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Mathematical model

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.

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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 unforseeable 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, 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

Mathematical modelling problems are often classified into white-box or black-box models, according to how much a prior information is available of the system. A black-box model is a system of which there is no a prior information available, and a white-box model is a system where all necessary information is available. Practically all systems are somewhere between the white-box and black-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 decaying function. But we are still left with several unknown paramters; 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.

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

An important part of the modelling process is the evaluation of an acquired model. How do we know if a mathematical model describes the system well? This is not an easy question to answer. Usually the engineer has a set of measurements from the system which are used in creating the model. Then, if the model was built well, the model will adequately show the relations between system variables for the measurements at hand. The question then becomes: How do we know that the measurement data is a representative set of possible values? Does the model describe well the properties of the system between the measurement data (interpolation)? Does the model describe well events outside the measurement data (extrapolation)? A common approach is to split the measured data into two parts; training data and verification data. The training data is used to train the model, that is, to estimate the model parameters. The verification data is used to evaluate model performance. Assuming that the training data and verification data are not the same, we can assume that if the model describes the verification data well, then the model describes the real system well.

However, this still leaves the extrapolation question open. How well does this model describe events outside the measured data? Consider again Newtoninan classical mechanics -model. Newton made his measurements without advanced equipment, so he could not measure properties of particles travelling at speeds close to the speed of light. Likewise, he did not measure the movements of molecules and other small particles, but macro particles only. It is then not surprising that his model does not extrapolate well into these domains, even though his model is quite sufficient for ordinary life physics.


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


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