Local Influence Diagnostics of Replicated Data with Measurement Errors

Transcription

1 ISSN Eglad UK Joural of Iformaio ad Compuig Sciece Vol. No. 8 pp.7-8 Local Ifluece Diagosics of Replicaed Daa wih Measureme Errors Jigig Lu Hairog Li Chuzheg Cao School of Mahemaics ad Saisics Naig Uiversiy of Iformaio Sciece & echology Naig Chia (Received Sepember 7 acceped December 7) Absrac. Replicaed daa wih measureme errors frequely exis i various scieific fields. I his work we propose a replicaed measureme error model for such daa uder scale mixures of ormal disribuios. We cosider local ifluece diagosics o deec ad classify ouliers i he daa hrough differe perurbaio schemes. A simulaio sudy ad a applicaio cofirm he effeciveess ad robusess of he diagosic mehod. Keywords: scale mixures of ormal disribuios measureme error local ifluece aalysis robusess ouliers. Iroducio Local ifluece aalysis [] is oe of he effecive ways o deec ad classify ouliers. hrough various perurbaio schemes o he esablished saisical model i ca deec he iflueial observaios ad make oulier discrimiaio. he laes research o his area ca be see for example i [-5]. I his paper we focus o oulier deecio i replicaed daa wih measureme errors. A firs we eed o esablish a appropriae model o depic he correlaio bewee repeaed measuremes daa as well as o characerize he effec of measureme errors o he daa. Geerally he model is based o he assumpio of ormal disribuio [67]. Recely Cao e al. [89] proposed a replicaed measureme error model uder heavy-ailed disribuio which ca brig us robus ifereces. I his paper we sudy local ifluece aalysis o he heavy-ailed replicaed measureme error model uder differe perurbaio schemes. We aim o achieve a effecive ad robus diagosic mehod for oulier deecio i replicaed measureme daa. he paper is orgaized as follows. I Secio we give he diagosic mehodology icludig a brief descripio of he heavy-ailed replicaed measureme error model ad he local ifluece approach. I Secio we carry ou umerical simulaio. I Secio we display a applicaio o a real daa. We give a brief coclusio i he las secio.. Mehodology. he model Le ad ( ) represe he rue values of he explaaory variable ad he respose variable i he observaios respecively. heir correspodig acual repeaed measureme daa are ad ( y ) q which saisfy a replicaed measureme error model x i p ( i) ( i) y q ( ) ( ) x () i i p () ( ) () ( ) Where ad are measureme errors. Le Z ( p q ) x x y y be he acual observaios. Ulike he radiioal ormaliy assumpio here we propose a hierarchical disribuio srucure for Z : () Correspodig auhor. address: caochuzheg@6.com. Published by World Academic Press World Academic Uio

2 Joural of Iformaio ad Compuig Sciece Vol. (8) No. pp id Z ~ N ( a b ( v ) D( φ)) m id ~ N( ( v ) ) v ~ H( v; ) where m p q a( ) b ( ) q p q p ad iid () q represe p-dimesioal ad q-dimesioal vecor of oes respecively = ( ) D () deoes he diagoal rasformaio ha rasforms a vecor o a δ p ε q diagoal marix. he lae variable ca adus he weigh of he ifluece of differe samples o he parameer esimaio so as o obai a robus iferece effec. Saisical iferece of his model ca be foud i [8].. he local ifluece approach he purpose of local ifluece is o summarize he behavior of some ifluece measure whe small perurbaios ake place i he daa or model where ( g ) is a g-dimesioal perurbaio vecor. Le θ ( ) be he parameer vecor of model () be he complee-daa where v Z ( Z ξ v) Z ( Z Z ) ξ ( ) ( v v ) v ˆ E[ ( ) v θ Z ] ˆ ˆ ˆ / ( ˆ ( ˆ) ˆ b D φ b). Le be he log-likelihood of he perurbed model for he complee-daa. We assume ha here is a l ( ) c θ Z c such ha lc ( θ Zc ) lc ( θ Z c) for all θ. Le θ ˆ( ) be he maximum likelihood esimaio of θ uder he fucio ( ˆ) E{ ( ) θ θ l c θ Zc θz }. We defie α( ) ( f ( )) as he ifluece graph where f ( ) { ( θˆ θˆ ) ( θˆ ( ) θ ˆ)} is he -displaceme fucio which ca describe he differece bewee θ ˆ( ) ˆθ ad. he curvaure C f d d d of i he direcio of he ui vecor d a ca ivesigae he behavior of he -displaceme fucio where { ( ˆ)} Δ θ θ Δ is he value of ( Δ θ θ) / θ wihou disurbace. he Hessia marix (θ) ˆ ( ) / θ θ θ θ θ has elemes give by (he elemes o show are all. he followig is he same) ˆ ( ˆ ( ˆ q )) ˆ q ˆ ˆ α( ) q [ ˆ ( y ˆ )] q ˆ ˆ ˆ [ ˆ ( y )] q c q q ˆ ˆ ˆ Δ ( ) ˆ [ ˆ ( ) ˆ] p q p ˆ ˆ p q [ ˆ ( ) ] x i ˆ ˆ ˆ [ ( y ) ] q. i I his secio we cosider hree differe perurbaio schemes: case-weigh perurbaio respose variable perurbaio ad variace raio perurbaio. he key sep is o calculae he elemes of he marix. Δ i) Case-weigh perurbaio We cosider a arbirary aribuio of weighs for he expeced complee-daa log-likelihood fucio called perurbed -fucio which is preseed by ( θ θˆ ) E[ ˆ lc ( θ Zc ) θz ] where ( ) is a vecor wih ( ). he marix Δ has elemes give by ˆ ( θ θ) ˆ q [ ˆ ( ˆ ( θ θ) )] [ ˆ ( ˆ )] y ˆ q ( θ θ) ˆ ˆ ˆ [ ˆ ( y )] q ( θ θˆ ) ˆ [ ( ) ] JIC for subscripio: publishig@wau.org.uk

3 76 Jigig Lu e al.: Local Ifluece Diagosics of Replicaed Daa wih Measureme Errors ( θ θˆ ) p ˆ ˆ p p ( i) [ ( ) ] x i ( θ θˆ ) q ˆ ˆ q. q ( ) [ ( ) ] y ii) Respose variable perurbaio his perurbaio scheme which ca diagose observaios wih grea ifluece o heir predicio is ( ) ( ) represeed by replacig wih y y wih ( ) meaig o-perurbed model. I his scheme he marix Δ is give by ( θ θˆ ) q ˆ y ( ) ( θ θ) q ( ˆ ˆ ) iii) Variace raio perurbaio his perurbaio is iroduced by wriig case ca be acquired whe ( ). he marix ( q θ θ) ( ˆ ( y ˆ )). Δ. I his case he marix is give by ( θ θˆ ) [ ( ˆ )] q ˆ y ˆ q ( θ θ) ˆ ˆ ˆ [ ˆ ( y )] q ( θ θˆ ) q q ˆ ˆ ( ( ) ) y q. Δ he o-perurbed. Simulaio sudy We coduc umerical sudies o evaluae he performace of several diagosic mehods. he proposed local ifluece will be compared wih he mehod based o adused Pearso residuals. o geerae daa of model () we cosider wo ypes of disribuio: ormal disribuio (N) ad sude- disribuio (). he rue values of he parameers are se as follows:.5. he umber of repeaed measuremes is. Here we choose he respose variable perurbaio as a example. I p q order o add a oulier we shif he value of he -h sample o x x. he parameer represes he degree of perurbaio. We will calculae ype I error ad ype II error o compare he efficiecy of he wo diagosic mehods. he hypohesis es ca be expressed as H : he -h sample is a ormal oe H : he -h sample is a oulier. () Here ype I error meas he probabiliy of he h sample is he ormal oe bu is classified as he oulier ad ype II error is he probabiliy of he -h sample is he oulier bu is classified as he ormal oe. Firsly he daa is geeraed from model () uder he ormal disribuio. We do N replicaios wih sample size. able repors he resuls of ype I ad II errors based o he samplig daa usig he wo diagosic mehods. For local ifluece ype I ad ype II errors are boh ideally small ( Iloc _ if. II ); while for he adused Pearso residual aalysis ype I error is very high ( I.87 ) o keep he loc _ if ype II error as. he resuls idicae ha local ifluece is more effecive ha he residual based mehod whe he daa comes from ormal disribuio. able : he es resuls of wo diagosic mehods uder he ormal disribuio Local ifluece Adused Pearso residual ype I error ype II error ype I error ype II error N..87 Secodly we geerae daa from model () uder he disribuio. All he desiged values of his sceario are he same as before. he es resuls are lised i able. Sice saisical iferece based o heavyailed model will be more robus ha he ormal model he ifluece of he ouliers o he diagosic resul is reasoably sligh. his is he reaso why he ype II error uder local ifluece is larger ha i uder residual mehod. However he ype I error uder local ifluece is sill smaller ha i uder residual mehod. Overall x res = JIC for coribuio: edior@ic.org.uk

4 Joural of Iformaio ad Compuig Sciece Vol. (8) No. pp he local ifluece aalysis based o he heavy-ailed disribuio is more effecive ad more robus ha he ormal oe. able : he es resuls of wo diagosic mehods uder he disribuio Local ifluece Adused Pearso residual ype I error ype II error ype I error ype II error.89. Applicaio I his secio we cosider he CSFII daa [] as a illusraive example. he daa coais he -h recall measures ad hree addiioal -h recall phoe ierviews of 7 wome s die habis. We cosider he calorie iake/5 as ad he sauraed fa iake/ as. Isead of ad he uriio variables ad are recorded by four -h recalls which are assumed o follow model () wih. We cosider four disribuios for compariso which iclude he ormal disribuio (N) he disribuio () he slash disribuio (SL) ad he coamiaed ormal disribuio (CN). y pq.. Local ifluece g Accordig o he specral aalysis we have k k e k e k where ( e) ( g e g ) are he eigevalues-eigevecors wih eigevalues ad orhogoal eigevecors. h h g e eg h Le k k / ( h ) e k ( ek ekg ) M () k ke k. Le ad SM () deoe respecively he mea ad sadard deviaio of { M() l g}. Observaios wih value of sigificaly greaer ha c l M() c SM () ( ) are cosidered as poeial ouliers []. i) Case-weigh perurbaio he local ifluece iesiy of each observaios uder his perurbaio are ploed i Figure. he umbers of ouliers uder differe disribuios are lised i able. he values of baselies are calculaed based o he variace of he iesiy. I is o difficul o fid ha he iesiies of ouliers uder he ormal disribuio is much higher ha hose uder he heavy-ailed disribuios. For example he iesiy of he sroges iflueial uder he ormal disribuio is greaer ha. while he iesiy of he sroges iflueial 569 uder ad SL is less ha.8. From his viewpoi he diagoses uder he heavy-ailed disribuios are much slighly affeced by he ouliers ha hose uder he ormal disribuio. M() M() l x.5 () N 8 x - () x - () SL. () CN Fig. : Local ifluece uder case-weigh perurbaio able : Oulier iformaio uder case-weigh perurbaio JIC for subscripio: publishig@wau.org.uk

5 78 Jigig Lu e al.: Local Ifluece Diagosics of Replicaed Daa wih Measureme Errors he umber of ouliers N SL CN ii) Respose variable perurbaio Figure shows he ifluece iesiy uder differe disribuios. Uder he ormal disribuio he iesiy of he sroges iflueial 5 is abou.5. Bu he sroges iflueial uder ad SL are 7 ad 98 respecively wih iesiy abou.5. he sroges iflueial 78 uder CN is abou.. Furhermore he umber of poeial ouliers are lised i able. We fid ha boh he umber ad he iesiy of he iflueial observaios are clearly reduced whe we use heavy-ailed disribuios. his resuls coicide wih he coclusio draw from case-weigh perurbaio. 6 x - () N x - () x - () SL x - () CN Fig. : Local ifluece uder respose variable perurbaio able : Oulier iformaio uder respose variable perurbaio he umber of ouliers N SL CN iii) Variaces raio perurbaio Figure ad able 5 give respecively he iesiy of he iflueial ad he iformaio of he ouliers uder his perurbaio. he diagosic resuls uder N ad CN are similar. However hey are obviously differe from he oher wo heavy-ailed disribuios. For example uder he ormal disribuio he iesiy of he sroges iflueial is abou.7 while uder disribuio he sroges iflueial 9 is oly.5 ad he sroges iflueial 75 uder he slash disribuio is less ha.. hese resuls help us have a more obecive udersadig of he ouliers ad avoid he misdiagosis caused by misclassificaio of he disribuio. JIC for coribuio: edior@ic.org.uk

6 Joural of Iformaio ad Compuig Sciece Vol. (8) No. pp () N 6 x - () x () SL () CN Fig. : Local ifluece uder variaces raio perurbaio able 5: Oulier iformaio uder variaces raio perurbaio he umber of ouliers N SL CN Global ifluece hrough he local iflueces above we rea he followig eigh cases as he mos poeial ouliers: ad 75. I order o reveal he impac of hese eigh observaios o he parameer esimaio we remove hem o obai he maximum likelihood esimaio. Lu ad Sog [] sugges ha he followig wo quaiies ca measure he differece bewee he origial maximum likelihood esimaio ad θ ˆ p ˆ ˆ ( )/ ˆ RC θ θ θ MRC max ( θˆ ˆ ) / ˆ θ θ where p p is he umber of parameers. he scale parameers i differe disribuios cao be compared direcly so we calculae RC ad MRC oly for he locaio parameers ad lis he resuls i able 6. We ca discover ha he greaes chages ake place uder he ormal disribuio. he RC ad MRC are he smalles uder. he diagosic similariy bewee SL ad shows ha he heavy-ailed disribuio o oly has a good robusess o ouliers bu also ca overcome misspecificaio of disribuio o some exe. able 6: RC ad MRC of global ifluece uder he four disribuios RC MRC N SL CN θˆ 5. Coclusio he deecio ad classificaio of ouliers play impora roles i daa miig. I his work we cosruc diagosics of local iflueces based o a proposed replicaed measureme error model. Numerical aalysis cofirms he effec of diagosic mehods uder he heavy-ailed disribuios. By comparig he resuls of differe heavy-ailed disribuios we ca obai a more obecive udersadig of he daa. he mehod proposed i his work ca be exeded o a wider rage of use. 6. Refereces R. D. Cook. Assessme of local ifluece (wih discussio). Joural of he Royal Saisical Sociey Series B 986 JIC for subscripio: publishig@wau.org.uk

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