# Uncertainty Quantification for a Dynamical Model#

This article is part of a series: Statistical Inference for Dynamical Models. The following aspects are covered

Model framework

Statistical inference and coping with uncertainty

Dynamical models encompass a range of objects that are used to describe real-world phenomena in terms of mathematical equations. The term “dynamical” refers to the fact that these systems evolve over time. The mathematical representation simplifies a system such that its essential characteristics can be inferred. Often, a model is parameterized by a set of unknown parameters. These parameters can be estimated statistically based on measured observations. The results may provide valuable insights about underlying processes. Moreover, an estimated model can be used to predict unobserved outcomes.

Dynamical models are applied in various fields, such as mathematics, physics, biology, chemistry, economics and engineering. This article series originates from modelling the progression of cancer and its treatment. It provides a model framework for systems that are characterised by elements that react and interact with each other in the course of time. Nevertheless, such a process does not necessarily need to occur on a microscopic level as for cells or genes. Also systems on a higher scale exhibit similar patterns. For example, predator-prey dynamics describe how the populations of two species, predator and prey, change over time. In epidemiology, the spread of infectious diseases can be modelled by the interaction of individuals. All these systems have in common that their dynamics usually evolve non-deterministically. For a deterministic process, based on a current system state that is known, all future states can be exactly determined according to underlying mechanistic laws. But in this context, different kinds of elements rather react and interact randomly in time. This behaviour requires stochastic processes to model the kinetics. However, stochastic dynamics are in general hard to be inferred statistically. It is often infeasible to calculate their likelihood function analytically. In this case, deterministic approximations can facilitate statistical inference. Since a real data set at hand is rarely “perfect” to determine model parameters, the problem of parameter identifiability needs to be taken into account. Experimental design analysis can resolve parameter non-identifiabilities, reduce the uncertainty of parameter estimates and, in practice, it can save resources.

This series collects articles that provide a model framework for dynamical reaction systems as well as methods to infer a model statistically. Moreover, articles show how the uncertainty of parameter estimates can be quantified and how the design of experiments can be optimized.

## Contributors#

Jordan Gault