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A mathematical model 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
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.
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 stochastic (randomness is present, even when given an identical set of initial
conditions). They can also be continuous or discrete and implemented with differential
equations or delay differential equations.
A priori information
Mathematical modelling problems are often classified into white-box or black-box models, according to how much a priori information is available of the system. A black-box model is a system of which
there is no a priori 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 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.
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 appropriate 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 Newtonian 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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See also
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