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Encyclopedia > Mathematical model

A mathematical model uses mathematical language to describe a system. Mathematical models are used particularly in the natural sciences and engineering disciplines (such as physics, biology, meteorology, and electrical engineering) but also in the social sciences (such as economics, sociology and political science); physicists, engineers, computer scientists, and economists use mathematical models most extensively. For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... For other uses, see System (disambiguation). ... The Michelsonâ€“Morley experiment was used to disprove that light propagated through a luminiferous aether. ... Engineering is the discipline and profession of applying scientific knowledge and utilizing natural laws and physical resources in order to design and implement materials, structures, machines, devices, systems, and processes that realize a desired objective and meet specified criteria. ... A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ... For the song by Girls Aloud see Biology (song) Biology studies the variety of life (clockwise from top-left) E. coli, tree fern, gazelle, Goliath beetle Biology (from Greek: Î’Î¹Î¿Î»Î¿Î³Î¯Î± - Î²Î¯Î¿Ï‚, bio, life; and Î»ÏŒÎ³Î¿Ï‚, logos, speech lit. ... // Meteorology (from Greek: Î¼ÎµÏ„Î­Ï‰ÏÎ¿Î½, meteoron, high in the sky; and Î»ÏŒÎ³Î¿Ï‚, logos, knowledge) is the interdisciplinary scientific study of the atmosphere that focuses on weather processes and forecasting. ... Electrical Engineers design power systems. ... The social sciences are groups of academic disciplines that study the human aspects of the world. ... Face-to-face trading interactions on the New York Stock Exchange trading floor. ... Sociology (from Latin: socius, companion; and the suffix -ology, the study of, from Greek Î»ÏŒÎ³Î¿Ï‚, lÃ³gos, knowledge [1]) is the scientific or systematic study of society, including patterns of social relationships, social interaction, and culture[2]. Areas studied in sociology can range from the analysis of brief contacts between anonymous... The Politics series Politics Portal This box:      Political Science is the field concerning the theory and practice of politics and the description and analysis of political systems and political behaviour. ... Not to be confused with physician, a person who practices medicine. ... Look up engineer in Wiktionary, the free dictionary. ... Computer science, or computing science, is the study of the theoretical foundations of information and computation and their implementation and application in computer systems. ... Alan Greenspan, former chairman, United States Federal Reserve. ...

Eykhoff (1974) defined a mathematical model as 'a representation of the essential aspects of an existing system (or a system to be constructed) which presents knowledge of that system in usable form'. For other uses, see System (disambiguation). ...

Mathematical models can take many forms, including but not limited to dynamical systems, statistical models, differential equations, or game theoretic models. These and other types of models can overlap, with a given model involving a variety of abstract structures. In engineering and mathematics, a dynamical system is a deterministic process in which a functions value changes over time according to a rule that is defined in terms of the functions current value. ... A statistical model is used in applied statistics. ... In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables. ... For other uses, see Game theory (disambiguation) and Game (disambiguation). ...

## Examples of mathematical models

• Population Growth. A simple (though approximate) model of population growth is the Malthusian growth model. The preferred population growth model is the logistic function.
• Model of a particle in a potential-field. In this model we consider a particle as being a point of mass m which describes a trajectory which is modeled by a function x: RR3 given its coordinates in space as a function of time. The potential field is given by a function V:R3R and the trajectory is a solution of the differential equation
$m frac{d^2}{dt^2} x(t) = - operatorname{grad} left( V right) (x(t)).$
Note this model assumes the particle is a point mass, which is certainly known to be false in many cases 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 p1, p2,..., pn. The consumer is assumed to have a cardinal utility function U (cardinal in the sense that it assigns numerical values to utilities), depending on the amounts of commodities x1, x2,..., xn consumed. The model further assumes that the consumer has a budget M which she uses to purchase a vector x1, x2,..., xn in such a way as to maximize U(x1, x2,..., xn). The problem of rational behavior in this model then becomes an optimization problem, that is:
$max U(x_1,x_2,ldots, x_n)$
subject to:
$sum_{i=1}^n p_i x_i leq M.$
$x_{i} geq 0 ; ; ; forall i in {1, 2, ldots, n }$
This model has been used in general equilibrium theory, particularly to show existence and Pareto optimality of economic equilibria. However, the fact that this particular formulation assigns numerical values to levels of satisfaction is the source of criticism (and even ridicule). However, it is not an essential ingredient of the theory and again this is an idealization.
• Neighbour-sensing model explains the mushroom formation from the initially chaotic fungal network.

The Malthusian growth model, sometimes called the simple exponential growth model, is essentially exponential growth based on a constant rate of compound interest. ... Logistic curve, specifically the sigmoid function A logistic function or logistic curve models the S-curve of growth of some set P. The initial stage of growth is approximately exponential; then, as competition arises, the growth slows, and at maturity, growth stops. ... Visualization of airflow into a duct modelled using the Navier-Stokes equations, a set of partial differential equations. ... In mathematics, the term optimization, or mathematical programming, refers to the study of problems in which one seeks to minimize or maximize a real function by systematically choosing the values of real or integer variables from within an allowed set. ... General Equilibrium (linear) supply and demand curves. ... Pareto efficiency, or Pareto optimality, is a central concept in game theory with broad applications in economics, engineering and the social sciences. ... Development of the cone like structure (view above, slice below) Neighbour-Sensing model - the proposed hypothesis of the fungal morphogenesis. ... For other uses, see Mushroom (disambiguation). ... Divisions Chytridiomycota Zygomycota Ascomycota Basidiomycota The Fungi (singular: fungus) are a large group of organisms ranked as a kingdom within the Domain Eukaryota. ...

## Background

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 simulations. This article is about the general term. ...

A mathematical model usually describes a system by a set of variables and a set of equations that establish relationships between the variables. The values of the variables can be practically anything; real or integer numbers, boolean values or strings, 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. In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... Not to be confused with Natural number. ... In computer science, the Boolean datatype, sometimes called the logical datatype, is a primitive datatype having one of two values: non-zero (often 1, or -1) and zero (which are equivalent to true and false, respectively). ... In information theory, a signal is the sequence of states of a communications channel that encodes a message. ...

## Building blocks

There are six basic groups of variables: decision variables, input variables, state variables, exogenous variables, random variables, and output variables. Since there can be many variables of each type, the variables are generally represented by vectors.

Decision variables are sometimes known as independent variables. Exogenous variables are sometimes known as parameters or constants. The variables are not independent of each other as the state variables are dependent on the decision, input, random, and exogenous variables. Furthermore, the output variables are dependent on the state of the system (represented by the state variables).

Objectives and constraints of the system and its users can be represented as functions of the output variables or state variables. The objective functions will depend on the perspective of the model's user. Depending on the context, an objective function is also known as an index of performance, as it is some 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).

## Classifying mathematical models

Many mathematical models can be classified in some of the following ways:

1. Linear vs. nonlinear: Mathematical models are usually composed by variables, which are abstractions of quantities of interest in the described systems, and operators that act on these variables, which can be algebraic operators, functions, differential operators, etc. If all the operators in a mathematical model present linearity, the resulting mathematical model is defined as linear. A model is considered to be nonlinear otherwise.
The question of linearity and nonlinearity is dependent on context, and linear models may have nonlinear expressions in them. For example, in a statistical linear model, it is assumed that a relationship is linear in the parameters, but it may be nonlinear in the predictor variables. Similarly, a differential equation is said to be linear if it can be written with linear differential operators, but it can still have nonlinear expressions in it. In a mathematical programming model, if the objective functions and constraints are represented entirely by linear equations, then the model is regarded as a linear model. If one or more of the objective functions or constraints are represented with a nonlinear equation, then the model is known as a nonlinear model.
Nonlinearity, even in fairly simple systems, is often associated with phenomena such as chaos and irreversibility. 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, but this can be problematic if one is trying to study aspects such as irreversibility, which are strongly tied to nonlinearity.
1. Deterministic vs. probabilistic (stochastic): A deterministic 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, deterministic models perform the same way for a given set of initial conditions. Conversely, in a stochastic model, randomness is present, and variable states are not described by unique values, but rather by probability distributions.
2. Static vs. dynamic: A static model does not account for the element of time, while a dynamic model does. Dynamic models typically are represented with difference equations or differential equations.
3. Lumped parameters vs. distributed parameters: If the model is homogeneous (consistent state throughout the entire system) the parameters are lumped. If the model is heterogeneous (varying state within the system), then the parameters are distributed. Distributed parameters are typically represented with partial differential equations.

In computer science and mathematics, a variable (pronounced ) (sometimes called an object or identifier in computer science) is a symbolic representation used to denote a quantity or expression. ... In mathematics, an operator is a function that performs some sort of operation on a number, variable, or function. ... For other uses, see Linear (disambiguation). ... In statistics the linear model is given by where Y is an nÃ—1 column vector of random variables, X is an nÃ—p matrix of known (i. ... In mathematics, a differential operator is an operator defined as a function of the differentiation operator. ... In mathematics, the term optimization, or mathematical programming, refers to the study of problems in which one seeks to minimize or maximize a real function by systematically choosing the values of real or integer variables from within an allowed set. ... Graph sample of linear equations A linear equation is an algebraic equation in which each term is either a constant or the product of a constant times the first power of a variable. ... In mathematics, a nonlinear system is one whose behavior cant be expressed as a sum of the behaviors of its parts (or of their multiples. ... For other uses, see Chaos Theory (disambiguation). ... Irreversibility is that property of an event which makes reverting back to the state before the occurrence of the event impossible. ... Linearization in mathematics and its applications in general refers to finding the linear approximation to a function at a given point. ... In mathematics, a deterministic system is a system in which no randomness is involved in the development of future states of the system. ... In the mathematics of probability, a stochastic process is a random function. ... In mathematics, a recurrence relation, also known as a difference equation, is an equation which defines a sequence recursively: each term of the sequence is defined as a function of the preceding terms. ... Distributed parameter system (as opposed to a lumped parameter system) refers to system whose state-space is infinite-dimensional. ... In mathematics, a partial differential equation (PDE) is a relation involving an unknown function of several independent variables and its partial derivatives with respect to those variables. ...

## A priori information

Mathematical modeling problems are often classified into black box or white box models, according to how much a priori information is available of the system. 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 only works as an intuitive guide for approach. Black box is technical jargon for a device or system or object when it is viewed primarily in terms of its input and output characteristics. ... In software engineering, white box, in contrast to a black box, is a subsystem whose internals are visible to view, but usually cannot be altered. ... The terms a priori and a posteriori are used in philosophy to distinguish between two different types of propositional knowledge. ...

Usually it is preferable to use as much a priori information as possible to make the model more accurate. Therefore the white-box models are usually considered easier, because if you have used the information correctly, then the model will behave correctly. Often the a priori information comes in forms of knowing the type of functions relating different variables. For example, if we make a model of how a medicine works in a human system, we know that usually the amount of medicine in the blood is an exponentially decaying function. But we are still left with several unknown parameters; how rapidly does the medicine amount decay, and what is the initial amount of medicine in blood? This example is therefore not a completely white-box model. These parameters have to be estimated through some means before one can use the model. A quantity is said to be subject to exponential decay if it decreases at a rate proportional to its value. ...

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 which usually do not assume almost anything about the incoming data. The problem with using a large set of functions to describe a system is that estimating the parameters becomes increasingly difficult when the amount of parameters (and different types of functions) increases. A neural network is an interconnected group of neurons. ...

### Subjective Information

Sometimes it is useful to incorporate subjective information into a mathematical model. This can be done based on intuition, experience, or expert opinion, or based on convenience of mathematical form. Bayesian statistics provides a theoretical framework for incorporating such subjectivity into a rigorous analysis: one specifies a prior probability distribution (which can be subjective) and then updates 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 an arbitrary decision (perhaps by looking at the shape of the coin) about what prior distribution to use. Incorporation of the subjective information is necessary in this case to get an accurate prediction of the probability, since otherwise one would guess 1 or 0 as the probability of the next flip being heads, which would be almost certainly wrong.[1] Look up Intuition in Wiktionary, the free dictionary. ... Look up Experience in Wiktionary, the free dictionary This article discusses the general concept of experience. ... Bayesian inference is statistical inference in which probabilities are interpreted not as frequencies or proportions or the like, but rather as degrees of belief. ...

## 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; the essential idea being that among models with roughly equal predictive power, the simplest one is the most desirable. While added complexity usually improves the fit of a model, it can make the model difficult to understand and work with, and can also pose computational problems, including Numerical instability. Thomas Kuhn argues that as science progresses, explanations tend to become more complex before a Paradigm shift offers radical simplification. For the House television show episode, see Occams Razor (House episode). ... In the mathematical subfield of numerical analysis, numerical stability is a desirable property of numerical algorithms. ... Thomas Samuel Kuhn (July 18, 1922 – June 17, 1996) was an American intellectual who wrote extensively on the history of science and developed several important notions in the philosophy of science. ... Paradigm shift is the term first used by Thomas Kuhn in his 1962 book The Structure of Scientific Revolutions to describe a change in basic assumptions within the ruling theory of science. ...

For example, when modeling the flight of an aircraft, we could embed each mechanical part of the aircraft into our model and would thus acquire an almost white-box model of the system. However, the computational cost of adding such a huge amount of detail would effectively inhibit the usage of such a model. Additionally, the uncertainty would increase due to an overly complex system, because each separate part induces some amount of variance into the model. It is therefore usually appropriate to make some approximations to reduce the model to a sensible size. Engineers often can accept some approximations in order to get a more robust and simple model. For example Newton's classical mechanics 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, and we study macro-particles only. Sir Isaac Newton in Knellers portrait of 1689. ... Classical mechanics (commonly confused with Newtonian mechanics, which is a subfield thereof) is used for describing the motion of macroscopic objects, from projectiles to parts of machinery, as well as astronomical objects, such as spacecraft, planets, stars, and galaxies. ... A line showing the speed of light on a scale model of Earth and the Moon, taking about 1â…“ seconds to traverse that distance. ...

## Training

Any model which is not pure white-box contains some parameters that can be used to fit the model to the system it shall describe. If the modelling is done by a neural network, the optimization of parameters is called training. In more conventional modelling through explicitly given mathematical functions, parameters are determined by curve fitting. The factual accuracy of this article is disputed. ... // Traditionally, the term neural network had been used to refer to a network or circuitry of biological neurons. ... Curve fitting is finding a curve which matches a series of data points and possibly other constraints. ...

## 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 this data was not used to set the model's parameters. This practice is referred to as cross-validation in statistics. In statistics cross-validation is the practice of partitioning a sample of data into subsamples such that analysis is initially performed on a single subsample, while further subsamples are retained blind in order for subsequent use in confirming and validating the initial analysis. ...

Defining a metric to measure distances between observed and predicted data is a useful tool of assessing model fit. In statistics, decision theory, and some economic models, a loss function plays a similar role. In mathematics a metric or distance function is a function which defines a distance between elements of a set. ... In statistics, decision theory and economics, a loss function is a function that maps an event (technically an element of a sample space) onto a real number representing the economic cost or regret associated with the event. ...

While it is rather straightforward to test the appropriateness of parameters, it can be more difficult to test the validity of the general mathematical form of a model. In general, more mathematical tools have been developed to test the fit of statistical models than models involving Differential equations. Tools from nonparametric statistics can sometimes be used to evaluate how well data fits a known distribution or to come up with a general model that makes only minimal assumptions about the model's mathematical form. A statistical model is used in applied statistics. ... In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables. ...

### 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 what systems or situations the data is a typical set of data from.

The question of whether the model describes well the properties of the system between data points is called interpolation, and the same question for events or data points outside the observed data is called extrapolation. For other uses, see Interpolation (disambiguation). ... In mathematics, extrapolation is the process of constructing new data points outside a discrete set of known data points. ...

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. Classical mechanics (commonly confused with Newtonian mechanics, which is a subfield thereof) is used for describing the motion of macroscopic objects, from projectiles to parts of machinery, as well as astronomical objects, such as spacecraft, planets, stars, and galaxies. ...

### Philosophical Considerations

Many types of modeling implicitly involve claims about causality. 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 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. Causality or causation denotes the relationship between one event (called cause) and another event (called effect) which is the consequence (result) of the first. ... In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables. ... Purpose in its most general sense is the anticipated aim which guides action. ...

An example of such criticism is the argument that the mathematical models of Optimal foraging theory do not offer insight that goes beyond the common-sense conclusions of evolution and other basic principles of ecology.[2]. A central concern of ecology has traditionally been foraging behavior. ... This article is about evolution in biology. ...

An abstract model (or conceptual model) is a theoretical construct that represents something, with a set of variables and a set of logical and quantitative relationships between them. ... This article is about the general term. ... This article is about computer modeling within a scientific medium. ... A statistical model is used in applied statistics. ... In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables. ... In engineering and mathematics, a dynamical system is a deterministic process in which a functions value changes over time according to a rule that is defined in terms of the functions current value. ... Biologically-inspired computing (also bio-inspired computing) is a field of study that loosely knits together subfields related to the topics of connectionism, social behaviour and emergence. ... A diagram of the IS/LM model In economics, a model is a theoretical construct that represents economic processes by a set of variables and a set of logical and quantitative relationships between them. ... Mathematical models are of great importance in physics. ... Mathematical biology or biomathematics is an interdisciplinary field of academic study which aims at modeling natural, biological processes using mathematical techniques and tools. ... Mathematical Psychology is an approach to psychological research that is based on mathematical modeling of perceptual, cognitive and motor processes, and on the establishment of law-like rules that relate quantifiable stimulus characteristics with quantifiable behavior. ... Mathematical sociology is the usage of mathematics to construct social theories. ...

## References

1. ^ MacKay, D.J. Information Theory, Inference, and Learning Algorithms, Cambridge, (2003-2004). ISBN: 0521642981
2. ^ Optimal Foraging Theory: A Critical Review - Annual Review of Ecology and Systematics, 15(1):523 - First Page Image

### General References

Books
• Aris, Rutherford [ 1978 ] ( 1994 ). Mathematical Modelling Techniques, New York : Dover. ISBN 0-486-68131-9
• Lin, C.C. & Segel, L.A. ( 1988 ). Mathematics Applied to Deterministic Problems in the Natural Sciences, Philadelphia : SIAM. ISBN 0-89871-229-7
• Gershenfeld, N., The Nature of Mathematical Modeling, Cambridge University Press, (1998).ISBN 0521570956
• Yang, X.-S., Mathematical Modelling for Earth Sciences, Dudedin Academic, (2008). ISBN 1903765927
Specific applications
• Korotayev A., Malkov A., Khaltourina D. ( 2006 ). Introduction to Social Macrodynamics: Compact Macromodels of the World System Growth., Moscow : Editorial URSS. ISBN 5-484-00414-4

Andrey Korotayev (born in 1961) is an anthropologist, economic historian, and sociologist. ...

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 Mathematical model - Wikipedia, the free encyclopedia (1749 words) Mathematical models are used particularly in the natural sciences and engineering disciplines (such as physics, biology, and electrical engineering) but also in the social sciences (such as economics, sociology and political science); physicists, engineers, computer scientists, and economists use mathematical models most extensively. Note this model assumes the particle is a point mass, which is certainly known to be false in many cases we use this model, for example, as a model of planetary motion. Mathematical modelling problems are often classified into fl box or white box models, according to how much a priori information is available of the system.
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