mathematical model

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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 purpo ...

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 used by humans, including speech (spoken language), gestures (Signed language, sign language) and writing. Most languages have a writing system composed of glyphs to inscribe the original soun ...
. 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 ...

s (such as
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. "Physical scien ...

,
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, Developmenta ...
,
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 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 composed of atoms, molecules and i ...
) 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 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 Algorithm, algorithmic proc ...
,
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 la ...
), as well as in non-physical systems such as the
social science Social science is the Branches of science, branch of science devoted to the study of society, societies and the Social relation, relationships among individuals within those societies. The term was formerly used to refer to the field of sociol ...

s (such as
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 services. ...
,
psychology Psychology is the science of mind and behavior. Psychology includes the study of consciousness, conscious and Unconscious mind, unconscious phenomena, as well as feeling and thought. It is an academic discipline of immense scope. Psychologis ...

,
sociology Sociology is a social science Social science is the Branches of science, branch of science devoted to the study of society, societies and the Social relation, relationships among individuals within those societies. The term was formerly ...
,
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 art of arranging sounds in time through the elements of melody, harmony, rhythm, and timbre. It is one of the universal cultural aspects of all human societies. General definitions of music include common elements such as pit ...

,
linguistics Linguistics is the science, scientific study of language. It encompasses the analysis of every aspect of language, as well as the methods for studying and modeling them. The traditional areas of linguistic analysis include phonetics, phonet ...

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

(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 reason, Metaphysics, existence, Epistemology, knowledge, Ethics, values, Philosophy o ...
). 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 modelA 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, su ...
s,
differential equations In mathematics, a differential equation is an equation In mathematics, an equation is a statement that asserts the equality (mathematics), equality of two Expression (mathematics), expressions, which are connected by the equals sign "=". The wo ...
, 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, or a paragraph that is larger than the rest of the text. The word is derived from the Latin Latin (, or , ) is a classical language bel ...
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 ...

# 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 spa ...
'', such as algebraic operators, functions, differential operators, etc. Variables are abstractions of system
parameters A parameter (from the Ancient Greek language, Ancient Greek wikt:παρά#Ancient Greek, παρά, ''para'': "beside", "subsidiary"; and wikt:μέτρον#Ancient Greek, μέτρον, ''metron'': "measure"), generally, is any characteristic that ...
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 ...

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, a linear equation is an equation that may be put in the form :a_1x_1+\cdots +a_nx_n+b=0, where x_1, \ldots, x_n are the variable (mathematics), variables (or unknown (mathematics), unknowns), and b, a_1, \ldots, a_n are the coeffic ...

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 that is not reversible is called irreversible. This concept arises frequently in thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relatio ...
. 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 equation that relates one or more function (mathematics), functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates ...
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 ...

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 '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 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 modelA 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, su ...
; 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 view that all events are determined completely by previously existing causes. Deterministic theories throughout the history of philosophy have sprung from diverse and sometimes overlapping motives and consider ...
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 modelA 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, su ...
"—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 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 models of strategic interactions among Rational agent, rational agents.Roger B. Myerson, Myerson, Roger B. (1991). ''Game Theory: Analysis of Conflict,'' Harvard University Press, p.&nbs1 Chapter-preview ...
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 Forbes Nash Jr., is the most common way to define the solution concept, solution of a non-cooperative game involving two or more players. In a Nash equilibrium, each player ...
. 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 trade ...
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 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 In probability and 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 problem, it is conventio ...
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 whol ...

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 ...

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 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 field there are m ...
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 (math ...
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 softwa ...
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 term ...
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 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 ( ...

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, awareness, or understanding of someone or something, such as facts ( descriptive knowledge), skills (procedural knowledge), or objects (Knowledge by acquaintance, acqu ...
,
experience Experience refers to conscious , an English Paracelsian physician Consciousness, at its simplest, is " sentience or awareness of internal and external existence". Despite millennia of analyses, definitions, explanations and debates by philosoph ...

, 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 statistical inference, a prior probability distribution, often simply called the prior, of an uncertain quantity is the probability distribution In probability theory and statistics Statistics is the discipline that concern ...
(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 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 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 identified by the American physicist and philosopher Thomas Kuhn Thomas Samuel Kuhn (; July 18, 1922 – June 17, 1996) was an American philosopher of science whose 1962 book '' The Structure of Scientific Revo ...
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, 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, commonly denoted , is a universal physical constant important in many areas of physics. Its exact value is defined as (approximately ). It is exact because, by international agreement, a Metre#Speed of light def ...
, 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 function. Although the polynomial function is a perfect fit, the linear function can be expected to generalize better: if the two functions were used to extrapolate beyond the fitted data, the linear function should make better predictions. In ...

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 whol ...

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 brain A brain is an organ (anatomy), organ that serves as the ...

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, ...

, 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 ma ...

''.

## 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, a model is a theory, theoretical construct representing economic wikt:process, processes by a set of Variable (mathematics), variables and a set of logical and/or quantitative relationships between them. The economic Conceptual 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 modelA 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, su ...
s than models involving
differential equation In mathematics, a differential equation is an equation that relates one or more function (mathematics), functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates ...
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 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 populatio ...

, 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 ...

. 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, ...

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 Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical scien ...

. Physical
theories A theory is a rational type of abstract thinking about a phenomenon A phenomenon (; plural phenomena) is an observable fact or event. The term came into its modern philosophical usage through Immanuel Kant Immanuel Kant (, ; ; 22 April ...

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: special relativity and general relativity. Special relativity applies to all physical phenomena in the absence of gravity. General relativity explains th ...
and
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 physics including quan ...
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 ...

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 equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. The equations provide a mathematic ...
and the
Schrödinger equation The Schrödinger equation is a 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 ...
. 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 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 composed of atoms, ...
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 widely used method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, ...
. 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 Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method consists in assuming a smal ...
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 Spacetime, space and time. In Albert Einstein's original treatment, the theory is based on two Postulates of ...
and
general relativity General relativity, also known as the general theory of relativity, is the differential geometry, geometric scientific theory, theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern ph ...
are examples of theories that use 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 process, wh ...

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 belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of t ...
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 Algorithm, algorithmic proc ...
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-sea ...
1*( 0 (1*) 0 (1*) )*, where "*" is the
Kleene star In mathematical logic Mathematical logic, also called formal logic, is a subfield of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (alg ...
, 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: l ...
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 ce ...

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: :: :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 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= Mathematical optimization (alter ...
problem, that is: :: $\max U\left(x_1,x_2,\ldots, x_n\right)$ :: 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 ove ...
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 formation from the initially chaotic fungus, 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 Algorithm, algorithmic proc ...
, mathematical models may be used to simulate computer networks. * In mechanics, mathematical models may be used to analyze the movement of a rocket model.

* Agent-based model * All models are wrong * Cliodynamics * Computer simulation * Conceptual model * Decision engineering * Grey box model * International Mathematical Modeling Challenge * Mathematical biology * Mathematical diagram * Mathematical economics * Mathematical modelling of infectious disease * Mathematical finance * Mathematical psychology * Mathematical sociology * Microscale and macroscale models * Model inversion * Scientific model * Sensitivity analysis * Statistical model * System identification * TK Solver - Rule-based modeling

# References

## 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.