A mathematical model is a description of a
system
A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole. A system, surrounded and influenced by its environment (systems), environment, is described by its boundaries, ...
using
mathematical
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
concepts and
language
Language is a structured system of communication. The structure of a language is its grammar and the free components are its vocabulary. Languages are the primary means by which humans communicate, and may be conveyed through a variety of met ...
. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used in the
natural science
Natural science is one of the branches of science concerned with the description, understanding and prediction of natural phenomena, based on empirical evidence from observation and experimentation. Mechanisms such as peer review and repeatab ...
s (such as
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
,
biology
Biology is the scientific study of life. It is a natural science with a broad scope but has several unifying themes that tie it together as a single, coherent field. For instance, all organisms are made up of cells that process hereditary i ...
,
earth science
Earth science or geoscience includes all fields of natural science related to the planet Earth. This is a branch of science dealing with the physical, chemical, and biological complex constitutions and synergistic linkages of Earth's four spheres ...
,
chemistry
Chemistry is the science, scientific study of the properties and behavior of matter. It is a natural science that covers the Chemical element, elements that make up matter to the chemical compound, compounds made of atoms, molecules and ions ...
) and
engineering
Engineering is the use of scientific method, scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad rang ...
disciplines (such as
computer science
Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical discipli ...
,
electrical engineering
Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems which use electricity, electronics, and electromagnetism. It emerged as an identifiable occupation in the l ...
), as well as in non-physical systems such as the
social science
Social science is one of the branches of science, devoted to the study of societies and the relationships among individuals within those societies. The term was formerly used to refer to the field of sociology, the original "science of soc ...
s (such as
economics
Economics () is the social science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services.
Economics focuses on the behaviour and intera ...
,
psychology
Psychology is the scientific study of mind and behavior. Psychology includes the study of conscious and unconscious phenomena, including feelings and thoughts. It is an academic discipline of immense scope, crossing the boundaries betwe ...
,
sociology
Sociology is a social science that focuses on society, human social behavior, patterns of Interpersonal ties, social relationships, social interaction, and aspects of culture associated with everyday life. It uses various methods of Empirical ...
,
political science
Political science is the scientific study of politics. It is a social science dealing with systems of governance and power, and the analysis of political activities, political thought, political behavior, and associated constitutions and la ...
). The use of mathematical models to solve problems in business or military operations is a large part of the field of
operations research
Operations research ( en-GB, operational research) (U.S. Air Force Specialty Code: Operations Analysis), often shortened to the initialism OR, is a discipline that deals with the development and application of analytical methods to improve deci ...
. Mathematical models are also used in
music
Music is generally defined as the art of arranging sound to create some combination of form, harmony, melody, rhythm or otherwise expressive content. Exact definitions of music vary considerably around the world, though it is an aspect ...
,
linguistics
Linguistics is the scientific study of human language. It is called a scientific study because it entails a comprehensive, systematic, objective, and precise analysis of all aspects of language, particularly its nature and structure. Linguis ...
, and
philosophy
Philosophy (from , ) is the systematized study of general and fundamental questions, such as those about existence, reason, knowledge, values, mind, and language. Such questions are often posed as problems to be studied or resolved. Some ...
(for example, intensively in
analytic philosophy
Analytic philosophy is a branch and tradition of philosophy using analysis, popular in the Western world and particularly the Anglosphere, which began around the turn of the 20th century in the contemporary era in the United Kingdom, United Sta ...
).
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, including
dynamical systems
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a p ...
,
statistical model
A statistical model is a mathematical model that embodies a set of statistical assumptions concerning the generation of Sample (statistics), sample data (and similar data from a larger Statistical population, population). A statistical model repres ...
s,
differential equations
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
, or
game 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 include
logical 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 the
physical sciences
Physical science is a branch of natural science that studies non-living systems, in contrast to life science. It in turn has many branches, each referred to as a "physical science", together called the "physical sciences".
Definition
Phy ...
, a traditional mathematical model contains most of the following elements:
#
Governing equation The governing equations of a mathematical model describe how the values of the unknown variables (i.e. the dependent variables) change when one or more of the known (i.e. independent) variables change.
Mass balance
A mass balance, also called a ...
s
# Supplementary sub-models
##
Defining equations
##
Constitutive equation
In physics and engineering, a constitutive equation or constitutive relation is a relation between two physical quantities (especially kinetic quantities as related to kinematic quantities) that is specific to a material or substance, and approx ...
s
# Assumptions and constraints
##
Initial
In a written or published work, an initial capital, also referred to as a drop capital or simply an initial cap, initial, initcapital, initcap or init or a drop cap or drop, is a letter at the beginning of a word, a chapter, or a paragraph that ...
and
boundary condition
In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to th ...
s
##
Classical constraints and
kinematic equations
Kinematics equations are the constraint equations of a mechanical system such as a robot manipulator that define how input movement at one or more joints specifies the configuration of the device, in order to achieve a task position or end-effect ...
Classifications
Mathematical models are of different types:
* Linear vs. nonlinear: If all the operators in a mathematical model exhibit
linear
Linearity is the property of a mathematical relationship (''function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear r ...
ity, 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 a
statistical 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 linear
differential operators, but it can still have nonlinear expressions in it. In a
mathematical programming
Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfi ...
model, if the objective functions and constraints are represented entirely by
linear equations, then the model is regarded as a linear model. If one or more of the objective functions or constraints are represented with a
nonlinear
In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many othe ...
equation, 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 as
chaos
Chaos or CHAOS may refer to:
Arts, entertainment and media Fictional elements
* Chaos (''Kinnikuman'')
* Chaos (''Sailor Moon'')
* Chaos (''Sesame Park'')
* Chaos (''Warhammer'')
* Chaos, in ''Fabula Nova Crystallis Final Fantasy''
* Cha ...
and
irreversibility
In science, a process that is not reversible is called irreversible. This concept arises frequently in thermodynamics. All complex natural processes are irreversible, although a phase transition at the coexistence temperature (e.g. melting of i ...
. Although there are exceptions, nonlinear systems and models tend to be more difficult to study than linear ones. A common approach to nonlinear problems is
linearization
In mathematics, linearization is finding the linear approximation to a function at a given point. The linear approximation of a function is the first order Taylor expansion around the point of interest. In the study of dynamical systems, lineariz ...
, but this can be problematic if one is trying to study aspects such as irreversibility, which are strongly tied to nonlinearity.
* Static vs. dynamic: A ''dynamic'' model accounts for time-dependent changes in the state of the system, while a ''static'' (or steady-state) model calculates the system in equilibrium, and thus is time-invariant. Dynamic models typically are represented by
differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
s or
difference 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 be ''explicit''. But sometimes it is the ''output'' parameters which are known, and the corresponding inputs must be solved for by an iterative procedure, such as
Newton's method
In numerical analysis, Newton's method, also known as the Newton–Raphson method, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the roots (or zeroes) of a real-valu ...
or
Broyden's method
In numerical analysis, Broyden's method is a quasi-Newton method for finding roots in variables. It was originally described by C. G. Broyden in 1965.
Newton's method for solving uses the Jacobian matrix, , at every iteration. However, compu ...
. In such a case the model is said to be ''implicit''. For example, a
jet engine
A jet engine is a type of reaction engine discharging a fast-moving jet of heated gas (usually air) that generates thrust by jet propulsion. While this broad definition can include rocket, Pump-jet, water jet, and hybrid propulsion, the term ...
's physical properties such as turbine and nozzle throat areas can be explicitly calculated given a design
thermodynamic cycle
A thermodynamic cycle consists of a linked sequence of thermodynamic processes that involve transfer of heat and work into and out of the system, while varying pressure, temperature, and other state variables within the system, and that eventual ...
(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: A
discrete model treats objects as discrete, such as the particles in a
molecular model
A molecular model is a physical model of an atomistic system that represents molecules and their processes. They play an important role in understanding chemistry and generating and testing hypotheses. The creation of mathematical models of molecu ...
or the states in a
statistical model
A statistical model is a mathematical model that embodies a set of statistical assumptions concerning the generation of Sample (statistics), sample data (and similar data from a larger Statistical population, population). A statistical model repres ...
; while a
continuous model
Continuous modelling is the mathematical practice of applying a model to continuous data (data which has a potentially infinite number, and divisibility, of attributes). They often use differential equation
In mathematics, a differential equati ...
represents 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): A
deterministic
Determinism is a philosophical view, where all events are determined completely by previously existing causes. Deterministic theories throughout the history of philosophy have developed from diverse and sometimes overlapping motives and consi ...
model 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
A statistical model is a mathematical model that embodies a set of statistical assumptions concerning the generation of Sample (statistics), sample data (and similar data from a larger Statistical population, population). A statistical model repres ...
"—randomness is present, and variable states are not described by unique values, but rather by
probability
Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and ...
distributions.
* Deductive, inductive, or floating: A is 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 of
catastrophe theory in science has been characterized as a floating model.
* Strategic vs non-strategic Models used in
game theory
Game theory is the study of mathematical models of strategic interactions among rational agents. Myerson, Roger B. (1991). ''Game Theory: Analysis of Conflict,'' Harvard University Press, p.&nbs1 Chapter-preview links, ppvii–xi It has appli ...
are 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 computing
solution concepts such as
Nash equilibrium
In game theory, the Nash equilibrium, named after the mathematician John Nash, is the most common way to define the solution of a non-cooperative game involving two or more players. In a Nash equilibrium, each player is assumed to know the equili ...
. An interesting property of strategic models is that they separate reasoning about rules of the game from reasoning about behavior of the players.
Construction
In
business
Business is the practice of making one's living or making money by producing or Trade, buying and selling Product (business), products (such as goods and Service (economics), services). It is also "any activity or enterprise entered into for pr ...
and
engineering
Engineering is the use of scientific method, scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad rang ...
, 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 variable
A state variable is one of the set of variables that are used to describe the mathematical "state" of a dynamical system. Intuitively, the state of a system describes enough about the system to determine its future behaviour in the absence of a ...
s,
exogenous
In a variety of contexts, exogeny or exogeneity () is the fact of an action or object originating externally. It contrasts with endogeneity or endogeny, the fact of being influenced within a system.
Economics
In an economic model, an exogeno ...
variables, and
random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
s.
Decision variables are sometimes known as independent variables. Exogenous variables are sometimes known as
parameter
A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
s or
constants.
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).
Objective
Objective may refer to:
* Objective (optics), an element in a camera or microscope
* ''The Objective'', a 2008 science fiction horror film
* Objective pronoun, a personal pronoun that is used as a grammatical object
* Objective Productions, a Brit ...
s and
constraints of the system and its users can be represented as
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
s of the output variables or state variables. The
objective function
In mathematical optimization and decision theory, a loss function or cost function (sometimes also called an error function) is a function that maps an event or values of one or more variables onto a real number intuitively representing some "cos ...
s 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 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,
economist
An economist is a professional and practitioner in the social sciences, social science discipline of economics.
The individual may also study, develop, and apply theories and concepts from economics and write about economic policy. Within this ...
s often apply
linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrices.
...
when using
input-output model
In computing, input/output (I/O, or informally io or IO) is the communication between an information processing system, such as a computer, and the outside world, possibly a human or another information processing system. Inputs are the signals ...
s. Complicated mathematical models that have many variables may be consolidated by use of
vectors where one symbol represents several variables.
''A priori'' information
![Blackbox3D-withGraphs](https://upload.wikimedia.org/wikipedia/commons/7/7c/Blackbox3D-withGraphs.png)
Mathematical modeling problems are often classified into
black box
In science, computing, and engineering, a black box is a system which can be viewed in terms of its inputs and outputs (or transfer characteristics), without any knowledge of its internal workings. Its implementation is "opaque" (black). The te ...
or
white box models, according to how much
a priori
("from the earlier") and ("from the later") are Latin phrases used in philosophy to distinguish types of knowledge, justification, or argument by their reliance on empirical evidence or experience. knowledge is independent from current ...
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, 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
A neural network is a network or circuit of biological neurons, or, in a modern sense, an artificial neural network, composed of artificial neurons or nodes. Thus, a neural network is either a biological neural network, made up of biological ...
which 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 of
nonlinear system identification System identification is a method of identifying or measuring the mathematical model of a system from measurements of the system inputs and outputs. The applications of system identification include any system where the inputs and outputs can be me ...
[Billings S.A. (2013), ''Nonlinear System Identification: NARMAX Methods in the Time, Frequency, and Spatio-Temporal Domains'', Wiley.] can 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 on
intuition
Intuition is the ability to acquire knowledge without recourse to conscious reasoning. Different fields use the word "intuition" in very different ways, including but not limited to: direct access to unconscious knowledge; unconscious cognition; ...
,
experience
Experience refers to conscious events in general, more specifically to perceptions, or to the practical knowledge and familiarity that is produced by these conscious processes. Understood as a conscious event in the widest sense, experience involv ...
, or
expert opinion
An expert witness, particularly in common law countries such as the United Kingdom, Australia, and the United States, is a person whose opinion by virtue of education, training, certification, skills or experience, is accepted by the judge a ...
, or based on convenience of mathematical form.
Bayesian statistics
Bayesian statistics is a theory in the field of statistics based on the Bayesian interpretation of probability where probability expresses a ''degree of belief'' in an event. The degree of belief may be based on prior knowledge about the event, ...
provides a theoretical framework for incorporating such subjectivity into a rigorous analysis: we specify a
prior probability distribution
In Bayesian statistical inference, a prior probability distribution, often simply called the prior, of an uncertain quantity is the probability distribution that would express one's beliefs about this quantity before some evidence is taken int ...
(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 razor
Occam's razor, Ockham's razor, or Ocham's razor ( la, novacula Occami), also known as the principle of parsimony or the law of parsimony ( la, lex parsimoniae), is the problem-solving principle that "entities should not be multiplied beyond neces ...
is 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, including
numerical instability
In the mathematics, mathematical subfield of numerical analysis, numerical stability is a generally desirable property of numerical algorithms. The precise definition of stability depends on the context. One is numerical linear algebra and the oth ...
.
Thomas Kuhn
Thomas Samuel Kuhn (; July 18, 1922 – June 17, 1996) was an American philosopher of science whose 1962 book ''The Structure of Scientific Revolutions'' was influential in both academic and popular circles, introducing the term '' paradigm ...
argues that as science progresses, explanations tend to become more complex before a
paradigm shift
A paradigm shift, a concept brought into the common lexicon by the American physicist and philosopher Thomas Kuhn, is a fundamental change in the basic concepts and experimental practices of a scientific discipline. Even though Kuhn restricted t ...
offers 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's classical mechanics
Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classical ...
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
The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...
, and we study macro-particles only.
Note that better accuracy does not necessarily mean a better model.
Statistical models are prone to
overfitting
mathematical modeling, overfitting is "the production of an analysis that corresponds too closely or exactly to a particular set of data, and may therefore fail to fit to additional data or predict future observations reliably". An overfitt ...
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.
Training and tuning
Any model which is not pure white-box contains some
parameter
A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
s that can be used to fit the model to the system it is intended to describe. If the modeling is done by an
artificial neural network
Artificial neural networks (ANNs), usually simply called neural networks (NNs) or neural nets, are computing systems inspired by the biological neural networks that constitute animal brains.
An ANN is based on a collection of connected unit ...
or other
machine learning
Machine learning (ML) is a field of inquiry devoted to understanding and building methods that 'learn', that is, methods that leverage data to improve performance on some set of tasks. It is seen as a part of artificial intelligence.
Machine ...
, the optimization of parameters is called ''training'', while the optimization of model hyperparameters is called ''tuning'' and often uses
cross-validation. In more conventional modeling through explicitly given mathematical functions, parameters are often determined by ''
curve fitting
Curve fitting is the process of constructing a curve, or mathematical function, that has the best fit to a series of data points, possibly subject to constraints. Curve fitting can involve either interpolation, where an exact fit to the data is ...
''.
Model evaluation
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.
Fit to empirical data
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 referred to as
cross-validation in statistics.
Defining a
metric
Metric or metrical may refer to:
* Metric system, an internationally adopted decimal system of measurement
* An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement
Mathematics
In mathem ...
to measure distances between observed and predicted data is a useful tool for assessing model fit. In statistics, decision theory, and some
economic model
In economics, a model is a theoretical construct representing economic processes by a set of variables and a set of logical and/or quantitative relationships between them. The economic model is a simplified, often mathematical, framework desig ...
s, a
loss function
In mathematical optimization and decision theory, a loss function or cost function (sometimes also called an error function) is a function that maps an event or values of one or more variables onto a real number intuitively representing some "cost ...
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 model
A statistical model is a mathematical model that embodies a set of statistical assumptions concerning the generation of Sample (statistics), sample data (and similar data from a larger Statistical population, population). A statistical model repres ...
s than models involving
differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
s. Tools from
nonparametric statistics
Nonparametric statistics is the branch of statistics that is not based solely on parametrized families of probability distributions (common examples of parameters are the mean and variance). Nonparametric statistics is based on either being distr ...
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.
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 called
interpolation, and the same question for events or data points outside the observed data is called
extrapolation
In mathematics, extrapolation is a type of estimation, beyond the original observation range, of the value of a variable on the basis of its relationship with another variable. It is similar to interpolation, which produces estimates between know ...
.
As an example of the typical limitations of the scope of a model, in evaluating Newtonian
classical mechanics
Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classical ...
, 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 about
causality
Causality (also referred to as causation, or cause and effect) is influence by which one event, process, state, or object (''a'' ''cause'') contributes to the production of another event, process, state, or object (an ''effect'') where the cau ...
. 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 of
optimal foraging theory
Optimal foraging theory (OFT) is a behavioral ecology model that helps predict how an animal behaves when searching for food. Although obtaining food provides the animal with energy, searching for and capturing the food require both energy and t ...
do not offer insight that goes beyond the common-sense conclusions of
evolution
Evolution is change in the heritable characteristics of biological populations over successive generations. These characteristics are the expressions of genes, which are passed on from parent to offspring during reproduction. Variation ...
and other basic principles of ecology.
Significance in the natural sciences
Mathematical models are of great importance in the natural sciences, particularly in
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
. Physical
theories
A theory is a rational type of abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with such processes as observational study or research. Theories may be ...
are almost invariably expressed using mathematical models.
Throughout history, more and more accurate mathematical models have been developed.
Newton's laws
Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows:
# A body remains at rest, or in motion ...
accurately describe many everyday phenomena, but at certain limits
theory of relativity
The theory of relativity usually encompasses two interrelated theories by Albert Einstein: special relativity and general relativity, proposed and published in 1905 and 1915, respectively. Special relativity applies to all physical phenomena in ...
and
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
must be used.
It is common to use idealized models in physics to simplify things. Massless ropes, point particles,
ideal gases
An ideal gas is a theoretical gas composed of many randomly moving point particles that are not subject to interparticle interactions. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is a ...
and the
particle in a box
In quantum mechanics, the particle in a box model (also known as the infinite potential well or the infinite square well) describes a particle free to move in a small space surrounded by impenetrable barriers. The model is mainly used as a hypo ...
are among the many simplified models used in physics. The laws of physics are represented with simple equations such as Newton's laws,
Maxwell's equations
Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits.
...
and the
Schrödinger equation
The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of th ...
. These laws are a basis for making mathematical models of real situations. Many real situations are very complex and thus modeled approximate 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 by
molecular orbital
In chemistry, a molecular orbital is a mathematical function describing the location and wave-like behavior of an electron in a molecule. This function can be used to calculate chemical and physical properties such as the probability of findin ...
models that are approximate solutions to the Schrödinger equation. In
engineering
Engineering is the use of scientific method, scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad rang ...
, physics models are often made by mathematical methods such as
finite element analysis
The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat ...
.
Different mathematical models use different geometries that are not necessarily accurate descriptions of the geometry of the universe.
Euclidean geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small ...
is much used in classical physics, while
special relativity
In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates:
# The laws o ...
and
general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
are examples of theories that use
geometries
This is a list of geometry topics.
Types, methodologies, and terminologies of geometry.
* Absolute geometry
* Affine geometry
* Algebraic geometry
* Analytic geometry
* Archimedes' use of infinitesimals
* Birational geometry
* Complex geomet ...
which 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 in
simulation
A simulation is the imitation of the operation of a real-world process or system over time. Simulations require the use of Conceptual model, models; the model represents the key characteristics or behaviors of the selected system or proc ...
s.
A mathematical model usually describes a system by a
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
of variables and a set of equations that establish relationships between the variables. Variables may be of many types;
real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010) ...
or
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
numbers,
boolean values or
strings
String or strings may refer to:
*String (structure), a long flexible structure made from threads twisted together, which is used to tie, bind, or hang other objects
Arts, entertainment, and media Films
* ''Strings'' (1991 film), a Canadian anim ...
, for example. The variables represent some properties of the system, for example, the measured system outputs often in the form of
signals
In signal processing, a signal is a function that conveys information about a phenomenon. Any quantity that can vary over space or time can be used as a signal to share messages between observers. The ''IEEE Transactions on Signal Processing'' ...
,
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 in
computer science
Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical discipli ...
is the mathematical models of various machines, an example is the
deterministic finite automaton
In the theory of computation, a branch of theoretical computer science, a deterministic finite automaton (DFA)—also known as deterministic finite acceptor (DFA), deterministic finite-state machine (DFSM), or deterministic finite-state autom ...
(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:
![DFAexample](https://upload.wikimedia.org/wikipedia/commons/9/9d/DFAexample.svg)
:: ''M'' = (''Q'', Σ, δ, ''q''
0, ''F'') where
::*''Q'' = ,
::*Σ = ,
::*''q
0'' = ''S''
1,
::*''F'' = , and
::*δ is defined by the following
state transition table
State may refer to:
Arts, entertainment, and media Literature
* ''State Magazine'', a monthly magazine published by the U.S. Department of State
* ''The State'' (newspaper), a daily newspaper in Columbia, South Carolina, United States
* ''Our S ...
:
::::
: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 the
regular language
In theoretical computer science and formal language theory, a regular language (also called a rational language) is a formal language that can be defined by a regular expression, in the strict sense in theoretical computer science (as opposed to ...
given by the
regular expression
A regular expression (shortened as regex or regexp; sometimes referred to as rational expression) is a sequence of characters that specifies a search pattern in text. Usually such patterns are used by string-searching algorithms for "find" or ...
1*( 0 (1*) 0 (1*) )*, where "*" is the
Kleene star
In mathematical logic and computer science, the Kleene star (or Kleene operator or Kleene closure) is a unary operation, either on sets of strings or on sets of symbols or characters. In mathematics,
it is more commonly known as the free monoid ...
, 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 geographical
map projection
In cartography, map projection is the term used to describe a broad set of transformations employed to represent the two-dimensional curved surface of a globe on a plane. In a map projection, coordinates, often expressed as latitude and longitud ...
of 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 as
dead reckoning
In navigation, dead reckoning is the process of calculating current position of some moving object by using a previously determined position, or fix, and then incorporating estimates of speed, heading direction, and course over elapsed time. ...
when used more formally. Mathematical modeling in this way does not necessarily require formal mathematics; animals have been shown to use dead reckoning.
* ''
Population
Population typically refers to the number of people in a single area, whether it be a city or town, region, country, continent, or the world. Governments typically quantify the size of the resident population within their jurisdiction using a ...
Growth''. A simple (though approximate) model of population growth is the
Malthusian growth model
A Malthusian growth model, sometimes called a simple exponential growth model, is essentially exponential growth based on the idea of the function being proportional to the speed to which the function grows. The model is named after Thomas Robert ...
. A slightly more realistic and largely used population growth model is the
logistic function
A logistic function or logistic curve is a common S-shaped curve (sigmoid curve) with equation
f(x) = \frac,
where
For values of x in the domain of real numbers from -\infty to +\infty, the S-curve shown on the right is obtained, with the ...
, 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
and the trajectory, that is a function
, is the solution of the differential equation:
::
:that can be written also as:
::
: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,...,''n'' each with a market price ''p''
1, ''p''
2,..., ''p''
''n''. The consumer is assumed to have an
ordinal utility In economics, an ordinal utility function is a function representing the preferences of an agent on an ordinal scale. Ordinal utility theory claims that it is only meaningful to ask which option is better than the other, but it is meaningless to a ...
function ''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,..., ''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,..., ''x''
''n'' in such a way as to maximize ''U''(''x''
1, ''x''
2,..., ''x''
''n''). The problem of rational behavior in this model then becomes a
mathematical optimization
Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfi ...
problem, that is:
::
:: subject to:
::
::
: This model has been used in a wide variety of economic contexts, such as in
general equilibrium theory
In economics, general equilibrium theory attempts to explain the behavior of supply, demand, and prices in a whole economy with several or many interacting markets, by seeking to prove that the interaction of demand and supply will result in an ov ...
to show existence and
Pareto efficiency
Pareto efficiency or Pareto optimality is a situation where no action or allocation is available that makes one individual better off without making another worse off. The concept is named after Vilfredo Pareto (1848–1923), Italian civil engine ...
of economic equilibria.
* ''
Neighbour-sensing model
The Neighbour-Sensing mathematical model of hyphal growth is a set of interactive computer models that simulate the way fungi hyphae grow in three-dimensional space. The three-dimensional simulation is an experimental tool which can be used to stu ...
'' is a model that explains the
mushroom
A mushroom or toadstool is the fleshy, spore-bearing fruiting body of a fungus, typically produced above ground, on soil, or on its food source. ''Toadstool'' generally denotes one poisonous to humans.
The standard for the name "mushroom" is t ...
formation from the initially chaotic
fungal
A fungus ( : fungi or funguses) is any member of the group of eukaryotic organisms that includes microorganisms such as yeasts and molds, as well as the more familiar mushrooms. These organisms are classified as a kingdom, separately from th ...
network.
* In
computer science
Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical discipli ...
, mathematical models may be used to simulate computer networks.
* In
mechanics
Mechanics (from Ancient Greek: μηχανική, ''mēkhanikḗ'', "of machines") is the area of mathematics and physics concerned with the relationships between force, matter, and motion among physical objects. Forces applied to objects r ...
, mathematical models may be used to analyze the movement of a rocket model.
See also
*
Agent-based model
*
All models are wrong
All or ALL may refer to:
Language
* All, an indefinite pronoun in English
* All, one of the English determiners
* Allar language (ISO 639-3 code)
* Allative case (abbreviated ALL)
Music
* All (band), an American punk rock band
* ''All'' (All ...
*
Cliodynamics
Cliodynamics () is a transdisciplinary area of research that integrates cultural evolution, economic history/cliometrics, macrosociology, the mathematical modeling of historical processes during the ''longue durée'', and the construction and analy ...
*
Computer simulation
Computer simulation is the process of mathematical modelling, performed on a computer, which is designed to predict the behaviour of, or the outcome of, a real-world or physical system. The reliability of some mathematical models can be dete ...
*
Conceptual model
A conceptual model is a representation of a system. It consists of concepts used to help people knowledge, know, understanding, understand, or simulation, simulate a subject the model represents. In contrast, physical models are physical object su ...
*
Decision engineering
Decision intelligence is an engineering discipline that augments data science with theory from social science, decision theory, and managerial science. Its application provides a framework for best practices in organizational decision-making an ...
*
Grey box model
In mathematics, statistics, and computational modelling, a grey box modelKroll, Andreas (2000). Grey-box models: Concepts and application. In: New Frontiers in Computational Intelligence and its Applications, vol.57 of Frontiers in artificial int ...
*
International Mathematical Modeling Challenge
The International Mathematical Modeling Challenge (IMMC or sometimes IM2C) is an international mathematical modeling competition for high school students in team mathematical modeling co-sponsored by COMAP and the NeoUnion ESC Organization. Teams ...
*
Mathematical biology
*
Mathematical diagram
Mathematical diagrams, such as charts and graphs, are mainly designed to convey mathematical relationships—for example, comparisons over time.
Specific types of mathematical diagrams
Argand diagram
A complex number can be visually repres ...
*
Mathematical economics
*
Mathematical modelling of infectious disease
Mathematical models can project how infectious diseases progress to show the likely outcome of an epidemic (including in plants) and help inform public health and plant health interventions. Models use basic assumptions or collected statistics alo ...
*
Mathematical finance
Mathematical finance, also known as quantitative finance and financial mathematics, is a field of applied mathematics, concerned with mathematical modeling of financial markets.
In general, there exist two separate branches of finance that require ...
*
Mathematical psychology
Mathematical psychology is an approach to psychological research that is based on mathematical modeling of perceptual, thought, cognitive and motor processes, and on the establishment of law-like rules that relate quantifiable stimulus characte ...
*
Mathematical sociology
Mathematical sociology or the sociology of mathematics is an interdisciplinary field of research concerned both with the use of mathematics within sociological research as well as research into the relationships that exist between maths and socie ...
*
Microscale and macroscale models
Microscale models form a broad class of computational models that simulate fine-scale details, in contrast with macroscale models, which amalgamate details into select categories. Microscale and macroscale models can be used together to understan ...
*
Model inversion
An inverse problem in science is the process of calculating from a set of observations the causal factors that produced them: for example, calculating an image in X-ray computed tomography, source reconstruction in acoustics, or calculating the ...
*
Scientific model
Scientific modelling is a scientific activity, the aim of which is to make a particular part or feature of the world easier to understand, define, quantify, visualize, or simulate by referencing it to existing and usually commonly accepted ...
*
Sensitivity analysis
Sensitivity analysis is the study of how the uncertainty in the output of a mathematical model or system (numerical or otherwise) can be divided and allocated to different sources of uncertainty in its inputs. A related practice is uncertainty anal ...
*
Statistical model
A statistical model is a mathematical model that embodies a set of statistical assumptions concerning the generation of Sample (statistics), sample data (and similar data from a larger Statistical population, population). A statistical model repres ...
*
System identification
*
TK Solver
TK Solver (originally TK!Solver) is a mathematical modeling and problem solving software system based on a declarative, rule-based language, commercialized by Universal Technical Systems, Inc.
History
Invented by Milos Konopasek in the late ...
- Rule-based modeling
References
Further reading
Books
* Aris, Rutherford
1978 ( 1994 ). ''Mathematical Modelling Techniques'', New York: Dover.
* Bender, E.A.
1978 ( 2000 ). ''An Introduction to Mathematical Modeling'', New York: Dover.
*
Gary Chartrand
Gary Theodore Chartrand (born 1936) is an American-born mathematician who specializes in graph theory. He is known for his textbooks on introductory graph theory and for the concept of a
highly irregular graph.
Biography
Gary Chartrand was born ...
(1977) ''Graphs as Mathematical Models'', Prindle, Webber & Schmidt
* Dubois, G. (2018
"Modeling and Simulation" Taylor & Francis, CRC Press.
* Gershenfeld, N. (1998) ''The Nature of Mathematical Modeling'', Cambridge University Press .
* Lin, C.C. & Segel, L.A. ( 1988 ). ''Mathematics Applied to Deterministic Problems in the Natural Sciences'', Philadelphia: SIAM.
Specific applications
* Papadimitriou, Fivos. (2010). Mathematical Modelling of Spatial-Ecological Complex Systems: an Evaluation. Geography, Environment, Sustainability 1(3), 67-80.
*
*
An Introduction to Infectious Disease Modelling' by Emilia Vynnycky and Richard G White.
External links
;General reference
* Patrone, F
with critical remarks.
Brings together all articles on mathematical modeling from ''Plus 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?
{{DEFAULTSORT:Mathematical Model
Applied mathematics
Conceptual modelling
Knowledge representation
Mathematical modeling,
Mathematical terminology
Mathematical and quantitative methods (economics)