Abstract
Scientific analysis often relies on the ability to make accurate predictions of a system’s dynamics. Mechanistic models, parameterized by a number of unknown parameters, are often used for this purpose. Accurate estimation of the model state and parameters prior to prediction is necessary, but may be complicated by issues such as noisy data and uncertainty in parameters and initial conditions. At the other end of the spectrum exist nonparametric methods, which rely solely on data to build their predictions. While these nonparametric methods do not require a model of the system, their performance is strongly influenced by the amount and noisiness of the data. In this article, we consider a hybrid approach to modeling and prediction which merges recent advancements in nonparametric analysis with standard parametric methods. The general idea is to replace a subset of a mechanistic model’s equations with their corresponding nonparametric representations, resulting in a hybrid modeling and prediction scheme. Overall, we find that this hybrid approach allows for more robust parameter estimation and improved short-term prediction in situations where there is a large uncertainty in model parameters. We demonstrate these advantages in the classical Lorenz-63 chaotic system and in networks of Hindmarsh-Rose neurons before application to experimentally collected structured population data.
Highlights
Parametric modeling involves defining an underlying set of mechanistic equations which describe a system’s dynamics
Where x = [x1, x2, . . ., xn]T is an n-dimensional vector of model state variables and p = [p1, p2, . . ., pl]T is an l-dimensional vector of model parameters which may be known from first principles, partially known or completely unknown. f represents our system dynamics which describe the evolution of the state x over time and h is an observation function which maps x to an m-dimensional vector of model observations, y = [y1, y2, . . ., ym]T
The second type, error in the parameters, takes the form of an uncertainty in the initial parameter values used by the unscented Kalman filter (UKF) for parameter estimation
Summary
Parametric modeling involves defining an underlying set of mechanistic equations which describe a system’s dynamics These mechanistic models often contain a number of unknown parameters as well as an uncertain state, both of which need to be quantified prior to use of the model for prediction. The success of parametric prediction is tied closely to the ability to construct accurate estimates of the model parameters and state. There is often a degree of model error, or a discrepancy between the structure of the model and that of the system, further complicating the estimation process and hindering prediction accuracy. Despite these potential issues, mechanistic models are frequently utilized in data analysis. As we will see in the subsequent examples, a large uncertainty in the initial parameter values often leads to inaccurate estimates resulting in poor model-based predictions
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