This article breaks down all the options available when running mbg models. For a summary of these terms, see the documentation for the mbg::MBGModelRunner$new() method
The model always requires two terms: input_data, which includes all point observations of the outcome to be estimated, and id_raster, which lays out the study area.
Formatted as a data.frame or data.table::data.table. Should contain at least the following fields:
indicator: Contains the outcome to be modeled. For binomial or Poission MBG models, this is the numerator. For Gaussian models, this is the observed mean.samplesize: Only required for binomial or Poisson data. This is the denominator for each observations.sd: Only required for Gaussian models. This is the observed standard deviation for each observation.x: The point’s x position, in the same coordinate reference system as the ID raster. For unprojected data, this is the longitudey: The points y position, in the same coordinate reference system as the ID raster. For unprojected data, this is the latitudecluster_id: A unique identifier for each row in the dataA terra::SpatRaster object meeting the following requirements:
An ID raster can be created using the mbg::build_id_raster function.
Before running a model, you could use the terra::extract function to ensure that all points in your input data overlap a non-NA pixel in the ID raster.
The arguments inla_family, inla_link, and inverse_link give the relationship between the observed data and the linear combination of effects that make up the model. The model defaults specify a binomial likelihood:
inla_family = 'binomial'inla_link = 'logit'inverse_link = 'plogis'For binomial data, each data point with numerator \(y_i\) and denominator \(N_i\) is evaluated against a probability \(p_i\), which is governed by a logit-linear combination of model effects:
\[ y_i \sim Binomial(N_i, p_i) \\ logit(p_i) = \ ... \]
The actual model effects (\(...\)) are described in the following section.
The model currently has four effect types which can be toggled and controlled via settings passed to mbg::MbgModelRunner.
Relevant settings:
use_covariates (default TRUE)covariate_rasters (default NULL)use_stacking (default FALSE)A covariate effect will only be included if use_covariates is TRUE (the default) and covariate_rasters are passed. The covariate_rasters are an optional list of terra::SpatRaster pixel-level predictive covariates. They can be incorporated into the model in two different ways depending on the value of use_stacking:
Only applied if a covariate effect is included and use_stacking is FALSE (the default).
The covariate effect at observation \(i\) is \(\gamma^{covariates}_i = \vec{\beta}X_{s_i}\), where \(\vec{\beta}\) are linear effects on the matrix of covariate values \(X\) evaluated at the location of observation \(i\) (\(s_i\)).
Note that an intercept is not included by default. If you want a model with no covariate effects other than an intercept, pass a covariate_rasters with an intercept raster containing all 1s.
A prior is applied to the variance of effects on all covariates other than the intercept: prior_covariate_effect (default list(threshold = 3, prob_above = 0.05)) is a penalized complexity prior that can be expressed as a level of certainty about the standard deviation on each fixed effect \(\beta\). For example, the default prior corresponds to \(P(\sigma_{\beta} > 3) = 0.05\).
Only applied if a covariate effect is included and use_stacking is TRUE.
For a stacked ensemble model, the covariate effect for observation \(i\) is: \[ \gamma^{covariates}_i = \sum_{j=1}^{J}\left[ w_{j} f_j(X_{s_i}) \right] \\ Constraints: w_j > 0 \ \forall \ j, \ \textstyle \sum_{j=1}^{J}(w_j) = 1 \] Where:
Relevant model settings:
stacking_model_settings (default list(gbm = NULL, treebag = NULL, rf = NULL)): Defines the list of component models \(f_j(X)\) to be fitted to the covariates. A named list—each name corresponds to a regression model in the caret package, and each value stores optional settings that can be passed to that model.stacking_cv_settings (default list(method = 'repeatedcv', number = 5, repeats = 5)): These are used by caret::traincontrol to cross-validate each regression modelstacking_use_admin_bounds (default FALSE), admin_bounds (default NULL), admin_bounds_id (default NULL): If stacking_use_admin_bounds is TRUE and the other two values are set, adds administrative fixed effects to each of the component models.stacking_prediction_range (default NULL): Can be used to restrict the prediction range of each component regression model. For binomial data, a reasonable limit is to not predict outside of c(0, 1).If the setting use_gp is TRUE (the default), adds a spatially correlated effect:
\[ Z \sim GP(0, \Sigma_s) \] Where \(Z\) is a Gaussian process with mean zero and stationary isotropic Matern covariance over space (\(\Sigma_s\)).
The Gaussian process is informed by priors on the range and variance:
prior_spde_range (default list(threshold = 0.1, prob_below = 0.05)) Prior on the geostatistical range of the Gaussian process, the distance beyond which there is little or no spatial autocorrelation between pairs of points on the GP. This is a penalized complexity prior expressed as a named list with two items. The threshold is a distance expressed relative to the diameter of the study area: for example, the default threshold of 0.1 corresponds to a geostatistical range equivalent to one-tenth the diameter of the study area. The prob_below is the probability that the true range falls below this threshold. In other words, the default prior is \(P(range < \frac{diameter}{10}) = 0.05\)prior_spde_sigma (default list(threshold = 3, prob_above = 0.05)) Penalized complexity prior on the variance of the Gaussian process, expressed in terms of the standard deviation \(\sigma_Z\). The default corresponds to a prior belief that \(P(\sigma_Z > 3) = 0.05\)To simplify estimation, the R-INLA package represents the continuous Gaussian process on a 2D spatial mesh. Three more settings control the mesh:
mesh_max_edge (default c(0.2, 5.0)): The maximum length of a mesh edge in areas where there is little to no data. Expressed in the same units of measurement as the projection used for the id_raster (for unprojected data, this is decimal degrees). The first term is the maximum edge length within the study area, and the second term is the maximum edge length outside the study area (the mesh extends beyond the study area to mitigate edge effects).mesh_cutoff (default 0.04): The minimum length of a mesh edge in areas where data is dense. Expressed in the same units of measurement as mesh_max_edge.spde_integrate_to_zero (default FALSE): Should the volume under the fitted mesh integrate to zero?For more details about the INLA approach to approximate Gaussian process regression, see the papers at the bottom of this page.
This effect is a random intercept grouped by administrative unit. The administrative level (polygon boundaries) of interest can be set by the user. If the effect is on, then the following term is added:
\[ \gamma^{admin}_{a_i} \sim N(0, \sigma^2_{admin}) \] In other words, \(\vec\gamma^{admin}\) is an vector of random intercepts with length equal to the total number of administrative units, IID normal with mean 0 and variance \(\sigma^2_{admin}\). All observations \(i\) in the same administrative division \(a\) share the same intercept \(\gamma^{admin}_{a_i}\).
Relevant settings:
use_admin_effect (default FALSE): Should the administrative-level effect be included in the model?prior_admin_effect (default list(threshold = 3, prob_above = 0.05)): A prior applied to the administrative effect variance, expressed in terms of the standard deviation. The default settings correspond to the prior belief that \(P(\sigma_{admin} > 3) = 0.05\)admin_bounds (default NULL): Administrative bounds that will be used to group observations.admin_bounds_id (default NULL): Unique identifier field for admin_boundsThe nugget is an independently and identically distributed (IID) normal effect applied to each observation. It corresponds to “irreducible variation” not captured by any other model effect:
\[ \gamma^{Nugget}_i \sim N(0, \sigma^2_{nugget}) \]
Relevant settings:
use_nugget (default TRUE): Should the nugget effect be included in model fitting?prior_nugget (default list(threshold = 3, prob_above = 0.05)): A prior applied to the nugget variance, expressed in terms of the standard deviation. The default settings correspond to the prior belief that \(P(\sigma_{nugget} > 3) = 0.05\)nugget_in_predict (default TRUE): If TRUE, independent samples from \(N(0, \sigma^2_{nugget})\) are added to each pixel-level predictive draw.As shown in the introductory tutorial, the mbg::MbgModelRunner object can automatically aggregate predictions to administrative boundaries. The following three objects are required to perform aggregation:
aggregation_table: A table created by mbg::build_aggregation_table. Contains information about the proportional area of each pixel within each administrative boundary polygon.aggregation_levels: A named list, where each name corresponds to the administrative aggregation level, and each value is a character vector with corresponding grouping fields in the aggregation_table.population_raster: A raster with the same dimensions as id_raster that contains population estimates for each pixel. Aggregation from pixels to administrative boundaries accounts for varying pixel-level populations as well as fractional pixel areas.Finally, the setting verbose (default TRUE) governs whether the model will perform detailed logging. You can access model logs afterwards by running mbg::logging_get_timer_log.
Bakka, H., et al. (2018). Spatial modeling with R‐INLA: A review. Wiley Interdisciplinary Reviews: Computational Statistics, 10(6), e1443. https://doi.org/10.1002/wics.1443
Bhatt, S., Cameron, E., Flaxman, S. R., Weiss, D. J., Smith, D. L., & Gething, P. W. (2017). Improved prediction accuracy for disease risk mapping using Gaussian process stacked generalization. Journal of The Royal Society Interface, 14(134), 20170520. https://doi.org/10.1098/rsif.2017.0520
Freeman, M. (2017). An introduction to hierarchical modeling. http://mfviz.com/hierarchical-models/
Moraga, Paula. (2019). Geospatial Health Data: Modeling and Visualization with R-INLA and Shiny. Chapman & Hall/CRC Biostatistics Series. ISBN 9780367357955. https://www.paulamoraga.com/book-geospatial/index.html
Opitz, T. (2017). Latent Gaussian modeling and INLA: A review with focus on space-time applications. Journal de la société française de statistique, 158(3), 62-85. https://www.numdam.org/article/JSFS_2017__158_3_62_0.pdf