LGCPs - Distance sampling • inlabru

Introduction

We’re going to estimate distribution and abundance from a line transect survey of dolphins in the Gulf of Mexico. These data are also available in the R package dsm (where they go under the name mexdolphins). In inlabru the data are called mexdolphin for sp format, and mexdolphin_sf for sf format.

Setting things up

Load libraries

library(inlabru)
library(fmesher)
library(INLA)
library(ggplot2)

Get the data

We’ll start by loading the data, renaming it, and extracting the mesh (for convenience).

mexdolphin <- mexdolphin_sf
mesh <- mexdolphin$mesh

Plot the data (the initial code below is just to get rid of tick marks, if desired)

noyticks <- theme(
  axis.text.y = element_blank(),
  axis.ticks = element_blank()
)

noxticks <- theme(
  axis.text.x = element_blank(),
  axis.ticks = element_blank()
)

ggplot() +
  gg(mexdolphin$ppoly) +
  gg(mexdolphin$samplers, color = "grey") +
  gg(mexdolphin$points, size = 0.2, alpha = 1) +
  theme(
    legend.key.width = unit(x = 0.2, "cm"),
    legend.key.height = unit(x = 0.3, "cm")
  ) +
  theme(legend.text = element_text(size = 6)) +
  coord_sf(datum = fm_crs(mexdolphin$ppoly))

Spatial model with a half-normal detection function

The samplers in this dataset are lines, not polygons, so we need to tell inlabru about the strip half-width, W, which in the case of these data is 8. We start by plotting the distances and histogram of frequencies in distance intervals:

W <- 8
ggplot(mexdolphin$points) +
  geom_histogram(aes(x = distance),
    breaks = seq(0, W, length.out = 9),
    boundary = 0, fill = NA, color = "black"
  ) +
  geom_point(aes(x = distance), y = 0, pch = "|", cex = 4)

We need to define a half-normal detection probability function. This must take distance as its first argument and the linear predictor of the sigma parameter as its second:

hn <- function(distance, sigma) {
  exp(-0.5 * (distance / sigma)^2)
}

To control the prior distribution for the sigma parameter, we use a transformation mapper that converts a \(N(0, 1)\) latent variable into an exponentially distributed variable with expectation 8 (this is a somewhat arbitrary value, but motivated by the maximum observation distance W):

bru_mapper_marginal(qexp, pexp, dexp, rate = 1 / 8)

Specify and fit an SPDE model to these data using a half-normal detection function form. We need to define a (Matérn) covariance function for the SPDE:

matern <- inla.spde2.pcmatern(mexdolphin$mesh,
  prior.sigma = c(2, 0.01),
  prior.range = c(50, 0.01)
)

Here, the range is probabilistically limited by \(P(\text{range}\leq 50)=0.01\) and the standard deviation of the spatial field is limited by \(P(\text{sd}\leq 2)=0.01\).

We need to now separately define the components of the model (the SPDE, the Intercept and the detection function parameter sigma)

cmp <- ~ mySPDE(main = geometry, model = matern) +
  sigma(1,
    prec.linear = 1,
    marginal = bru_mapper_marginal(qexp, pexp, dexp, rate = 1 / 8)
  ) +
  Intercept(1)

The marginal argument in the sigma component specifies the transformation function taking N(0,1) to Exponential(1/8).

The formula, which describes how these components are combined to form the linear predictor (remembering that we need an offset due to the unknown direction of the detections!):

form <- geometry + distance ~ mySPDE +
  log(hn(distance, sigma)) +
  Intercept + log(2)

Before version 2.9.0.9004, a less compact approach had to be used, by applying the transformation between N(0,1) and Exponential(1/8) directly in the predictor expression, requiring corresponding adjustments to the later predict() calls, etc:

# sigma_transf <- function(x) {
#   bru_forward_transformation(qexp, x, rate = 1 / 8)
# }
# cmp <- ~ mySPDE(main = geometry, model = matern) +
#   sigma_theta(1, prec.linear = 1) +
#   Intercept(1)
# form <- geometry + distance ~ mySPDE +
#   log(hn(distance, sigma_transf(sigma_theta))) +
#   Intercept + log(2)

Then we fit the model, passing both the components and the formula (previously the formula was constructed invisibly by inlabru), and specify integration domains for the spatial and distance dimensions:

fit <- lgcp(
  components = cmp,
  mexdolphin$points,
  samplers = mexdolphin$samplers,
  domain = list(
    geometry = mesh,
    distance = fm_mesh_1d(seq(0, 8, length.out = 30))
  ),
  formula = form
)

Look at the SPDE parameter posteriors

spde.range <- spde.posterior(fit, "mySPDE", what = "range")
plot(spde.range)

spde.logvar <- spde.posterior(fit, "mySPDE", what = "log.variance")
plot(spde.logvar)

Predict spatial intensity, and plot it:

pxl <- fm_pixels(mesh, dims = c(200, 100), mask = mexdolphin$ppoly)
pr.int <- predict(fit, pxl, ~ exp(mySPDE + Intercept))

ggplot() +
  gg(pr.int, geom = "tile") +
  gg(mexdolphin$ppoly, linewidth = 1, alpha = 0) +
  gg(mexdolphin$samplers, color = "grey") +
  gg(mexdolphin$points, size = 0.2, alpha = 1) +
  theme(
    legend.key.width = unit(x = 0.2, "cm"),
    legend.key.height = unit(x = 0.3, "cm")
  ) +
  theme(legend.text = element_text(size = 6))

Predict the detection function and plot it, to generate a plot like the one below. Here, we should make sure that it doesn’t try to evaluate the effects of components that can’t be evaluated using the given input data. From version 2.8.0, inlabru automatically detects which components are involved. See ?predict.bru for more information.

distdf <- data.frame(distance = seq(0, 8, length.out = 100))
dfun <- predict(fit, distdf, ~ hn(distance, sigma))
plot(dfun)

The average detection probability within the maximum detection distance is estimated to be 0.7025877.

We can look at the posterior for expected number of dolphins as usual:

predpts <- fm_int(mexdolphin$mesh, mexdolphin$ppoly)
Lambda <- predict(fit, predpts, ~ sum(weight * exp(mySPDE + Intercept)))
Lambda
#>       mean       sd   q0.025     q0.5   q0.975   median sd.mc_std_err
#> 1 242.7117 51.42936 158.9018 240.5585 348.6077 240.5585      3.752695
#>   mean.mc_std_err
#> 1        5.893475

and including the randomness about the expected number. In this case, it turns out that you need lots of posterior samples, e.g. 2,000 to smooth out the Monte Carlo error in the posterior, and this takes a little while to compute:

Ns <- seq(50, 450, by = 1)
Nest <- predict(fit, predpts,
  ~ data.frame(
    N = Ns,
    density = dpois(
      Ns,
      lambda = sum(weight * exp(mySPDE + Intercept))
    )
  ),
  n.samples = 2000
)

Nest <- dplyr::bind_rows(
  cbind(Nest, Method = "Posterior"),
  data.frame(
    N = Nest$N,
    mean = dpois(Nest$N, lambda = Lambda$mean),
    mean.mc_std_err = 0,
    Method = "Plugin"
  )
)
ggplot(data = Nest) +
  geom_line(aes(x = N, y = mean, colour = Method)) +
  geom_ribbon(
    aes(
      x = N,
      ymin = mean - 2 * mean.mc_std_err,
      ymax = mean + 2 * mean.mc_std_err,
      fill = Method,
    ),
    alpha = 0.2
  ) +
  geom_line(aes(x = N, y = mean, colour = Method)) +
  ylab("Probability mass function")

Hazard-rate Detection Function

Try doing this all again, but use this hazard-rate detection function model, with the same prior for the sigma parameter as for the half-Normal model (such parameters aren’t always comparable, but in this example it’s a reasonable choice):

hr <- function(distance, sigma) {
  1 - exp(-(distance / sigma)^-1)
}

Solution:

formula1 <- geometry + distance ~ mySPDE +
  log(hr(distance, sigma)) +
  Intercept + log(2)

fit1 <- lgcp(
  components = cmp,
  mexdolphin$points,
  samplers = mexdolphin$samplers,
  domain = list(
    geometry = mesh,
    distance = fm_mesh_1d(seq(0, 8, length.out = 30))
  ),
  formula = formula1
)

Plots:

spde.range <- spde.posterior(fit1, "mySPDE", what = "range")
plot(spde.range)

spde.logvar <- spde.posterior(fit1, "mySPDE", what = "log.variance")
plot(spde.logvar)

pr.int1 <- predict(fit1, pxl, ~ exp(mySPDE + Intercept))

ggplot() +
  gg(pr.int1, geom = "tile") +
  gg(mexdolphin$ppoly, linewidth = 1, alpha = 0) +
  gg(mexdolphin$samplers, color = "grey") +
  gg(mexdolphin$points, size = 0.2, alpha = 1) +
  theme(
    legend.key.width = unit(x = 0.2, "cm"),
    legend.key.height = unit(x = 0.3, "cm")
  ) +
  theme(legend.text = element_text(size = 6))

distdf <- data.frame(distance = seq(0, 8, length.out = 100))
dfun1 <- predict(fit1, distdf, ~ hr(distance, sigma))
plot(dfun1)

predpts <- fm_int(mexdolphin$mesh, mexdolphin$ppoly)
Lambda1 <- predict(fit1, predpts, ~ sum(weight * exp(mySPDE + Intercept)))
Lambda1
#>       mean       sd   q0.025     q0.5   q0.975   median sd.mc_std_err
#> 1 295.1062 83.32696 153.0904 285.0364 471.5856 285.0364       8.49171
#>   mean.mc_std_err
#> 1        10.03104

Ns <- seq(50, 650, by = 1)
Nest1 <- predict(
  fit1,
  predpts,
  ~ data.frame(
    N = Ns,
    density = dpois(
      Ns,
      lambda = sum(weight * exp(mySPDE + Intercept))
    )
  ),
  n.samples = 2000
)

Nest1 <- dplyr::bind_rows(
  cbind(Nest1, Method = "Posterior"),
  data.frame(
    N = Nest1$N,
    mean = dpois(Nest1$N, lambda = Lambda1$mean),
    mean.mc_std_err = 0,
    Method = "Plugin"
  )
)

ggplot(data = Nest1) +
  geom_line(aes(x = N, y = mean, colour = Method)) +
  geom_ribbon(
    aes(
      x = N,
      ymin = mean - 2 * mean.mc_std_err,
      ymax = mean + 2 * mean.mc_std_err,
      fill = Method
    ),
    alpha = 0.2
  ) +
  geom_line(aes(x = N, y = mean, colour = Method)) +
  ylab("Probability mass function")

Comparing the models

Look at the goodness-of-fit of the two models in the distance dimension

bc <- bincount(
  result = fit,
  observations = mexdolphin$points$distance,
  breaks = seq(0, max(mexdolphin$points$distance), length.out = 9),
  predictor = distance ~ hn(distance, sigma)
)
attributes(bc)$ggp


bc1 <- bincount(
  result = fit1,
  observations = mexdolphin$points$distance,
  breaks = seq(0, max(mexdolphin$points$distance), length.out = 9),
  predictor = distance ~ hn(distance, sigma)
)
attributes(bc1)$ggp

Fit Models only to the distance sampling data

Half-normal first

formula <- distance ~ log(hn(distance, sigma)) + Intercept
cmp <- ~ sigma(1,
  prec.linear = 1,
  marginal = bru_mapper_marginal(qexp, pexp, dexp, rate = 1 / 8)
) +
  Intercept(1)
dfit <- lgcp(
  components = cmp,
  mexdolphin$points,
  domain = list(distance = fm_mesh_1d(seq(0, 8, length.out = 30))),
  formula = formula,
  options = list(bru_initial = list(sigma = 1, Intercept = 3))
)
detfun <- predict(dfit, distdf, ~ hn(distance, sigma))

Hazard-rate next

formula1 <- distance ~ log(hr(distance, sigma)) + Intercept
cmp <- ~ sigma(1,
  prec.linear = 1,
  marginal = bru_mapper_marginal(qexp, pexp, dexp, rate = 1 / 8)
) +
  Intercept(1)
dfit1 <- lgcp(
  components = cmp,
  mexdolphin$points,
  domain = list(distance = fm_mesh_1d(seq(0, 8, length.out = 30))),
  formula = formula1
)
detfun1 <- predict(dfit1, distdf, ~ hr(distance, sigma))

Plot both lines on histogram of observations. First scale lines to have same area as that of histogram.

Half-normal:

hnline <- data.frame(
  distance = detfun$distance,
  p = detfun$mean,
  lower = detfun$q0.025,
  upper = detfun$q0.975
)
wts <- diff(hnline$distance)
wts[1] <- wts[1] / 2
wts <- c(wts, wts[1])
hnarea <- sum(wts * hnline$p)
n <- length(mexdolphin$points$distance)
scale <- n / hnarea
hnline$En <- hnline$p * scale
hnline$En.lower <- hnline$lower * scale
hnline$En.upper <- hnline$upper * scale

Hazard-rate:

hrline <- data.frame(
  distance = detfun1$distance,
  p = detfun1$mean,
  lower = detfun1$q0.025,
  upper = detfun1$q0.975
)
wts <- diff(hrline$distance)
wts[1] <- wts[1] / 2
wts <- c(wts, wts[1])
hrarea <- sum(wts * hrline$p)
n <- length(mexdolphin$points$distance)
scale <- n / hrarea
hrline$En <- hrline$p * scale
hrline$En.lower <- hrline$lower * scale
hrline$En.upper <- hrline$upper * scale

Combine lines in a single object for plotting

dlines <- rbind(
  cbind(hnline, model = "Half-normal"),
  cbind(hrline, model = "Hazard-rate")
)

Plot without the 95% credible intervals

ggplot(data.frame(mexdolphin$points)) +
  geom_histogram(aes(x = distance),
    breaks = seq(0, 8, length.out = 9),
    alpha = 0.3
  ) +
  geom_point(aes(x = distance), y = 0.2, shape = "|", size = 3) +
  geom_line(
    data = dlines,
    aes(x = distance, y = En, group = model, col = model)
  )

Plot with the 95% credible intervals (without taking the count rescaling into account)

ggplot(data.frame(mexdolphin$points)) +
  geom_histogram(aes(x = distance),
    breaks = seq(0, 8, length.out = 9),
    alpha = 0.3
  ) +
  geom_point(aes(x = distance), y = 0.2, shape = "|", size = 3) +
  geom_line(
    data = dlines,
    aes(x = distance, y = En, group = model, col = model)
  ) +
  geom_ribbon(
    data = dlines,
    aes(
      x = distance,
      ymin = En.lower,
      ymax = En.upper,
      group = model,
      col = model,
      fill = model
    ),
    alpha = 0.2, lty = 2
  )

LGCPs - Distance sampling

David Borchers and Finn Lindgren