# Parameterization (DRAFT)#

Part of a series: Data-induced Uncertainties.

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This article is under construction.

## Introduction#

The goal of mathematical models is to replicate the state and dynamics of real-world systems as accurately as possible in order to generate useful insights into the problem being studied. This goal can only be achieved if the model incorporates a sufficiently accurate description of the actual system. As a counter-example, consider physical processes which are often described by simplified schemes known as parameterizations (REF). The values and constraints of these parameters, in turn, are often quite poorly known due to a lack of theory or limited observation. In that sense, parameter uncertainty is related to other kinds of uncertainties such as structural and measurement uncertainty.

## What is a parameter?#

This question might seem trivial but is crucial for the adequate handling of the associated uncertainty. Thus, we decided to dedicate a section two the two main interpretations of parameters

Frequentistic approach

In the Bayesian interpretation, any model parameter is by itself a random variable and has its own probability (or density) distribution. Deciding on an appropriate distribution and incorporating the information from the data are the key aspects of this approach.

Frequentists, on the other side, see all parameters as fixed values from the population. By sampling just enough data, we should manage to get arbitrarily close to this unknown true parameter. The key aspect here is to find a good function which translates the data into an estimate of the underlying parameter. These so-called estimators will again be random variables and hence also be described via probability (or density) functions.

Therefore, the starting point to characterize parametric uncertainty is independent of the chosen approach. Ideally, one would have a pdf for each uncertain parameter.

## Quantifying/dealing with parameter uncertainty#

The types and amounts of uncertainty associated with each parameter of the model must be determined and classified. Furthermore, it is necessary to estimate and quantify the evolution of uncertainty within the system. The uncertainty propagation is constructed by estimating the distribution of model outputs from a set of uncertain input parameters. Where for each given input noise, propagation of uncertainty measures output noise of the system.

Many mathematical models of complex system involve numerous parameters which need adjustment and tailored configuration before each application case to calibrate the model using observational data, either manually or through optimization algorithms. Because of resource constrain it is crucially important to focus on the parameters that are most influential for the model outputs. One way of doing this is to carry out a global sensitivity analysis (GSA). GSA aims to identify the parameters whose uncertainty has the largest impact on the variability of a output.