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Note: The term model has a different meaning in model theory, a branch of mathematical logic. In Mathematics, model theory is the study of (classes of mathematical structures such as groups, Fields graphs or even models Mathematical logic is a subfield of Logic and Mathematics with close connections to Computer science and Philosophical logic. An artifact which is used to illustrate a mathematical idea is also called a mathematical model and this usage is the reverse of the sense explained below.

A mathematical model uses mathematical language to describe a system. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and System (from Latin systēma, in turn from Greek systēma is a set of interacting or interdependent Entities, real or abstract 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. In Science, the term natural science refers to a naturalistic approach to the study of the Universe, which is understood as obeying rules or law of Engineering is the Discipline and Profession of applying technical and scientific Knowledge and Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. Foundations of modern biology There are five unifying principles Meteorology (from Greek grc μετέωρος metéōros, "high in the sky" and grc -λογία -logia) is the Interdisciplinary Electrical engineering, sometimes referred to as electrical and electronic engineering, is a field of Engineering that deals with the study and application of The social sciences comprise academic disciplines concerned with the study of the social life of human groups and individuals including Anthropology, Communication studies Economics is the social science that studies the production distribution, and consumption of goods and services. Sociology (from Latin: socius "companion" and the suffix -ology "the study of" from Greek λόγος lógos "knowledge" Political science is a branch of Social sciences that deals with the theory and practice of Politics and the description and analysis of Political systems A physicist is a Scientist who studies or practices Physics. Physicists study a wide range of physical phenomena in many branches of physics spanning An engineer is a person professionally engaged in a field of Engineering. Computer science (or computing science) is the study and the Science of the theoretical foundations of Information and Computation and their An economist is an expert in the Social science of Economics.

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'. System (from Latin systēma, in turn from Greek systēma is a set of interacting or interdependent Entities, real or abstract

Mathematical models can take many forms, including but not limited to dynamical systems, statistical models, differential equations, or game theoretic models. Dynamical systems theory is an area of Applied mathematics used to describe the behavior of complex Dynamical systems usually by employing Differential Statistical models are used in Applied statistics. Three notions are sufficient to describe all statistical models A differential equation is a mathematical Equation for an unknown function of one or several variables that relates the values of the Game theory is a branch of Applied mathematics that is used in the Social sciences (most notably Economics) Biology, Engineering, These and other types of models can overlap, with a given model involving a variety of abstract structures.

Contents

Examples of mathematical models

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. A differential equation is a mathematical Equation for an unknown function of one or several variables that relates the values of the
\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. 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 General equilibrium theory is a branch of theoretical Microeconomics. Pareto efficiency, or Pareto optimality, is an important concept in Economics with broad applications in Game theory, Engineering and the 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.

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. Simulation is the imitation of some real thing state of affairs or process

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. In Mathematics, the real numbers may be described informally in several different ways The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French In Computer science, the Boolean datatype, sometimes called the logical datatype, is a Primitive datatype having one of two values 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). In the fields of communications, Signal processing, and in Electrical engineering more generally a signal is any time-varying or spatial-varying quantity The actual model is the set of functions that describe the relations between the different variables.

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. A variable (ˈvɛərɪəbl is an Attribute of a physical or an abstract System which may change its Value while it is under Observation. In Mathematics, an operator is a function which operates on (or modifies another function If all the operators in a mathematical model present linearity, the resulting mathematical model is defined as linear. The word linear comes from the Latin word linearis, which means created by lines. 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. In Statistics the linear model is given by Y = X \beta + \varepsilon where Y is an n ×1 column vector of random 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 Mathematics, a differential operator is an Operator defined as a function of the differentiation operator 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. 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 A linear equation is an Algebraic equation in which each term is either a Constant or the product of a constant and (the first power of a single Variable 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. This article describes the use of the term nonlinearity in mathematics
Nonlinearity, even in fairly simple systems, is often associated with phenomena such as chaos and irreversibility. In Mathematics, chaos theory describes the behavior of certain dynamical systems – that is systems whose state evolves with time – that may exhibit dynamics that In science a Process that is not reversible is called irreversible. 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. In Mathematics and its applications linearization refers to finding the Linear approximation to a function at a given point
  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. In Mathematics, a deterministic system is a system in which no Randomness is involved in the development of future states of the system 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. A stochastic process, or sometimes random process, is the counterpart to a deterministic process (or Deterministic system) in Probability theory.
  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. "Difference equation" redirects here It should not be confused with a Differential equation.
  3. Lumped parameters vs. distributed parameters: If the model is homogeneous (consistent state throughout the entire system) the parameters are lumped. A distributed parameter system (as opposed to a lumped parameter system) is a System whose State space is infinite- dimensional. 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 Mathematics, partial differential equations ( PDE) are a type of Differential equation, i

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. Black box is a technical term 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 can be viewed but usually cannot be altered "A priori" redirects here For other uses see A priori. 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.

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. A quantity is said to be subject to exponential decay if it decreases at a rate proportional to its value 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 which usually do not assume almost anything about the incoming data. Traditionally the term neural network had been used to refer to a network or circuit of biological neurons. 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.

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. Experience as a general concept comprises Knowledge of or skill in or Observation of some thing or some event gained through involvement in or An expert witness is a Witness, who by virtue of Education, Training, Skill, or Experience, is believed to have Knowledge 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. Bayesian inference is Statistical inference in which evidence or observations are used to update or to newly infer the Probability that a hypothesis may be true 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]

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. Occam's razor (sometimes spelled Ockham's razor) is a principle attributed to the 14th-century English Logician and Franciscan Friar, 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. In the mathematical subfield of Numerical analysis, numerical stability is a desirable property of numerical Algorithms The precise definition of stability Thomas Kuhn argues that as science progresses, explanations tend to become more complex before a Paradigm shift offers radical simplification. Thomas Samuel Kuhn (surname ˈkuːn July 18, 1922  &ndash June 17, 1996) was an American intellectual who wrote extensively Paradigm shift, sometimes known as extraordinary science or revolutionary science, is the term first used by Thomas Kuhn in his influential

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. Sir Isaac Newton, FRS (ˈnjuːtən 4 January 1643 31 March 1727) Biography Early years See also Isaac Newton's early life and achievements Classical mechanics is used for describing the motion of Macroscopic objects from Projectiles to parts of Machinery, as well as Astronomical objects 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.

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. In Mathematics, Statistics, and the mathematical Sciences a parameter ( G auxiliary measure) is a quantity that defines certain characteristics If the modelling is done by a neural network, the optimization of parameters is called training. Traditionally the term neural network had been used to refer to a network or circuit of biological neurons. In more conventional modelling through explicitly given mathematical functions, parameters are determined by curve fitting. Curve fitting is finding a curve which has the best fit to 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. Cross-validation, sometimes called rotation estimation, is the statistical practice of partitioning a sample of Data into subsets

Defining a metric to measure distances between observed and predicted data is a useful tool of assessing model fit. In Mathematics, a metric or distance function is a function which defines a Distance between elements of a set. In statistics, decision theory, and some economic models, a loss function plays a similar role. In Statistics, Decision theory and Economics, a loss function is a function that maps an event (technically an element of a Sample space

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. Statistical models are used in Applied statistics. Three notions are sufficient to describe all statistical models A differential equation is a mathematical Equation for an unknown function of one or several variables that relates the values of the 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. Non-parametric statistics is a branch of Statistics concerned with non-parametric Statistical models and non-parametric inference, including non-parametric

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. In the mathematical subfield of Numerical analysis, interpolation is a method of constructing new data points within the range of a Discrete set of 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. Classical mechanics is used for describing the motion of Macroscopic objects from Projectiles to parts of Machinery, as well as Astronomical objects 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 (but not causation) denotes a necessary relationship between one event (called cause and another event (called effect) which is the direct consequence This is usually (but not always) true of models involving differential equations. A differential equation is a mathematical Equation for an unknown function of one or several variables that relates the values of the 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. Purpose is the Cognitive Awareness in Cause and effect linking for achieving a Goal in a given System, whether 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.

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. eVolution is the third Album by eLDee, it was due to be released in 2008 [2].

See also

References

  1. ^ MacKay, D. Scientific modelling is the process of generating abstract, conceptual, Graphical and or mathematical models. Simulation is the imitation of some real thing state of affairs or process A computer simulation, a computer model or a computational model is a Computer program, or network of computers that attempts to simulate an Statistical models are used in Applied statistics. Three notions are sufficient to describe all statistical models A differential equation is a mathematical Equation for an unknown function of one or several variables that relates the values of the Dynamical systems theory is an area of Applied mathematics used to describe the behavior of complex Dynamical systems usually by employing Differential Biologically inspired (often hyphenated as biologically-inspired computing (also bio-inspired computing) is a field of study that loosely knits together subfields Mathematical models are of great importance in Physics. Physical theories are almost invariably expressed using Mathematical models and the mathematics For use of basic artimethics in Biology see relevant topic such as Serial dilution. Mathematical psychology is an approach to psychological research that is based on Mathematical modeling of perceptual cognitive and motor processes and on the establishment Mathematical sociology is the usage of mathematics to construct social theories 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

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Specific applications

External links

General reference material
Software
See Computer simulation Open Source ASCEND (open source NLA/DAE modelling environment Computational Infrastructure for Operations

Dictionary

mathematical model

-noun

  1. An abstract mathematical representation of a process, device or concept; it uses a number of variables to represent inputs, outputs and internal states, and sets of equations and inequalities to describe their interaction
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