GEV responses

GEV responses#

In this tutorial, we illustrate how to set up a distributional regression model with the generalized extreme value distribution as a response distribution. First, we simulate some data in R:

  • The location parameter (\(\mu\)) is a function of an intercept and a non-linear covariate effect.

  • The scale parameter (\(\sigma\)) is a function of an intercept and a linear effect and uses a log-link.

  • The shape or concentration parameter (\(\xi\)) is a function of an intercept and a linear effect.

After simulating the data, we can configure the model with a single call to the rliesel::liesel() function.

library(rliesel)

Please make sure you are using a virtual or conda environment with Liesel installed, e.g. using `reticulate::use_virtualenv()` or `reticulate::use_condaenv()`. See `vignette("versions", "reticulate")`.

After setting the environment, check if the installed versions of RLiesel and Liesel are compatible with `check_liesel_version()`.

library(VGAM)

Loading required package: stats4

Loading required package: splines

set.seed(13)

n <- 1000

x0 <- runif(n)
x1 <- runif(n)
x2 <- runif(n)

y <- rgev(
  n,
  location = 0 + sin(2 * pi * x0),
  scale = exp(-3 + x1),
  shape = 0.1 + x2
)

plot(y)


model <- liesel(
  response = y,
  distribution = "GeneralizedExtremeValue",
  predictors = list(
    loc = predictor(~ s(x0)),
    scale = predictor(~ x1, inverse_link = "Exp"),
    concentration = predictor(~ x2)
  )
)

Installed Liesel version 0.2.10-dev is compatible, continuing to set up model

Now, we can continue in Python and use the lsl.dist_reg_mcmc() function to set up a sampling algorithm with IWLS kernels for the regression coefficients (\(\boldsymbol{\beta}\)) and a Gibbs kernel for the smoothing parameter (\(\tau^2\)) of the spline.

The support of the GEV distribution changes with the parameter values (compare Wikipedia). To ensure that the initial parameters support the observed data we set \(xi = 0.1\) and disable jittering of the the variance and regression parameters. For the latter, we supply user-defined jitter functions to lsl.dist_reg_mcmc that are essentially the identity function w.r.t. the parameter value.

import liesel.model as lsl
import jax.numpy as jnp

model = r.model

# concentration == 0.0 seems to break the sampler
model.vars["concentration_p0_beta"].value = jnp.array([0.1, 0.0])

builder = lsl.dist_reg_mcmc(model, seed=42, num_chains=4, tau2_jitter_fn=lambda key, val: val, beta_jitter_fn=lambda key, val: val)
builder.set_duration(warmup_duration=1000, posterior_duration=1000)

engine = builder.build()
engine.sample_all_epochs()

Some tabular summary statistics of the posterior samples:

import liesel.goose as gs

results = engine.get_results()
gs.Summary(results)

Parameter summary:

kernel mean sd q_0.05 q_0.5 q_0.95 sample_size ess_bulk ess_tail rhat
parameter index
concentration_p0_beta (0,) kernel_00 0.102 0.053 0.017 0.101 0.193 4000 371.025 840.414 1.015
(1,) kernel_00 0.972 0.099 0.811 0.975 1.132 4000 161.693 449.305 1.038
loc_np0_beta (0,) kernel_03 0.486 0.214 0.140 0.492 0.844 4000 81.248 246.038 1.037
(1,) kernel_03 -0.168 0.106 -0.343 -0.168 0.003 4000 125.877 351.505 1.028
(2,) kernel_03 -0.462 0.143 -0.684 -0.469 -0.211 4000 107.938 201.522 1.019
(3,) kernel_03 -0.022 0.068 -0.132 -0.024 0.100 4000 131.218 231.624 1.026
(4,) kernel_03 0.477 0.062 0.374 0.478 0.575 4000 118.509 253.829 1.022
(5,) kernel_03 0.457 0.027 0.410 0.456 0.500 4000 122.648 230.152 1.016
(6,) kernel_03 -5.906 0.031 -5.956 -5.905 -5.856 4000 105.971 257.752 1.031
(7,) kernel_03 -0.384 0.062 -0.482 -0.384 -0.284 4000 149.321 292.330 1.005
(8,) kernel_03 -1.788 0.026 -1.831 -1.788 -1.746 4000 103.978 245.266 1.027
loc_np0_tau2 () kernel_02 6.048 4.345 2.356 4.932 13.018 4000 3980.546 3781.071 1.001
loc_p0_beta (0,) kernel_04 -0.027 0.003 -0.031 -0.027 -0.022 4000 21.449 188.444 1.157
scale_p0_beta (0,) kernel_01 -3.100 0.061 -3.200 -3.100 -2.997 4000 42.713 245.213 1.107
(1,) kernel_01 1.201 0.080 1.070 1.201 1.335 4000 158.839 438.542 1.045

Error summary:

count relative
kernel error_code error_msg phase
kernel_00 90 nan acceptance prob warmup 88 0.022
posterior 1 0.000
kernel_01 90 nan acceptance prob warmup 45 0.011
posterior 0 0.000
kernel_03 90 nan acceptance prob warmup 27 0.007
posterior 0 0.000
kernel_04 90 nan acceptance prob warmup 22 0.005
posterior 0 0.000

And the corresponding trace plots:

fig = gs.plot_trace(results, "loc_p0_beta")


fig = gs.plot_trace(results, "loc_np0_tau2")


fig = gs.plot_trace(results, "loc_np0_beta")


fig = gs.plot_trace(results, "scale_p0_beta")


fig = gs.plot_trace(results, "concentration_p0_beta")

We need to reset the index of the summary data frame before we can transfer it to R.

summary = gs.Summary(results).to_dataframe().reset_index()

After transferring the summary data frame to R, we can process it with packages like dplyr and ggplot2. Here is a visualization of the estimated spline vs. the true function:

library(dplyr)

Attaching package: 'dplyr'

The following objects are masked from 'package:stats':

    filter, lag

The following objects are masked from 'package:base':

    intersect, setdiff, setequal, union

library(ggplot2)
library(reticulate)

summary <- py$summary

beta <- summary %>%
  filter(variable == "loc_np0_beta") %>%
  group_by(var_index) %>%
  summarize(mean = mean(mean)) %>%
  ungroup()

beta <- beta$mean
X <- py_to_r(model$vars["loc_np0_X"]$value)
estimate <- X %*% beta

true <- sin(2 * pi * x0)

ggplot(data.frame(x0 = x0, estimate = estimate, true = true)) +
  geom_line(aes(x0, estimate), color = palette()[2]) +
  geom_line(aes(x0, true), color = palette()[4]) +
  ggtitle("Estimated spline (red) vs. true function (blue)") +
  ylab("f") +
  theme_minimal()