A mathematical model is the use of mathematical language to describe the behaviour of a system. Mathematical models are used in particularly in the sciences such biology, electrical engineering, physics but also in other fields such as economics, sociology and political science.

1 Background

Often when an engineer analyses a system or is supposed to control a system, he uses a mathematical model. In analysis, the engineer can build a descriptive model of the system as a hypothesis of how the system could work, or try to estimate how an unforeseeable event could affect the system. Similarly, in control of a system the engineer can try out different control approaches in simulations.

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 or strings, for example. The variables represent some properties of the system, for example, measured system outputs often in the form of signals, timing data, counters, event occurrence (yes/no). The actual model is the set of functions that describe the relations between the different variables.

2 Types of mathematical models

Mathematical models can be divided up several ways, they can be deterministic (that is perform the same way for a given set of initial conditions), or stochasticStochastic from the Greek "stochos" or "goal", means of, relating to, or characterized by conjecture; conjectural; random. A stochastic process is one whose behavior is non-deterministic in that the next state of the environment is not fully determined by (randomness is present, even when given an identical set of initial conditions). They can also be continuous or discreteThe word discrete comes from the latin word discretus which means separate''. It is used with different meanings in different contexts: In perception a discrete entity is something that can be perceived individually and not as connected to, or part of som and implemented with differential equationIn mathematics, a differential equation is an equation that describes a prescribed relationship between a set of unknowns which are to be regarded as an unknown function and its (ordinary or partial) derivatives. In practice the "unknown function" is usuas or delay differential equation s.

3 A priori information

Mathematical modelling problems are often classified into black-box or white-box models, according to how much a prioriA priori is a Latin phrase meaning "from the former". In much of the modern Western tradition, the term a priori is considered to mean propositional knowledge that can be had without, or "prior to", experience. It is usually contrasted with a posteriori k information is available of the system. A black-box model is a system of which there is no a priori information available. A white-box model (also called glass box or clear box) is a system where all necessary information is available. Practically all systems are somewhere between the black-box and white-box models, so this concept only works as an intuitive guide for approach.

Usually it is preferable to use as much a priori information as possible to make the model more accurate. Therefore the white-box models are usually considered easier, because if you have used the information correctly, then the model will behave correctly. Often the a priori information comes in forms of knowing the type of functions relating different variables. For example, if we make a model of how a medicine works in a human system, we know that usually the amount of medicine in the blood is an exponentially decayingA quantity is said to be subject to exponential decay if it decreases at a rate proportional to its value. Symbolically, this can be expressed as the following differential equation, where N is the quantity and is a positive number called the decay consta function. But we are still left with several unknown parameters; how rapidly does the medicine amount decay, and what is the initial amount of medicine in blood? This example is therefore not a completely white-box model. These parameters have to be estimated through some means before one can use the model.

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.