) + N r( Y AP ) M { I AAMA ( ) AM} AP + N r( AP) M { I A( A MA) A M} ( Y AP) + N r( AP) M { I A( A MA) A M} ( AP) (A8) By he roery of race, we have N r( Y AP) M { I A( A MA) A M} ( Y AP) = ( vec( Y AP) ) N M { I A( A MA) A M} I vec( Y AP) = { vec( Y AP) } C { I A( A CA) A C} I + o( ) vec( Y AP) * By lemma A (b), vec( Y A P ) = O ( m ) Sce I A( A CA) A C s he roeco marx, we have N r Y A P M I A A MA { A M} Y A P = * O m ( ) ( ) ( ) ( ) Thus he frs erm of he equao (A8) s eglgble For he remag erms of he equao (A8) whe θ = θ, we ge N r ( Y A P M ( I A A MA A M) A P ) ( ) A= A

54 5 = N r Y A P M ( I A A MA A M) A ( P ) ( ) A= A by lemma A (a), N r ( A P M ( I A A MA A M) Y A P ) ( ) ( ) A= A by lemma A (b), ad A = N rp M{ I A A MA A M} Y A P ( ) ( ) A= A N r ( A P M ( I A A MA A M) A P P P = = ) ( ) ( ) A= A q q by lemma A (c) I follows ha gy (, N M, θ ) = { } N r Y A P M I A A MA A M A ( P ) ( ) A= A A N rp M( I A A MA A M) Y A P ( ) ( ) + O ( * m ) (A9) A= A Usg he roery of race, we ge for he frs erm of (A9), N r Y A P M I A A MA A M A { } P ( ) ( ) A= A A = N rp A = N r I MA ( A MA ) A M Y A P ( ) { } A= A A= A I MA A MA A M Y A P P ( ) ( ) { } = N vec A [{ I MA( AMA) A} M P] vec( Y AP), A= A ad for he secod erm of (A9),

55 53 A N rp Thus, M{ I A ( A MA ) A M} Y A P ( ) A= A A = N r A= A M{ I A A MA A M} Y A P P ( ) ( ) = N vec A [ M{ I A( AMA) AM} P] vec( Y AP) A= A gy (, N M, θ ) = [ ] N vec A M ( I A ( A MA ) AM ) P vec( Y AP )+ O ( * m ) A= A [ ] = vec A C ( I A( ACA) AC ) P vec( Y AP )+ o ( * m ) A= A by he assumo ha m N c = o( m ), =, L, I follows ha [ ( ) ] ( ) + gy (, N M, θ ) Q C I A ( A CA ) A C P vec Y A P o( m = ) veca where Q = (b) For elemes θ ad θ of θ, gy (, N M, θ) = N r ( Y AP) M ( I A( A MA) A M) ( Y AP) + N r ( Y AP ) M ( I AAMA ( ) AM) AP + N r ( AP) M ( I A( A MA) A M) ( Y AP)

56 54 + N r ( AP) M ( I A( A MA) A M)( AP) (A) Noe ha { I A ACA AC} ( ) s bouded over Θ sce I A( ACA) AC s he roeco marx We have for he frs erm of he equao (A), N r( Y AP) M { I A( A MA) A M} ( Y AP) = r( Y A P ) C { I A A CA A C} ( Y A P ) O m + * ( ) ( ) = O ( * m ), for he secod erm of (A), N r( Y AP ) M { I AAMA ( ) AM} AP = r( Y AP) C { I A( A CA) A C} AP + O m ( * ) = O ( * m ), ad for he hrd erm of (A), Thus N r( AP) M { I A( A MA) A M} ( Y AP) = O m ( * ) gy (, N M, θ) = N r( AP) M { I A( A MA) A M}( AP) + O m ( * ) For θ = θ, gy (, N M, θ) = N r ( A P ) M ( I A A MA A M) ( A P ) ( ) + O ( * m ) A= A

57 55 = rp A C{ I A A CA A C} A P + P A C{ I A A CA A C} A P + ( ) ( ) o( ) = { } + A { r C I A A CA A A C A PP r A C I A A CA A C } PP + ( ) ( ) o ( ) = vec A C I A A CA A C P P vec A [ { } ( ) ] + vec A C I A A CA A C P P vec A { ( ) } + o () [ ] where we have used lemma A (d) ad he roeres of race I follows ha [ ] gy (, N M, θ ) =Q C I A A CA A C PP Q { ( ) } + o () Proof of rooso By he same argume as ha heorem, ca be show ha ˆθ I(Θ) wh robably aroachg oe as m * creases The res of roof s based o Taylor s heorem ad he asymoc ormaly of Y The same argume he roof of heorem 4B Fuller (987) ca be used for our case oo We oly eed o calculae he frs ad secod dervaves of he obecve fuco g( Y, N M; θ ) wh resec o θ sce he obecve fuco whch we mmze s dffere from ha of Fuller From (4B9) Fuller (987), we oba wh robably aroachg oe as m *, ˆ (, ; θ ) (, ; θ ) θ θ = gy N M gy N M where he elemes of θ * are evaluaed a os o he le segme ogθ ad ˆθ By lemma A3 (a), [ ] gy (, N M, θ ) Q C I A ( A CA ) A C P vec Y A P o( m = { } ( ) + ) By lemma A3 (b),