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In science, a parameter space is the set of all possible combinations of values for all the different parameters contained in a particular mathematical model. The ranges of values of the parameters may form the axes of a 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 inputs of a function, in which case the technical term for the parameter space is domain of a function.

Parameter spaces are particularly useful for describing families of probability distributions 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."[1]

In the context of statistics, parameter spaces form the background for parameter estimation. As Ross describes in his book:[2]

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 constrained, say to the positive quadrant or the unit square, or in case of symmetry, to the triangular region where, say $$\Theta_1 \le \Theta_2$$.

The idea of intentionally truncating the parameter space has also been advanced elsewhere.[3]

Examples

A simple model of health deterioration after developing lung cancer could include the two parameters gender[4] and smoker/non-smoker, in which case the parameter space is the following set of four possibilities: {(Male, Smoker), (Male, Non-smoker), (Female, Smoker), (Female, Non-smoker)} .

The logistic map $$\qquad x_{n+1} = r x_n (1-x_n)$$ 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 $$y(t) = A \cdot \sin(\omega t + \phi)$$, the parameters are amplitude A > 0, angular frequency ω > 0, and phase φ ∈ S1. Thus the parameter space is

$$R^+ \times R^+ \times S^1 .$$

In complex dynamics, the parameter space is the complex plane C = { z = x + y i : x, y ∈ R }, where i2 = −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 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".[5]

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 Klein has stressed, as the geometry of a four-dimensional quadric in a five-dimensional space.[6]

Thus the Klein quadric describes the parameters of lines in space.

Configuration space
Data analysis
Parametric equation
Parametric surface
Phase space

Notes and references

Cook & Goff (2006)
Gavin J.S. Ross (1990) Nonlinear Estimation, page 94, Springer-Verlag
Van Eeden, C. (2006)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."
Gasperino, J.; Rom, W. N. (2004). "Gender and lung cancer". Clinical Lung Cancer 5 (6): 353–359. doi:10.3816/CLC.2004.n.013.
Dirk Struik (1967) A Concise History of Mathematics, 3rd edition, page 165, Dover Books

Struik (1967) 168

Brandon G. Cook & John Eric Goff (2006) 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.