Spatial Misalignment

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Gervase Anthony
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1 November 4, 2009

2 Objectives Land use in Madagascar Example with WinBUGS code

3 Objectives Learn how to realign nonnested block level data Become familiar with ways to deal with misaligned data which are embedded in regression modeling

4 Breaking it down: from blocks to atoms B i are blocks on the response grid - atoms in B i are B ik C j are blocks on the explanatory grid - atoms in C j are C jl "- Oll = B = B22 B12 C 13 B = B 23 B ll B21 Figure 6.10 Illustrative representation of areal data misalignment.

5 The Model - basic components Y - response (measured on the response grid, B i ) W - covariates on the response grid (B i ) X - covariates on the explanatory grid (C j ) µ i - random effects (spatial association among Y i s) ω i - random effects (spatial association among X j s)

6 The Model - Assumptions Y i s are aggregated measurements W i s are aggregated measurements or inheritable X j s are aggregated measurements µ i s are inherited by latent Y ik ω i s are inherited by latent X jl

7 From apples to oranges (and back) For all non-edge atoms X jl can also be labeled as X ik What about edge atoms? Use neighboring nonedge atoms to determine the distribution of the edge atom E.g., if X is a count variable: X ie ω i P o(e ω i BiE ) (Note: B ie is the area of B ie and ω i is a new set of random effects) The same basic procedure can be used for Y je as well.

8 Modeling latent variables - X jl and Y ik Assuming X and Y are counts: X jl ω j P o(e ω j C jl ) ( ) (X j1,..., X jlj X j, ω j ) Mult X j ; C j1 C j,..., C jl j C j (ω j, ω i ) CAR(λ ω ) ( ( Y ik µ i, θ ik P o e µ i X B ik h ik ( ) iid µ i N η µ, 1 τ µ B ik ; θ ik )) h (z; θ ik ) is a preselected function used for model specfication

9 The big picture / " {wi} () "- X J > X jl > X'k --+-> {Yi} ---.> {Yik} > {Yj} 2. Figure 6.11 Graphical version of the model, with variables as described in the text. Boxes indicate data nodes, while circles indicate unknowns.

Introduction FMPC data I I What is the population in each cell of the windrose? What is the population in each cell of the windrose broken down into categories for age and sex? a u a.lon ensl... > 500.

10 Introduction FMPC data I I What is the population in each cell of the windrose? What is the population in each cell of the windrose broken down into categories for age and sex? a u a.lon ensl... > _ ,...-.r,",", :.:.:.:.:.:.:.:.:.:.:.;.;.:.:.:-:.:.....,.....,., --, :.:.:...:::.:;:: ::::~::::::~:~::~~~:::~~:;: ::::: :::..:.::::::~::::::::: :.:... :.::::~:::...:.:.:;:~:~::....:... :... ;:... :...: :':':':;':':-:X':':'::':':':-"':'... ;.:.::-:.:.: r <= 40, rue ure ensl > o - 10 I o o 1 3 4km,,2 J I Figure 6.9 Census block groups and 10-km windro se near the FMPC site, w ith 1990 population density by block group and 1980 structure density by cell both tn

11 Step 1 - model the number of structures in each atom, X jl X jl P o(e ω j C jl ) ( ) (X j1,..., X jlj X j., ω j ) Mult X j. ; C j1 C j,..., C jl j C j (ω j, ω i ) CAR(λ ω ) λ ω = 10

12 Step 2 - estimate the population in each atom ( )) Y ik P o e µ i (X ik + θ Ki ( ) (Y i1,..., Y iki Y i. ) Mult Y i. ; X i1 +θ/k i X i. +θ,..., X ik +θ/k i i X i. +θ ( ) iid µ i N η µ, 1 τ µ η µ = 1.1 τ µ = 0.5 θ = 1

13 Introduction I Step 3 - Aggregate the population totals for all the atoms in each cell of the windrose

14 Step 4 - Model the population count in each cell by sex and age group ( [ ] 17 ( ) ) Y ikga P o exp µ i + gα + β a I a X ik + θ K i g = 0 for males, 1 for female a=1 I a = indicator for age group ( ) iid µ i N η µ, 1 τ µ η µ = 2.5 log(3/36) τ µ = 0.5 θ = 1

15 Land use in Madagascar Example with WinBUGS code We have learned discrete methods for realigning each type and combination of spatially misaligned data. This section will provide examples of dealing with misalignment directly within a regression model. First example (theory) - land use in Madagascar Second example (actual data!) - flowers in South Africa

16 Land use in Madagascar Example with WinBUGS code Madagascar Data: Population - collected by town P i : population in the ith town Land Use - collected by pixels (4 km x 4 km) L ij : jth pixel of the ith town Elevation - pixel Slope - pixel Spatial effects parameters: ϕ ij - pixel-level spatial effects δ i - town-level spatial effects

17 Land use in Madagascar Example with WinBUGS code The joint distribution of land use and population: p(l, P E ij, S ij, ϕ ij, δ i ) Rearranged to examine the effect of population on land use: p(p E ij, S ij, δ i ) }{{} P ij P i Mult(P i. ;λ ij /λ i. ) p(l P, E ij, S ij, ϕ ij ) }{{} L ij Bin(16,q ij ) Where, log λ ij = β 0 + β 1 E ij + β 2 S ij + δ i ( ) qij log = α 0 + α 1 E ij + α 2 S ij + α 3 P ij + ϕ ij 1 q ij

18 Land use in Madagascar Example with WinBUGS code Model 1: ( ) qij log = α 0 + α 1 E ij + α 2 S ij + α 3 P ij 1 q ij Model 2: log ( qij ) = α 0 + α 1 E ij + α 2 S ij + α 3 P ij + ϕ ij 1 q ij Conclusion: There is a relationship between population and land use.

19 Land use in Madagascar Example with WinBUGS code Flowers in South Africa s Cape Floristic Region Observations were made at a number of locations regarding whether the Grand Protea flower was present at that location. Information on a number of environmental covariates is available for each of 476 one minute by one minute grid cells in the study region. The point level response data is converted to grid level by modeling the number of times a flower is observed in a cell given the number of sampling locations in that grid.

20 Land use in Madagascar Example with WinBUGS code Y i Bin(n i, p i ) ( ) pi log = w 1 p iβ + µ + ρ i i w i is a vector of environmental covariates µ is the non-spatial random effects ρ i is the spatial random effects

Sources: Introduction Land use in Madagascar Example with WinBUGS code Banerjee, Carlin, and Gelfand (2004) Hierarchical Modeling and Analysis for

21 Sources: Introduction Land use in Madagascar Example with WinBUGS code Banerjee, Carlin, and Gelfand (2004) Hierarchical Modeling and Analysis for Spatial Data. Latimer, Wu, Gelfand, and Silander (2006) Building Statistical Models to Analyze Species Distributions. 16 Ecological Applications

Spatial Misalignment

Spatial Misalignment Jamie Monogan University of Georgia Spring 2013 Jamie Monogan (UGA) Spatial Misalignment Spring 2013 1 / 28 Objectives By the end of today s meeting, participants should be able to: