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Modelling Multivariate Spatial Data

Sudipto Banerjee

1 Biostatistics, School of Public Health, University of Minnesota, Minneapolis, Minnesota, U.S.A.

June 20th, 2014

1

Multivariate spatial modelling

Point-referenced spatial data often come as multivariate measurements at each location.

Examples: Environmental monitoring: stations yield measurements on ozone, NO, CO, and PM2.5. Community ecology: assembiages of plant species due to water availibility, temerature, and light requirements. Forestry: measurements of stand characteristics age, total biomass, and average tree diameter. Atmospheric modeling: at a given site we observe surface temperature, precipitation and wind speed

We anticipate dependence between measurements at a particular location across locations

2 Bayesian Multivariate Spatial Regression Models

Multivariate spatial modelling

Point-referenced spatial data often come as multivariate measurements at each location. Examples:

Environmental monitoring: stations yield measurements on ozone, NO, CO, and PM2.5.

Community ecology: assembiages of plant species due to water availibility, temerature, and light requirements. Forestry: measurements of stand characteristics age, total biomass, and average tree diameter. Atmospheric modeling: at a given site we observe surface temperature, precipitation and wind speed

We anticipate dependence between measurements at a particular location across locations

2 Bayesian Multivariate Spatial Regression Models

Multivariate spatial modelling

Point-referenced spatial data often come as multivariate measurements at each location. Examples:

Environmental monitoring: stations yield measurements on ozone, NO, CO, and PM2.5. Community ecology: assembiages of plant species due to water availibility, temerature, and light requirements.

Forestry: measurements of stand characteristics age, total biomass, and average tree diameter. Atmospheric modeling: at a given site we observe surface temperature, precipitation and wind speed

We anticipate dependence between measurements at a particular location across locations

2 Bayesian Multivariate Spatial Regression Models

Multivariate spatial modelling

Point-referenced spatial data often come as multivariate measurements at each location. Examples:

Environmental monitoring: stations yield measurements on ozone, NO, CO, and PM2.5. Community ecology: assembiages of plant species due to water availibility, temerature, and light requirements. Forestry: measurements of stand characteristics age, total biomass, and average tree diameter.

Atmospheric modeling: at a given site we observe surface temperature, precipitation and wind speed

We anticipate dependence between measurements at a particular location across locations

2 Bayesian Multivariate Spatial Regression Models

Multivariate spatial modelling

Point-referenced spatial data often come as multivariate measurements at each location. Examples:

Environmental monitoring: stations yield measurements on ozone, NO, CO, and PM2.5. Community ecology: assembiages of plant species due to water availibility, temerature, and light requirements. Forestry: measurements of stand characteristics age, total biomass, and average tree diameter. Atmospheric modeling: at a given site we observe surface temperature, precipitation and wind speed

We anticipate dependence between measurements at a particular location across locations

2 Bayesian Multivariate Spatial Regression Models

Multivariate spatial modelling

Point-referenced spatial data often come as multivariate measurements at each location. Examples:

Environmental monitoring: stations yield measurements on ozone, NO, CO, and PM2.5. Community ecology: assembiages of plant species due to water availibility, temerature, and light requirements. Forestry: measurements of stand characteristics age, total biomass, and average tree diameter. Atmospheric modeling: at a given site we observe surface temperature, precipitation and wind speed

We anticipate dependence between measurements

at a particular location across locations

2 Bayesian Multivariate Spatial Regression Models

Multivariate spatial modelling

Point-referenced spatial data often come as multivariate measurements at each location. Examples:

Environmental monitoring: stations yield measurements on ozone, NO, CO, and PM2.5. Community ecology: assembiages of plant species due to water availibility, temerature, and light requirements. Forestry: measurements of stand characteristics age, total biomass, and average tree diameter. Atmospheric modeling: at a given site we observe surface temperature, precipitation and wind speed

We anticipate dependence between measurements at a particular location

across locations

2 Bayesian Multivariate Spatial Regression Models

Multivariate spatial modelling

Point-referenced spatial data often come as multivariate measurements at each location. Examples:

We anticipate dependence between measurements at a particular location across locations

2 Bayesian Multivariate Spatial Regression Models

Multivariate spatial modelling

Bivariate Linear Spatial Regression

A single covariate X(s) and a univariate response Y (s)

At any arbotrary point in the domain, we conceive a linear spatial relationship:

E[Y (s) |X(s)] = β0 + β1X(s);

where X(s) and Y (s) are spatial processes.

Regression on uncountable sets:

Regress {Y (s) : s ∈ D} on {X(s) : s ∈ D} .

Inference: Estimate β0 and β1. Estimate spatial surface {X(s) : s ∈ D}. Estimate spatial surface {Y (s) : s ∈ D}.

3 Bayesian Multivariate Spatial Regression Models

Multivariate spatial modelling

Bivariate spatial process

A bivariate distribution [Y,X] will yield regression [Y |X].

So why not start with a bivariate process?

Z(s) = [ X(s) Y (s)

] ∼ GP2

([ µX(s) µY (s)

] ,

[ CXX(·;θZ) CXY (·;θZ) CY X(·;θZ) CY Y (·;θZ)

]) The cross-covariance function:

CZ(s, t;θZ) = [ CXX(s, t;θZ) CXY (s, t;θZ) CY X(s, t;θZ) CY Y (s, t;θZ)

] ,

where CXY (s, t) = cov(X(s), Y (t)) and so on.

4 Bayesian Multivariate Spatial Regression Models

Multivariate spatial modelling

Cross-covariance functions satisfy certain properties:

CXY (s, t) = cov(X(s), Y (t)) = cov(Y (t), X(s)) = CY X(t,s).

Caution: CXY (s, t) 6= CXY (t,s) and CXY (s, t) 6= CY X(s, t) .

In matrix terms, CZ(s, t;θZ)> = CZ(t,s;θZ)

Positive-definiteness for any finite collection of points:

n∑ i=1

n∑ j=1

a>i CZ(si, tj ;θZ)aj > 0 for all ai ∈

Multivariate spatial modelling

Bivariare Spatial Regression from a Separable Process

To ensure E[Y (s) |X(s)] = β0 + β1X(s), we must have

Z(s) = [ X(s) Y (s)

] ∼ N

([ µ1 µ2

] ,

[ T11 T12 T12 T22

]) for every s ∈ D

Simplifying assumption :

CZ(s, t) = ρ(s, t)T =⇒ ΣZ = {ρ(si,sj)T} = R(φ)⊗ T .

6 Bayesian Multivariate Spatial Regression Models

Multivariate spatial modelling

Then, p(Y (s) |X(s)) = N(Y (s) |β0 + β1X(s), σ2), where

β0 = µ2 − T12 T11

µ1,

β1 = T12 T11

,

σ2 = T22 − T 212 T11

.

Regression coefficients are functions of process parameters.

Estimate {µ1, µ2, T11, T12, T22} by sampling from

p(φ)×N(µ | δ,Vµ)× IW (T | r,S)×N(Z |µ,R(φ)⊗ T)

Immediately obtain posterior samples of {β0, β1, σ2}.

7 Bayesian Multivariate Spatial Regression Models

Multivariate spatial modelling

Misaligned Spatial Data

8 Bayesian Multivariate Spatial Regression Models

Multivariate spatial modelling

Bivariate Spatial Regression with Misalignment

Rearrange the components of Z to Z̃ = (X(s1), X(s2), . . . , X(sn), Y (s1), Y (s2), . . . , Y (sn))> yields[

X Y

] ∼ N

([ µ11 µ21

] , T⊗ R (φ)

) .

Priors: Wishart for T−1, normal (perhaps flat) for (µ1, µ2), discrete prior for φ or perhaps a uniform on (0, .5max dist).

Estimation: Markov chain Monte Carlo (Gibbs, Metropolis, Slice, HMC/NUTS); Integrated Nested Laplace Approximation (INLA).

9 Bayesian Multivariate Spatial Regression Models

Multivariate spatial modelling

Dew-Shrub Data from Negev Desert in Israel

Negev desert is very arid

Condensation can contribute to annual water levels

Analysis: Determine impact of shrub density on dew duration

1129 locations with UTM coordinates

X(s) : Shrub density at location s (within 5m× 5m blocks)

Y (s) : Dew duration at location s (in 100-th of an hour)

Separable model with an exponential correlation function, ρ(‖s