**Mathematical modelling**is the use of mathematical language to describe the behaviour of a system, be it biological, economic, electrical, mechanical, thermodynamic, or one of many other examples.

## Background

A mathematical model usually describes a system by means of variables.

The values of the variables can be practically anything; real or integer numbers, boolean values, strings etc.

The variables represent some properties of the system, for example, measured system outputs often in the from of signals, timing data, counters, event occurrence (yes/no), etc.

The actual model is the set of functions that describe the relations between the different variables.

## A priori information

In black-box models one tries to estimate both the functional form of relations between variables and the numerical parameters in those functions. Using a priori information we could end up, for example, with a set of functions that probably could describe the system adequately. If there is no a priori information we would try to use functions as general as possible to cover all different models. An often used approach for black-box models are neural networks which usually do not assume almost anything about the incoming data. The problem with using a large set of functions to describe a system is that estimating the parameters becomes increasingly difficult when the amount of parameters (and different types of functions) increases.

## Complexity

Another basic issue is the complexity of a model. If we were, for example, modelling the flight of an airplane, we could embed each mechanical part of the airplane into our model and would thus acquire an almost white-box model of the system. However, the computational cost of adding such a huge amount of detail would effectively inhibit the usage of such a model. Additionally, the uncertainty would increase due to an overly complex system, because each separate part induces some amount of variance into the model. It is therefore usually apropiate to make some approximations to reduce the model to a sensible size. The engineer often can accept some approximations in order to get a more robust and simple model. For example Newton's classical mechanics is an approximated model of the real world. Still, Newton's model is quite sufficient for most ordinary-life situations, that is, as long as particle speeds are well below the speed of light, and we study macro-particles only.

## Model evaluation

*Note:* The term 'model' is also given a formal meaning in model theory, a part of axiomatic set theory.

See also: