Combining temporal data for Multi-Year Baysian Disaggregation

A practical guide to pooling annual data for Bayesian disaggregation regression in R

This post is a guest post by my predoctoral fellow, Samana.

Bayesian spatial disaggregation gives you a high resolution risk surface from low-resolution case counts ….. but only one year at a time. This post shows how to pool several years into a single model fit.

Hi all, my name is Samana and I’m a predoctoral fellow and I’ve been working on the problem of a multi-year disaggregation model!

In this blog I will be presenting a recipe of how to do a multi-year disaggregation model fit.

Currently, using the disaggregation package we can only really fit one model per time slice of data we might have, but this blog aims to change that and let you fit a disag model with multiple years of data.

Part A demonstrates the mechanics without a time trend; Part B adds a time spline to the six-year model. This blog is part A.

This post shows how to combine annual disag_data objects and fit one pooled spatial disaggregation model. It demonstrates shared effects across years and it is not a full spatio-temporal model with separate spatial field for each year.

The resolution mismatch problem:

• Simply put, there is a resolution mismatch between satellite covariates (which are very high resolution) and public health data, like case counts of a particular disease in a specific county (this is low resolution).

• Public health data is often reported as annual case counts for administrative areas such as counties and districts whereas environmental information that may explain risk, such as temperature, land cover, vegetation is often available on a fine spatial grid.

• The problem is that we can’t just combine these together and then fit a model for risk predictions to get a high resolution risk surface. This is because of the ecological fallacy.

These figures above show the resolution mismatch; on the left we have the low resolution public health case counts and on the right we have the high resolution environmental satellite covariates (Nandi et al. 2023, https://doi.org/10.18637/jss.v106.i11).

What is disaggregation and why you’d want to do it:

• So these resolutions are linked by modelling a latent disease risk at the pixel level and THEN aggregating the pixel level predictions to the polygon level so they’re consistent with the observed disease counts. all of this is done by Bayesian spatial disaggregation (implemented in the R package disaggregation )

• The package fits a Bayesian disaggregation regression model (similar to a GLM) with an INLA style spatial random field via TMB, using the function prepare_data() to line up polygons, covariates and population, and disag_model() to fit the model.

• The current single year workflow is like this:

  1. Read in the inputs: a sf object containing the area boundaries and case counts, a SpatRaster of fine-scale covariates, and a population raster when modelling counts with population as exposure.

  2. Prepare these inputs into a coherent data object with prepare_data().

  3. Fit the model with disag_model(), choosing the likelihood and link and whether to include a spatial field and an independent (iid) effect.

  4. Inspect the fit with summary() and plot(), then use predict() to produce fine-scale prediction and uncertainty maps.

• For multiple years, the missing step is between 2 and 3: prepare each year separately, then combine the resulting disag_data objects carefully before fitting one pooled model.

Why multi-year disaggregation?

• When you have many years of polygon level counts, a natural question is: fit each year separately, or pool several years into one model?

• Pooling helps when you want the spatial field and covariate effects to borrow strength across years, while still letting a time trend move through a flexible term (we’ve used a natural spline on centered year in part B of this blog).

• I’ve been calling this “squishing” years together, stacking each year’s prepared data into one long object before handing it to disag_model().

• Bayesian spatial disaggregation is straight forward for one year of data but multi-year models are harder because the disag_data object contains many components that need to be in the right shape, format and size.

• This object contains polygon-level outcomes, pixel-level covariates, aggregation weights, spatial coordinates, start end indices and the spatial mesh. These objects must remain correctly aligned when the years are combined.

This tutorial develops a reproducible workflow for combining yearly disag_data objects informally, “squishing” or “combining” them before fitting a multi-year Bayesian disaggregation model.

Model components

The disaggregation regression models demonstrated here include fixed effects of covariates and a number of random effects, for various different modelling purposes.

• In this example, the pooled model estimates shared climate effects. We have different covariate observations for each year, but want to fit a model using all years of data in one go.

• The spatial random effect operates at high resolution. If you model each year separately, then each time period has its own spatial field, defined over the mesh and projected to the pixels. It captures smooth residual spatial variation that remains after accounting for measured environmental covariates. However, we often want to pool estimates for this random effect across years.

• The iid random effect operates at polygon level, the lower resolution. It gives each polygon-time observation an additional independent residual term, helping account for overdispersion and unmeasured area-level factors.

• When yearly data are squished, the model is fitted jointly across all periods. The workflow must preserve the relationships between each polygon, its pixels, its aggregation weights and its appropriate spatial field.

• The iid term allows an additional residual effect for each polygon-year observation. Stacking the data does not automatically create a separate spatial field for each year or model how fields evolve over time.

THE DATA:

The example region is Madagascar, deliberately chosen because it’s the same country used in the disaggregation package’s own founding methods paper (Nandi et al. 2020, malaria case counts across Madagascar, [Nandi et al](https://www.jstatsoft.org/article/view/v106i11)) so the mesh settings and covariate naming are the same as that original worked example.

The geometry, climate covariates AND population is real public data; only the case counts are simulated (real disease data can’t be shared here for data-permission reasons)

The dataset pairs real, public district geometry, real climate covariates and real population with simulated case counts:

• Real public administrative level-2 (district) boundaries for Madagascar, from the GADM database (https://gadm.org), downloaded via the R geodata package.names.

• Year varying covariates (temp_wc, precip_wc): annual mean temperature and annual total precipitation for each simulated year, built from REAL monthly TerraClimate records (Abatzoglou et al. 2018) not long-run climate normals, so these differ from one year to the next. Centred and scaled (mean 0, sd 1, using one mean/sd computed across all years combined so real year-to-year differences are preserved). Source: https://www.climatologylab.org/terraclimate.html.

• Real population: Gridded Population of the World (GPWv4) population density for year 2005 (closest available census year to the 2001-2006 simulation window). Source: https://sedac.ciesin.columbia.edu/data/collection/gpw-v4

• Simulated case counts drawn from known truth parameters (shipped alongside the data as true_parameters.csv), using the real covariates and population above plus a simulated spatial random field and district-level noise. So this post can check whether the fitted model recovers the true coefficients, a much better teaching device than fitting to arbitrary noise.

  

• None of the numbers in the dataset correspond to any real disease case count.

Lets get started!

Download the example data from [here](https://www.dropbox.com/scl/fo/j78acp5t6p4ncyswi8o1p/AAeCjTGqGiiT7QaVzuQJc-A?rlkey=eyim14onxbe63oewzsn9tx1k0&st=n92ocd57&dl=0).

The dropbox folder contains two raster tiff files, case counts in a csv, the true parameters used for simulation and the geography in a gpkg file.

## Packages

library(dplyr)
library(sf)
library(terra)
library(splines)
library(disaggregation)   

# Point this at wherever you downloaded/cloned the dataset described above
data_dir <- "C:/My_drive/simdata_madagascar"

Step 1 : Read the dataset

madagascar <- st_read(file.path(data_dir, "madagascar_districts.gpkg"), quiet = TRUE)

covariate_stack   <- rast(file.path(data_dir, "real_covariates.tif"))
population_raster <- rast(file.path(data_dir, "real_population.tif"))

case_counts <- read.csv(file.path(data_dir, "simulated_case_counts.csv")) |>
  mutate(district_id = as.character(district_id))

true_parameters <- read.csv(file.path(data_dir, "true_parameters.csv"))
true_parameters   # ground truth used to simulate cases - compare the

       parameter true_value
1      intercept  -10.88137
2   beta_temp_wc    0.35000
3 beta_precip_wc   -0.20000
4     beta_year1    0.40000
5     beta_year2   -0.20000
6     beta_year3    0.15000
7       field_sd    0.40000
8         iid_sd    0.15000

# fitted model against these at the end

Step 2: pick the window and build the shapes

years <- 2001:2003

shapes_2001 <- madagascar |> left_join(filter(case_counts, Year == years[1]), by = "district_id")
shapes_2002 <- madagascar |> left_join(filter(case_counts, Year == years[2]), by = "district_id")
shapes_2003 <- madagascar |> left_join(filter(case_counts, Year == years[3]), by = "district_id")

Step 3: covariate stack per year

The published covariate raster holds year-varying layers: temp_wc_2001, precip_wc_2001, …, temp_wc_2006, precip_wc_2006.

So each year below pulls out its OWN pair of layers by name instead of reusing the same values for every year.

cov_stack_2001 <- covariate_stack[[c("temp_wc_2001", "precip_wc_2001")]]
names(cov_stack_2001) <- c("temp_wc", "precip_wc")

cov_stack_2002 <- covariate_stack[[c("temp_wc_2002", "precip_wc_2002")]]
names(cov_stack_2002) <- c("temp_wc", "precip_wc")

cov_stack_2003 <- covariate_stack[[c("temp_wc_2003", "precip_wc_2003")]]
names(cov_stack_2003) <- c("temp_wc", "precip_wc")

Step 4: prepare data for each year

Mesh args here match those used in the disaggregation package’s own Madagascar malaria worked example (Nandi et al. 2020) a nice side benefit of picking the same country is that these don’t need retuning from scratch.

Always retune these to your own study area if you swap in a different region.

The details for what these parameters mean can be found in the inla tutorials for example.

mesh_args_mdg <- list(max.edge = c(0.7, 8), cutoff = 0.05, offset = c(1, 2))

dis_2001 <- prepare_data(
  polygon_shapefile  = shapes_2001,
  covariate_rasters  = cov_stack_2001,
  aggregation_raster = population_raster,
  mesh_args          = mesh_args_mdg,
  id_var             = "district_id",
  response_var       = "cases",
  na_action          = TRUE
)

Warning: [rast] CRS do not match

Warning: [extract] transforming vector data to the CRS of the raster

dis_2002 <- prepare_data(
  polygon_shapefile  = shapes_2002,
  covariate_rasters  = cov_stack_2002,
  aggregation_raster = population_raster,
  mesh_args          = mesh_args_mdg,
  id_var             = "district_id",
  response_var       = "cases",
  na_action          = TRUE
)

Warning: [rast] CRS do not match
Warning: [extract] transforming vector data to the CRS of the raster

dis_2003 <- prepare_data(
  polygon_shapefile  = shapes_2003,
  covariate_rasters  = cov_stack_2003,
  aggregation_raster = population_raster,
  mesh_args          = mesh_args_mdg,
  id_var             = "district_id",
  response_var       = "cases",
  na_action          = TRUE
)

Warning: [rast] CRS do not match
Warning: [extract] transforming vector data to the CRS of the raster

Step 5: The squish: stack the 3 years of data into one disag data object

To stack these objects together correctly, we handle different parts of the data differently. polygon_data and covariate_data are just stacked with a year label attached. We offset start_end_index: each prepare_data() call starts its pixel-row numbering again, so the second and third years’ indices must be shifted to point to their new positions in the combined covariate table.

# polygon_data and covariate_data just stack with a year label attached
poly_0103 <- bind_rows(
  transform(dis_2001$polygon_data, year = 2001),
  transform(dis_2002$polygon_data, year = 2002),
  transform(dis_2003$polygon_data, year = 2003)
)

cov_0103 <- bind_rows(
  transform(dis_2001$covariate_data, year = 2001),
  transform(dis_2002$covariate_data, year = 2002),
  transform(dis_2003$covariate_data, year = 2003)
)

# aggregation_pixels is just concatenated
agg_0103 <- c(
  dis_2001$aggregation_pixels,
  dis_2002$aggregation_pixels,
  dis_2003$aggregation_pixels
)

# start_end_index needs a running offset added, since row indices in
# covariate_data restart at 1 for every prepare_data() call
se_2001 <- as.matrix(dis_2001$start_end_index)
off_2001 <- nrow(dis_2001$covariate_data)

se_2002 <- as.matrix(dis_2002$start_end_index) + off_2001
off_2002 <- off_2001 + nrow(dis_2002$covariate_data)

se_2003 <- as.matrix(dis_2003$start_end_index) + off_2002

se_0103 <- rbind(se_2001, se_2002, se_2003)

coords_0103 <- rbind(
  dis_2001$coords_for_fit,
  dis_2002$coords_for_fit,
  dis_2003$coords_for_fit
)

# Build the pooled object, using one year's output as a template for the
# mesh and any other slots that don't need to change
dis_0103 <- dis_2001
dis_0103$polygon_data       <- poly_0103
dis_0103$covariate_data     <- cov_0103
dis_0103$aggregation_pixels <- agg_0103
dis_0103$start_end_index    <- se_0103
dis_0103$coords_for_fit     <- coords_0103

step 6: fit the pooled model

fit_0103 <- disaggregation::disag_model(
  data       = dis_0103,
  iterations = 1000,
  field      = TRUE,
  iid        = TRUE,
  family     = "poisson",
  link       = "log"
)

Fitting model. This may be slow.

summary(fit_0103)

Likelihood function: poisson 
Link function: log 
Model parameters:
                     Estimate   Std. Error
intercept         -10.7437746 6.495870e-02
temp_wc             0.4263034 3.892773e-02
precip_wc          -0.1798346 3.071823e-02
iideffect_log_tau  21.0592828 6.294932e+03
log_sigma          -2.7303956 7.641095e-01
log_rho             2.0763569 6.676009e-01

Model convergence: 0 (relative convergence (4))
Negative log likelihood:  -2265.96648299655 

In sample performance:
      RMSE      MAE   pearson  spearman log_pearson
1 3.451929 2.594429 0.9107692 0.9018133   0.9027995

true_parameters  

       parameter true_value
1      intercept  -10.88137
2   beta_temp_wc    0.35000
3 beta_precip_wc   -0.20000
4     beta_year1    0.40000
5     beta_year2   -0.20000
6     beta_year3    0.15000
7       field_sd    0.40000
8         iid_sd    0.15000

Compare that against three separate single-year fits (disag_model() on 2001, 2002, 2003 individually, no squishing) to see whether pooling changes the covariate estimates, tightens their standard errors, or shifts the spatial field. This comparison is the actual point of the exercise the code above just gets you to a model you can compare against.

fit_2001 <- disaggregation::disag_model(dis_2001, iterations = 1000, field = TRUE,
                                        iid = TRUE, family = "poisson", link = "log")

Fitting model. This may be slow.

fit_2002 <- disaggregation::disag_model(dis_2002, iterations = 1000, field = TRUE,
                                        iid = TRUE, family = "poisson", link = "log")

Fitting model. This may be slow.

fit_2003 <- disaggregation::disag_model(dis_2003, iterations = 1000, field = TRUE,
                                        iid = TRUE, family = "poisson", link = "log")

Fitting model. This may be slow.

summary(fit_2001)

Likelihood function: poisson 
Link function: log 
Model parameters:
                     Estimate   Std. Error
intercept         -10.8944795   0.09285636
temp_wc             0.3834646   0.06309135
precip_wc          -0.2368024   0.06681629
iideffect_log_tau  13.9161222 147.10593484
log_sigma          -2.5340003   0.85083992
log_rho             2.1368333   0.73794055

Model convergence: 0 (relative convergence (4))
Negative log likelihood:  -1807.57628310386 

In sample performance:
     RMSE     MAE   pearson  spearman log_pearson
1 2.64717 1.83092 0.9328089 0.9470736   0.9214865

summary(fit_2002)

Likelihood function: poisson 
Link function: log 
Model parameters:
                     Estimate   Std. Error
intercept         -10.6482023 7.651025e-02
temp_wc             0.4048466 5.784000e-02
precip_wc          -0.2217375 3.907827e-02
iideffect_log_tau  21.9405819 9.302321e+03
log_sigma          -2.8582050 9.327172e-01
log_rho             2.1126865 7.852839e-01

Model convergence: 0 (relative convergence (4))
Negative log likelihood:  -2051.7868081772 

In sample performance:
      RMSE      MAE   pearson  spearman log_pearson
1 3.136982 2.411554 0.9409313 0.9344445   0.9243826

summary(fit_2003)

Likelihood function: poisson 
Link function: log 
Model parameters:
                     Estimate   Std. Error
intercept         -10.6896813 9.838397e-02
temp_wc             0.4714723 7.021382e-02
precip_wc          -0.1614230 4.803833e-02
iideffect_log_tau  21.7370233 8.171264e+03
log_sigma          -2.3157917 9.227456e-01
log_rho             1.9333046 6.661438e-01

Model convergence: 0 (relative convergence (4))
Negative log likelihood:  -1676.17190547005 

In sample performance:
     RMSE      MAE   pearson  spearman log_pearson
1 2.89901 2.240364 0.9423111 0.9185083   0.9319642

true_parameters

       parameter true_value
1      intercept  -10.88137
2   beta_temp_wc    0.35000
3 beta_precip_wc   -0.20000
4     beta_year1    0.40000
5     beta_year2   -0.20000
6     beta_year3    0.15000
7       field_sd    0.40000
8         iid_sd    0.15000

Results:

The pooled 2001–2003 model converged successfully and recovered the simulated climate effects reasonably well.

Table: estimated versus simulated coefficients

Parameter	Simulated value	Pooled estimate (SE)	2001	2002	2003
Temperature	0.350	0.426 (0.039)	0.383 (0.063)	0.405 (0.058)	0.471 (0.070)
Precipitation	-0.200	-0.180 (0.031)	-0.237 (0.067)	-0.222 (0.039)	-0.161 (0.048)

Pooling the three years reduced uncertainty: the pooled model had smaller standard errors for both temperature and precipitation than any separate annual model. The temperature estimate was positive and the precipitation estimate negative, consistent with the values used to simulate the data.

Figure 2. Estimated associations of annual temperature and precipitation. Points show fitted coefficients and horizontal lines show approximate 95% intervals. Dashed vertical lines show the coefficients used to simulate the data. The pooled 2001–2003 model estimated both associations more precisely than models fitted separately to each year.

Refrences

• Nandi, A., Lucas, T., Arambepola, R., Python, A. (2023). disaggregation: An R Package for Bayesian Spatial Disaggregation Modeling. Journal of Statistical Software, 106(11). https://doi.org/10.18637/jss.v106.i11

• Nandi, A., Lucas, T., Arambepola, R., Gething, P., Weiss, D. (2020). disaggregation: An R Package for Bayesian Spatial Disaggregation Modelling (original methods paper, Madagascar malaria worked example). https://arxiv.org/abs/2001.04847

• disaggregation package vignette: https://cran.r project.org/web/packages/disaggregation/vignettes/disaggregation.html

• disaggregation GitHub: https://github.com/aknandi/disaggregation

• GADM database of global administrative boundaries: https://gadm.org

• Abatzoglou, J.T., Dobrowski, S.Z., Parks, S.A., Hegewisch, K.C. (2018). TerraClimate, a high-resolution global dataset of monthly climate and climatic water balance from 1958-2015. Scientific Data, 5, 170191. https://doi.org/10.1038/sdata.2017.191

• TerraClimate data access: https://www.climatologylab.org/terraclimate.html

• CIESIN (2018). Gridded Population of the World, Version 4 (GPWv4): Population Density, Revision 11. NASA SEDAC. https://sedac.ciesin.columbia.edu/data/collection/gpw-v4

• geodata R package (Hijmans et al.): https://github.com/rspatial/geodata