29 research outputs found
Constructing Priors that Penalize the Complexity of Gaussian Random Fields
Priors are important for achieving proper posteriors with physically
meaningful covariance structures for Gaussian random fields (GRFs) since the
likelihood typically only provides limited information about the covariance
structure under in-fill asymptotics. We extend the recent Penalised Complexity
prior framework and develop a principled joint prior for the range and the
marginal variance of one-dimensional, two-dimensional and three-dimensional
Mat\'ern GRFs with fixed smoothness. The prior is weakly informative and
penalises complexity by shrinking the range towards infinity and the marginal
variance towards zero. We propose guidelines for selecting the hyperparameters,
and a simulation study shows that the new prior provides a principled
alternative to reference priors that can leverage prior knowledge to achieve
shorter credible intervals while maintaining good coverage.
We extend the prior to a non-stationary GRF parametrized through local ranges
and marginal standard deviations, and introduce a scheme for selecting the
hyperparameters based on the coverage of the parameters when fitting simulated
stationary data. The approach is applied to a dataset of annual precipitation
in southern Norway and the scheme for selecting the hyperparameters leads to
concervative estimates of non-stationarity and improved predictive performance
over the stationary model
Spatial Aggregation with Respect to a Population Distribution
Spatial aggregation with respect to a population distribution involves
estimating aggregate quantities for a population based on an observation of
individuals in a subpopulation. In this context, a geostatistical workflow must
account for three major sources of `aggregation error': aggregation weights,
fine scale variation, and finite population variation. However, common practice
is to treat the unknown population distribution as a known population density
and ignore empirical variability in outcomes. We improve common practice by
introducing a `sampling frame model' that allows aggregation models to account
for the three sources of aggregation error simply and transparently.
We compare the proposed and the traditional approach using two simulation
studies that mimic neonatal mortality rate (NMR) data from the 2014 Kenya
Demographic and Health Survey (KDHS2014). For the traditional approach,
undercoverage/overcoverage depends arbitrarily on the aggregation grid
resolution, while the new approach exhibits low sensitivity. The differences
between the two aggregation approaches increase as the population of an area
decreases. The differences are substantial at the second administrative level
and finer, but also at the first administrative level for some population
quantities. We find differences between the proposed and traditional approach
are consistent with those we observe in an application to NMR data from the
KDHS2014.Comment: main manuscript: 33 pages, 5 figures, 5 tables; supplemental
materials: 15 pages, 2 figures, 15 table
Exploring a New Class of Non-stationary Spatial Gaussian Random Fields with Varying Local Anisotropy
Gaussian random fields (GRFs) constitute an important part of spatial
modelling, but can be computationally infeasible for general covariance
structures. An efficient approach is to specify GRFs via stochastic partial
differential equations (SPDEs) and derive Gaussian Markov random field (GMRF)
approximations of the solutions. We consider the construction of a class of
non-stationary GRFs with varying local anisotropy, where the local anisotropy
is introduced by allowing the coefficients in the SPDE to vary with position.
This is done by using a form of diffusion equation driven by Gaussian white
noise with a spatially varying diffusion matrix. This allows for the
introduction of parameters that control the GRF by parametrizing the diffusion
matrix. These parameters and the GRF may be considered to be part of a
hierarchical model and the parameters estimated in a Bayesian framework. The
results show that the use of an SPDE with non-constant coefficients is a
promising way of creating non-stationary spatial GMRFs that allow for physical
interpretability of the parameters, although there are several remaining
challenges that would need to be solved before these models can be put to
general practical use
Impact of Jittering on Raster- and Distance-based Geostatistical Analyses of DHS Data
Fine-scale covariate rasters are routinely used in geostatistical models for
mapping demographic and health indicators based on household surveys from the
Demographic and Health Surveys (DHS) program. However, the geostatistical
analyses ignore the fact that GPS coordinates in DHS surveys are jittered for
privacy purposes. We demonstrate the need to account for this jittering, and we
propose a computationally efficient approach that can be routinely applied. We
use the new method to analyse the prevalence of completion of secondary
education for 20--49 year old women in Nigeria in 2018 based on the 2018 DHS
survey. The analysis demonstrates substantial changes in the estimates of
spatial range and fixed effects compared to when we ignore jittering. Through a
simulation study that mimics the dataset, we demonstrate that accounting for
jittering reduces attenuation in the estimated coefficients for covariates and
improves predictions. The results also show that the common approach of
averaging covariate values in windows around the observed locations does not
lead to the same improvements as accounting for jittering
Estimating Under Five Mortality in Space and Time in a Developing World Context
Accurate estimates of the under-5 mortality rate (U5MR) in a developing world
context are a key barometer of the health of a nation. This paper describes new
models to analyze survey data on mortality in this context. We are interested
in both spatial and temporal description, that is, wishing to estimate U5MR
across regions and years, and to investigate the association between the U5MR
and spatially-varying covariate surfaces. We illustrate the methodology by
producing yearly estimates for subnational areas in Kenya over the period 1980
- 2014 using data from demographic health surveys (DHS). We use a binomial
likelihood with fixed effects for the urban/rural stratification to account for
the complex survey design. We carry out smoothing using Bayesian hierarchical
models with continuous spatial and temporally discrete components. A key
component of the model is an offset to adjust for bias due to the effects of
HIV epidemics. Substantively, there has been a sharp decline in U5MR in the
period 1980 - 2014, but large variability in estimated subnational rates
remains. A priority for future research is understanding this variability.
Temperature, precipitation and a measure of malaria infection prevalence were
candidates for inclusion in the covariate model.Comment: 36 pages, 11 figure
Does non-stationary spatial data always require non-stationary random fields?
A stationary spatial model is an idealization and we expect that the true
dependence structures of physical phenomena are spatially varying, but how
should we handle this non-stationarity in practice? We study the challenges
involved in applying a flexible non-stationary model to a dataset of annual
precipitation in the conterminous US, where exploratory data analysis shows
strong evidence of a non-stationary covariance structure.
The aim of this paper is to investigate the modelling pipeline once
non-stationarity has been detected in spatial data. We show that there is a
real danger of over-fitting the model and that careful modelling is necessary
in order to properly account for varying second-order structure. In fact, the
example shows that sometimes non-stationary Gaussian random fields are not
necessary to model non-stationary spatial data.Comment: Minor change from previous version. arXiv admin note: text overlap
with arXiv:1306.040