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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, is described by its boundaries, structure and purp ...

system
using
mathematical Mathematics (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approximately 10.7 ...
concepts and
language A language is a structured system of communication Communication (from Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the ...
. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used in the
natural science Natural science is a Branches of science, branch of science concerned with the description, understanding and prediction of Phenomenon, natural phenomena, based on empirical evidence from observation and experimentation. Mechanisms such as peer r ...

natural science
s (such as
physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succession of eve ...

physics
,
biology Biology is the natural science that studies life and living organisms, including their anatomy, physical structure, Biochemistry, chemical processes, Molecular biology, molecular interactions, Physiology, physiological mechanisms, Development ...

biology
,
earth science Earth science or geoscience includes all fields of natural science Natural science is a branch of science Science (from the Latin word ''scientia'', meaning "knowledge") is a systematic enterprise that Scientific method, builds and Ta ...
,
chemistry Chemistry is the study of the properties and behavior of . It is a that covers the that make up matter to the composed of s, s and s: their composition, structure, properties, behavior and the changes they undergo during a with other . ...

chemistry
) and
engineering Engineering is the use of 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 range of more specializ ...

engineering
disciplines (such as
computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application. Computer science is the study of , , and . Computer science ...
,
electrical engineering Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems which use electricity, electronics The field of electronics is a branch of physics and electrical enginee ...

electrical engineering
), as well as in non-physical systems such as the
social science Social science is the branch A branch ( or , ) or tree branch (sometimes referred to in botany Botany, also called , plant biology or phytology, is the science of plant life and a branch of biology. A botanist, plant scientist o ...

social science
s (such as
economics Economics () is a social science Social science is the branch A branch ( or , ) or tree branch (sometimes referred to in botany Botany, also called , plant biology or phytology, is the science of plant life and a bran ...

economics
,
psychology Psychology is the scientific Science () is a systematic enterprise that builds and organizes knowledge Knowledge is a familiarity or awareness, of someone or something, such as facts A fact is an occurrence in the real world. ...

psychology
,
sociology Sociology is a social science Social science is the branch The branches and leaves of a tree. A branch ( or , ) or tree branch (sometimes referred to in botany Botany, also called , plant biology or phytology, is the scie ...
,
political science Political science is the scientific study of politics Politics (from , ) is the set of activities that are associated with making decisions in groups, or other forms of power relations between individuals, such as the distribution of ...
). The use of mathematical models to solve problems in business or military operations is a large part of the field of operations research. Mathematical models are also used in
music Music is the of arranging s in time through the of melody, harmony, rhythm, and timbre. It is one of the aspects of all human societies. General include common elements such as (which governs and ), (and its associated concepts , , and ...

music
,
linguistics Linguistics is the scientific study of language, meaning that it is a comprehensive, systematic, objective, and precise study of language. Linguistics encompasses the analysis of every aspect of language, as well as the methods for studying ...

linguistics
, and
philosophy Philosophy (from , ) is the study of general and fundamental questions, such as those about Metaphysics, existence, reason, Epistemology, knowledge, Ethics, values, Philosophy of mind, mind, and Philosophy of language, language. Such questio ...

philosophy
(for example, intensively in
analytic philosophy Analytic philosophy is a branch and tradition of philosophy Philosophy (from , ) is the study of general and fundamental questions, such as those about existence Existence is the ability of an entity to interact with physical reality ...
). 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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
,
statistical model A statistical model is a mathematical model 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 ...
s,
differential equations In mathematics, a differential equation is an equation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), ...
, 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 abiotic component, 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". ...
, a traditional mathematical model contains most of the following elements: #
Governing equationThe governing equations of a mathematical model 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 ...
s # Supplementary sub-models ## Defining equations ##
Constitutive equation In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior through Sp ...
s # Assumptions and constraints ##
Initial In a written or published work, an initial or drop cap is a letter at the beginning of a word, a chapter (books), chapter, or a paragraph that is larger than the rest of the text. The word is derived from the Latin ''initialis'', which means ''sta ...
and
boundary condition Boundary or Boundaries may refer to: * Border, in political geography Entertainment * ''Boundaries'' (2016 film), a 2016 Canadian film * ''Boundaries'' (2018 film), a 2018 American-Canadian road trip film Mathematics and physics * Boundary (top ...
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-effecto ...

kinematic equations


Classifications

Mathematical models are usually composed of relationships and '' variables''. Relationships can be described by ''
operators Operator may refer to: Mathematics * A symbol indicating a mathematical operation * Logical operator or logical connective in mathematical logic * Operator (mathematics), mapping that acts on elements of a space to produce elements of another sp ...
'', such as algebraic operators, functions, differential operators, etc. Variables are abstractions of system
parameters A parameter (), generally, is any characteristic that can help in defining or classifying a particular system A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified wh ...
of interest, that can be quantified. Several classification criteria can be used for mathematical models according to their structure: * Linear vs. nonlinear: If all the operators in a mathematical model exhibit
linear Linearity is the property of a mathematical relationship (''function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out se ...

linear
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 operator 300px, A harmonic function defined on an annulus. Harmonic functions are exactly those functions which lie in the kernel of the Laplace operator, an important differential operator. In mathematics, a differential operator is an Operator (mathe ...
s, but it can still have nonlinear expressions in it. In a
mathematical programming Nelder-Mead minimum search of Simionescu's function. Simplex vertices are ordered by their values, with 1 having the lowest ( best) value., alt= Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming i ...
model, if the objective functions and constraints are represented entirely by
linear equation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

linear equation
s, 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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
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 (''Kinnikuman'') * Chaos (Sailor Moon), Chaos (''Sailor Moon'') * Chaos (Sesame Park), Chaos (''Sesame Park'') * Chaos (Warhammer), Chaos ('' ...
and
irreversibility In science, a process A process is a series or set of Action (philosophy), activities that interact to produce a result; it may occur once-only or be recurrent or periodic. Things called a process include: Business and management *Business proc ...
. 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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
, 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 functional equation, equation that relates one or more function (mathematics), functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives ...

differential equation
s or
difference equation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s. * 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 Sir Isaac Newton (25 December 1642 – 20 March Old Style and New Style dates, 1726/27) was an English mathematician, physic ...

Newton's method
or
Broyden's method In numerical analysis, Broyden's method is a quasi-Newton method for root-finding algorithm, finding roots in variables. It was originally described by Charles George Broyden, C. G. Broyden in 1965. Newton's method for solving uses the Jacobian m ...
. 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 A reaction engine is an engine or motor that produces thrust Thrust is a described quantitatively by . When a system expels or in one direction, the accelerated mass will cause a force of ...

jet engine
'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 es that involve and into and out of the system, while varying pressure, temperature, and other within the system, and that eventually returns the to its initial state. In the process of p ...
(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 modelA molecular model, in this article, is a physical model that represents molecules A scanning tunneling microscopy image of pentacene molecules, which consist of linear chains of five carbon rings. A molecule is an electrically neutral group ...
or the states in a
statistical model A statistical model is a mathematical model 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 ...
; while a continuous model 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 the philosophical Philosophy (from , ) is the study of general and fundamental questions, such as those about existence Existence is the ability of an entity to interact with physical or mental reality Reality is the ...
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 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 ...
"—randomness is present, and variable states are not described by unique values, but rather by
probability Probability is the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained ...

probability
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 mathematics, catastrophe theory is a branch of bifurcation theory in the study of dynamical systems; it is also a particular special case of more general singularity theory in geometry. Bifurcation theory studies and classifies phenomena char ...
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 model 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. ...
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 Game theory is the study of mathematical model 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 ...
. 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 activity of making one's living or making money by producing or buying and selling products (such as goods and services). Simply put, it is "any activity or enterprise entered into for profit." Having a business name A trad ...

business
and
engineering Engineering is the use of 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 range of more specializ ...

engineering
, 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 In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of ...
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 An economy (; ) ...
variables, and
random variable A random variable is a variable whose values depend on outcomes of a random In common parlance, randomness is the apparent or actual lack of pattern or predictability in events. A random sequence of events, symbols or steps often has no ...
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 A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified wh ...

parameter
s or
constant Constant or The Constant may refer to: Mathematics * Constant (mathematics) In mathematics, the word constant can have multiple meanings. As an adjective, it refers to non-variance (i.e. unchanging with respect to some other Value (mathematics ...
s. 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 Briti ...

Objective
s and constraints of the system and its users can be represented as
function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
s of the output variables or state variables. The
objective functionIn mathematical optimization and decision theory, a loss function or cost function (sometimes also called an error function) is a function that maps an event (probability theory), event or values of one or more variables onto a real number intuitive ...
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 science Social science is the branch A branch ( or , ) or tree branch (sometimes referred to in botany Botany, also called , plant biology or phytology, is the s ...

economist
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 matrix (mat ...
when using
input-output model In computing Computing is any goal-oriented activity requiring, benefiting from, or creating computing machinery. It includes the study and experimentation of algorithmic processes and development of both computer hardware , hardware and soft ...
s. Complicated mathematical models that have many variables may be consolidated by use of
vectors Vector may refer to: Biology *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism; a disease vector *Vector (molecular biology), a DNA molecule used as a vehicle to artificially carr ...
where one symbol represents several variables.


''A priori'' information

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 ter ...

black box
or white box models, according to how much
a priori ''A priori'' and ''a posteriori'' ('from the earlier' and 'from the later', respectively) are Latin phrases used in philosophy Philosophy (from , ) is the study of general and fundamental questions, such as those about reason, Metaph ...
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
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#REDIRECT Artificial neural network Artificial neural networks (ANNs), usually simply called neural networks (NNs), are computing systems vaguely inspired by the biological neural networks that constitute animal brain A brain is an organ ( ...

neural networks
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 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 rul ...
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 Knowledge is a familiarity or awareness, of someone or something, such as facts A fact is something that is truth, true. The usual test for a statement of fact is verifiability—that is ...
,
experience Experience refers to conscious , an English Paracelsian Paracelsianism (also Paracelsism; German: ') was an early modern History of medicine, medical movement based on the theories and therapies of Paracelsus. It developed in the second half ...

experience
, or
expert opinion An expert witness, particularly in common law countries such as the United Kingdom The United Kingdom of Great Britain and Northern Ireland, commonly known as the United Kingdom (UK) or Britain,Usage is mixed. The Guardian' and Telegrap ...
, or based on convenience of mathematical form.
Bayesian statistics Bayesian statistics is a theory in the field of statistics based on the Bayesian probability, Bayesian interpretation of probability where probability expresses a ''degree of belief'' in an Event (probability theory), event. The degree of belief m ...
provides a theoretical framework for incorporating such subjectivity into a rigorous analysis: we specify a
prior probability distribution In Bayesian Thomas Bayes (/beɪz/; c. 1701 – 1761) was an English statistician, philosopher, and Presbyterian minister. Bayesian () refers to a range of concepts and approaches that are ultimately based on a degree-of-belief interpretation o ...
(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, 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 necessi ...
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 mathematical Mathematics (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population ...
.
Thomas Kuhn Thomas Samuel Kuhn (; July 18, 1922 – June 17, 1996) was an American whose 1962 book ' was influential in both academic and popular circles, introducing the term ', which has since become an English-language idiom. Kuhn made several cla ...
argues that as science progresses, explanations tend to become more complex before a
paradigm shift A paradigm shift, a concept identified by the American physicist and philosopher Thomas Kuhn Thomas Samuel Kuhn (; July 18, 1922 – June 17, 1996) was an American whose 1962 book ' was influential in both academic and popular circles, ...
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
Newton's
classical mechanics is an approximated model of the real world. Still, Newton's model is quite sufficient for most ordinary-life situations, that is, as long as particle speeds are well below the
speed of light The speed of light in vacuum A vacuum is a space Space is the boundless three-dimensional Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called paramet ...
, and we study macro-particles only. Note that better accuracy does not necessarily mean a better model.
Statistical modelsA 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 represen ...
are prone to
overfitting In statistics, 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 additional data or predict future observations reliably". An overfitted model is a ...

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.


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 A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified wh ...

parameter
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), are computing systems vaguely inspired by the biological neural networks that constitute animal brains. An ANN is based on a collection of connected units or nod ...

artificial neural network
or other
machine learning Machine learning (ML) is the study of computer algorithms that can improve automatically through experience and by the use of data. It is seen as a part of artificial intelligence. Machine learning algorithms build a model based on sample data ...

machine learning
, 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 In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve m ...

curve fitting
''.


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 (Mapping EvapoTranspiration at high Resolution with Internalized Calibration) is a computer model Computer simulation is the process of mathematical modelling, performed on a computer, which is designed to predict the behaviour of or th ...
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 Economics () is a social science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services. Economics focuses on the behav ...

economic model
s, a
loss functionIn mathematical optimization File:Nelder-Mead Simionescu.gif, Nelder-Mead minimum search of Test functions for optimization, Simionescu's function. Simplex vertices are ordered by their values, with 1 having the lowest ( best) value., alt= Math ...
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 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 ...
s than models involving
differential equation In mathematics, a differential equation is an functional equation, equation that relates one or more function (mathematics), functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives ...

differential equation
s. Tools from
nonparametric statisticsNonparametric statistics is the branch of statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social prob ...
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 In the mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantitie ...

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 known ...

extrapolation
. As an example of the typical limitations of the scope of a model, in evaluating Newtonian classical mechanics, we can note that Newton made his measurements without advanced equipment, so he could not measure properties of particles travelling at speeds close to the speed of light. Likewise, he did not measure the movements of molecules and other small particles, but macro particles only. It is then not surprising that his model does not extrapolate well into these domains, even though his model is quite sufficient for ordinary life physics.


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 (relativity), event, process, state or object (a ''cause'') contributes to the production of another event, process, state or object (an ''effect'') ...
. 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 ti ...
do not offer insight that goes beyond the common-sense conclusions of
evolution Evolution is change in the heritable Heredity, also called inheritance or biological inheritance, is the passing on of Phenotypic trait, traits from parents to their offspring; either through asexual reproduction or sexual reproduction, ...

evolution
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 that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succession of eve ...

physics
. Physical
theories A theory is a reason, rational type of abstraction, 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 researc ...

theories
are almost invariably expressed using mathematical models. Throughout history, more and more accurate mathematical models have been developed.
Newton's laws In classical mechanics, Newton's laws of motion are three law Law is 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, surround ...
accurately describe many everyday phenomena, but at certain limits
theory of relativity The theory of relativity usually encompasses two interrelated theories by Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born , widely acknowledged to be one of the greatest physicists of all time ...
and
quantum mechanics Quantum mechanics is a fundamental theory A theory is a reason, rational type of abstraction, abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with ...
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 Gas is one of the four fundamental states of matter (the others being solid, liquid A liquid is a nearly incompressible fluid In physics, a fluid is a substance that continually Deformation (mecha ...
and the
particle in a box In quantum mechanics Quantum mechanics is a fundamental Scientific theory, 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 ...

particle in a box
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 are a set of coupled partial differential equation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), ...
and the
Schrödinger equation The Schrödinger equation is a linear Linearity is the property of a mathematical relationship (''function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a ma ...
. 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 Function (mathematics), mathematical function describing the location and Matter wave, wave-like behavior of an electron in a molecule. This function can be used to calculate chemical and physical propertie ...
models that are approximate solutions to the Schrödinger equation. In
engineering Engineering is the use of 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 range of more specializ ...

engineering
, physics models are often made by mathematical methods such as
finite element analysis The finite element method (FEM) is a widely used method for numerically solving differential equations In mathematics, a differential equation is an equation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the stud ...
. 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 Alexandria Alexandria ( or ; ar, الإسكندرية ; arz, اسكندرية ; Coptic Coptic may refer to: Afro-Asia * Copts, an ethnoreligious group mainly in the area of modern ...
is much used in classical physics, while
special relativity In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force ...
and
general relativity General relativity, also known as the general theory of relativity, is the geometric Geometry (from the grc, γεωμετρία; '' geo-'' "earth", '' -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathema ...
are examples of theories that use
geometries
geometries
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 models; the model represents the key characteristics or behaviors of the selected system or process, whereas the simulat ...

simulation
s. A mathematical model usually describes a system by a set of variables and a set of equations that establish relationships between the variables. Variables may be of many types;
real Real may refer to: * Reality Reality is the sum or aggregate of all that is real or existent within a system, as opposed to that which is only Object of the mind, imaginary. The term is also used to refer to the ontological status of things, ind ...
or
integer An integer (from the Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share" or "to ...
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), ''Strings'' (1991 fil ...
, for example. The variables represent some properties of the system, for example, measured system outputs often in the form of signals, timing data, counters, and event occurrence (yes/no). 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 deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application. Computer science is the study of , , and . Computer science ...
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 An artistic representation of a Turing machine. Turing machines are used to model general computing devices. Theoretical computer science (TCS) is a subset of general com ...
(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: :: ''M'' = (''Q'', Σ, δ, ''q''0, ''F'') where ::*''Q'' = , ::*Σ = , ::*''q0'' = ''S''1, ::*''F'' = , and ::*δ is defined by the following
state transition table In automata theory Automata theory is the study of abstract machines and automata, as well as the computational problem In theoretical computer science An artistic representation of a Turing machine. Turing machines are used to model general ...
: :::: :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 An artistic representation of a Turing machine. Turing machines are used to model general computing devices. Theoretical computer science (TCS) is a subset of general computer science that focuses on mathematica ...
given by the
regular expression A regular expression (shortened as regex or regexp; also referred to as rational expression) is a sequence of Character (computing), characters that specifies a ''search pattern matching, pattern''. Usually such patterns are used by string-se ...
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 Set (mathematics), sets of string (computer science), strings or on sets of symbols or characters. In mathematics it ...
, 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, a map projection is a way to flatten a globe's surface into a plane in order to make a map. This requires a systematic transformation of the latitudes and longitudes of locations from the Surface (mathematics), surface of the globe ...
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 Navigation is a field of study that focuses on the process of monitoring and controlling the movement of a craft or vehicle from one place to another.Bowditch, 2003:799. The field of navigation includes four general categories: ...
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 the number of people in a single area whether it be a city or town, region, country, or the world. Governments typically quantify the size of the resident population within their jurisdiction by a process called a ...

Population
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 Exponential growth is a process that increases quantity over time. It occurs when the instantaneous rate of change (that is, the deri ...
. 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 A sigmoid function is a function (mathematics), mathematical function having a characteristic "S"-shaped curve or sigmoid curve. A common example of a sigmoid f ...
, 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^3\!\rightarrow\mathbb and the trajectory, that is a function \mathbf\!:\mathbb\rightarrow\mathbb^3, is the solution of the differential equation: :: -\fracm=\frac\mathbf+\frac\mathbf+\frac\mathbf, :that can be written also as: :: m\frac=-\nabla V mathbf(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,...,''n'' each with a market price ''p''1, ''p''2,..., ''p''''n''. The consumer is assumed to have an
ordinal utilityIn economics Economics () is the social science that studies how people interact with value; in particular, the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and ...
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. Optimization problems of sorts arise i ...
problem, that is: :: \max U(x_1,x_2,\ldots, x_n) :: subject to: :: \sum_^n p_i x_i \leq M. :: x_ \geq 0 \; \; \; \forall i \in \ : 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 individual or preference criterion can be better off without making at least one individual or preference criterion worse off or without any loss thereof. The concept is named after Vi ...
of economic equilibria. * '' Neighbour-sensing model'' is a model that explains the
mushroom A mushroom or toadstool is the fleshy, spore )'', growing on a thinning, thinned hybrid black poplar ''(populus, Populus x canadensis)''. The last stage of the moss#Life cycle, moss lifecycle is shown, where the sporophytes are visible befor ...

mushroom
formation from the initially chaotic
fungal A fungus (plural The plural (sometimes list of glossing abbreviations, abbreviated ), in many languages, is one of the values of the grammatical number, grammatical category of number. The plural of a noun typically denotes a quantity great ...

fungal
network. * In
computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application. Computer science is the study of , , and . Computer science ...
, mathematical models may be used to simulate computer networks. * In
mechanics Mechanics (Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approximat ...

mechanics
, mathematical models may be used to analyze the movement of a rocket model.


See also

*
Agent-based model An agent-based model (ABM) is a class of computational models A computational model uses computer programs to simulate and study complex systems using an algorithmic or mechanistic approach and is widely used in a diverse range of fields spanning ...
*
All models are wrong "All models are wrong" is a common aphorism An aphorism (from Greek ἀφορισμός: ''aphorismos'', denoting 'delimitation', 'distinction', and 'definition') is a concise, terse, laconic, or memorable expression of a general truth or princ ...
*
Cliodynamics Cliodynamics () is a transdisciplinary area of research that integrates cultural evolution, economic history Economic history is the academic study of economies or economic events of the past. Research is conducted using a combination of histor ...
*
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 determ ...
*
Conceptual model A conceptual model is a depiction, representation of a system. It consists of concepts used to help people knowledge, know, understanding, understand, or simulation, simulate a subject the model represents. It is also a set of concepts. In contras ...

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 * Scientific model * Sensitivity analysis * Statistical model * System identification * TK Solver - 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 (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)