Mathematical model

For other uses, seeMathematical model (disambiguation).

Amathematical modelis anabstractdescription of a concretesystemusingmathematicalconcepts andlanguage.The process of developing a mathematicalmodelis termedmathematical modeling.Mathematical models are used inapplied mathematicsand in thenatural sciences(such asphysics,biology,earth science,chemistry) andengineeringdisciplines (such ascomputer science,electrical engineering), as well as in non-physical systems such as thesocial sciences(such aseconomics,psychology,sociology,political science). It can also be taught as a subject in its own right.

The use of mathematical models to solve problems in business or military operations is a large part of the field ofoperations research.Mathematical models are also used inmusic,linguistics,and philosophy(for example, intensively inanalytic philosophy). A model may help to explain a system and to study the effects of different components, and to make predictions about behavior.

Elements of a mathematical model

Mathematical models can take many forms, includingdynamical systems,statistical models,differential equations,orgame theoretic models.These and other types of models can overlap, with a given model involving a variety of abstract structures. In general, mathematical models may includelogical models.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. In thephysical sciences,a traditional mathematical model contains most of the following elements:

1. Governing equations
2. Supplementary sub-models
   1. Defining equations
   2. Constitutive equations
3. Assumptions and constraints
   1. Initialandboundary conditions
   2. Classical constraintsandkinematic equations

Classifications

Mathematical models are of different types:

• Linear vs. nonlinear. If all the operators in a mathematical model exhibitlinearity,the resulting mathematical model is defined as linear. A model is considered to be nonlinear otherwise. The definition of linearity and nonlinearity is dependent on context, and linear models may have nonlinear expressions in them. For example, in astatistical linear model,it is assumed that a relationship is linear in the parameters, but it may be nonlinear in the predictor variables. Similarly, a differential equation is said to be linear if it can be written with lineardifferential operators,but it can still have nonlinear expressions in it. In amathematical programmingmodel, if the objective functions and constraints are represented entirely bylinear equations,then the model is regarded as a linear model. If one or more of the objective functions or constraints are represented with anonlinearequation, then the model is known as a nonlinear model.
  Linear structure implies that a problem can be decomposed into simpler parts that can be treated independently and/or analyzed at a different scale and the results obtained will remain valid for the initial problem when recomposed and rescaled.
  Nonlinearity, even in fairly simple systems, is often associated with phenomena such aschaosandirreversibility.Although there are exceptions, nonlinear systems and models tend to be more difficult to study than linear ones. A common approach to nonlinear problems islinearization,but this can be problematic if one is trying to study aspects such as irreversibility, which are strongly tied to nonlinearity.
• Static vs. dynamic. Adynamicmodel accounts for time-dependent changes in the state of the system, while astatic(or steady-state) model calculates the system in equilibrium, and thus is time-invariant. Dynamic models typically are represented bydifferential equationsordifference equations.
• Explicit vs. implicit. If all of the input parameters of the overall model are known, and the output parameters can be calculated by a finite series of computations, the model is said to beexplicit.But sometimes it is theoutputparameters which are known, and the corresponding inputs must be solved for by an iterative procedure, such asNewton's methodorBroyden's method.In such a case the model is said to beimplicit.For example, ajet engine's physical properties such as turbine and nozzle throat areas can be explicitly calculated given a designthermodynamic cycle(air and fuel flow rates, pressures, and temperatures) at a specific flight condition and power setting, but the engine's operating cycles at other flight conditions and power settings cannot be explicitly calculated from the constant physical properties.
• Discrete vs. continuous. Adiscrete modeltreats objects as discrete, such as the particles in amolecular modelor the states in astatistical model;while acontinuous modelrepresents the objects in a continuous manner, such as the velocity field of fluid in pipe flows, temperatures and stresses in a solid, and electric field that applies continuously over the entire model due to a point charge.
• Deterministic vs. probabilistic (stochastic). Adeterministicmodel is one in which every set of variable states is uniquely determined by parameters in the model and by sets of previous states of these variables; therefore, a deterministic model always performs the same way for a given set of initial conditions. Conversely, in a stochastic model—usually called a "statistical model"—randomness is present, and variable states are not described by unique values, but rather byprobabilitydistributions.
• Deductive, inductive, or floating. Adeductive modelis a logical structure based on a theory. An inductive model arises from empirical findings and generalization from them. The floating model rests on neither theory nor observation, but is merely the invocation of expected structure. Application of mathematics in social sciences outside of economics has been criticized for unfounded models.Application ofcatastrophe theoryin science has been characterized as a floating model.
• Strategic vs. non-strategic. Models used ingame theoryare different in a sense that they model agents with incompatible incentives, such as competing species or bidders in an auction. Strategic models assume that players are autonomous decision makers who rationally choose actions that maximize their objective function. A key challenge of using strategic models is defining and computingsolution conceptssuch asNash equilibrium.An interesting property of strategic models is that they separate reasoning about rules of the game from reasoning about behavior of the players.

Construction

Inbusinessandengineering,mathematical models may be used to maximize a certain output. The system under consideration will require certain inputs. The system relating inputs to outputs depends on other variables too:decision variables,state variables,exogenousvariables, andrandom variables.Decision variables are sometimes known as independent variables. Exogenous variables are sometimes known asparametersorconstants.The variables are not independent of each 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).

Objectivesandconstraintsof the system and its users can be represented asfunctionsof the output variables or state variables. Theobjective functionswill depend on the perspective of the model's user. Depending on the context, an objective function is also known as anindex of performance,as it is some measure 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,economistsoften applylinear algebrawhen usinginput–output models.Complicated mathematical models that have many variables may be consolidated by use ofvectorswhere one symbol represents several variables.

A prioriinformation

To analyse something with a typical "black box approach", only the behavior of the stimulus/response will be accounted for, to infer the (unknown)box.The usual representation of thisblack box systemis adata flow diagramcentered in the box.

Mathematical modeling problems are often classified intoblack boxorwhite boxmodels, according to how mucha prioriinformation 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, 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 anexponentially decayingfunction, 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 areneural networkswhich usually do not make assumptions about incoming data. Alternatively, the NARMAX (Nonlinear AutoRegressive Moving Average model with eXogenous inputs) algorithms which were developed as part ofnonlinear system identificationcan be used to select the model terms, determine 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 written down and related to the underlying process, whereas neural networks produce an approximation that is opaque.

Subjective information

Sometimes it is useful to incorporate subjective information into a mathematical model. This can be done based onintuition,experience,orexpert opinion,or based on convenience of mathematical form.Bayesian statisticsprovides a theoretical framework for incorporating such subjectivity into a rigorous analysis: we specify aprior probability distribution(which can be subjective), and then update this distribution based on empirical data.

An example of when such approach would be necessary is a situation in which an experimenter bends a coin slightly and tosses it once, recording whether it comes up heads, and is then given 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 shape of the coin) about what prior distribution to use. Incorporation of such subjective information might be important to get an accurate estimate of the probability.

Complexity

In general, model complexity involves a trade-off between simplicity and accuracy of the model.Occam's razoris a principle particularly relevant to modeling, its essential idea being that among models with roughly equal predictive power, the simplest one is the most desirable. While added complexity usually improves the realism of a model, it can make the model difficult to understand and analyze, and can also pose computational problems, includingnumerical instability.Thomas Kuhnargues that as science progresses, explanations tend to become more complex before aparadigm shiftoffers radical simplification.

For example, when modeling the flight of an aircraft, we could embed each mechanical part of the aircraft 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. Engineers often can accept some approximations in order to get a more robust and simple model. For example,Newton'sclassical mechanicsis 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 thespeed of light,and we study macro-particles only. Note that better accuracy does not necessarily mean a better model.Statistical modelsare prone tooverfittingwhich 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.

Training, tuning, and fitting

Any model which is not pure white-box contains someparametersthat can be used tofit the modelto the system it is intended to describe. If the modeling is done by anartificial neural networkor othermachine learning,the optimization of parameters is calledtraining,while the optimization of model hyperparameters is calledtuningand often usescross-validation.In more conventional modeling through explicitly given mathematical functions, parameters are often determined bycurve fitting.

Evaluation and assessment

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 to answer as it involves several different types of evaluation.

Prediction of empirical data

Usually, the easiest part of model evaluation is checking whether a model predicts experimental measurements or other empirical data not used in the model development. In models with parameters, a common approach 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 referred to ascross-validationin statistics.

Defining ametricto measure distances between observed and predicted data is a useful tool for assessing model fit. In statistics, decision theory, and someeconomic models,aloss functionplays 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 ofstatistical modelsthan models involvingdifferential equations.Tools fromnonparametric statisticscan 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.

Scope of the model

Assessing the scope of a model, that is, determining what situations the model is applicable to, can be less straightforward. If the model was constructed based on a set of data, one must determine for which systems or situations 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 calledinterpolation,and the same question for events or data points outside the observed data is calledextrapolation.

As an example of the typical limitations of the scope of a model, in evaluating Newtonianclassical 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. 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.

Philosophical considerations

Many types of modeling implicitly involve claims aboutcausality.This is usually (but not always) true of models involving differential equations. As the purpose of modeling is to increase 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 described 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 argument that the mathematical models ofoptimal foraging theorydo not offer insight that goes beyond the common-sense conclusions ofevolutionand other basic principles of ecology.It should also be noted that while mathematical modeling uses mathematical concepts and language, it is not itself a branch of mathematics and does not necessarily conform to anymathematical logic,but is typically a branch of some science or other technical subject, with corresponding concepts and standards of argumentation.

Significance in the natural sciences

Mathematical models are of great importance in the natural sciences, particularly inphysics.Physicaltheoriesare almost invariably expressed using mathematical models. Throughout history, more and more accurate mathematical models have been developed.Newton's lawsaccurately describe many everyday phenomena, but at certain limitstheory of relativityandquantum mechanicsmust be used.

It is common to use idealized models in physics to simplify things. Massless ropes, point particles,ideal gasesand theparticle in a boxare among the many simplified models used in physics. The laws of physics are represented with simple equations such as Newton's laws,Maxwell's equationsand theSchrödinger equation.These laws are a basis for making mathematical models of real situations. Many real situations are very complex and thus modeled approximately on a computer, a model that is computationally feasible to compute is made from the basic laws or from approximate models made from the basic laws. For example, molecules can be modeled bymolecular orbitalmodels that are approximate solutions to the Schrödinger equation. Inengineering,physics models are often made by mathematical methods such asfinite element analysis.

Different mathematical models use different geometries that are not necessarily accurate descriptions of the geometry of the universe.Euclidean geometryis much used in classical physics, whilespecial relativityandgeneral relativityare examples of theories that usegeometrieswhich are not Euclidean.

Some applications

Often when engineers analyze a system to be controlled or optimized, they use a mathematical model. In analysis, engineers 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, engineers can try out different control approaches insimulations.

A mathematical model usually describes a system by asetof variables and a set of equations that establish relationships between the variables. Variables may be of many types;realorintegernumbers,Booleanvalues orstrings,for example. The variables represent some properties of the system, for example, the measured system outputs often in the form ofsignals,timing data,counters, and event occurrence. The actual model is the set of functions that describe the relations between the different variables.

Examples

• One of the popular examples incomputer scienceis the mathematical models of various machines, an example is thedeterministic finite automaton(DFA) which is defined as an abstract mathematical concept, but due to the deterministic nature of a DFA, it is implementable in hardware and software for solving various specific problems. For example, the following is a DFA M with a binary alphabet, which requires that the input contains an even number of 0s:

Thestate diagramfor$M$

$M=(Q,\Sigma,\delta,q_{0},F)$where

• $Q=\{S_{1},S_{2}\},$
• $\Sigma =\{0,1\},$
• $q_{0}=S_{1},$
• $F=\{S_{1}\},$and
• $\delta$is defined by the followingstate-transition table:

	0	1
S1	$S_{2}$	$S_{1}$
S2	$S_{1}$	$S_{2}$

The state$S_{1}$represents that there has been an even number of 0s in the input so far, while$S_{2}$signifies an odd number. A 1 in the input does not change the state of the automaton. When the input ends, the state will show whether the input contained an even number of 0s or not. If the input did contain an even number of 0s,$M$will finish in state$S_{1},$an accepting state, so the input string will be accepted.

The language recognized by$M$is theregular languagegiven by theregular expression1*( 0 (1*) 0 (1*) )*, where "*" is theKleene star,e.g., 1* denotes any non-negative number (possibly zero) of symbols "1".

• Many everyday activities carried out without a thought are uses of mathematical models. A geographicalmap projectionof a region of the earth onto a small, plane surface is a model which can be used for many purposes such as planning travel.
• Another simple activity is predicting the position of a vehicle from its initial position, direction and speed of travel, using the equation that distance traveled is the product of time and speed. This is known asdead reckoningwhen used more formally. Mathematical modeling in this way does not necessarily require formal mathematics; animals have been shown to use dead reckoning.
• PopulationGrowth.A simple (though approximate) model of population growth is theMalthusian growth model.A slightly more realistic and largely used population growth model is thelogistic function,and its extensions.
• Model of a particle in a potential-field.In this model we consider a particle as being a point of mass which describes a trajectory in space which is modeled by a function giving its coordinates in space as a function of time. The potential field is given by a function$V\!:\mathbb {R} ^{3}\!\to \mathbb {R}$and the trajectory, that is a function$\mathbf {r} \!:\mathbb {R} \to \mathbb {R} ^{3},$is the solution of the differential equation:$$-{\frac {\mathrm {d} ^{2}\mathbf {r} (t)}{\mathrm {d} t^{2}}}m={\frac {\partial V[\mathbf {r} (t)]}{\partial x}}\mathbf {\hat {x}} +{\frac {\partial V[\mathbf {r} (t)]}{\partial y}}\mathbf {\hat {y}} +{\frac {\partial V[\mathbf {r} (t)]}{\partial z}}\mathbf {\hat {z}},$$that can be written also as$$m{\frac {\mathrm {d} ^{2}\mathbf {r} (t)}{\mathrm {d} t^{2}}}=-\nabla V[\mathbf {r} (t)].$$

Note this model assumes the particle is a point mass, which is certainly known to be false in many cases in which we use this model; for example, as a model of planetary motion.

• Model of rational behavior for a consumer.In this model we assume a consumer faces a choice of$n$commodities labeled$1,2,\dots,n$each with a market price$p_{1},p_{2},\dots,p_{n}.$The consumer is assumed to have anordinal utilityfunction$U$(ordinal in the sense that only the sign of the differences between two utilities, and not the level of each utility, is meaningful), depending on the amounts of commodities$x_{1},x_{2},\dots,x_{n}$consumed. The model further assumes that the consumer has a budget$M$which is used to purchase a vector$x_{1},x_{2},\dots,x_{n}$in such a way as to maximize$U(x_{1},x_{2},\dots,x_{n}).$The problem of rational behavior in this model then becomes amathematical optimizationproblem, that is:$$\max \,U(x_{1},x_{2},\ldots,x_{n})$$subject to:$$\sum _{i=1}^{n}p_{i}x_{i}\leq M,$$$$x_{i}\geq 0\;\;\;{\text{ for all }}i=1,2,\dots,n.$$This model has been used in a wide variety of economic contexts, such as ingeneral equilibrium theoryto show existence andPareto efficiencyof economic equilibria.
• Neighbour-sensing modelis a model that explains themushroomformation from the initially chaoticfungalnetwork.
• Incomputer science,mathematical models may be used to simulate computer networks.
• Inmechanics,mathematical models may be used to analyze the movement of a rocket model.

See also

• Agent-based model
• All models are wrong
• Cliodynamics
• Computer simulation
• Conceptual model
• Decision engineering
• Grey box model
• International Mathematical Modeling Challenge
• Mathematical biology
• Mathematical diagram
• Mathematical economics
• Mathematical modelling of infectious disease
• Mathematical finance
• Mathematical psychology
• Mathematical sociology
• Microscale and macroscale models
• Model inversion
• Resilience (mathematics)
• Scientific model
• Sensitivity analysis
• Statistical model
• Surrogate model
• System identification

Further reading

Books

• Aris, Rutherford [ 1978 ] ( 1994 ).Mathematical Modelling Techniques,New York: Dover.ISBN0-486-68131-9
• Bender, E.A. [ 1978 ] ( 2000 ).An Introduction to Mathematical Modeling,New York: Dover.ISBN0-486-41180-X
• Gary Chartrand(1977)Graphs as Mathematical Models,Prindle, Webber & SchmidtISBN0871502364
• Dubois, G. (2018)"Modeling and Simulation",Taylor & Francis, CRC Press.
• Gershenfeld, N. (1998)The Nature of Mathematical Modeling,Cambridge University PressISBN0-521-57095-6.
• Lin, C.C. & Segel, L.A. ( 1988 ).Mathematics Applied to Deterministic Problems in the Natural Sciences,Philadelphia: SIAM.ISBN0-89871-229-7

Specific applications

• Papadimitriou, Fivos. (2010). Mathematical Modelling of Spatial-Ecological Complex Systems: an Evaluation. Geography, Environment, Sustainability 1(3), 67-80.doi:10.24057/2071-9388-2010-3-1-67-80
• Peierls, R. (1980). "Model-making in physics".Contemporary Physics.21:3–17.Bibcode:1980ConPh..21....3P.doi:10.1080/00107518008210938.
• An Introduction to Infectious Disease Modellingby Emilia Vynnycky and Richard G White.

External links

General reference

• Patrone, F.Introduction to modeling via differential equations,with critical remarks.
• Plus teacher and student package: Mathematical Modelling.Brings together all articles on mathematical modeling fromPlus Magazine,the online mathematics magazine produced by the Millennium Mathematics Project at the University of Cambridge.

Philosophical

• Frigg, R. and S. Hartmann,Models in Science,in: The Stanford Encyclopedia of Philosophy, (Spring 2006 Edition)
• Griffiths, E. C. (2010)What is a model?