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Encyclopedia > Model (abstract)

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. Models in this sense are constructed to enable reasoning within an idealized logical framework about these processes and are an important component of scientific theories. Idealized here means that the model may make explicit assumptions that are known to be false (or incomplete) in some detail. Such assumptions may be justified on the grounds that they simplify the model while, at the same time, allowing the production of acceptably accurate solutions, as is illustrated below. Reasoning is the act of using reason to derive a conclusion from certain premises. ... The word theory has a number of distinct meanings in different fields of knowledge, depending on their methodologies and the context of discussion. ...

Mathematical models

Main article: Mathematical model A mathematical model is an abstract model that uses mathematical language to describe the behaviour of a system. ...

• Model of a particle in a potential field. In this model we consider a particle as being a point of mass m that describes a trajectory modelled 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
$m frac{d^2}{dt^2} x(t) = - operatorname{grad} V(x(t)).$
Note this model assumes that the particle is a point mass, which is certainly known to be false in many cases where we use the model, e.g. when we use it 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 labelled 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 one of constrained maximization, that is maximize
$U(x_1,x_2,ldots, x_n)$
subject to
$sum_{i=1}^n p_i x_i = M.$
This model has been used in models of 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 a source of criticism. But this is not an essential ingredient of the theory and again, the model is an idealization.

Pareto efficiency, or Pareto optimality, is a central concept in game theory with broad applications in economics, engineering and the social sciences. ...

Other types of models

These two models are examples of mathematical models; following are examples of models that are not mathematical (or at least not numerical).

• Myers-Briggs personality type. Myers-Briggs Type Indicator® is a technique that claims to produce a representation of a person's preferences, using four scales. These scales can be combined in various ways to produce 16 personality types. Types are typically denoted by four letters — for example, INTJ (introverted intuition with extroverted thinking) — to represent a person's preferences. This model is claimed by CPP (formerly known as Consulting Psychologists Press, Inc.) to produce a good predictor of a person's career and marriage partner preference. It should be pointed out, see [1], that there is considerable disagreement among psychologists on whether this assessment technique (and the implied idealized personality model) is of any value.
• Model of political contagion. Some versions of this model are sometimes referred to as the domino theory. In the broadest possible terms, according to this model, political movements that take hold in one country are likely to spread to geographically neighboring ones. This model is surprisingly popular, although as it stands, it is extremely impoverished conceptually, saying nothing about the type of political movement, the degree of geographical proximity, the time scale at which these events take place, etc.

The Center for Applications of Psychological Type is a non-profit organization co-founded by Isabel Myers in 1975 for MBTI development, research and training. ... The Center for Applications of Psychological Type is a non-profit organization co-founded by Isabel Myers in 1975 for MBTI development, research and training. ... This article does not cite any references or sources. ...

Use of models

The purpose of a model is to provide an argumentative framework for applying logic and mathematics that can be independently evaluated (for example by testing) and that can be applied for reasoning in a range of situations. Models are used throughout the natural and social sciences, psychology and the philosophy of science. Some models are predominantly statistical (for example portfolio models used in finance); others use calculus, linear algebra or convexity, see mathematical model. Of particular political significance are models used in economics, since they are used to justify decisions regarding taxation and government spending. This often leads to hotly contested debates in the academic world as well as in the political arena; see for instance supply side economics. In logic, an argument is a set of statements, consisting of a number of premises, a number of inferences, and a conclusion, which is said to have the following property: if the premises are true, then the conclusion must be true, or highly likely to be true. ... A statistical model is used in applied statistics. ... A mathematical model is an abstract model that uses mathematical language to describe the behaviour of a system. ... 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. ... Academia is a collective term for the scientific and cultural community engaged in higher education and research, taken as a whole. ... Supply-side economics is a school of macroeconomic thought which emphasizes the importance of tax cuts and business incentives in encouraging economic growth, in the belief that businesses and individuals will use their tax savings to create new businesses and expand old businesses, which in turn will increase productivity, employment...

Abstract models are used primarily as a reusable tool for discovering new facts, for providing systematic logical arguments as explicatory or pedagogical aids, for evaluating hypotheses theoretically, and for devising experimental procedures to test them. Reasoning within models is determined by a set of logical principles, although rarely is the reasoning used completely mathematical. In logic, an argument is a set of statements, consisting of a number of premises, a number of inferences, and a conclusion, which is said to have the following property: if the premises are true, then the conclusion must be true, or highly likely to be true. ...

In some cases, abstract models can be used to implement computer simulations that illustrate the behavior of a system over time. Simulations are used everywhere in science, especially in economics, engineering, biology, ecology etc., to discover the effects of changing a variable. The validity of different simulation methodologies is a subject of debate in the philosophy and methodology of science. A computer simulation or a computer model is a computer program that attempts to simulate an abstract model of a particular system. ... This article needs additional references or sources for verification. ... Engineering is the design, analysis, and/or construction of works for practical purposes. ... This article or section does not cite any references or sources. ... This article or section does not cite any references or sources. ...

The automated use of modeling has been identified as a significant issue in the creation of artificial intelligence. Some researchers argue a system without a model cannot achieve understanding, while others argue that running full, consistent models is too computationally costly for either machines or animals, and that much intelligent behavior is reactive or instinctive. Garry Kasparov playing against Deep Blue, the first machine to win a chess game against a reigning world champion. ... In computer science, computational complexity theory is the branch of the theory of computation that studies the resources, or cost, of the computation required to solve a given computational problem. ... In artificial intelligence, reactive planning denotes a group of techniques for action selection of autonomous agents. ... Instinct is the word used to describe inherent dispositions towards particular actions. ...

Structure of models

Main article: Conceptual schema

A conceptual model is a representation of some phenomenon, data or theory by logical and mathematical objects such as functions, relations, tables, stochastic processes, formulas, axiom systems, rules of inference etc. A conceptual model has an ontology, that is the set of expressions in the model which are intended to denote some aspect of the modeled object. Here we are deliberately vague as to how expressions are constructed in a model and particularly what the logical structure of formulas in a model actually is. In fact, we have made no assumption that models are encoded in any formal logical system at all, although we briefly address this issue below. Moreover, the definition given here is oblivious about whether two expressions really should denote the same thing. Note that this notion of ontology is different from (and weaker than) ontology as is sometimes understood in philosophy; in our sense there is no claim that the expressions actually denote anything which exists physically or spatio-temporally (to use W. Quine's formulation). A conceptual schema, or High-level data model or conceptual data model, is a map of concepts and their relationships, for example, a conceptual schema for a karate studio would include abstractions such as student, belt, grading and tournament. ... Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A... In mathematics, a finitary relation is defined by one of the formal definitions given below. ... In the mathematics of probability, a stochastic process is a random function. ... // In philosophy, ontology (from the Greek , genitive : of being (part. ... Logic (from ancient Greek &#955;&#8057;&#947;&#959;&#962; (logos), meaning reason) is the study of arguments. ...

For example, a stochastic model of stock prices includes in its ontology a sample space, random variables, the mean and variance of stock prices, various regression coefficients etc. Models of quantum mechanics in which pure states are represented as unit vectors in a Hilbert space include in their ontologies observables, dynamics, measurement operators etc. It is possible that observables and states of quantum mechanics are as physically real as the electrons they model, but by adopting this purely formal notion of ontology we avoid altogether this question. Fig. ... The term pure state refers to several related concepts in physics, particularly quantum mechanics and in functional analysis. ... In physics, particularly in quantum physics, a system observable is a property of the system state that can be determined by some sequence of physical operations. ... In physics, dynamics is the branch of classical mechanics that is concerned with the effects of forces on the motion of objects. ... Various meters Measurement is the estimation of a physical quantity such as length, temperature, or time. ...

Modeling

Modeling, especially scientific modeling refers to the process of generating a model as a conceptual representation of some phenomenon as discussed above. Typically a model will refer only to some aspects of the phenomenon in question, and two models of the same phenomenon may be essentially different, that is in which the difference is more than just a simple renaming. This may be due to differing requirements of the model's end users or to conceptual or esthetic differences by the modellers and decisions made during the modeling process. Esthetic considerations that may influence the structure of a model might be the modeller's preference for a reduced ontology, preferences regarding probabilistic models vis-a-vis deterministic ones, discrete vs continuous time etc. For this reason users of a model need to understand the model's original purpose and the assumptions of its validity. Having found a model for some desired aspect of reality, it can serve as the basis for simulation, the only way for non-invasive examination of physical reality besides real-world experiments. Scientific modeling is the process of generating abstract or conceptual models. ... // In philosophy, ontology (from the Greek , genitive : of being (part. ... Look up simulation in Wiktionary, the free dictionary. ... In the scientific method, an experiment (Latin: ex-+-periri, of (or from) trying), is a set of actions concerning phenomena. ...

References

• I. Briggs Myers with P. Myers, Gifts Differing. Understanding Personality Type, CPP Books, 1993.
• R. Frigg and S. Hartmann, Models in Science. Entry in the Stanford Encyclopedia of Philosophy.
• K. Lancaster, Mathematical Economics, Dover Publications, 1968.
• W. Quine, From a Logical Point of View, Harper Torchbooks, 1961.
• Examples of models: — Interactive models for biological, physical and social systems

A causal model is an abstract model that uses cause and effect logic to describe the behaviour of a system. ... EconMult is a general fleet model to be used in fisheries modelling. ... Ecosystem models, or ecological models, are mathematical representations of ecosystems. ... Note: The term model is also given a formal meaning in model theory, a part of axiomatic set theory. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... The Meta-Object Facility (MOF), is an Object Management Group (OMG) standard for Model Driven Engineering. ... // Presentation MDE (Model Driven Engineering) is an emerging technique in software, system and data engineering, based on the systematic use of models. ... // The notion of Model transformation is of central importance to Information Technology. ... Morphological analysis is a technique developed by Fritz Zwicky (1966, 1969) for exploring all the possible solutions to a multi-dimensional, non-quantified problem complex. ...

Modeling languages

• UML for software systems
• Role Activity Diagram and IDEF for processes
• VRML for 3-D computer graphics models designed particularly with the World Wide Web in mind.
• MA/Casper A proprietary software system which supports an extended form of Morphological modelling. From the Swedish Morphological Society

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