P-spline ANOVA-type interaction models - for spatio-temporal

28 Pages·2008·1.76 MB·English

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