A probabilistic treatment of uncertainty in nonlinear dynamical systems

In this work, computationally efficient approximate methods are developed for analyzing uncertain dynamical systems. Uncertainties in both the excitation and the modeling are considered and examples are presented illustrating the accuracy of the proposed approximations.

For nonlinear systems under uncertain excitation, methods are developed to approximate the stationary probability density function and statistical quantities of interest. The methods are based on approximating solutions to the Fokker-Planck equation for the system and differ from traditional methods in which approximate solutions to stochastic differential equations are found. The new methods require little computational effort and examples are presented for which the accuracy of the proposed approximations compare favorably to results obtained by existing methods. The most significant improvements are made in approximating quantities related to the extreme values of the response, such as expected outcrossing rates, which are crucial for evaluating the reliability of the system.

Laplace's method of asymptotic approximation is applied to approximate the probability integrals which arise when analyzing systems with modeling uncertainty. The asymptotic approximation reduces the problem of evaluating a multidimensional integral to solving a minimization problem and the results become asymptotically exact as the uncertainty in the modeling goes to zero. The method is found to provide good approximations for the moments and outcrossing rates for systems with uncertain parameters under stochastic excitation, even when there is a large amount of uncertainty in the parameters. The method is also applied to classical reliability integrals, providing approximations in both the transformed (independently, normally distributed) variables and the original variables. In the transformed variables, the asymptotic approximation yields a very simple formula for approximating the value of SORM integrals. In many cases, it may be computationally expensive to transform the variables, and an approximation is also developed in the original variables. Examples are presented illustrating the accuracy of the approximations and results are compared with existing approximations.

A meta-analytic approach to quantifying scientific uncertainty in stock assessments

Quantifying scientific uncertainty when setting total allowable catch limits for fish stocks is a major challenge, but it is a requirement in the United States since changes to national fisheries legislation. Multiple sources of error are readily identifiable, including estimation error, model specification error, forecast error, and errors associated with the definition and estimation of reference points. Our focus here, however, is to quantify the influence of estimation error and model specification error on assessment outcomes. These are fundamental sources of uncertainty in developing scientific advice concerning appropriate catch levels and although a study of these two factors may not be inclusive, it is feasible with available information. For data-rich stock assessments conducted on the U.S. west coast we report approximate coefficients of variation in terminal biomass estimates from assessments based on inversion of the assessment of the model’s Hessian matrix (i.e., the asymptotic standard error). To summarize variation “among” stock assessments, as a proxy for model specification error, we characterize variation among multiple historical assessments of the same stock. Results indicate that for 17 groundfish and coastal pelagic species, the mean coefficient of variation of terminal biomass is 18%. In contrast, the coefficient of variation ascribable to model specification error (i.e., pooled among-assessment variation) is 37%. We show that if a precautionary probability of overfishing equal to 0.40 is adopted by managers, and only model specification error is considered, a 9% reduction in the overfishing catch level is indicated.

Use of genetic data to assess the uncertainty in stock assessments due to the assumed stock structure: the case of albacore (Thunnus alalunga) from the Atlantic Ocean

Stock assessments can be problematic because of uncertainties associated with the data or because of simplified assumptions made when modeling biological processes (Rosenberg and Restrepo, 1995). For example, the common assumption in stock assessments that stocks are homogeneous and discrete (i.e., there is no migration between the stocks) is not necessarily true (Kell et al., 2004a, 2004b).

Near optimal combination of sensory and motor uncertainty in time during a naturalistic perception-action task.

Most behavioral tasks have time constraints for successful completion, such as catching a ball in flight. Many of these tasks require trading off the time allocated to perception and action, especially when only one of the two is possible at any time. In general, the longer we perceive, the smaller the uncertainty in perceptual estimates. However, a longer perception phase leaves less time for action, which results in less precise movements. Here we examine subjects catching a virtual ball. Critically, as soon as subjects began to move, the ball became invisible. We study how subjects trade-off sensory and movement uncertainty by deciding when to initiate their actions. We formulate this task in a probabilistic framework and show that subjects' decisions when to start moving are statistically near optimal given their individual sensory and motor uncertainties. Moreover, we accurately predict individual subject's task performance. Thus we show that subjects in a natural task are quantitatively aware of how sensory and motor variability depend on time and act so as to minimize overall task variability.

Representations of uncertainty in sensorimotor control.

Uncertainty is ubiquitous in our sensorimotor interactions, arising from factors such as sensory and motor noise and ambiguity about the environment. Setting it apart from previous theories, a quintessential property of the Bayesian framework for making inference about the state of world so as to select actions, is the requirement to represent the uncertainty associated with inferences in the form of probability distributions. In the context of sensorimotor control and learning, the Bayesian framework suggests that to respond optimally to environmental stimuli the central nervous system needs to construct estimates of the sensorimotor transformations, in the form of internal models, as well as represent the structure of the uncertainty in the inputs, outputs and in the transformations themselves. Here we review Bayesian inference and learning models that have been successful in demonstrating the sensitivity of the sensorimotor system to different forms of uncertainty as well as recent studies aimed at characterizing the representation of the uncertainty at different computational levels.

Handling uncertainty in housing stock models

Housing stock models can be useful tools in helping to assess the environmental and socio-economic impacts of retrofits to residential buildings; however, existing housing stock models are not able to quantify the uncertainties that arise in the modelling process from various sources, thus limiting the role that they can play in helping decision makers. This paper examines the different sources of uncertainty involved in housing stock models and proposes a framework for handling these uncertainties. This framework involves integrating probabilistic sensitivity analysis with a Bayesian calibration process in order to quantify uncertain parameters more accurately. The proposed framework is tested on a case study building, and suggestions are made on how to expand the framework for retrofit analysis at an urban-scale. © 2011 Elsevier Ltd.