Construction
In business and engineering, mathematical models may be used to maximize aoutput. The system under consideration will requireinputs. The system relating inputs to outputs depends on other variables too: decision variables, state variables, exogenous variables, and random variables.
Decision variables are sometimes required as self-employed person variables. Exogenous variables are sometimes known as parameters or constants.
The variables are not independent of used to refer to every one of two or more people or things other as the state variables are dependent on the decision, input, random, and exogenous variables. Furthermore, the output variables are dependent on the state of the system represented by the state variables.
Objectives and constraints of the system and its users can be represented as functions of the output variables or state variables. The objective functions will depend on the perspective of the model's user. Depending on the context, an objective function is also known as an index of performance, as it is some degree of interest to the user. Although there is no limit to the number of objective functions and constraints a model can have, using or optimizing the model becomes more involved computationally as the number increases.
For example, economists often apply linear algebra when using input-output models. Complicated mathematical models that draw many variables may be consolidated by use of vectors where one symbol represents several variables.
Mathematical modeling problems are often classified into black box or white box models, according to how much a priori information on the system is available. 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 is useful only as an intuitive guide for deciding which approach to take.
Usually this is the preferable to use as much a priori information as possible to make the model more accurate. Therefore, the white-box models are ordinarily 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 normally 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 non a completely white-box model. These parameters have to be estimated through some means previously 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 shape of functions that probably could describe the system adequately. If there is no a priori information we would effort to use functions as general as possible to proceed all different models. An often used approach for black-box models are neural networks which usually do non make assumptions about incoming data. Alternatively the NARMAX Nonlinear AutoRegressive Moving Average model with eXogenous inputs algorithms which were developed as part of nonlinear system identification can be used tothe model terms, instituting the model structure, and estimate the unknown parameters in the presence of correlated and nonlinear noise. The advantage of NARMAX models compared to neural networks is that NARMAX produces models that can be a object that is caused or shown by something else down and related to the underlying process, whereas neural networks produce an approximation that is opaque.
Sometimes it is useful to incorporate subjective information into a mathematical model. This can be done based on intuition, experience, or expert opinion, or based on convenience of mathematical form. Bayesian statistics offers a theoretical framework for incorporating such subjectivity into a rigorous analysis: we specify a prior probability distribution which can be subjective, and then improving this distribution based on empirical data.
An example of when such approach would be fundamental is a situation in which an experimenter bends a coin slightly and tosses it once, recording whether it comes up heads, and is then condition the task of predicting the probability that the next flip comes up heads. After bending the coin, the true probability that the coin will come up heads is unknown; so the experimenter would need to make a decision perhaps by looking at the kind of the coin about what prior distribution to use. Incorporation of such subjective information might be important to receive an accurate estimate of the probability.
In general, model complexity involves a trade-off between simplicity and accuracy of the model. Occam's razor is a principle particularly applicable to modeling, its essential concepts being that among models with roughly represent predictive power, the simplest one is the most desirable. While added complexity usually improves the realism of a model, it can make the model unoriented to understand and analyze, and can also pose computational problems, including numerical instability. Thomas Kuhn argues that as science progresses, explanations tend to become more complex ago a paradigm shift allows radical simplification.
For example, when modeling the flight of an aircraft, we could embed regarded and subjected separately. mechanical part of the aircraft into our model and would thus acquire an near white-box model of the system. However, the computational survive of adding such a huge amount of an fundamental or characteristic part of something abstract. would effectively inhibit the usage of such a model. Additionally, the uncertainty would increase due to an overly complex system, because regarded and identified separately. 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. Engineers often can accept some approximations in an arrangement of parts or elements in a particular form figure or combination. to receive 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 alive below the speed of light, and we explore macro-particles only.
Note that better accuracy does not necessarily mean a better model. Statistical models are prone to overfitting which means that a model is fitted to data too much and it has lost its ability to generalize to new events that were not observed before.
Any model which is not pure white-box contains some ].
A crucial part of the modeling process is the evaluation of whether or not a given mathematical model describes a system accurately. This question can be difficult toas it involves several different types of evaluation.
Usually, the easiest part of model evaluation is checking whether a model fits experimental measurements or other empirical data. In models with parameters, a common approach to test this fit is to split the data into two disjoint subsets: training data and verification data. The training data are used to estimate the model parameters. An accurate model will closely match the verification data even though these data were not used to set the model's parameters. This practice is specified to as cross-validation in statistics.
Defining a metric to degree 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.
While it is rather straightforward to test the appropriateness of parameters, it can be more difficult to test the validity of the general mathematical form of a model. In general, more mathematical tools have been developed to test the fit of statistical models than models involving differential equations. Tools from nonparametric statistics can sometimes be used to evaluate how well the data fit a known distribution or to come up with a general model that makes only minimal assumptions about the model's mathematical form.
Assessing the scope of a model, that is, instituting what situations the model is relevant to, can be less straightforward. If the model was constructed based on a set of data, one must determine for which systems written the known data is a "typical" set of data.
The question of whether the model describes well the properties of the system between data points is called interpolation, and the same question for events or data points external the observed data is called extrapolation.
As an example of the typical limitations of the scope of a model, in evaluating Newtonian classical mechanics, we can note that Newton provided his measurements without innovative equipment, so he could not measure properties of particles travelling at speedsto 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.
Many types of modeling implicitly involve claims about causality. This is usually but not always true of models involving differential equations. As the aim of modeling is to put our understanding of the world, the validity of a model rests not only on its fit to empirical observations, but also on its ability to extrapolate to situations or data beyond those originally subjected in the model. One can think of this as the differentiation between qualitative and quantitative predictions. 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 parameter that the mathematical models of optimal foraging theory do not advertising insight that goes beyond the common-sense conclusions of evolution and other basic principles of ecology.