p-spline mixed models for spatio-temporal...

P-spline mixed models for spatio-temporal data

María Durbánjoint work with Dae-Jin Lee

DEPARTMENT OF STATISTICSUNIVERSIDAD CARLOS III DE MADRID

June 2009

Uc3m/ Dept. of Statistics 1First Workshop on Spatio-temporal Disease Mapping, Valencia 2009

María Durbán

Outline

1 P-splinesMixed models approachMultidimensional P-splines

2 P-splines for spatial count dataSpatial smoothingSmooth-CAR modelApplication: Scottish Lip Cancer data

3 Spatio-temporal data Smoothing with P-splinesANOVA-Type Interaction ModelsApplication Environmental spatio-temporal data

4 Spatio-temporal Disease Mapping


María Durbán

Outline






María Durbán

P-splines“The Flexible Smoother”

I Penalized Likelihood splines (Eilers & Marx, 1996):

• Given the data (xi,yi), i = 1, ...,n

• Fit a sum of local basis functions:

yi = f (xi) + εi, ε ∼ N (0,σ2)

where f (xi) = Bθ and

I B = B(x) is a Regression Basis, and

I θ is a vector of coefficients.

• Control the fit through a smoothing parameter (λ).

» Regression Basis


María Durbán


I B-splines Basis:

• p + 1 Piece-wise polynomialsof degree p.

• Connected by knots.• In general the choice is p=3,

cubic spline.

B-splines of degree p:

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María Durbán


I B-splines Basis:

• y = f (xi) = Bθ

• B-splines Regression:

min S(θ; y) = ‖y− Bθ‖2

θ = (B′B)−1B′y

I Optimal selection ofknots (Complex).

• P-Splines: add a penalty tocontrol smoothness.

Example:

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» Methodology


María Durbán


Methodology:

• Minimize the penalized sum of squares (PSS):

S(θ; y, λ)p = ‖y− Bθ‖2 + PENALTY

• The PENALTY term, controls the smoothness of the fit by λ.

I Eilers & Marx (1996):⇒ (discrete) Penalty over adjacent coefficients θ.

I Lang & Brezger (2004):⇒ “Bayesian P-splines”: random walk priors for θ, e.g.:

θ|θm−1 ∼ N (θm−1, τ2), or

θ|θm−1,θm−2 ∼ N (2θm−1 − θm−2, τ2)


María Durbán


• PSS becomes:S(θ; y, λ)p = ‖y− Bθ‖2 + θ′Pθ

I P = λD′D.I λ is the smoothing parameter.I D are difference matrices.

• For given λ , min S(θ; y, λ)p

θ =(B′B + λD′D

)−1 B′y

I λ can be selected by CV, GCV, AIC or BIC.


María Durbán


I 1d P-splines:

• No penalty over coefficients.

• Penalty over coefficients.

Example:

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B-splines basis and θ without penalty

» Advantages


María Durbán


I 1d P-splines:

• No penalty over coefficients.

• Penalty over coefficients.

Example:

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B-splines basis and θ with penalty

» Advantages


María Durbán


Advantages over other smoothers:• Low-Rank : “dim(B) < dim(data)”.

• Computationally efficient: “# knots ≤ 40”.

• Selection of number and Location of knots is NOT and issue.

• Discrete Penalties over the θ, not over the fitted curve.

• Easy extension to:I Mixed models,I non-gaussian data (GLM’s) andI Multidimensional smoothing.I Spatial and Spatio-temporal smoothing.

» Mixed models


María Durbán

P-splinesA mixed model approach

I Reformulate:

• Model y = Bθ + ε, into

y = Xβ + Zα + ε, ε ∼ N (0,σ2I)

I where X and Z are “fixed” and “random” effects matrices.

I with coefficients β and α ∼ N (0,G), and G = σ2αR

I λ = σ2

σ2α

» Reparameterization


María Durbán


I Reparameterization:

B ≡ [ X : Z ]⇒ Bθ = Xβ + Zα

IWe use the Singular Value Decomposition (SVD) on D′D

» SVD


María Durbán


I Singular Value Decomposition (SVD)

D′D = UΣU′

• with U = [Un : Us]

D′D = [Un : Us]

[0d

Σ

] [ U′nU′s

]

I Σ ≡ non-null eigenvalues.I Un ≡ eigenvectors corresponding to the null eigenvalues.I Us ≡ eigenvectors corresponding to the non-null eigenvalues.


María Durbán


• The fix effects (β) are unpenalized and

• The Penalty θ′Pθ becomesα′Fα

where F = λΣ is diagonal.

• And the random effects (α) covariance matrix G:

G = σ2F−1

• Mixed Model Basis:

X = [ 1 : x ]

Z = BUs


María Durbán


Advantages:

• Flexibility:

I Easy incorporation of smoothing in complex models (“spatial” randomeffects and/or correlated errors).

• Mixed Models Theory:

I Estimation and Inference.

• Software Implementation.

I R, Splus, MATLAB or SAS.

• Extension to non-gaussian data:

I Generalized Linear Mixed Models (GLMM)


María Durbán

Multidimensional P-splines

Example: 2d-array

• Data Y = yij, i = 1, ..., n1 and j = 1, ..., n2

• Array structure: n1 rows and n2columns

Y =

y11 y12 · · · y1n2y21 y22 · · · y2n2...

.... . .

...yn11 · · · · · · yn1n2

• Regressors:

x1 = (x11, · · · , x1n1 )′

x2 = (x21, · · · , x2n2 )′

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(a) Simulated data


María Durbán


I Use of Tensor Products of B-splines (Durbán et al, 2002):

Example: 2d-array

• Marginal Basis:

• B1 = B1(x1), of dim. n1 × c1.• B2 = B2(x2), of dim. n2 × c2.

• 2d B-splines Basis:

• Kronecker Product (⊗) ofmarginal basis:

B = B2⊗B1, of dim. n1n2×c1c2x2

x1

2d

−B

sp

line

x2x1

2d

−B

sp

line


María Durbán


Model:

y = f (x1, x2) + ε,

with yn1n2×1

• In matrix form, y = Bθ can be written as:

Y = B1AB2, of dim n1 × n2

where A is a matrix c1 × c2 of coefficients θ of length c1c2 × 1.

IDEA:

• Set penalties over Θ.

• Row-wise Penalty: θ′(Ic2 ⊗D′1D1

)θ

• Column-wise Penalty: θ′(D′2D2 ⊗ Ic1

)θ


María Durbán


I Penalty Matrix in 2d:

P = λ1 Ic2 ⊗D′1D1︸︷︷︸P1

+λ2 D′2D2 ⊗ Ic1︸︷︷︸P2

• λ1 and λ2 are the smoothing parameters in each dimension.

• Anisotropy: (λ1 6= λ2)


María Durbán

Multidimensional P-splinesMixed Models Representation

I As in 1d Case:

Example:

I The Mixed Model consists of:

y = Xβ + Zα(Linear/Fixed) (Non-Linear/Random)

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(b) Fitted Surface

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(a) Linear/Fixed part

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(b) Non−linear/Random part


María Durbán


I Mixed Models Representation:

• As in 1d case, the aim is:

B ≡ [ X : Z ] =⇒ Bθ = Xβ + Zα

• The SVD over P allows the simultaneous diagonalization of D′1D1 and D′2D2

• The penalty P becomes F (block diagonal matrix):

F =

λ2Σ2 ⊗ I2

λ1I2 ⊗ Σ1

λ1Ic2−2 ⊗ Σ1 + λ2Σ2 ⊗ Ic1−2

» Model


María Durbán


ANOVA-type Decomposition of Smooth Surfaces:

y = f (x1) + f (x2) + f (x1, x2)(additive term for x1 ) (additive term for x2) (interaction term for x1, x2)

X1

X2

Y

Fitted Surface

X1

X2

Y

Additive term for x1

X1X2

Y

Additive term for x2

X1

X2

Y

Non−additive term

» Advantages


María Durbán


Advantages:

• Extension to d-dimensions:

B = B2 ⊗ B1 ⊗ · · · ⊗ Bd

• Efficient algorithms:

• Currie et al (2006): Generalized Linear Array Models (GLAM)

• Anisotropy (different smoothing for each dimension):

• Complex models: spatial data smoothing


María Durbán

Outline






María Durbán

P-splines for spatial count dataP-splines for spatial smoothing

IWe propose:

• 2d P-splines:

• Geostatistics: at sampling locations.

• Regional/areal: at the centroids.

I Models of the form:y = f (lon, lat) + ε

where

• f (lon, lat) is a large-scale spatial smooth trend: Xβ + Zα.

• The mixed model allows the simultaneous estimation of smoothing andspatial correlation.

» Spatial count data


María Durbán


IWe propose:

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● • 2d P-splines:




where





María Durbán


IWe propose:

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• 2d P-splines:




where





María Durbán

P-splines for spatial count dataBasis for Spatial Data

I B-spline Basis for spatial data:

• Given that data are NOT in an array

B = B2 ⊗ B1 replace by B2�B1

� denotes the “Row-wise Kronecker” or Box-Product.

Def. Box-Product:

B2�B1 = (B2 ⊗ 1c1)� (1c2 ⊗ B1)

� is the “element-wise” product.


María Durbán

P-splines for spatial count dataI In many applications:

• Collect count data observed in regions or areas.• E.g.: # of cases of disease or deaths

• Counts are Poisson distributed.

y ∼ P(µ)

Penalized-GLMM

• P-splines as mixed models:

I Linear Predictor:η = Bθ =⇒ Xβ + Zα

I Penalized log-Likelihood:

`p(β,α; y) = `(β,α; y)− 12α′Fα

I Estimation via PQL


María Durbán

Smooth-CAR modelCAR model

I Most popular approach:

• Conditional AutoregressiveModels (CAR), Besag (1991)

• Spatial Dependence across“neighbours”.

• Different neighbourhoodcriteria.I Common border.I Centroids distance, 4-nearest

neighbours.


María Durbán


I Most popular approach:

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• Conditional AutoregressiveModels (CAR), Besag (1991)

• Spatial Dependence across“neighbours”.

• Different neighbourhoodcriteria.I Common border.I Centroids distance, 4-nearest

neighbours.


María Durbán


I Formulation:y = Xβ + b,

where b = (b1, b2, ..., bn)′ is a vector for the spatial effects

• Impose a spatial dependency structure by a prior distribution for b:

b ∼ N (0,Gb)

where Gb depends on the “neighbourhood structure”:

I defined by Contiguity matrix (Q)


María Durbán


X We follow an Empirical Bayes approach:

I Intrinsic CAR:Gb = σ2

b Q− + κ−1I (Besag, 1991)

• Two independent and separate variance components:

I Spatially-structured variation: σ2b Q−

I Unstructured non-spatial correlation: κ−1I

I Alternative CAR models structures:

Gb = σ2b (φQ + (1− φ)I)−1 (Leroux et al, 1999)

Gb = σ2b (φQ− + (1− φ)I) (Dean et al, 2001)

where

I φ measures the relative weight between structured and unstructured variability

I 0 ≤ φ ≤ 1


María Durbán

Smooth-CAR modelLee and Durban (2009)

IWe propose a “hybrid” model:

• Spatial P-spline with CAR structure: “Smooth-CAR” model

• Model:η = Xβ + Zα + b ,

where b ∼ N (0,Gb)

Our approach:

η = Spatial Trend︸︷︷︸Xβ + Zα

+ Local area-level spatial correlation︸︷︷︸Spatial Random Effects

(Large-scale) (Small-scale)


María Durbán

Smooth-CAR model

I Summary:

Model Linear Predictor Area-level var.Poisson Xβ + Zα −

CAR Xβ + b b ∼ N (0,Gb)Smooth-CAR Xβ + Zα + b b ∼ N (0,Gb)

I The Smooth-CAR:

I Allow us model the spatial trend (Xβ + Zα) along large geographical distances and

I Local area-level correlation by a CAR component (b).


María Durbán

Application: Scottish Lip Cancer data

Example: Scottish Lip Cancer

• Breslow and Clayton (1993)

• Observed (y) and Expected (e)cases of lip cancer

• 56 counties in Scotland

• Period: 1975− 1980.

SCOTTISH LIP CANCER

OBSERVED EXPECTED

0

20

40

60

80


María Durbán

Application: Scottish Lip Cancer dataFitted Models

IWe fit several models:

• Smooth P-splines models:

η = log(e) + Xβ + Zα (Poisson)

log(e) is the offset term.

• CAR models:η = log(e) + Xβ + b , b ∼ N (0,Gb),

with:

Gb = σ2b (φQ− + (1− φ)I) (Dean)

• Smooth-CAR model:

η = log(e) + Xβ + Zα + b , b ∼ N (0,Gb)


María Durbán

Application: Scottish Lip Cancer dataModels comparison criteria

I In order to compare the proposed models we use:

AIC = Dev + 2× dfBIC = Dev + log(n)× df

where:

• df is the effective dimension of the model (“degrees of freedom”).

I is a measure of the complexity of the fitted model,I Calculated as the trace(H),

y = Hy


María Durbán

Application: Scottish Lip Cancer dataComparisons of fitted models

ParametersModel λ1 λ2 σ2

s κ−1 φ AIC BIC dfSmooth: Poisson 11.75 3.63 - - - 114.04 228.46 15.90

CAR: Dean - - 0.78 - 0.99 89.36 179.56 32.78Smooth-CAR: Dean 30.11 16.37 0.53 - 0.97 87.46 175.70 30.64

I Observations:

• φ ≈ 1 −→ Overdispersion is due to “structured” spatial correlation (σ2b Q−).

• Smooth-CAR performs better in terms of the selected criteria.


María Durbán

I Dean’s CAR model:

(a) Linear Trend (b) CAR random effect (c) CAR

−1.0 −0.5 0.0 0.5 1.0 1.5

(a) Large-scale linear trend: Xβ

(b) CAR structured random effects: b ∼ N (0,Gb)

(c) Xβ + b

I Smooth-CAR model:

(a) Smooth Trend (b) CAR component (c) Trend+CAR

−1.0 −0.5 0.0 0.5 1.0 1.5

(a) Smooth large-scale spatial trend: Xβ + Zα

(b) CAR structured random effects: b ∼ N (0,Gb)

(c) Xβ + Zα + b

Spatio-temporal data Smoothing with P-splines

Outline






María Durbán


Spatio-temporal data

• Response variable, yijt

• measured over geographical locations, s = (xi, xj), with i, j = 1, .., n

• and over time periods, xt, for t = 1, ....,T

• ISSUE: huge amount of data available• e.g. : Environmental data, epidemiologic studies, disease mapping

applications, ...

• Smoothing techniques:

• Study spatial and temporal trends.

• Space and time interactions.

X 3-dimensional smoothing: P-splines and GLAM.


María Durbán


Example of GLAM in 3dCurrie et. al (2006)

• 3d-case:f (x1, x2, x3) = Bθ

• Basis: B = B1 ⊗ B2 ⊗ B3

• θ can be expressed as a 3d-array A = {θ}ijk of dim. c1 × c2 × c3

θ(1,1,c3)θ(1,c2,c3)

θ(1,1,1)1,...,c2

columns

rows 1,...,c1

layer

1,...,c3

uuuuuuuuuθ(1,c2,1)

ttttttttt

θ(c1,1,c3)θ(c1,c2,c3)

θ(c1,1,1) θ(c1,c2,1)

ttttttttt


María Durbán


• 3d-Penalty matrix:

• Set penalties over the 3d-array A:

P = λ1 D′1D1 ⊗ Ic2 ⊗ Ic3︸︷︷︸row-wise

+λ2 Ic1 ⊗ D′2D2 ⊗ Ic3︸︷︷︸column-wise

+λt Ic1 ⊗ Ic2 ⊗ D′tDt︸︷︷︸layer-wise

• For spatio-temporal data:

f ( longitude, latitude︸︷︷︸Space

, time)

• Spatial anisotropy (λ1 6= λ2), different amount of smoothing for latitude andlongitude.

• Temporal smoothing (λt)

• Space-time interaction.


María Durbán


• For spatio-temporal data, we propose:

B-splines Basis:

B = Bs ⊗ Bt,

whereBs ≡ is the spatial B-spline basis (B1� B2) and

Bt ≡ is the B-spline basis for time of dim. t× c3.

X as GLAM:Given yijt = Yt×n, and θijt = Act×cs , we have

E[Y] = BtAB′s

X as Mixed models Bθ = Xβ + Zα


María Durbán

Spatio-temporal data Smoothing with P-splines ANOVA-Type Interaction Models

Smooth-ANOVA decomposition models

• Chen (1993), Gu (2002):

• “Smoothing-Spline ANOVA” (SS-ANOVA).

• Interpretation as “main effects” and “interactions”.

• Models of type:

y = f (x1) + f (x2) + f (xt) “Main/additive effects”+f (x1, x2) + f (x1, xt) + f (x2, xt) “2-way interactions”+f (x1, x2, xt) “3-way interactions”

• PROBLEMS:• identifiability, and• basis dimension (“curse of dimensionality”)


María Durbán


P-spline ANOVA modelfor spatio-temporal smoothing

• Lee and Durbán (2009a), consider:

y = γ + fs(x1, x2) + fs(time) + fst(x1, x2, time) + ε ,

wherefs(x1, x2) ≡ Spatial 2d smooth surface

ft(time) ≡ Smooth time trendfst(x1, x2, time) ≡ Space-time interaction

• We need to construct an identifiable model.

• Our approach is based on:• low-rank basis (P-splines)• the mixed model representation and SVD properties.


María Durbán


Basis, Coefficients and Penalty

• For each smooth term f (·), in spatio-temporal ANOVA model we have

• B−spline basis:B = [1nt : Bs ⊗ 1t : 1n ⊗ Bt : Bs ⊗ Bt]

• vector of coefficients:

θ = (γ,θ(s)′ ,θ(t)′ ,θ(st)′)′

• and a blockdiagonal Penalty:

P =

0

Ps

Pt

Pst

,

where Ps = 2d-spatial penaltyPt = 1d-penalty for timePst = 3d space-time penalty


María Durbán


X However, B is NOT full column-rank (“linear dependency”)

X Model is NOT identifiable

Solution:

• Reparameterize as a mixed model (using SVD).

• For each term we have:Basis [ X : Z ]

fs(x1, x2) ≡ x1 : x2 (1)

ft(xt) ≡ xt (2)

fst(x1, x2, xt) ≡ x1 : x2 : xt (3)

• Some terms in (1) and (2) also appear in (3).


María Durbán


I The mixed model representation, allow us to identify the columns toremove in order to maintain the identifiability of the model.

and obtain a blockdiagonal penalty F

F =

08

FsFt

Fst

,

withλ1, λ2λtτ1, τ2, τt

I In P-splines context, this is equivalent toX apply constraints over regression coefficients θi,j,k


María Durbán


I For the ANOVA spatio-temporal model, the resultant mixed modelreparameterization is equivalent to apply the next constraints:

• time effect coefficient: ∑ctt=1 θ

(t)t = 0,

• constraints over the spatio-temporal array of coefficients, Θ(st), ofdimensions ct × cs:

c1∑i

θ(st)t,ij =

c2∑j

θ(st)t,ij =

c1∑i

c2∑j

θ(st)t,ij = 0.


María Durbán


In practiceI We only need to construct the matrices X, Z and penalty F

fs(x1, x2) ft(xt) fst(x1, x2, xt)

X ≡ by columns x1 : x2 xt (x1, x2, xt)

Z ≡ by blocks ′′ ′′ ′′

F ≡ blockdiagonal Fs Ft Fst(λ1, λ2) λt (τ1, τ2, τt)


María Durbán

Spatio-temporal data Smoothing with P-splines Application Environmental spatio-temporal data

Ozone pollution in EuropeLee and Durbán (2009a)

• Sample of 45 monitoring stations

• Monthly averages of O3 levels (in µg/m3 units)

• from january 1999 to december 2005 (t = 1, ..., 84)

Models:

• Additive:fs(x1, x2) + ft(xt)

• Spatio-temporal Interaction:X ANOVA:

fs(x1, x2) + ft(xt) + fst(x1, x2, xt)


María Durbán


Spatial 2d + time

fs(x1, x2) + ft(xt)

0 5 10 15 20 25

4045

5055

6065

Latitude

Long

itude

40

50

60

70

80

90

1999 2000 2001 2002 2003 2004 2005

−20

−10

010

20year

f(tim

e)

X Space-time interaction is not considered

X time smooth trend is additive


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Spatio-temporal ANOVA model

Play animation =

+ +

y f(space)

f(time)

1999 : 1

f(space,time)


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Comparison of fitted valuesAdditive VS ANOVA

I Additive model fit I ANOVA model fitfs(x1, x2) + fs(xt) fs(x1, x2) + ft(xt) + fst(x1, x2, xt)

1999 2000 2001 2002 2003 2004 2005 2006

2040

6080

100

120

140

year

O3

SpainSwedenAustriaUK

1999 2000 2001 2002 2003 2004 2005 2006

2040

6080

100

120

140

year

O3

SpainSwedenAustriaUK

X Additive model assumes a spatial smooth surface over all monitoring stations that remainsconstant over time.

X ANOVA model captures individual characteristics of the stations throughout time.


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Comparison of ModelsANOVA and Additive

Model AIC dfANOVA 14280.73 366.03

Additive 16506.28 65.98

I Observations:

• Best overall performance of ANOVA in terms of AIC• ANOVA model is more realistic than Additive, and easier to decompose

and interpret in terms of the fit.


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Outline






María Durbán

Spatio-temporal Disease Mapping

P-spline ANOVA model for disease mapping

• Y and E are t× n arrays of observed and expected cases of disease over t timeperiods, and M = log( Y

E ).

• Consider an ANOVA model for η

fs(x1, x2) + ft(xt) + fst(x1, x2, xt)

−0.5

0.0

0.5

1.0

1.5

t1 t2 t3 t4 t5

t6 t7 t8 t9 t10

0

2

4

6

8


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Summary

I New flexible approach for spatial and spatio-temporal data smoothing:

• based on P-splines as mixed models and• ANOVA decomposition

I Methodology also extensible for disease mapping applications.

I Computationally efficient algorithms (GLAM)


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Bibliography

Lee, D.-J. and Durbán, M. (2009)Smooth-CAR mixed models for spatial count data.Computational Statistics and Data Analysis 53(8):2968-2979.

Lee, D.-J. and Durbán, M. (2009)P-spline ANOVA-Type interaction models for spatio-temporal smoothing.Submitted.

Eilers, PHC., Currie, ID. and Durbán, M. (2006)Fast and compact smoothing on large multidimensional grids.Computational Statistics and Data Analysis, 50(1):61-76.

Currie, ID., Durbán M. and Eilers, PHC. (2006)Generalized linear array models with applications to multidimensional smoothing.Journal of the Royal Statistical Society B, 68:1-22.


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p-spline mixed models for spatio-temporal...

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