P-spline ANOVA-type interaction models for spatio-temporal smoothing Dae-Jin Lee(cid:63) and Mar´ıa Durb´an Universidad Carlos III de Madrid Department of Statistics IWSM Utrecht 2008 D.-J.LeeandM.Durban (UC3M) ’P-splineANOVA-typemodels’ IWSM2008 1/26 Outline 1 Motivation 2 Penalized splines for Spatio-Temporal data 3 ANOVA-Type Interaction Models 4 Application to O pollution in Europe 3 5 Conclusions D.-J.LeeandM.Durban (UC3M) ’P-splineANOVA-typemodels’ IWSM2008 2/26 1. Motivation • Air pollution • Enviromental policies • Monitoring networks: (cid:73) European Environmental Agency (EEA) (cid:73) EMEP project (European Monitoring and Evaluation Programme) • Ozone (O ) is currently one of the air pollutants of most concern in 3 Europe. D.-J.LeeandM.Durban (UC3M) ’P-splineANOVA-typemodels’ IWSM2008 3/26 Monitoring stations across Europe l 65 l l l l 60 l l l l l 55 ll l lll l l l l 50 l l l ll l l lllll l lllll l 45 l l l l 40 l l −5 0 5 10 15 20 25 sampleof45monitoringstations Monitoringstation D.-J.LeeandM.Durban (UC3M) ’P-splineANOVA-typemodels’ IWSM2008 4/26 O time series plot for selected locations 3 (cid:73) Seasonal pattern: Spain Finland 140 FUrKance 120 100 O3 80 60 40 20 1999 2000 2001 2002 2003 2004 2005 time D.-J.LeeandM.Durban (UC3M) ’P-splineANOVA-typemodels’ IWSM2008 5/26 O3 level from 01/2004 to 12/2005 Playanimation ene.99 222000 444000 666000 888000 111000000 111222000 D.-J.LeeandM.Durban (UC3M) ’P-splineANOVA-typemodels’ IWSM2008 6/26 1. Motivation (cid:73) Spatio-temporal data • Response variable, y ijt (cid:73) measured over geographical locations, s=(x,x ), with i,j =1,..,n i j (cid:73) and over time periods, x , for t =1,....,T t • ISSUE: huge amount of data available (cid:73) e.g. : Environmental data, epidemiologic studies, disease mapping applications, ... • Smoothing techniques: (cid:73) Study spatial and temporal trends. (cid:73) Space and time interactions. (cid:73) “Penalized Splines” (Eilers and Marx, 1996). D.-J.LeeandM.Durban (UC3M) ’P-splineANOVA-typemodels’ IWSM2008 7/26 2. Penalized splines (cid:73) “The flexible smoother” • Methodology: (cid:73) Given the data (x,y ), i =1,...,n. i i (cid:73) Fit a sum of local basis functions: f(x)=Bθ i (cid:73) Minimize the Penalized Sum of Squares: (cid:107)y −f(x)(cid:107)2+Penalty i i (cid:73) The Penalty controls the smoothness of the fit. (cid:88) Smoothing parameter: λ (cid:88) Apply a discrete penalty over coefficients θ, e.g. in 1d: P=λD(cid:48)D where D is a difference matrix acting on θ. D.-J.LeeandM.Durban (UC3M) ’P-splineANOVA-typemodels’ IWSM2008 8/26 2. Penalized splines (cid:73) “The flexible smoother” • For array data (Currie et al., 2006): (cid:73) Generalized Linear Array Methods (GLAM): f(x ,...,x )=Bθ 1 d (cid:73) where B is the Kronecker product of d B-splines basis: B=B ⊗B ⊗....⊗B 1 2 d (cid:73) Efficient Algorithms for smoothing on multidimensional grids (e.g. mortality data, images, etc...). (cid:73) Easy representation as a Mixed Model: f(x ,...,x )=Xβ+Zα 1 d D.-J.LeeandM.Durban (UC3M) ’P-splineANOVA-typemodels’ IWSM2008 9/26 2. Penalized splines (cid:73) Example of GLAM: • 3d-case: f(x ,x ,x )=Bθ 1 2 3 • Basis: B=B ⊗B ⊗B 1 2 3 (cid:73) θ can be expressed as a 3d-array A={θ} of dim. c ×c ×c ijk 1 2 3 θ θ (cid:116)(cid:116)la(cid:116)y(cid:116)e(cid:116)r1,(cid:116).(cid:116)..(cid:116),c(cid:116)3(1,1,c3) (cid:115)(cid:115)(cid:115)(cid:115)(cid:115)(cid:115)(cid:115)(cid:115)(cid:115)(1,c2,c3) θ columns θ (1,1,1) 1,...,c2 (1,c2,1) rows 1,...,c1 θ(c1,1,c3) (cid:115)(cid:115)(cid:115)(cid:115)(cid:115)(cid:115)(cid:115)(cid:115)θ(cid:115)(c1,c2,c3) θ θ (c1,1,1) (c1,c2,1) D.-J.LeeandM.Durban (UC3M) ’P-splineANOVA-typemodels’ IWSM2008 10/26