Mathematical model

A model may help to explain a system and to study the effects of different components, and to make predictions about behavior.

In many cases, the quality of a scientific field depends on how well the mathematical models developed on the theoretical side agree with results of repeatable experiments.

Lack of agreement between theoretical mathematical models and experimental measurements often leads to important advances as better theories are developed.

Depending on the context, an objective function is also known as an index of performance, as it is some measure of interest to the user.

Practically all systems are somewhere between the black-box and white-box models, so this concept is useful only as an intuitive guide for deciding which approach to take.

Using a priori information we could end up, for example, with a set of functions that probably could describe the system adequately.

An often used approach for black-box models are neural networks which usually do not make assumptions about incoming data.

The advantage of NARMAX models compared to neural networks is that NARMAX produces models that can be written down and related to the underlying process, whereas neural networks produce an approximation that is opaque.

Thomas Kuhn argues that as science progresses, explanations tend to become more complex before a paradigm shift offers radical simplification.

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.

For example, Newton's classical mechanics is an approximated model of the real world.

[10] In more conventional modeling through explicitly given mathematical functions, parameters are often determined by curve fitting.

Defining a metric to measure distances between observed and predicted data is a useful tool for assessing model fit.

In statistics, decision theory, and some economic models, a loss function plays a similar role.

As an example of the typical limitations of the scope of a model, in evaluating Newtonian classical mechanics, we can note that Newton made his measurements without advanced equipment, so he could not measure properties of particles traveling at speeds close to the speed of light.

One can also argue that a model is worthless unless it provides some insight which goes beyond what is already known from direct investigation of the phenomenon being studied.

An example of such criticism is the argument that the mathematical models of optimal foraging theory do not offer insight that goes beyond the common-sense conclusions of evolution and other basic principles of ecology.

Newton's laws accurately describe many everyday phenomena, but at certain limits theory of relativity and quantum mechanics must be used.

In engineering, physics models are often made by mathematical methods such as finite element analysis.

Variables may be of many types; real or integer numbers, Boolean values or strings, for example.