en>Colonies Chris: sp, date & link fixes, typos fixed: indentified → identified using AWB

2010-04-18T13:42:07Z

sp, date & link fixes, typos fixed: indentified → identified using AWB

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{{More footnotes|date=August 2010}}
In [[science]], a '''parameter space''' is the [[set (mathematics)|set]] of all possible combinations of values for all the different [[parameter]]s contained in a particular [[mathematical model]]. The ranges of values of the parameters may form the axes of a [[Plot (graphics)|plot]], and particular outcomes of the model may be plotted against these axes to illustrate how different regions of the parameter space produce different types of behaviour in the model.

Often the parameters are [[input]]s of a [[function (mathematics)|function]], in which case the technical term for the parameter space is [[domain of a function]].{{Citation needed|date=August 2010}}

Parameter spaces are particularly useful for describing families of [[probability distribution]]s that depend on parameters. More generally in science, the term parameter space is used to describe experimental variables. For example, the concept has been used in the science of [[soccer]] in the article "Parameter space for successful soccer kicks." In the study, "Success rates are determined through the use of four-dimensional parameter space volumes."<ref>Cook & Goff (2006)</ref>

In the context of [[statistics]], parameter spaces form the background for [[parameter estimation]].
As Ross (1990){{Page needed|date=September 2011}} describes in his book:
:Parameter space is a subset of p-dimensional space consisting of the set of values of Θ which are allowable in a particular model. The values may sometimes be constrainted, say to the positive quadrant or the unit square, or in case of symmetry, to the triangular region where, say <math>\Theta_1 \le \Theta_2.</math>

The idea of intentionally truncating the parameter space has also been advanced elsewhere.<ref>Van Eeden, C. (2006)[http://books.google.com/books?id=m-MePzncfyAC&pg=PA3 ''Restricted parameter space estimation problems: admissibility and minimaxity properties''], Springer ISBN 0-387-33747-4 (p. 2) "Gains in the minimax value can be very substantial when the parameter space is bounded."</ref>

==Examples==
* A simple model of health deterioration after developing [[lung cancer]] could include the two parameters gender<ref name=one>{{Cite journal |last=Gasperino |first=J. |coauthors=Rom, W. N. |year=2004 |title=Gender and lung cancer |url=http://www.sciencedirect.com/science/article/pii/S1525730411701863 |journal=Clinical Lung Cancer |volume=5 |issue=6 |pages=353–359 |doi= 10.3816/CLC.2004.n.013|pmid= |bibcode=}}</ref> and smoker/non-smoker, in which case the parameter space is the following set of four possibilities: {{math|{(Male, Smoker), (Male, Non-smoker), (Female, Smoker), (Female, Non-smoker)} }}.

* The [[logistic map]] <math> \qquad x_{n+1} = r x_n (1-x_n) </math> has one parameter, ''r'', which can take any positive value. The parameter space is therefore the set of all positive numbers.
For some values of ''r'', this function ends up cycling round a few values, or fixed on one value. These long-term values can be plotted against ''r'' in a [[bifurcation diagram]] to show the different behaviours of the function for different values of ''r''.

* In a [[sine wave]] model <math>y(t) = A \cdot \sin(\omega t + \phi),</math> the parameters are [[amplitude]] ''A'' > 0, [[angular frequency]] ω > 0, and [[phase (waves)|phase]] φ ∈ S<sup>1</sup>. Thus the parameter space is
:<math>R^+ \times R^+ \times S^1 .</math>{{Citation needed|date=January 2011}}

* In [[complex dynamics]], the parameter space is the [[complex plane]] '''C''' = { ''z'' = ''x'' + ''y'' i : x, y ∈ '''R''' }, where i<sup>2</sup> = −1.

The famous [[Mandelbrot set]] is a [[subset]] of this parameter space, consisting of the points in the complex plane which give a [[bounded set]] of numbers when a particular [[iterated function]] is repeatedly applied from that starting point. The remaining points, which are not in the set, give an unbounded set of numbers (they tend to infinity) when this function is repeatedly applied from that starting point.

==History==
Parameter space contributed to the liberation of [[geometry]] from the confines of [[three-dimensional space]]. For instance, the parameter space of [[sphere (geometry)|spheres]] in three dimensions, has four dimensions—three for the sphere center and another for the radius. According to [[Dirk Struik]], it was the book ''Neue Geometrie des Raumes'' (1849) by [[Julius Plücker]] that showed
:...geometry need not solely be based on points as basic elements. Lines, planes, circles, spheres can all be used as the elements (''Raumelemente'') on which a geometry can be based. This fertile conception threw new light on both synthetic and algebraic geometry and created new forms of duality. The number of dimensions of a particular form of geometry could now be any positive number, depending on the number of parameters necessary to define the "element".<ref>Struik (1967) 165</ref>
The requirement for higher dimensions is illustrated by Plücker's [[line geometry]]. Struik writes
:[Plücker's] geometry of lines in three-space could be considered as a four-dimensional geometry, or, as [[Felix Klein|Klein]] has stressed, as the geometry of a four-dimensional [[quadric]] in a five-dimensional space.<ref>Struik (1967) 168</ref>

==See also==
*[[Configuration space]]
*[[Data analysis]]
*[[Parametric equation]]
*[[Parametric surface]]
*[[Phase space]]

==Notes and references==
{{reflist}}
* Brandon G. Cook & John Eric Goff (2006) [http://stacks.iop.org/EJP/27/865 Parameter Space for Successful Soccer Kicks] [[European Journal of Physics]] 27:865.
* Constance van Eeden (2006) ''Restricted Parameter Space Estimation Problems: Admissibility and Minimaxity Properties'', Lecture Notes in Statistics #188, [[Springer Science+Business Media]].
* Gavin J.S. Ross (1990) ''Nonlinear Estimation'', page 94, [[Springer-Verlag]].
* [[Dirk Struik]] (1967) ''A Concise History of Mathematics'', 3rd edition, page 165, [[Dover Books]].

{{DEFAULTSORT:Parameter Space}}
[[Category:Statistical theory]]
[[Category:Mathematical terminology]]

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en>Colonies Chris: sp, date & link fixes, typos fixed: indentified → identified using AWB