Carlo Albert

General Unified Threshold Model of Survival - a Toxicokinetic- Toxicodynamic Framework for Ecotoxicology

Toxicokinetic-toxicodynamic models (TKTD models) simulate the time-course of processes leading to toxic effects on organisms. Even for an apparently simple endpoint as survival, a large number of very different TKTD approaches exist. These differ in their underlying hypotheses and assumptions, although often the assumptions are not explicitly stated. Thus, our first objective was to illuminate the underlying assumptions (individual tolerance or stochastic death, speed of toxicodynamic damage recovery, threshold distribution) of various existing modeling approaches for survival and show how they relate to each other (e.g., critical body residue, critical target occupation, damage assessment, DEBtox survival, threshold damage). Our second objective was to develop a general unified threshold model for survival (GUTS), from which a large range of existing models can be derived as special cases. Specific assumptions to arrive at these special cases are made and explained. Finally, we illustrate how special cases of GUTS can be fitted to survival data. We envision that GUTS will help increase the application of TKTD models in ecotoxicological research as well as environmental risk assessment of chemicals. It unifies a wide range of previously unrelated approaches, clarifies their underlying assumptions, and facilitates further improvement in the modeling of survival under chemical stress.

Toxicokinetic‐toxicodynamic modeling of quantal and graded sublethal endpoints: A brief discussion of concepts

We report on the advantages and problems of using toxicokinetic-toxicodynamic (TKTD) models for the analysis, understanding, and simulation of sublethal effects. Only a few toxicodynamic approaches for sublethal effects are available. These differ in their effect mechanism and emphasis on linkages between endpoints. We discuss how the distinction between quantal and graded endpoints and the type of linkage between endpoints can guide model design and selection. Strengths and limitations of two main approaches and possible ways forward are outlined.

Model bias and complexity - Understanding the effects of structural deficits and input errors on runoff predictions

Oversimplified models and erroneous inputs play a significant role in impairing environmental predictions. To assess the contribution of these errors to model uncertainties is still challenging. Our objective is to understand the effect of model complexity on systematic modeling errors. Our method consists of formulating alternative models with increasing detail and flexibility and describing their systematic deviations by an autoregressive bias process. We test the approach in an urban catchment with five drainage models. Our results show that a single bias description produces reliable predictions for all models. The bias decreases with increasing model complexity and then stabilizes. The bias decline can be associated with reduced structural deficits, while the remaining bias is probably dominated by input errors. Combining a bias description with a multimodel comparison is an effective way to assess the influence of structural and rainfall errors on flow forecasts. We investigate how a random bias process behaves as a function of model complexity.We analyze 5 model structures to simulate a stormwater system.The reduction of systematic deviations is associated with decreasing structural deficits.In this study the remaining bias is likely to be dominated by input errors.The method provides sound probabilistic predictions in a relatively efficient way.

Modelling survival : exposure pattern, species sensitivity and uncertainty

The General Unified Threshold model for Survival (GUTS) integrates previously published toxicokinetic-toxicodynamic models and estimates survival with explicitly defined assumptions. Importantly, GUTS accounts for time-variable exposure to the stressor. We performed three studies to test the ability of GUTS to predict survival of aquatic organisms across different pesticide exposure patterns, time scales and species. Firstly, using synthetic data, we identified experimental data requirements which allow for the estimation of all parameters of the GUTS proper model. Secondly, we assessed how well GUTS, calibrated with short-term survival data of Gammarus pulex exposed to four pesticides, can forecast effects of longer-term pulsed exposures. Thirdly, we tested the ability of GUTS to estimate 14-day median effect concentrations of malathion for a range of species and use these estimates to build species sensitivity distributions for different exposure patterns. We find that GUTS adequately predicts survival across exposure patterns that vary over time. When toxicity is assessed for time-variable concentrations species may differ in their responses depending on the exposure profile. This can result in different species sensitivity rankings and safe levels. The interplay of exposure pattern and species sensitivity deserves systematic investigation in order to better understand how organisms respond to stress, including humans.

A simulated annealing approach to approximate Bayes computations

Approximate Bayes computations (ABC) are used for parameter inference when the likelihood function of the model is expensive to evaluate but relatively cheap to sample from. In particle ABC, an ensemble of particles in the product space of model outputs and parameters is propagated in such a way that its output marginal approaches a delta function at the data and its parameter marginal approaches the posterior distribution. Inspired by Simulated Annealing, we present a new class of particle algorithms for ABC, based on a sequence of Metropolis kernels, associated with a decreasing sequence of tolerances w.r.t. the data. Unlike other algorithms, our class of algorithms is not based on importance sampling. Hence, it does not suffer from a loss of effective sample size due to re-sampling. We prove convergence under a condition on the speed at which the tolerance is decreased. Furthermore, we present a scheme that adapts the tolerance and the jump distribution in parameter space according to some mean-fields of the ensemble, which preserves the statistical independence of the particles, in the limit of infinite sample size. This adaptive scheme aims at converging as close as possible to the correct result with as few system updates as possible via minimizing the entropy production of the process. The performance of this new class of algorithms is compared against two other recent algorithms on two toy examples as well as on a real-world example from genetics.

A mechanistic dynamic emulator

Abstract In applied sciences, we often deal with deterministic simulation models that are too slow for simulation-intensive tasks such as calibration or real-time control. In this paper, an emulator for a generic dynamic model, given by a system of ordinary nonlinear differential equations, is developed. The nonlinear differential equations are linearized and Gaussian white noise is added to account for the nonlinearities. The resulting linear stochastic system is conditioned on a set of solutions of the nonlinear equations that have been calculated prior to the emulation. A path-integral approach is used to derive the Gaussian distribution of the emulated solution. The solution reveals that most of the computational burden can be shifted to the conditioning phase of the emulator and the complexity of the actual emulation step only scales like O ( N n ) in multiplications of matrices of the dimension of the state space. Here, N is the number of time-points at which the solution is to be emulated and n is the number of solutions the emulator is conditioned on. The applicability of the algorithm is demonstrated with the hydrological model logSPM.

Fast mechanism-based emulator of a slow urban hydrodynamic drainage simulator

Gaussian process (GP) emulation is a data-driven method that substitutes a slow simulator with a stochastic approximation. It is then typically orders of magnitude faster than the simulator at the costs of introducing interpolation errors. Our approach, the mechanism-based GP emulator, uses knowledge of the simulator mechanisms in addition to the information gained from previous simulator runs, so called design data. In this study, we investigate how the degree of incorporating mechanisms into the design of the GP emulator influences emulation accuracy. Similarly to the previous results, we get a significant gain in accuracy already when using the simplest approximation of the mechanisms by a single linear reservoir. However, in this case, we again considerably improve emulation accuracy when using the next two approximations. This allows us to decreases the required number of design data to achieve a similar accuracy as a non-mechanistic emulator. We substitute a hydrological model with a faster mechanism-based GP emulator.We compare the gains in emulation accuracy by considering various mechanisms.Already the simplest mechanisms lead to an improvement of emulation accuracy.The emulator leads to a good accuracy already for very small design data sets.The suggested emulator can shorten, e.g., computation time for model calibration.

Emulation of dynamic simulators with application to hydrology

Many simulation-intensive tasks in the applied sciences, such as sensitivity analysis, parameter inference or real time control, are hampered by slow simulators. Emulators provide the opportunity of speeding up simulations at the cost of introducing some inaccuracy. An emulator is a fast approximation to a simulator that interpolates between design input-output pairs of the simulator. Increasing the number of design data sets is a computationally demanding way of improving the accuracy of emulation. We investigate the complementary approach of increasing emulation accuracy by including knowledge about the mechanisms of the simulator into the formulation of the emulator. To approximately reproduce the output of dynamic simulators, we consider emulators that are based on a system of linear, ordinary or partial stochastic differential equations with a noise term formulated as a Gaussian process of the parameters to be emulated. This stochastic model is then conditioned to the design data so that it mimics the behavior of the nonlinear simulator as a function of the parameters. The drift terms of the linear model are designed to provide a simplified description of the simulator as a function of its key parameters so that the required corrections by the conditioned Gaussian process noise are as small as possible. The goal of this paper is to compare the gain in accuracy of these emulators by enlarging the design data set and by varying the degree of simplification of the linear model. We apply this framework to a simulator for the shallow water equations in a channel and compare emulation accuracy for emulators based on different spatial discretization levels of the channel and for a standard non-mechanistic emulator. Our results indicate that we have a large gain in accuracy already when using the simplest mechanistic description by a single linear reservoir to formulate the drift term of the linear model. Adding some more reservoirs does not lead to a significant improvement in accuracy. However, the transition to a spatially continuous linear model leads again to a similarly large gain in accuracy as the transition from the non-mechanistic emulator to that based on one reservoir.

The effect of ambiguous prior knowledge on Bayesian model parameter inference and prediction

Environmental modeling often requires combining prior knowledge with information obtained from data. The robust Bayesian approach makes it possible to consider ambiguity in this prior knowledge. Describing such ambiguity using sets of probability distributions defined by the Density Ratio Class has important conceptual advantages over alternative robust formulations. Earlier studies showed that the Density Ratio Class is invariant under Bayesian inference and marginalization. We prove that (i) the Density Ratio Class is also invariant under propagation through deterministic models, whereas (ii)?predictions of a stochastic model with parameters defined by a Density Ratio Class are embedded in a Density Ratio Class. These invariance properties make it possible to describe sequential learning and prediction under a unified framework. We developed numerical algorithms to minimize the additional computational burden relative to the use of single priors. Practical feasibility of these methods is demonstrated by their application to a simple ecological model. Display Omitted There is often ambiguity about the choice of Bayesian prior probability distribution.The Density Ratio Class represents such ambiguity using sets of densities.We show that this class is invariant under inference, marginalization, and prediction.Such properties are conceptually satisfying and enable computational efficiency.We demonstrate concepts and new algorithms using a simple ecological model.

Appraisal of data-driven and mechanistic emulators of nonlinear simulators

System identification, sensitivity analysis, optimization and control, require a large number of model evaluations. Accurate simulators are too slow for these applications. Fast emulators provide a solution to this efficiency demand, sacrificing unneeded accuracy for speed. There are many strategies for developing emulators but selecting one remains subjective. Herein we compare the performance of two kinds of emulators: mechanistic emulators that use knowledge of the simulators equations, and purely data-driven emulators using matrix factorization. We borrow simulators from urban water management, because more stringent performance criteria on water utilities have made emulation a crucial tool within this field. Results suggest that naive data-driven emulation outperforms mechanistic emulation. We discuss scenarios in which mechanistic emulation seems favorable for extrapolation in time and dealing with sparse and unevenly sampled data. We also point to advances in Machine Learning that have not permeated yet into the environmental science community.