Spatial regression in R part 2: INLA

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Ten months after part 1 of spatial regression in R (oh my gosh the place did these months go?), here’s a (hopefully long-awaited) second half this time utilizing INLA, a bundle that’s useful in lots of conditions.

What this will probably be about

There are a lot of several types of spatial information, and all include particular fashions. Right here we’ll deal with so-called geostatistical or point-reference fashions. Principally, you collected some informations in several places and need to account for the truth that places nearer collectively usually tend to present related values than places additional appart.

Partially 1, we noticed learn how to match spatial regression of the next type:

[ y_i sim mathcal{N}(mu_i, sigma) ]

the place (i) index the totally different traces in your dataset, (y) is the response variable, (mu) is a vector of anticipated values and (sigma) is the residual normal deviation.

[ mu_i = X_i * beta + u(s_i) ]

the anticipated values depend upon some covariates which are within the design matrix X, these get multiplied by some coefficient to estimate (the (beta)’s) plus a spatial random impact (u) that depend upon location (s_i) of the info level.

The essential half in these fashions is in estimating the spatial random impact, more often than not it’s modelled as a multivariate regular distribution:

[ u(s_i) sim mathcal{MVN}(0, Sigma) ]

and the variance-covariance matrix ((Sigma)) is populated utilizing the Matern correlation operate:

[ cov(i, j) = delta * Matern(d_{ij}, kappa) ]
the covariance between any two places within the dataset depend upon their distance ((d)), the vary of the Matern operate ((kappa)) and the spatial variance ((delta)). The parameters to estimate in a majority of these fashions are subsequently: (i) the coefficient of the covariates results ((beta)), the variation within the spatial impact ((delta)), the vary of the spatial impact ((kappa)) and the residual variation ((sigma)).

Like within the first half we’ll match this mannequin to an instance dataset from the geoR bundle:

library(geoR)

# the ca20 dataset
# load the instance dataset,
# calcium content material in soil samples in Brazil
information(ca20)
# put this in an information body
dat <- information.body(x = ca20$coords[,1], 
                  y = ca20$coords[,2], 
                  calcium = ca20$information, 
                  elevation = ca20$covariate[,1], 
                  area = issue(ca20$covariate[,2]))

The mannequin that we need to match is:

[ calcium_i = intercept + elevation_i + region_i + u(s_i) ]

INLA

INLA is a bundle that enables to suit a broad vary of mannequin, it makes use of Laplace approximation to suit Bayesian fashions a lot, a lot sooner than algorithms resembling MCMC. INLA permits for becoming geostatistical fashions through stochastic partial differential equation (SPDE), place for extra background informations on this are these two gitbooks: spde-gitbook and inla-gitbook.

Becoming a spatial mannequin in INLA require a set of explicit steps:

  1. Create a mesh to approximate the spatial impact
  2. Create a projection matrix to hyperlink the observations to the mesh
  3. Set the stochastic partial differential equation
  4. optionally specify a dataset to derive mannequin predictions
  5. Put every part collectively in a stack oject
  6. Match the mannequin

Let’s undergo these steps:

The mesh

With a view to estimate the spatial random impact INLA makes use of a mesh, that may be simply outlined as comply with:

library(INLA)
# meshes in 2D house may be created as comply with:
mesh <- inla.mesh.second(loc = dat[,c("x", "y")],
                     max.edge = c(50, 5000))

inla.mesh.second must location of the samples plus some informations on how exact the mesh ought to be. A extra exact mesh will present a greater estimation of the spatial impact (the prediction will probably be smoother) however this comes at the price of longer computational instances. Right here we specified the mesh by saying that the utmost distance between two nodes is between 50 and 5000 meters. You may interactively discover learn how to set your mesh utilizing the operate meshbuilder from INLA.

The projection matrix

The projection matrix makes the hyperlink between your noticed information and the spatial impact estimated by the mannequin. It’s easy to create:

Amat <- inla.spde.make.A(mesh, loc = as.matrix(dat[,c("x", "y")]))

Set the SPDE

The spatial impact in INLA is being estimated utilizing some advanced mathematical equipment named stochastic partial differential equation. The essential thought is that we will estimate a steady spatial impact utilizing a set of discrete level (the nodes outlined within the mesh) and foundation operate, simlar to regression splines. This makes the estimation of the spatial fields a lot simpler. Now the difficult half in setting the SPDE is in defining the priors for the vary of the spatial impact (how far do it’s essential to go for 2 factors to be unbiased, the (kappa) parameter) and the variation within the spatial impact (how variable is the sector from a degree to the subsequent, the (delta) parameter).


spde <- inla.spde2.pcmatern(mesh, 
                            prior.vary = c(500, 0.5),
                            prior.sigma = c(2, 0.05))

Setting these priors may be difficult since it should have a big influence on the fitted mannequin. Fortuitously, since becoming fashions in INLA is comparatively quick, it’s straightforward to do sensitivity evaluation and get a grasp on what works. The prior for the vary correspond to the next system:

[ P(kappa < kappa_0) = p ]

the place (kappa) is the vary and (kappa_0) and (p) are the 2 values to cross to the operate. Above we mentioned: the likelihood that the vary of the spatial impact is beneath 500 meters is 0.5. For the variation parameter the prior is about similarly:

[ P(delta > delta_0) = p ]

So within the traces above we mentioned: the likelihood that the variation within the spatial impact is bigger than 2 is 0.05.

Setting these priors may be very complicated at first, I like to recommend that you just check out some values, test the fashions and browse round to get some inspiration. I like to recommend to set relatively robust priors for the (delta), if the prior is just too imprecise (p near 0.5), then the mannequin can run into some drawback, particularly for extra advanced fashions.

As soon as we have now all of this we will put every part in a stack:

# create the info stack
dat_stack <- inla.stack(information = record(calcium = dat$calcium), # the response variable
                        A = record(Amat, 1, 1, 1), # the projection matrix
                        results = record(i = 1:spde$n.spde, # the spatial impact
                                       Intercept = rep(1, nrow(dat)), # the intercept
                                       elevation = dat$elevation,
                                       area = issue(dat$area)))

The important thing half right here is that within the (A) argument we specify the projection of the totally different results. The spatial impact is called i (however we will title it something we would like) and is listed by the variety of mesh nodes. Bear in mind the finer the mesh the finer the estimation of the spatial impact. This spatial impact i is linked to the info through the projection matrix (Amat). The opposite impact are all immediately linked to the info so no want for projection matrices there.

Set the predictions

In INLA it’s often simpler to derive mannequin predictions by passing the brand new information that we need to use to make predictions on to the mannequin becoming. In different phrases, we have to outline these new information earlier than becoming the mannequin.

We’ll first derive a brand new information for prediction of simply the elevation and area impact:

# a newdata to get the predictions
modmat <- develop.grid(elevation = seq(min(dat$elevation), 
                                      max(dat$elevation), 
                                      size.out = 10),
                      area = distinctive(dat$area))

# the stack for these predictions
pred_stack_fixef <- inla.stack(information = record(calcium = NA),
                               A = record(1, 1, 1),
                               results = record(Intercept = rep(1, nrow(modmat)),
                                              elevation = modmat$elevation,
                                              area = issue(modmat$area)),
                               tag = "prd_fixef")

The important thing factor to notice right here is that we set calcium=NA within the information argument, the mannequin will then estimate these values primarily based on the consequences and the parameters of the fashions. On this stack we additionally use a tag prd_fixef that can enable us in a while to extra simply seize the expected values.

As a result of we fitted a spatial mannequin we additionally need to make prediction throughout house. We might do that both primarily based on the spatial discipline alone, however it’s extra attention-grabbing to derive these spatial prediction by additionally accounting for the opposite covariates (elevation and area). The derivation of this prediction stacks is a little more concerned since we’ll then want elevation and area values not simply on the noticed places however throughout house. This may require a few non-INLA associated steps which will look scarry, however the finish purpose is mainly to outline rasters with elevation and area infos from the info that we have now:

library(raster)
library(fields) # for Tps

## first we outline an empty raster to carry the coordinates of the predictions
r <- raster(xmn = min(dat$x), xmx = max(dat$x),
            ymn = min(dat$y), ymx = max(dat$y),
            decision = 25)

## the we use thin-plate spline to derive elevation throughout the info
elev_m <- Tps(dat[,c("x","y")], dat$elevation)
## put this right into a raster
elev <- interpolate(r, elev_m)

## for the area data we create a SpatialPolygons 
## primarily based on the coordinates given within the ca20 object
pp <- SpatialPolygons(record(Polygons(record(Polygon(ca20[[5]])), ID = "reg1"),
                           Polygons(record(Polygon(ca20[[6]])), ID = "reg2"),
                           Polygons(record(Polygon(ca20[[7]])), ID = "reg3")))
# flip the SpatialPolygon right into a raster object
area <- rasterize(pp, r)

# the brand new information body with coordinates from the raster
# plus elevation and area data
newdat <- as.information.body(xyFromCell(r, cell = 1:ncell(r)))
newdat$elevation <- values(elev)
newdat$area <- issue(values(area))
# take away NAs
newdat <- na.omit(newdat)

# create a brand new projection matrix for the factors
Apred <- inla.spde.make.A(mesh,
                          loc = as.matrix(newdat[,c("x", "y")]))

# put this in a brand new stack
pred_stack_alleff <- inla.stack(information = record(calcium = NA),
                               A = record(Apred, 1, 1, 1),
                               results = record(i = 1:spde$n.spde,
                                              Intercept = rep(1, nrow(newdat)),
                                              elevation = newdat$elevation,
                                              area = issue(newdat$area)),
                               tag = "prd_alleff")

Once more the concept there was to get elevation and area data from a raster of the area of curiosity, as soon as we had this within the newdat object we outlined a brand new projection matrix since this time we can even use the spatial discipline to derive predictions. Then we put all these infos into a brand new stack with a brand new tag.

Match the mannequin

We’re finely (virtually) prepared to suit the mannequin, we simply must put the the noticed information and the 2 prediction stacks collectively:

# put all of the stacks collectively
all_stack <- inla.stack(dat_stack, pred_stack_fixef,
                      pred_stack_alleff)

We will now match the mannequin:

# match the mannequin
m_inla <- inla(calcium ~ -1 + Intercept + elevation + area + f(i, mannequin = spde),
               information = inla.stack.information(all_stack),
               management.predictor = record(A = inla.stack.A(all_stack), compute = TRUE),
               quantiles = NULL)

This could take round 30 sec to run. To suit the mannequin the primary argument is a system describing the mannequin, we have to use -1 to take away the interior intercept and match it individually. Then we use the f() to specify a random impact that’s listed by i following the SPDE mannequin outlined above. The remaining is simply passing the info and the projection matrix. Notice that we set compute=TRUE to ensure that the mannequin to estimate the calcium values that got as NAs.

We will get the mannequin abstract:

abstract(m_inla)
## 
## Name:
##    c("inla(system = calcium ~ -1 + Intercept + elevation + area + 
##    ", " f(i, mannequin = spde), information = inla.stack.information(all_stack), 
##    quantiles = NULL, ", " management.predictor = record(A = 
##    inla.stack.A(all_stack), compute = TRUE))" ) 
## Time used:
##     Pre = 3.33, Working = 10.2, Publish = 0.179, Whole = 13.7 
## Fastened results:
##             imply     sd   mode kld
## Intercept 29.608 17.259 29.628   0
## elevation  1.336  1.711  1.311   0
## region1    2.353 16.212  2.350   0
## region2   10.665 16.107 10.668   0
## region3   16.587 16.211 16.589   0
## 
## Random results:
##   Title     Mannequin
##     i SPDE2 mannequin
## 
## Mannequin hyperparameters:
##                                            imply     sd    mode
## Precision for the Gaussian observations   0.031  0.009   0.027
## Vary for i                             202.389 47.472 187.915
## Stdev for i                               7.450  1.045   7.152
## 
## Anticipated variety of efficient parameters(stdev): 71.96(22.09)
## Variety of equal replicates : 2.47 
## 
## Marginal log-Probability:  -667.01 
## Posterior marginals for the linear predictor and
##  the fitted values are computed

A few vital components:

  • Fastened results: these are the mannequin estimate for the intercept, elevation and area coefficients
  • Random results: the definition of the spatial random impact
  • Mannequin hyperparameters: the residual deviation (given as precision), the vary of the spatial impact ((kappa)) and the deviation within the spatial impact ((delta))

Utilizing a useful operate from Greg Albery we will derive the expected correlation between two factors given there distance:

# devtools::install_github("https://github.com/gfalbery/ggregplot")
library(ggregplot)
library(dplyr)

INLARange(ModelList = m_inla, MaxRange = 1000, MeshList = mesh, WNames = "i") +
  labs(x = "Distance in meters") +
  theme(legend.place = "none")

And now we will take a look at the mannequin predictions first from the fastened results solely, so averaging over the spatial variations:

## first we create an index to simply discover these 
## prediction throughout the fitted mannequin
id_fixef <- inla.stack.index(all_stack, "prd_fixef")$information

## add to modmat the prediction and their sd
modmat$calcium <- m_inla$abstract.fitted.values[id_fixef, "mean"]
modmat$sd <- m_inla$abstract.fitted.values[id_fixef, "sd"]

## a plot with the unique information
ggplot(dat, aes(x = elevation, y = calcium)) +
  geom_ribbon(information = modmat, aes(ymin = calcium - 2 * sd,
                                 ymax = calcium + 2 * sd,
                                 fill = area),
              alpha = 0.2) +
  geom_line(information = modmat, aes(shade = area)) +
  geom_point(aes(shade = area)) 

Utilizing the tag we outlined earlier we will simply seize the related traces within the mannequin object which are saved throughout the abstract.fitted.values information body within the object. There we get the imply and the usual deviation that we then plot along with the unique information.

And now come the cool map:

# once more get the proper indices
id_alleff <- inla.stack.index(all_stack, "prd_alleff")$information

# now add the mannequin predictions
newdat$pred <- m_inla$abstract.fitted.values[id_alleff, "mean"]
newdat$sd <- m_inla$abstract.fitted.values[id_alleff, "sd"]
# get decrease and higher confidence interval
newdat$lower_ci <- with(newdat, pred - 2 * sd)
newdat$upper_ci <- with(newdat, pred + 2 * sd)

# some information wraggling
nn <- pivot_longer(newdat, cols = c("pred", "lower_ci", "upper_ci"))

ggplot(nn, aes(x=x, y=y, fill=worth)) +
  geom_raster() +
  facet_wrap(~title) +
  scale_fill_continuous(sort = "viridis")

There we see one of many benefit of Bayesian evaluation, we will get estimation of variation and uncertainty round any estimated portions from the mannequin.

Voila! That is all for as we speak, subsequent time we’ll proceed alongside the Bayesian traces with Gaussian Course of in greta, get excited!

Some useful refenrences for additional studying:



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