Small Area Estimation Under Spatial Nonstationarity

Transcription

1 Small Area Estmaton Under Spatal Nonstatonarty Hukum Chandra Indan Agrcultural Statstcs Research Insttute, New Delh Ncola Salvat Unversty of Psa Ray Chambers Unversty of Wollongong Nkos Tzavds Unversty of Southampton

2 Overvew Background Lnear mxed model (LMM) and the EBLUP of the small area mean Spatal non-statonary extenson to the LMM and the GWEBLUP MSE estmaton SAE for out of sample areas Emprcal evaluatons: model-based and desgn-based smulatons Concludng remarks 2

3 Background Mxed effects models are very popular for SAE A mxed effects model conssts of a fxed effects part and a random effects part wth the latter accountng for between area varatons beyond that explaned by the covarates ncluded n the model It s customary to assume that populaton unts n dfferent small areas are uncorrelated Extensons of the mxed effects model to allow for spatally correlated random effects usng smultaneous autoregressve (SAR) models (Sngh et al., 2005 and Prates and Salvat, 2008) SAR models defne the dependence between areas by usng a contguty matrx and allow for spatal correlaton n the error structure whle the fxed effects parameters are spatally nvarant Offer margnal gan n SAE 3

4 Background More flexble approaches based on nonparametrc extensons of the mxed effects model (Opsomer et al., 2008 and Ugarte et al., 2009) An alternatve approach for ncorporatng the spatal nformaton n the model s by assumng that the regresson coeffcents vary spatally across the geography of nterest - spatal nonstatonarty (Brunsdon et al., 1998, Fotherngham et al., 2002) Models are ftted usng geographcal weghted regresson (GWR) approach and are sutable for modellng spatal nonstatonarty GWR extenson to small area predctor based on M-quantle small area model nvestgated by Salvat et al. (2010) We propose a geographcally weghted extenson to the wdely used emprcal best lnear unbased predctor (EBLUP) that s often used for SAE under a lnear mxed model (LMM) 4

5 Lnear Mxed Effects Models for SAE y = x β + z a + ε, = 1,..., A; j = 1,..., N T T y denote the value of the varable of nterest y for unt j ( j = 1,, N ) = 1,, A n small area ( ) x and z denote the column vectors of auxlary varables of order p and q respectvely a, ε are ndependent random errors, both wth zero means and wth 2 Var ( a ) = Ω ( θ ), Var( ε ) σ a ε = ε, Model parameters β, θ, σ 2 e typcally estmated va ML/REML - leads to predcted values a ˆ 5

6 Usng the estmated fxed and random effects, EBLUP of the small 1 area mean of y ( Problem m = N y )(Henderson, 1975; Rao, 2003) j U 1 ˆ { ˆ EBLUP T T m ˆ = N y + xβ + za} j s j r LMM - the fxed effect parameters β are spatally nvarant There are stuatons, where the relatonshp between y and x s not constant over the study area, a phenomenon referred to as spatal nonstatonarty Soluton Geographcal weghted regresson (GWR) s a method that s wdely used for fttng data exhbtng spatal nonstatonarty (Brunsdon et al., 1998, Fotherngham et al., 2002) 6

7 Geographcally Weghted Mxed Effects Models for SAE We use the GWR concept to ft a local mxed model and consder SAE under ths model We refer to ths local mxed model as a geographcally weghted lnear mxed model (GWLMM), expressed as T T y = xβ ( u ) + za + ε, j = 1,, N, = 1,, A, u - the spatal locaton (longtude and lattude) of unt j n area β ( u ) - fxed effects at locaton u a and ε are the area-specfc and ndvdual-specfc random errors: 2 a N( 0, Ω ), ε N(0, σ e ) and a ε 7

8 Geographcal Weghted EBLUP (GWEBLUP) The GWEBLUP predctor of m GWEBLUP 1 T ˆ ˆ T m ( ) ˆ = N y + x u + ( u ) j s j r ( ) T 1 1 T 1 ( u ) = Xs Vss ( u ) Xs Xs Vss ( u ) y s { β z a } β ˆ ˆ ˆ s the GWEBLUE of β at locaton u wth ( ) 1 T Vˆ ( u ) = Z Ω ˆ Z + ˆ σ W ( u ) ss s s e s 1 ˆ( ) ˆ T ˆ a u ( )( ˆ( )) = ΩΖs Vss u y s Xs β u ΩΖ β s the GWEBLUP of a at locaton u ˆ ˆ 2 σ e (, ) ( u ) Ω are the estmated values of the varance components W s a spatal weghtng matrx that are specfc to locaton j n s area such that unts nearer to locaton j are gven greater weght 8

9 Geographcal Weghted EBLUP (GWEBLUP) We use a Gaussan specfcaton for ths weghtng functon j, k W { 2 }, ( u ) = w = exp 0.5( d / b) s k j k d denotes the Eucldean dstance between pont j and k and b s the bandwdth and we use a cross valdaton to compute the bandwdth Parameter Estmaton Under the GWLMM and followng Henderson et al. (1959) we maxmze the geographcally weghted jont maxmum lkelhood functon to obtan the GWEBLUE β ˆ( u ) at a locaton u and the GWEBLUP a ˆ( u ) at a locaton u An teratve procedure was mplemented for computng the ML estmates of model parameters 9

10 MSE Estmaton MSE of EBLUP predctor Prasad and Rao MSE estmator (PR) - uncondtonal approach to MSE estmator (Prasad and Rao,1990) Bas-Robust type MSE estmator (CCT) - condtonal approach to MSE estmaton (Chambers, Chandra and Tzavds, 2011) MSE of GWEBLUP predctor The uncondtonal MSE estmator (MSE_U): a second order approxmaton of the MSE, followed by the approxmatons proposed n Prasad and Rao (1990) and Datta and Lahr (2000) The condtonal MSE estmator (MSE_C): based on the pseudolnearzaton approach to MSE estmaton proposed by Chambers, Chandra and Tzavds (2011) 10

11 Geographcally Weghted Synthetc Predcton (GWSYN) In real applcatons of SAE domans may be unplanned Ths may result n target small areas wth zero sample szes also referred to as out of sample areas Under LMM, the synthetc EBLUP predctor (SYN) for the small area average for out of sample area m ˆ EBLUPSYN = x T β ˆ Under GWLMM, the GWSYN for the average of small area mˆ N ( u ) GWSYN 1 T ˆ = xβ j U 11

12 Emprcal Evaluatons Two types of smulaton studes are carred out Model based smulatons a synthetc populaton s generated at each smulaton run under alternatve model specfcatons and a sample s drawn from ths populaton Desgn based smulatons are based on realstc populaton structures obtaned from real survey data The survey data are frst used to generate a synthetc populaton. The synthetc populaton s then kept fxed and wthn area random samples of sze equal to the area-specfc szes n the orgnal sample, are drawn Performance Measures Average Relatve Bas (AvRBas) Average Relatve Root MSE (AvRRMSE) 12

13 Estmators Investgated n Smulaton Studes Estmator MSE Estmators Lnear mxed model EBLUP CCT MSE (MSE_C) and PR MSE (MSE_U) SYN CCT MSE (MSE_C) and PR MSE (MSE_U) Geographcally Weghted Lnear mxed model GWEBLUP MSE_C and MSE_U GWSYN MSE_C and MSE_U 13

14 Model Based Smulatons Statonary process - N = 15000, N randomly generated (average = 500) 1 - n = 400, n = nn N ( = 20) - average, A = 20 2 ~ (20), = 1,...,, = 1,..., = N σ u =, e x Ch j N A - a ~ (0, 23.52) e ~ N(0, σ = 94.09), a e - y = x + a + e - T = 1000 smulatons Nonstatonary process a y = β + β x + + e wth - β 0 = longtude lattude - β 1 = 0.2 longtude lattude - Locaton coordnates (, longtude lattude ) ( 0,50) nd U 14

15 Model Based Smulaton Results Predctor Indcator Summary of across small areas dstrbuton (%) Mn Q1 Medan Mean Q3 Max Statonary process EBLUP RB RRMSE GWEBLUP RB RRMSE Non statonary process EBLUP RB RRMSE GWEBLUP RB RRMSE

16 Area specfc values of actual RMSE (sold lne) and average estmated RMSE: MSE_U (dotted lne/green) and MSE_C (dashed lne/red) 16

17 Desgn Based Smulatons The dataset comes from the U.S. Envronmental Protecton Agency's Envronmental Montorng and Assessment Program (EMAP) Northeast lakes survey (Larsen et al., 2001) The varable of nterest s Acd Neutralzng Capacty (ANC), an ndcator of the acdfcaton rsk of water bodes Elevaton: covarate n the fxed part of the model Small areas: 113 Hydrologc Unt Codes (HUCs), of whch 64 have (<5) observatons and 27 dd not have any observatons Target: estmaton of small area mean of ANC for n (86 areas) and out (27 areas) of sample HUCs 17

18 Desgn Based Smulatons Survey data used to generate nonparametrcally a synthetc populaton of 21,026 ANC ndvdual values by usng a nearestneghbour mputaton algorthm that retans the spatal structure of the observed ANC values n the EMAP sample data (Salvat et al., 2010) A total of 1000 ndependent random samples of lake locatons are then taken from the populaton of 21,026 lake locatons by randomly selectng locatons n the 86 HUCs that contanng EMAP sampled lakes, wth sample szes n these HUCs set to the orgnal EMAP sample sze Lakes n HUCs not sampled by EMAP are also not sampled n the smulaton study 18

19 Spatal Nonstatonarty n EMAP data Intercepts Slopes ANOVA test of Brundson et al. (1999) rejected the null hypothess of statonarty of the model parameters when the workng model was ftted to the orgnal EMAP sample data 19

20 Predctor Desgn Based Smulaton Results Indcator Summary of across areas dstrbuton (%) Mn Q1 Medan Mean Q3 Max 86 sampled HUCs n EBLUP RB RRMSE GWEBLUP RB RRMSE non-sampled HUCs SYN RB RRMSE GWSYN RB RRMSE

21 Regon-specfc values of actual RMSE (sold lne) and average estmated RMSE: MSE_U (dotted lne/green) and MSE_C (dashed lne/red) 21

22 Concludng Remarks We examne a geographcally weghted extenson of the popular EBLUP, whch we refer to as the GWEBLUP We propose two methods for estmatng ts MSE The emprcal results show that the GWEBLUP can be used for effcently borrowng strength over space n the presence of spatal nonstatonarty n the data The GWEBLUP can sgnfcantly mprove synthetc estmaton for out of sample areas It s worth notng that all emprcal studes are carred out by usng the centrods of the small areas We expect that the gans from usng the GWEBLUP wll be further enhanced f nformaton on unt level spatal coordnates s avalable 22

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