Spatially-varying Coefficients Research Articles

Guidelines for the use of spatially varying coefficients in species distribution models

AbstractAimSpecies distribution models (SDMs) are increasingly applied across macroscales using detection‐nondetection data. These models typically assume that a single set of regression coefficients can adequately describe species–environment relationships and/or population trends. However, such relationships often show nonlinear and/or spatially varying patterns that arise from complex interactions with abiotic and biotic processes that operate at different scales. Spatially varying coefficient (SVC) models can readily account for variability in the effects of environmental covariates. Yet, their use in ecology is relatively scarce due to gaps in understanding the inferential benefits that SVC models can provide compared to simpler frameworks.InnovationHere we demonstrate the inferential benefits of SVC SDMs, with a particular focus on how this approach can be used to generate and test ecological hypotheses regarding the drivers of spatial variability in population trends and species–environment relationships. We illustrate the inferential benefits of SVC SDMs with simulations and two case studies: one that assesses spatially varying trends of 51 forest bird species in the eastern United States over two decades and a second that evaluates spatial variability in the effects of five decades of land cover change on grasshopper sparrow (Ammodramus savannarum) occurrence across the continental United States.Main conclusionsWe found strong support for SVC SDMs compared to simpler alternatives in both empirical case studies. Factors operating at fine spatial scales, accounted for by the SVCs, were the primary divers of spatial variability in forest bird occurrence trends. Additionally, SVCs revealed complex species–habitat relationships with grassland and cropland area for grasshopper sparrow, providing nuanced insights into how future land use change may shape its distribution. These applications display the utility of SVC SDMs to help reveal the environmental factors that drive species distributions across both local and broad scales. We conclude by discussing the potential applications of SVC SDMs in ecology and conservation.

A high order Cartesian grid, finite volume method for elliptic interface problems

We present a higher-order finite volume method for solving elliptic PDEs with jump conditions on interfaces embedded in a 2D Cartesian grid. Second, fourth, and sixth order accuracy is demonstrated on a variety of tests including problems with high-contrast and spatially varying coefficients, large discontinuities in the source term, and complex interface geometries. We include a generalized truncation error analysis based on cell-centered Taylor series expansions, which then define stencils in terms of local discrete solution data and geometric information. In the process, we develop a simple method based on Green's theorem for computing exact geometric moments directly from an implicit function definition of the embedded interface. This approach produces stencils with a simple bilinear representation, where spatially-varying coefficients and jump conditions can be easily included and finite volume conservation can be enforced.

Kernel well-posedness and computation by power series in backstepping output feedback for radially-dependent reaction–diffusion PDEs on multidimensional balls

Recently, the problem of boundary stabilization and estimation for unstable linear constant-coefficient reaction–diffusion equation on n-balls (in particular, disks and spheres) has been solved by means of the backstepping method. However, the extension of this result to spatially-varying coefficients is far from trivial. Some early success has been achieved under simplifying conditions, such as radially-varying reaction coefficients under revolution symmetry, on a disk or a sphere. These particular cases notwithstanding, the problem remains open. The main issue is that the equations become singular in the radius; when applying the backstepping method, the same type of singularity appears in the kernel equations. Traditionally, well-posedness of these equations has been proved by transforming them into integral equations and then applying the method of successive approximations. In this case, with the resulting integral equation becoming singular, successive approximations do not easily apply. This paper takes a different route and directly addresses the kernel equations via a power series approach (in the spirit of the method of Frobenius for ordinary differential equations), finding in the process the required conditions for the radially-varying reaction (namely, analyticity and evenness) and showing the existence and convergence of the series solution. This approach provides a direct numerical method that can be readily applied, despite singularities, to both control and observer boundary design problems.

Spatially varying coefficients can improve parsimony and descriptive power for species distribution models

Species distribution models (SDMs) are widely used to relate species occurrence and density to local environmental conditions, and often include a spatially correlated variable to account for spatial patterns in residuals. Ecologists have extended SDMs to include spatially varying coefficients (SVCs), where the response to a given covariate varies smoothly over space and time. However, SVCs see relatively little use perhaps because they remain less known relative to other SDM techniques. We therefore review ecological contexts where SVCs can improve the interpretability and descriptive power from SDMs, including local responses to regional indices that represent ecological teleconnections; density‐dependent habitat selection; spatially varying detectability; and context‐dependent covariate responses that represent interactions with unmeasured covariates. We then illustrate three additional examples in detail using the vector autoregressive spatio‐temporal (VAST) model. First, a spatially varying decadal trends model identifies decadal trends for arrowtooth flounder Atheresthes stomias density in the Bering Sea from 1982 to 2019. Second, a trait‐based joint SDM highlights the role of body size and temperature in spatial community assembly in the Gulf of Alaska. Third, an age‐structured SDM for walleye pollock Gadus chalcogrammus in the Bering Sea contrasts cohorts with broad spatial distributions (1996 and 2009) and those that are more spatially constrained (2002 and 2015). We conclude that SVCs extend SDMs to address a wide variety of ecological contexts and can be used to better understand a range of ecological processes, e.g. density dependence, community assembly and population dynamics.

A combined statistical and machine learning approach for spatial prediction of extreme wildfire frequencies and sizes

Motivated by the Extreme Value Analysis 2021 (EVA 2021) data challenge we propose a method based on statistics and machine learning for the spatial prediction of extreme wildfire frequencies and sizes. This method is tailored to handle large datasets, including missing observations. Our approach relies on a four-stage high-dimensional bivariate sparse spatial model for zero-inflated data, which is developed using stochastic partial differential equations(SPDE). In Stage 1, the observations are categorized in zero/nonzero categories and are modeled using a two-layered hierarchical Bayesian sparse spatial model to estimate the probabilities of these two categories. In Stage 2, before modeling the positive observations using spatially-varying coefficients, smoothed parameter surfaces are obtained from empirical estimates using fixed rank kriging. This approximate Bayesian method inference was employed to avoid the high computational burden of large spatial data modeling using spatially-varying coefficients. In Stage 3, the standardized log-transformed positive observations from the second stage are further modeled using a sparse bivariate spatial Gaussian process. The Gaussian distribution assumption for wildfire counts developed in the third stage is computationally effective but erroneous. Thus in Stage 4, the predicted values are rectified using Random Forests. The posterior inference is drawn for Stages 1 and 3 using Markov chain Monte Carlo (MCMC) sampling. A cross-validation scheme is then created for the artificially generated gaps, and the EVA 2021 prediction scores of the proposed model are compared to those obtained using certain natural competitors.

Nearly Ds-optimal assigned location design for a linear model with spatially varying coefficients

Spatially varying coefficient (SVC) models are an increasingly popular approach for modeling spatial heterogeneity. One topic that has not been well addressed is how best to design experiments when the data are to be used to estimate a SVC model. The applied problem that motivated this study is that agronomists have begun conducting whole-field experiments on farmers’ fields as an alternative to small-scale experiments on experiment stations. The goal is to guide the precision application of nutrients, such as nitrogen fertilizer. The research reported here seeks to optimally allocate treatments to the farm’s plots by leveraging information from model designs. Nearly Ds-Optimal allocation designs are obtained for an experiment that provides data for estimating the parameters of a linear SVC model. This nearly optimal design is far more informative than standard designs such as Latin square, simple random allocation, and randomized strip-plot designs; all of which could also be used to generate data for SVC models. Furthermore, the suggested method does not need a regular plot shape for the experimental design, which is necessary for Latin square or strip plot designs.

Spatio-Temporal Variability of the Impact of Population Mobility on Local Business Sales in Response to COVID-19 in Seoul, Korea

Social distancing is an effective method for controlling the COVID-19 pandemic by decreasing population mobility, but it has also negatively affected local business sales. This paper explores the spatio-temporal impact of population mobility on local business sales in response to COVID-19 in Seoul, South Korea. First, this study examined the temporal variability by analyzing statistical interaction terms in linear regression models. Second, the spatio-temporal variability was captured using Moran eigenvector spatial filtering (MESF)-based spatially varying coefficients (SVC) models with additional statistical interaction terms. Population mobility and local business sales were estimated from public transportation ridership and restaurant sales, respectively, which were both obtained from spatial big datasets. The analysis results show the existence of various relationships between changes in the population mobility and local business sales according to the corresponding period and region. This study confirms the usability of spatial big datasets and spatio-temporal varying coefficients models for COVID-19 studies and provides support for policy-makers in response to infectious disease.

Europe-wide air pollution modeling from 2000 to 2019 using geographically weighted regression.

Previous European land-use regression (LUR) models assumed fixed linear relationships between air pollution concentrations and predictors such as traffic and land use. We evaluated whether including spatially-varying relationships could improve European LUR models by using geographically weighted regression (GWR) and random forest (RF). We built separate LUR models for each year from 2000 to 2019 for NO2, O3, PM2.5 and PM10 using annual average monitoring observations across Europe. Potential predictors included satellite retrievals, chemical transport model estimates and land-use variables. Supervised linear regression (SLR) was used to select predictors, and then GWR estimated the potentially spatially-varying coefficients. We developed multi-year models using geographically and temporally weighted regression (GTWR). Five-fold cross-validation per year showed that GWR and GTWR explained similar spatial variations in annual average concentrations (average R2=NO2: 0.66; O3: 0.58; PM10: 0.62; PM2.5: 0.77), which are better than SLR (average R2=NO2: 0.61; O3: 0.46; PM10: 0.51; PM2.5: 0.75) and RF (average R2=NO2: 0.64; O3: 0.53; PM10: 0.56; PM2.5: 0.67). The GTWR predictions and a previously-used method of back-extrapolating 2010 model predictions using CTM were overall highly correlated (R2>0.8) for all pollutants. Including spatially-varying relationships using GWR modestly improved European air pollution annual LUR models, allowing time-varying exposure-health risk models.

Measuring spatial nonstationary effects of POI-based mixed use on urban vibrancy using Bayesian spatially varying coefficients model

Understanding the relationship between mixed land use and urban vibrancy is vital in advanced urban planning applications. This study presents a Bayesian spatially varying coefficient (SVC) model to explore the spatially nonstationary relationship between mixed land use and urban vibrancy after controlling for other factors. We first use the convolutional conditional autoregressive prior to accommodate the ecological bias resulting from unobserved confounders. Then we develop our approach in the case of a single predictor to allow the spatially varying coefficient process. We further introduce a type of the Bayesian SVC model that considers the stratified heterogeneity of the outcome, allowing the coefficients to simultaneously vary at the local and subregion level. We illustrate the proposed model by conducting a case study in Shenzhen using mobile phone data, an officially registered point-of-interest (POI) dataset, and several supplementary datasets. The model evaluation results show that including spatially unstructured and structured component combinations can improve the model's fitness and predictive ability; additionally, considering spatial stratified heterogeneity can further enhance the model's performance. Our findings provide an alternative for measuring the variable local-scale association between mixed-use and urban vibrancy and offer new insights that broaden the fields of environmental science and spatial statistics.

Exploring spatial nonstationarity for four mammal species reveals regional variation in environmental relationships

AbstractBroad‐scale ecological research on species distributions commonly presumes that the correlative relationships discovered are stationary over space. This is an assumption of most species distribution models (SDMs) that combine observations of species occurrence with environmental characteristics to understand current ecological correlates and to predict distributions based on those relationships. However, ecological relationships may vary spatially because of changes in the environment (i.e., resource availability) or the organism itself (i.e., local adaptation). Discovering this within‐species variation typically requires dense datasets over large geographic areas, which are now being provided by the recent proliferation of open‐access biodiversity occurrence records. Using nearly 4000 sampling locations from an open‐access, state‐wide camera‐trapping project, we explore the space‐varying effects of covariates on the distribution of four mammal species at two scales: region‐specific and fine resolution, with the latter estimated using spatially varying coefficients (SVC) models, to understand the scale of spatial variation in ecological relationships. Among the four species tested, the ecological relationships for two were best explained with the regional models, equivocal results for one species, while the SVC model had superior fit and predictive performance for the final species (white‐tailed deer, Odocoileus virginianus). Spatial nonstationarity was more common in relationships with landscape composition characteristics, such as housing density, than in landscape configuration metrics, such as patch richness density. One of the most appealing results of an SVC approach is not only the improved predictions across large landscapes but also understanding how animals are responding to the environment differently at the management unit level. For example, we found that deer's spatially varying relationship with forest cover was best explained by an interactive effect of deer management units (i.e., differences in deer populations) and predator pressure. These findings lead to a new hypothesis about how deer may be differentially using forested environments across space and could be a promising area of future research. Given sufficient data, accounting for nonstationarity in SDMs can show large‐scale ecological patterns while also detecting local level changes in animal ecology in areas small enough that management or protection can be readily implemented.

Boundary Control of Coupled Wave Systems with Spatially-Varying Coefficients

Boundary Control of Coupled Wave Systems with Spatially-Varying Coefficients

A zero-inflated mixture spatially varying coefficient modeling of cholera incidences

Spatial disease modeling remains an important public health tool. For cholera, the presence of zero counts is common. The Poisson model is inadequate to (1) capture over-dispersion, and (2) distinguish between excess zeros arising from non-susceptible and susceptible populations. In this study, we develop zero-inflated (ZI) mixture spatially varying coefficient (SVC) models to (1) distinguish between the sources of the excess zeros and (2) uncover the spatially varying effects of precipitation and temperature (LST) on cholera. We demonstrate the potential of the models using cholera data from Ghana. A striking observation is that the Poisson model outperformed the ZI mixture models in terms of fit. The ZI Negative Binomial (ZINB) outperformed the ZI Poisson (ZIP) model. Subject to our objectives, we make inferences using the ZINB model. The proportion of zeros estimated with the ZINB model is 0.41 and exceeded what would have been estimated using a Poisson model which is 0.35. We observed the spatial trends of the effects of precipitation and LST to have both increasing and decreasing gradients; an observation implying that the use of only the global coefficients would lead to wrong inferences. We conclude that (1) the use of ZI mixture models has epidemiological significance. Therefore, its choice over the Poisson model should be based on an epidemiological concept rather than model fit and, (2) the extension of ZI mixture models to accommodate spatially varying coefficients uncovered remarkable varying effects of the covariates. These findings have significant implications for public health monitoring of cholera.

Examining the vintage effect in hedonic pricing using spatially varying coefficients models: a case study of single-family houses in the Canton of Zurich

This article examines the spatially varying effect of age on single-family house (SFH) prices. Age has been shown to be a key driver for house depreciation and is usually associated with a negative price effect. In practice, however, there exist deviations from this behavior which are referred to as vintage effects. We estimate a spatially varying coefficients (SVC) model to investigate the spatial structures of vintage effects on SFH pricing. For SFHs in the Canton of Zurich, Switzerland, we find substantial spatial variation in the age effect. In particular, we find a local, strong vintage effect primarily in urban areas compared to pure depreciative age effects in rural locations. Using cross validation, we assess the potential improvement in predictive performance by incorporating spatially varying vintage effects in hedonic models. We find a substantial improvement in out-of-sample predictive performance of SVC models over classical spatial hedonic models.

Balancing Spatial and Non‐Spatial Variation in Varying Coefficient Modeling: A Remedy for Spurious Correlation

This study discusses the importance of balancing spatial and non‐spatial variation in spatial regression modeling. Unlike spatially varying coefficients (SVC) modeling, which is popular in spatial statistics, non‐spatially varying coefficients (NVC) modeling has largely been unexplored in spatial fields. Nevertheless, as we will explain, consideration of non‐spatial variation is needed not only to improve model accuracy but also to reduce spurious correlation among varying coefficients, which is a major problem in SVC modeling. We consider a Moran eigenvector approach modeling spatially and non‐spatially varying coefficients (S&NVC). A Monte Carlo simulation experiment comparing our S&NVC model with existing SVC models suggests both modeling accuracy and computational efficiency for our approach. Beyond that, somewhat surprisingly, our approach identifies true and spurious correlations among coefficients nearly perfectly, even when usual SVC models suffer from severe spurious correlations. It implies that S&NVC model should be used even when the analysis purpose is modeling SVCs. Finally, our S&NVC model is employed to analyze a residential land price data set. Its results suggest existence of both spatial and non‐spatial variation in regression coefficients in practice. The S&NVC model is now implemented in the R package spmoran.

Output Feedback Control of Coupled Linear Parabolic ODE–PDE–ODE Systems

This article deals with the backstepping design of observer-based compensators for parabolic ODE–PDE–ODE systems. The latter consist of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\boldsymbol {n}$</tex-math></inline-formula> coupled parabolic PDEs with distinct diffusion coefficients and spatially-varying coefficients, that are bidirectionally coupled to ODEs at both boundaries. The actuation and sensing appears through these ODEs resulting in a challenging control problem. For this setup a systematic backstepping approach is proposed, in order to determine a state feedback controller and an observer. In particular, the state feedback loop and the observer error dynamics are mapped into stable ODE–PDE–ODE cascades by making use of a sequence of transformations. With this, the design can be traced back to the solution of kernel equations already found in the literature as well as initial and boundary value problems, that can be solved numerically. Exponential stability of the closed-loop system is verified, wherein the decay rate can be directly specified in the design. The results of the article are illustrated by the output feedback control of an unstable ODE–PDE–ODE system with two coupled parabolic PDEs.

Adaptive event-triggered PDE control for load-moving cable systems

Motivated by lateral vibration suppression of a mining cable elevator, which is a load-moving cable system, where the load moves along a viscoelastic guideway whose stiffness and damping coefficients are unknown, we present event-triggered adaptive output-feedback boundary control design of a hyperbolic PDE–ODE coupled system using the measurements at the PDE actuated boundary and the ODE, where the PDE subsystem is a class of 2 × 2 coupled hyperbolic PDEs with spatially-varying coefficients and on a time-varying domain, and a high uncertainty exist in the system matrix of the ODE subsystem at the uncontrolled boundary of the PDE. A continuous-in-time observer-based adaptive backstepping control law is designed where the control gains can be self-tuned to adjust the system matrix of the ODE into a given target system matrix, based on which an observer-based dynamic event-triggering mechanism is built and the existence of a minimal dwell-time is proved. The asymptotic stability of the overall adaptive event-based output-feedback closed-loop system is proved via Lyapunov analysis. In numerical simulation, the performance of the proposed controller is verified in lateral vibration suppression of a mining cable elevator.

Modelling context-specific relationships between neighbourhood socioeconomic disadvantage and private car use

Car use generates negative externalities, which are responsible for many health, environmental and economic problems. To tackle this issue, more work is needed to identify better the correlates of car use, especially at the contextual level. In this study, a mobility-focused questionnaire involving 1722 working French adults living in the Paris area (France) was used to explore gender-stratified relationships between residential socioeconomic deprivation and car use as the main transport mode against public transport, after controlling for potential confounders. While the vast majority of similar studies have assumed linear and global statistical relationships, the present work involved a random slope hierarchical generalised additive modelling framework, which revealed both non-linear and territorially-varying relationships. Among women, living in a more deprived neighbourhood was associated with an increase in the odds of reporting car use up to a certain threshold, after which the relationship plateaued, while among men, this relationship is linearly negative. In the most deprived department of the Paris region (Seine-Saint-Denis), living in a more deprived neighbourhood was associated with a lower odds of car use among men while a more complex nonlinear bell-shaped relationship was observed among women. The opposite was found in the wealthiest department (Yvelines), with a negative relationship among women and a U-shaped one among men. In Paris inner city, again a strong opposite trend was distinguished according to sex, with a negative relationship among women. These findings suggest that spatial contexts, characterised by complex interactions between socioeconomic factors, the built environment and the distance to Paris, play the role of moderators in the relationship between residential deprivation and car use. In conclusion, this study reinforces the idea that environment-transport relationships should be understood through local analyses (e.g. random slope multilevel or spatially-varying coefficients models) rather than global ones only, in order to guide specific public policies more effectively.

Backstepping control of ODE/wave PDE cascaded systems with dynamic boundary conditions

We consider backstepping control for ODE/wave PDE cascaded systems with dynamic boundary conditions. The ODE state is driven by the uncontrolled boundary of the wave PDE. A backstepping compensator is designed such that the closed-loop system is globally asymptotically stable. The controller design and stability analysis are based on coordinate and backstepping transformations. Changes of coordinate recast the original system into a coupled first-order hyperbolic system with spatially-varying coefficients cascading into an ODE. Backstepping transformations are exploited to design a compensator for the resulting coupled system. The validity of the proposed design is illustrated by controlling a bit of the oil well drilling system to operate at a constant uniform rotary speed.

A backstepping approach to output regulation for coupled linear wave–ODE systems

This paper is concerned with the output regulation problem for coupled wave–ODE systems subject to spatially-varying coefficients. This problem is traced back to the regulator design for a coupled 2 × 2 hyperbolic PIDE–ODE system, for which a new and systematic backstepping design procedure is presented. The related state and disturbance estimation problem are solved by determining a natural disturbance observer of second order. The corresponding observer design relies on the backstepping stabilization of a coupled 2 × 2 hyperbolic ODE–PIDE–ODE system by dynamic output injection control. With this, a constructive solution of the output regulation problem is presented for coupled wave–ODE systems with disturbances acting in-domain, at a boundary or at the ODE. The output to be controlled can depend both on the distributed and lumped states, while in the former case in-domain and boundary outputs are allowed. A large class of exogenous signals is taken into account by considering a non-diagonalizable signal model. The new regulator design method is illustrated for a heavy chain system with a load of forth order.

Observer of Coupled Hyperbolic PDEs for Deep-sea Construction Vessel Vibrations

Motivated by vibration state estimation of a deep-sea construction vessel used to install oil drilling equipment on the seafloor, this paper presents state observer design of a 4 × 4 coupled heterodirectional hyperbolic PDE-ODE system, characterized by spatially-varying coefficients and a time-varying domain. The exponential stability of the observer error system is proved via Lyapunov analysis. Effective estimation of lateral-longitudinal coupled vibration states of the deep-sea construction vessel is verified in numerical simulation.