Unlocking ensemble ecosystem modelling for large and complex networks

This supplementary file provides additional implementation details for the Sequential Monte Carlo Ensemble Ecosystem Modelling (SMC-EEM) method described in the manuscript (Algorithm 2). It focuses on sampling ensembles of feasible and stable ecosystem models for large and complex networks. The approach builds on SMC-Approximate Bayesian Computation (SMC-ABC), adapting the implementation of Drovandi and Pettitt (2011). A key enhancement over the manuscript’s overview is the inclusion of a bijective parameter transform to ensure proposals respect original parameter constraints (e.g., uniform prior bounds) while enabling unconstrained reparameterisation during inference.

The implementation is adapted from Drovandi and Pettitt (2011, Biometrics) on SMC-ABC for parameter estimation in complex models. Additional context on reparameterisation and constraints is referenced (see Section S.4.2 of Vollert et al.). A related application on strategic model reduction using model sloppiness in coral calcification (Monsalve-Bravo, Adams, 2023, Environmental Modelling & Software) is also cited.

The SMC-EEM algorithm (Algorithm S1) and its embedded MCMC-ABC move step (Algorithm S2) are detailed as follows. Algorithm S1: SMC-EEM - Initialise: Define a discrepancy function ρ(θ). Specify a prior π(θ). Select tuning variables: number of particles M; retention fraction a (percentage of particles retained each step); desired probability c of particles remaining unmoved during MCMC-ABC; and the number of trial MCMC-ABC steps n_MCMC to gauge acceptance rate. Notes highlight three critical tuning parameters whose choices affect computation time and posterior quality; suggested values are provided in Table S1. - Sample: Generate M particles {θ_i} from π(θ). - Reweight: Evaluate discrepancies ρ(θ_i) for all particles. Sort particles by ascending ρ. Set the discrepancy threshold ε_t based on the number of particles to retain n_keep = floor(a × M). - While infeasible or unstable models remain (max(ρ) > 0): - Resample: Apply a bijective transform to map θ to an unconstrained space. Duplicate retained particles (those with lowest ρ) to replace those with ρ_i > ε_t. Compute sample covariance Σ = cov({θ_i}_{i=1}^{n_keep}). - Move (trial phase): For n_MCMC trial MCMC-ABC steps, apply Algorithm S2 to move particles; estimate the MCMC-ABC acceptance rate a_t. Determine the total number of MCMC-ABC iterations R_t = ceil(log(c) / log(1 − a_t)) and update n_MCMC = R_t / 2. - Move (remaining steps): For the remaining R_t − n_MCMC steps, apply Algorithm S2. - Reweight: Sort particles by ρ and set ε_t = ρ(θ_{n_keep}). If feasible and stable particles (ρ(θ_i) = 0) would be dropped (i > n_keep), adjust n_keep to retain all with ρ(θ_i) = 0. Transform all particles back from the unconstrained space to the original parameterisation. Algorithm S2: MCMC-ABC (within SMC-EEM) For each particle i among the retained set: - Propose θ_i* from a multivariate normal proposal: θ_i* ~ N(θ_i, Σ) in the transformed space. - Compute prior probabilities π(θ_i) and π(θ_i*) in the transformed space. - Map current and proposed points to the original parameter space, evaluate the discrepancy ρ(θ_i*). - Accept or reject using a Metropolis–Hastings step with acceptance probability α = min(1, π(θ_i*)/π(θ_i)) provided the proposal is within the discrepancy threshold (ρ(θ_i*) ≤ ε_t). Key implementation features include: a bijective transform to respect original parameter constraints while sampling in an unconstrained space; an adaptive determination of the number of MCMC-ABC iterations via the estimated acceptance rate; and resample–move steps with covariance estimated from retained particles.

This file presents algorithmic and implementation details rather than empirical results. Key contributions include: (1) incorporation of a bijective transform to enforce original parameter constraints during MCMC-ABC proposals while enabling unconstrained sampling; (2) explicit resample–move SMC-ABC scheme tailored to ensembles of feasible and stable ecosystem models, with feasibility/stability enforced via a discrepancy threshold and retaining all particles with ρ = 0; (3) adaptive calibration of the number of MCMC-ABC iterations using R_t = ceil(log(c) / log(1 − a_t)) based on trial acceptance rates; and (4) discussion of three critical tuning parameters (e.g., M, retention fraction a, probability c), including guidance and a recommendation to assess reproducibility through multiple independent runs. The tuning recommendations emphasize preserving sample diversity, potentially at increased computational cost.

By detailing SMC-EEM with a bijective parameter transform and an adaptive MCMC-ABC move step, the supplement addresses practical challenges in applying ensemble ecosystem modelling to large and complex networks. The transform guarantees proposals adhere to parameter constraints (e.g., bounded priors), while the resample–move framework efficiently focuses computation on particles with lower discrepancy. Retaining all feasible and stable models (ρ = 0) preserves the ensemble’s integrity. Collectively, these choices improve robustness and practicality of exploring ensembles of stable, feasible ecosystem models.

This S1 file provides concrete implementation guidance for SMC-EEM, including Algorithms S1 (SMC-EEM) and S2 (MCMC-ABC), a bijective transform to handle constrained parameters, and tuning strategies emphasizing diversity and reproducibility. The guidance is intended to make ensemble ecosystem modelling practical for large, complex networks. Users are advised to follow suggested tuning values (Table S1) and verify reproducibility via multiple independent runs.

The algorithm’s performance and the quality of posterior samples are sensitive to tuning choices (e.g., number of particles M, retention fraction a, and the probability c of particles remaining unmoved). The recommended values are conservative, prioritizing sample diversity over computational speed. The approach requires specification of a suitable discrepancy function and feasibility/stability criteria, which may be model- and application-specific. Computational cost can increase due to conservative tuning and adaptive MCMC-ABC steps.