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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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-54 -52 -50 -48

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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Important characteristics

1

3

2

4

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Important characteristics

range

nugget

sill

partial sill

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distance (km)

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Projected example data

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Rain

30

60

90

30

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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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Number of point pairs is important

3010

8473

11968

143751583916722

1597014653124159514

6920

44932781

1370546

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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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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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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))

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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))

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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))

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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

n

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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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x

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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

0.5

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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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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mean

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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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