Post on 21-Jun-2021
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INLA and inlabruwith spatialpatternsThierry Onkelinx
Overzicht
1 Checking spatial autocorrelationPearson residualsVariogram
2 Prepare the modelCreating a meshCreating an SPDE model
3 Fitting the modelOnly the dataPredictions
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Checkingspatial auto-correlation
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Checkingspatial auto-correlationPearson residuals
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Definition
▶ components?
▶ observed value (yi), fitted value (yi), mean squared error (MSE)▶ formula▶
pri =yi − yi√MSE
▶ MSE (variance) depends on distribution! Check it usinginla.doc("name_of_your_distribution").
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Definition
▶ components?▶ observed value (yi), fitted value (yi), mean squared error (MSE)
▶ formula▶
pri =yi − yi√MSE
▶ MSE (variance) depends on distribution! Check it usinginla.doc("name_of_your_distribution").
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Definition
▶ components?▶ observed value (yi), fitted value (yi), mean squared error (MSE)▶ formula
▶pri =
yi − yi√MSE
▶ MSE (variance) depends on distribution! Check it usinginla.doc("name_of_your_distribution").
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Definition
▶ components?▶ observed value (yi), fitted value (yi), mean squared error (MSE)▶ formula▶
pri =yi − yi√MSE
▶ MSE (variance) depends on distribution! Check it usinginla.doc("name_of_your_distribution").
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Definition
▶ components?▶ observed value (yi), fitted value (yi), mean squared error (MSE)▶ formula▶
pri =yi − yi√MSE
▶ MSE (variance) depends on distribution! Check it usinginla.doc("name_of_your_distribution").
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Example data: rainfall in Parana state, Brazil
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-26
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Longitude
Lati
tude
Rain
30
60
90
30
60
90
Rain
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Calculate Pearson residuals
model_iid <- inla(Rain ~ Xc + Yc, family = "gamma", data = dataset,control.compute = list(waic = TRUE))
dataset %>%mutate(
mu = model_iid$summary.fitted.values$mean,sigma2 = mu ^ 2 / model_iid$summary.hyperpar[1, "mean"],Pearson_iid = (Rain - mu) / sqrt(sigma2)
) -> dataset
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Challenge 1
▶ What is the mean for your model?▶ What is the variance for your model? Hint: inla.doc("your
distribution")▶ Calculate the Pearson residuals for your model
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Checkingspatial auto-correlationVariogram
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Definition
vg_default <- variogram(Pearson_iid ~ 1, locations = ~X + Y,data = as.data.frame(dataset), cressie = TRUE)
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distance (km)
vari
ance
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Important characteristics
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distance (km)
vari
ance
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Important characteristics
range
nugget
sill
partial sill
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distance (km)
vari
ance
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Projected example data
7000000
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7300000
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5000000 5100000 5200000 5300000 5400000 5500000 5600000
Rain
30
60
90
30
60
90
Rain
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Increased cutoff
vg_large <- variogram(Pearson_iid ~ 1, locations = ~X + Y, cressie = TRUE,data = as.data.frame(dataset), cutoff = 600e3)
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distance (km)
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Number of point pairs is important
3010
8473
11968
143751583916722
1597014653124159514
6920
44932781
1370546
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distance (km)
vari
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Too small width leads to unstable variograms
vg_small <- variogram(Pearson_iid ~ 1, locations = ~X + Y, cressie = TRUE,data = as.data.frame(dataset), width = 1e3)
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0.6
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distance (km)
vari
ance
number ofpoint pairs
(0,10]
(10,20]
(20,50]
(50,100]
(100,200]
(200,500]
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Sensible small width yields the most informative variogram
vg_final <- variogram(Pearson_iid ~ 1, locations = ~X + Y, cressie = TRUE,data = as.data.frame(dataset), width = 10e3)
150590
9451325
16942008
22322539272429483128316833723518
3762
3723
383639074014
40824211
4065423542114096
3882
2206
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distance (km)
vari
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Challenge 2
▶ What is the minimum binwidth for your data?▶ Calculate the variogram for your model▶ What is the approximate range of of the variogram?▶ What is the nugget, sill and partial sill?
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Prepare themodel
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Prepare themodelCreating a mesh
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Size of a mesh I
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Size of a mesh II
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Size of a mesh III
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Size of a mesh IV
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Size of a mesh V
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Guidelines
▶ equilateral triangles work best▶ edge length should be around a third to a tenth of the range▶ avoid narrow triangles▶ avoid small edges▶ add extra, larger triangles around the border▶ simplify the border
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Mesh only within the border
mesh <- inla.mesh.2d(boundary = border, max.edge = 0.15)ggplot() + gg(mesh) + coord_fixed() + theme_map() +ggtitle(paste("Vertices: ", mesh$n))
Vertices: 261
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Mesh going outside the border
mesh <- inla.mesh.2d(boundary = border, max.edge = c(0.15, 0.3))ggplot() + gg(mesh) + coord_fixed() + theme_map() +ggtitle(paste("Vertices: ", mesh$n))
Vertices: 417
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Mesh for rainfall data
mesh <- inla.mesh.2d(boundary = boundary, max.edge = c(30e3, 100e3))ggplot(dataset) + gg(mesh) + geom_sf() + ggtitle(paste("Vertices: ", mesh$n)) +coord_sf(datum = st_crs(5880))
7000000
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7300000
7400000
7500000
7600000
4800000 5000000 5200000 5400000 5600000
x
yVertices: 8531
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Use cutoff to simplify mesh
mesh1 <- inla.mesh.2d(boundary = boundary, max.edge = c(30e3, 100e3),cutoff = 10e3)
ggplot(dataset) + gg(mesh1) + geom_sf() +ggtitle(paste("Vertices: ", mesh1$n)) + coord_sf(datum = st_crs(5880))
7000000
7100000
7200000
7300000
7400000
7500000
7600000
4800000 5000000 5200000 5400000 5600000
x
y
Vertices: 844
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Finer mesh for final model run
mesh2 <- inla.mesh.2d(boundary = boundary, max.edge = c(10e3, 30e3),cutoff = 5e3)
ggplot(dataset) + gg(mesh2) + geom_sf() +ggtitle(paste("Vertices: ", mesh2$n)) + coord_sf(datum = st_crs(5880))
7000000
7100000
7200000
7300000
7400000
7500000
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4800000 5000000 5200000 5400000 5600000
x
y
Vertices: 5920
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Challenge 3
▶ What are the relevant max.edge and cutoff for a course mesh?▶ What are the relevant max.edge and cutoff for a smooth mesh?▶ Create a course and a smooth mesh for your data
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Prepare themodelCreating an SPDE model
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SPDE using penalised complexity priors
Stochastic Partial Differential Equations▶ prior.range = c(r, alpha_r): P(ρ < r) < αr▶ prior.sigma = c(s, alpha_s): P(σ > s) < αs
spde1 <- inla.spde2.pcmatern(mesh1, prior.range = c(100e3, 0.5),prior.sigma = c(0.9, 0.05))
spde2 <- inla.spde2.pcmatern(mesh2, prior.range = c(100e3, 0.5),prior.sigma = c(0.9, 0.05))
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Challenge 4
▶ What are relevant priors for the range and sigma for your data▶ Hint: see challenge 2
▶ Make the SPDE models for your data
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Fitting themodel
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Fitting themodelOnly the data
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The stack for the observed data
A1 <- inla.spde.make.A(mesh = mesh1, loc = st_coordinates(dataset))stack1 <- inla.stack(tag = "estimation", ## tagdata = list(Rain = dataset$Rain), ## responseA = list(A1, 1), ## projector matrices (SPDE and fixed effects)effects = list(
list(site = seq_len(spde1$n.spde)), ## random field indexdataset %>%
as.data.frame() %>%transmute(Intercept = 1, Xc, Yc) ## fixed effect covariates
))
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Model fit
INLA
model_spde1 <- inla(Rain ~ 0 + Intercept + Xc + Yc + f(site, model = spde1),family = "gamma", data = inla.stack.data(stack1),control.predictor = list(A = inla.stack.A(stack1)),control.compute = list(waic = TRUE)
)
inlabru
bru_spde1 <- bru(Rain ~ Xc + Yc + site(map = st_coordinates, model = spde1),family = "gamma", data = dataset)
bru_spde1 <- bru(Rain ~ Xc + Yc + site(map = coordinates, model = spde1),family = "gamma", data = as_Spatial(dataset))
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Comparison of fixed effect parameters
Intercept Xc Yc
INLA inlabru INLA inlabru INLA inlabru
-0.70
-0.65
-0.60
-0.55
-0.50
-0.45
-0.30
-0.25
-0.20
-0.15
-0.10
3.4
3.6
3.8
4.0
model
mea
n
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Comparing hyperparameters
Precision parameter for the Gamma observations Range for site Stdev for site
INLA inlabru INLA inlabru INLA inlabru
0.40
0.45
0.50
0.55
50000
75000
100000
125000
3.6
4.0
4.4
4.8
model
mea
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Correlation structure
spde.posterior(bru_spde1, "site", what = "matern.covariance") -> covplotspde.posterior(bru_spde1, "site", what = "matern.correlation") -> corplotmultiplot(plot(covplot), plot(corplot))
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Calculate Pearson residuals
dataset %>%mutate(
mu = model_spde1$summary.fitted.values$mean,sigma2 = mu ^ 2 / model_spde1$summary.hyperpar[1, "mean"],Pearson_iid = (Rain - mu) / sqrt(sigma2)
) -> dataset
## Error: Problem with `mutate()` input `mu`.## x Input `mu` can't be recycled to size 528.## i Input `mu` is `model_spde1$summary.fitted.values$mean`.## i Input `mu` must be size 528 or 1, not 1664.
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Using the stack index
si <- inla.stack.index(stack1, "estimation")$datadataset %>%mutate(
mu = model_spde1$summary.fitted.values$mean[si],sigma2 = mu ^ 2 / model_spde1$summary.hyperpar[1, "mean"],Pearson_spde = (Rain - mu) / sqrt(sigma2)
) -> dataset
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Using inlabru
fit <- predict(bru_spde1, as_Spatial(dataset), ~exp(Intercept + Xc + Yc + site))dataset %>%
mutate(mu = fit$mean,sigma2 = mu ^ 2 / model_spde1$summary.hyperpar[1, "mean"],Pearson_spde = (Rain - mu) / sqrt(sigma2)
) -> dataset
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Variogram
vg_fit <- variogram(Pearson_spde ~ 1, cressie = TRUE,data = as_Spatial(dataset), width = 10e3)
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distance (km)
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Interpolate GMRF
A1.grid <- inla.mesh.projector(mesh1, dims = c(41, 41))inla.mesh.project(A1.grid, model_spde1$summary.random$site) %>%as.matrix() %>%as.data.frame() %>%bind_cols(
expand.grid(x = A1.grid$x, y = A1.grid$y)) %>%filter(!is.na(ID)) -> eta_spde
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Plot GMRF
ggplot(dataset) + geom_tile(data = eta_spde, aes(x = x, y = y, fill = mean)) +geom_sf() + scale_fill_gradient2()
27°S
26°S
25°S
24°S
23°S
22°S
56°W 54°W 52°W 50°W 48°W
x
y
-1.0
-0.5
0.0
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mean
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Fitting themodelPredictions
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Prediction stack for SPDE grid + fixed effects
expand.grid(X = A1.grid$x, Y = A1.grid$y) %>%mutate(Intercept = 1, Xc = X / 1e5 - 53, Yc = Y / 1e5 - 71) -> grid_data
stack1_grid <- inla.stack(tag = "grid", ## tagdata = list(Rain = NA), ## responseA = list(A1.grid$proj$A, 1), ## projector matrices (SPDE and fixed effects)effects = list(
list(site = seq_len(spde1$n.spde)), ## random field indexgrid_data ## covariates at grid locations
))
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Refit the model with the combinated stack
stack_all <- inla.stack(stack1, stack1_grid)model_grid <- inla(Rain ~ 0 + Intercept + Xc + Yc + f(site, model = spde1),
family = "gamma", data = inla.stack.data(stack_all),control.predictor = list(A = inla.stack.A(stack_all),
link = 1),control.compute = list(waic = TRUE),control.mode = list(theta = model_spde1$mode$theta,
restart = FALSE),control.results = list(return.marginals.random = FALSE,
return.marginals.predictor = FALSE))
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Plot grid I
si <- inla.stack.index(stack_all, "grid")$datagrid_data %>%bind_cols(model_grid$summary.fitted.values[si, ]) %>%`coordinates<-`(~X + Y) %>%`proj4string<-`(CRS(SRS_string = "EPSG:5880")) -> gd
gd[!is.na(over(gd, boundary)), ] %>%as.data.frame() %>%ggplot() + geom_tile(aes(x = X, y = Y, fill = mean)) + coord_fixed()
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Plot grid II
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5000000 5200000 5400000 5600000
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Y
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mean
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Using inlabru
pred_mesh <- predict(bru_spde1, pixels(mesh1), ~exp(Intercept + Xc + Yc + site))ggplot() + gg(pred_mesh) + gg(boundary)
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Challenge 5
▶ Fit the model using the SPDE▶ Plot a map of the GMRF▶ Plot a map of the predictions and their credible interval
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