Practice Exercises

Extensions of Geostatistical Modelling with inlabru

Author

Olatunji Johnson

1 Introduction

These exercises are designed as follow-up practice for the workshop
“Geostatistical Modelling with inlabru”, using synthetic onchocerciasis data for Nigeria.

  • All data are in the oncho_data/ folder and documented in oncho_data/README.md.
  • You can download the onchocerciasis practice data as a zip: Onchocerciasis practice data (ZIP)
  • The data are documented in: oncho_data/README.md
  • You do not need the simulation code.
  • Many tasks are extensions of what was taught in the workshop.
  • They are marked as Extension and may require you to combine ideas, check documentation, or be creative in your modelling.

You are encouraged to:

  • Work in small groups,
  • Document your modelling choices,
  • Compare different model specifications,
  • Reflect on model assumptions and limitations.

You will use the following files:

  • Day 1:
    • spatial_oncho_binomial_data.csv
    • spatial_oncho_poisson_data.csv
    • spatiotemporal_oncho_binomial_data.csv
  • Day 2:
    • joint_oncho_processes.csv
    • nonstationary_oncho_data.csv
    • (Optionally) spatial_oncho_poisson_data.csv for multi-likelihood extensions
  • Day 3:
    • hybrid_ml_geostatistical_oncho_data.csv
    • oncho_prediction_grid_utm.csv

2 Day 1 – Spatial & Spatio-Temporal Modelling (Onchocerciasis)

2.1 1.1 Getting started with the spatial binomial data

Use: data/spatial_oncho_binomial_data.csv.

Tasks

  1. Load the data and inspect:

    • Plot locations (utm_x, utm_y) on a map (optionally overlay nga_shapefile).
    • Summarise n_examined, n_pos, and empirical prevalence.

  2. Fit a non-spatial binomial model:

    • Response: n_pos / n_examined.
    • Predictors: at least river_index, north_index, elevation, ndvi, urban, and 1–2 others of your choice (rainfall, temperature, pop_density).

  3. Inspect model residuals:

    • Map residuals.
    • Compute an empirical variogram of residuals.

  4. Fit a spatial binomial model in inlabru:

    • Use UTM coordinates to build a mesh and SPDE model.
    • Include the same fixed effects as in (2), plus a spatial random field.

  5. Compare:

    • How do coefficient estimates and uncertainties change when you add the spatial field?
    • Does the spatial field capture meaningful residual structure?

Extension A (not taught explicitly)

  1. Fit a binomial model with a non-standard link (e.g., complementary log–log) and compare:

    • Does this change the interpretation of covariate effects?
    • How do fitted probabilities change in high-risk vs low-risk areas?

2.2 1.2 Mesh design and sensitivity

Use: spatial_oncho_binomial_data.csv.

Tasks

  1. Construct two different meshes:

    • Mesh 1: coarser (fewer nodes, larger max edge).
    • Mesh 2: finer (more nodes, smaller max edge, better boundary resolution).

  2. Fit the same spatial binomial model under both meshes.

  3. Compare:

    • Posterior range and marginal variance of the spatial field.
    • Predicted prevalence surface on a grid (using oncho_prediction_grid_utm.csv).
    • Computation time and stability.

Extension B (not taught explicitly)

  1. Experiment with different PC priors for range and marginal standard deviation:

    • Make range smaller vs larger a priori.
    • Discuss how prior choices influence the spatial field and predictions.

2.3 1.3 Spatial Poisson incidence model

Use: spatial_oncho_poisson_data.csv.

Tasks

  1. Fit a Poisson spatial model:

    • Response: cases, offset: log(population).
    • Predictors: at least river_index, north_index, ndvi, rainfall, temperature, and 1–2 others.
    • Include a spatial SPDE field.

  2. Map the predicted incidence rate and compare to observed crude rates.

  3. Inspect posterior uncertainty (e.g., posterior SD) of the spatial field.

Extension C (not taught explicitly)

  1. Fit a negative binomial spatial model (if you know how to specify one in INLA/inlabru):

    • Compare fit (WAIC or similar) with the Poisson model.
    • Discuss whether overdispersion appears to matter for this dataset.

2.4 1.4 Spatio-temporal prevalence of onchocerciasis

Use: spatiotemporal_oncho_binomial_data.csv.

Tasks

  1. Explore:

    • Plot prevalence over time for a few randomly selected sites.
    • Map aggregated prevalence by time (e.g., mean per time point).

  2. Fit a separable spatio-temporal model:

    • Spatial SPDE field.
    • Temporal RW1 or RW2 effect (time index).
    • Fixed effects: a subset of the covariates (e.g., river_index, north_index, ndvi).

  3. Generate predicted prevalence surfaces for at least two time points
    (e.g., earliest and latest).

Extension D (not taught explicitly)

  1. Add a space–time interaction term:

    • For example, a group-indexed spatial field varying by a simple time grouping
      (early vs late), or another structure not covered in detail in the workshop.
    • Compare model fit and predicted dynamics to the separable model:
      • Do you see evidence that the spatial pattern changes over time?

3 Day 2 – Joint & Non-Stationary Processes (Onchocerciasis)

3.1 2.1 Joint modelling of multiple onchocerciasis processes

Use: joint_oncho_processes.csv.

Tasks

  1. Explore the data:

    • How do n_examined and n_pos differ between group levels?
    • Are there obvious spatial differences between groups?

  2. Fit a joint binomial model with:

    • Shared spatial field across groups,
    • Group-specific intercepts,
    • Common covariate effects (e.g., river_index, north_index, ndvi).

  3. Extend to a model with group-specific covariate effects for at least one key predictor
    (e.g., river_index × group).

  4. Map group-specific predicted prevalence and group-specific spatial fields (if used).

Extension E (not fully taught)

  1. Fit a model that includes a group-specific spatial field component in addition to the shared field:

    • Compare the magnitude of shared vs group-specific spatial variability.
    • Discuss when adding group-specific spatial fields is justified and when it might overfit.

3.2 2.2 Non-stationary spatial structure

Use: nonstationary_oncho_data.csv.

Tasks

  1. Explore:

    • Plot weight_north against latitude (if you convert UTM to lat/long or just inspect weight_north).
    • Map prevalence and S_nonstationary.

  2. Fit a stationary spatial model (single SPDE field):

    • Response: n_pos / n_examined,
    • Fixed effects: some covariates (e.g., river_index, north_index, ndvi),
    • One spatial field.

  3. Compare the model’s spatial residuals to S_nonstationary (roughly) and comment on where the stationary model seems to misrepresent the pattern.

Extension F (not taught explicitly)

  1. Propose and fit a non-stationary model, for example:

    • Two spatial fields with different ranges, combined with a spatially varying weight
      (e.g., function of north_index),
    • Or spatially varying coefficients for river_index (e.g., effect stronger in the north).

  2. Compare:

    • Predictions,
    • Uncertainty,
    • A model selection criterion (e.g., WAIC), between the stationary and non-stationary models.

3.3 2.3 Multi-likelihood modelling (binomial + Poisson)

Use: spatial_oncho_binomial_data.csv and spatial_oncho_poisson_data.csv.

These datasets were not simulated at identical coordinates, but you can link them approximately.

Tasks

  1. Construct an approximate joint dataset:

    • For each binomial location, find the closest Poisson location (e.g., using nearest neighbour based on UTM coordinates).
    • Create a combined data frame that records:
      • Binomial outcome (n_pos, n_examined),
      • Poisson outcome (cases, population),
      • Shared covariates (e.g., from the binomial dataset).

  2. Fit a joint model with:

    • Binomial likelihood for the prevalence data.
    • Poisson likelihood (with offset) for the count data.
    • A shared spatial field.

Extension G (not taught in detail)

  1. Compare:

    • Separate models for each outcome vs the joint model.
    • Spatial fields and covariate effects.
    • Discuss how the joint model “borrows strength” between data sources and when that might be beneficial or risky.

4 Day 3 – Hybrid Machine Learning + Geostatistics

4.1 3.1 ML-only models

Use: hybrid_ml_geostatistical_oncho_data.csv.

Tasks

  1. Split data:

    • Randomly assign ~70% of locations to a training set and the rest to a test set.

  2. Fit at least two ML models (outside inlabru), for example:

    • Random Forest,
    • XGBoost,
    • Gradient Boosting Machine, or others you know.

    Use covariates such as: elevation, ndvi, urban, rainfall, temperature, pop_density, river_index, north_index.

  3. Evaluate ML predictions on the test set:

    • Use appropriate metrics (e.g., RMSE on prevalence, Brier score, etc.).
    • Inspect variable importance (if available).

  4. Compute and map residuals (observed – predicted) on the full dataset.

Extension H (not taught explicitly)

  1. Compute a spatial variogram of residuals and assess:

    • Whether there is remaining spatial structure.
    • Which ML method leaves the least spatial structure.

4.2 3.2 Hybrid ML + spatial residuals in inlabru

Use: hybrid_ml_geostatistical_oncho_data.csv.

Tasks

  1. For your best ML model from 3.1:

    • Save the fitted values at all locations.
    • Add them to the data as a new covariate (e.g. ml_pred).

  2. Fit a hybrid model in inlabru:

    • Use ml_pred as an offset or fixed covariate.
    • Add a spatial SPDE residual field.

  3. Compare:

    • Spatial residual field from the hybrid model,
    • Residuals from the ML-only model.

  4. Map:

    • Hybrid model predictions on the oncho_prediction_grid_utm.csv grid.
    • Posterior SD of the spatial residual field.

Extension I (not taught explicitly)

  1. Try a two-stage hybrid:

    • Fit the ML model on the training set only.
    • Use the spatial residual model on top of out-of-sample ML predictions.
    • Evaluate on the test set whether the hybrid model systematically improves predictions.

4.3 3.3 Robustness and scenario experiments

Use: hybrid_ml_geostatistical_oncho_data.csv.

Tasks

  1. Create modified versions of the dataset with covariate issues:

    • Version A: randomly set 20–30% of rainfall values to NA.
    • Version B: add strong noise to ndvi in the north only (e.g., for high north_index).

  2. Refit:

    • ML-only model,
    • Hybrid model (ML + spatial),

    to each version.

  3. Compare:

    • Prediction performance (on the original uncorrupted test subset, if you keep one aside).
    • Stability of spatial fields and covariate effects.

Extension J (not taught explicitly)

  1. Write a short report (~1–2 pages) summarising:

    • When the spatial residual component helps buffer against covariate problems,
    • When it fails to compensate,
    • Implications for real-world onchocerciasis mapping in Nigeria.

5 Optional: Cross-Day Extension Projects

If you have time and interest, you can combine ideas from multiple days:

  1. Full pipeline for onchocerciasis risk mapping:

    • Start from a spatial binomial model,
    • Add non-stationarity or multiple groups,
    • Integrate ML for complex covariate effects,
    • Produce prediction and uncertainty maps for Nigeria,
    • Reflect on choices at each step.

  2. Model comparison and reporting:

    • Pick 3–4 models (e.g., stationary vs non-stationary, GLM vs spatial vs hybrid ML),
    • Compare them using a consistent set of metrics (WAIC, cross-validation, residual diagnostics),
    • Produce a short “modelling report” as if for a real onchocerciasis mapping project.