Spatial Capture-Recapture With Penalized Regression Splines to Flexibly Model Wildlife Density and Distribution

Technical Summary: Spatial Capture-Recapture With Penalized Regression Splines

Problem Statement
Spatial capture-recapture (SCR) models are standard tools for estimating wildlife abundance and distribution, relying on a latent spatial point process to describe animal activity centers. Traditional SCR models typically assume activity centers follow a homogeneous or inhomogeneous Poisson process. This assumption implies conditional independence between animal locations given the intensity function. However, in practice, this independence is often violated due to unobserved factors such as social clustering, territoriality, or preferential habitat selection driven by unmeasured covariates.

While Log-Gaussian Cox processes (LGCPs) are commonly used in spatial statistics to model such clustering and unobserved heterogeneity, existing SCR frameworks lack methods to fit LGCPs. Furthermore, current flexible approaches within SCR, such as unpenalized regression splines, require analysts to manually specify smoothness (e.g., via knot selection and basis function counts), often leading to issues like overfitting or under-smoothing when selected via information criteria like AIC. There is a need for a framework that can simultaneously model nonlinear covariate effects and spatial dependence (clustering) without requiring manual smoothness selection, while remaining computationally feasible despite the latent nature of the activity centers.

Methodology
The authors propose a new SCR framework that integrates penalized regression splines to model the intensity function of animal activity centers. This approach allows for the use of LGCPs within SCR, effectively extending the inhomogeneous Poisson process by adding a latent Gaussian random field.

Key methodological components include:

• Model Formulation: The intensity function, $\log\{D(s)\}$, is modeled using a combination of fixed effects for spatial covariates and penalized regression splines. These splines can model smooth, nonlinear effects of covariates or smooth spatial effects over $x$- and $y$-coordinates (the latter inducing an LGCP structure).
• Likelihood Construction: Because activity centers are latent, the likelihood requires integrating over all possible locations. In standard Poisson SCR models, this integral separates into $n$ two-dimensional integrals due to independence. In the proposed LGCP framework, the dependence on a latent Gaussian field prevents this separation, resulting in a complex, high-dimensional integral.
• Approximation and Optimization: To handle the intractable integration over the latent random field, the authors employ a Laplace-approximate penalized marginal maximum likelihood approach. The Gaussian random field is approximated using basis functions, reducing the infinite-dimensional integral to a finite-dimensional one over the spline coefficients.
• Implementation: The method utilizes the TMB (Template Model Builder) R package, which leverages automatic differentiation to efficiently compute the Laplace approximation and optimize the penalized likelihood. This allows for the estimation of both fixed effects and the smoothing parameters (variance components) within a single model.

Key Contributions

1. First Implementation of Penalized Splines in SCR: This paper presents the first implementation of penalized regression splines for the point process model within SCR, enabling the fitting of LGCPs.
2. Flexible Modeling: The framework allows for the flexible modeling of nonlinear effects of spatial covariates on density and accounts for spatial clustering that cannot be explained by observed covariates.
3. Computational Feasibility: By combining the Laplace approximation with automatic differentiation via TMB, the authors provide a computationally efficient method to fit these complex models, overcoming the challenges of latent point patterns.
4. Software Availability: The authors provide R functions (specifically fit.scr.smooth()) compatible with mgcv syntax, making the method accessible to practitioners.

Results
The paper evaluates the proposed method through a simulation study and two case studies:

• Simulation Study: The authors simulated data with a known nonlinear (Type III Holling) relationship between a covariate and density.

  • Traditional log-linear models (SCR-I) and homogeneous models (SCR-H) exhibited bias, with log-linear models overpredicting at extreme covariate values and underpredicting at mid-range values.
  • The proposed model with penalized splines (SCR-S) closely recovered the true nonlinear relationship and the true intensity surface, yielding the lowest Integrated Squared Error (ISE).
  • A model using only spatial smoothing without covariates (SCR-LGCP) captured large-scale trends but struggled with fine-scale variation compared to the covariate-informed spline model.
  • Crucially, while distribution estimates varied significantly between models, population size estimates remained robust across all models, reinforcing previous findings that abundance estimates are less sensitive to density model misspecification than distribution estimates.

• Case Study 1: Black Bear Hair-Snare Data:

  • Applied to data from Louisiana, USA. The model fitted a smooth spatial effect (LGCP) to account for observed spatial structure.
  • The estimated total population size ($\hat{N} = 50$) was identical to a previous analysis using a homogeneous Poisson process, confirming the robustness of abundance estimates.
  • However, the new model provided a more appropriate inference regarding the spatial distribution of the bears, reflecting the observed clustering which the homogeneous model could not capture.

• Case Study 2: Gibbon Acoustic Survey Data:

  • Applied to data from Phnom Prich Wildlife Sanctuary, Cambodia. The model included fixed effects for forest type and a penalized spline for distance to villages, alongside a spatial smooth.
  • Unlike a standard log-linear model (SCR-I) which forced a global linear effect of distance to villages, the penalized spline (SCR-S) revealed a local effect that flattened out beyond 10 km.
  • The spatial distribution estimates differed markedly: the log-linear model predicted high densities in dense forest patches where no gibbons were detected, whereas the penalized spline model correctly identified pockets of high density and areas of virtual uninhabitation, aligning better with the observed data.

Significance and Claims
The authors claim that their method offers a "promising route" for using more flexible spatial processes to model the dependence between activity center locations. The primary significance lies in the ability to:

1. Improve Distribution Inference: While traditional approaches yield robust population size estimates even with misspecified density models, the proposed method substantially improves the estimation of spatial animal distributions, which is critical for conservation planning and management.
2. Incorporate Biological Realism: The framework allows for the incorporation of biologically realistic, nonlinear relationships (e.g., local effects of human disturbance) that global linear models cannot capture.
3. Handle Unobserved Clustering: By utilizing LGCPs, the method accounts for spatial clustering arising from unobserved covariates or social mechanisms, reducing bias and spurious significance in fixed effect estimates.

The authors note that while the method improves distribution modeling, interpreting the resulting intensity surfaces requires care, as it is difficult to distinguish between systematic spatial variation (epistemic uncertainty) and stochastic clustering (aleatoric uncertainty) from a single realization. They suggest that domain knowledge or longitudinal data is necessary for full interpretation. The paper concludes that this approach is particularly valuable for surveys spanning large, heterogeneous regions, such as camera-trap and acoustic surveys.