Parameter space

In generative art people talk about parameter space as the set of possibleparameters for a generative system.

In statistics one can study the distribution of a random variable. Several models exist, the most common one being the normal distribution (or Gaussian distribution). When the distribution is known explicitly, it often depends on several parameters. A parameter space is simply the set of values that this parameter can take. For example, if we toss a coin, we can use the Bernoulli distribution of parameter $p$. In this case the parameter space is the intervall $[0,1]$.

More precisely, $Theta$ is a parameter space of dimension $pinmathbb\{N\}^*$ if there exists a $p$-dimensional vector space $E$ such that $Thetasubseteq\; E$. $p$ is called "number of parameters".

For example, $mathbb\{R\}\; imesmathbb\{R\}^+$ is a parameter space because it is included in $mathbb\{R\}^2$. It is the parameter space for the normal distribution.

The term parameter space as used in data-fitting (see for example "Data Reduction and Error Analysis for the Physical Sciences" by Bevington and Robinson), refers to the hypothetical space where a "location" is defined by the values of all optimizable parameters. For example, if we fit data using a function which has 10 optimizable parameters, each of these parameters is seen as a dimension and the parameter space in this case is 10-dimensional. Every "location" then corresponds to a χ² (chi-squared) value indicating the goodness-of-fit, hence we have a "field" in our 10-dimensional space. Following this "field" downwards leads us to the "location" in parameter space with the lowest χ², i.e. the optimum parameter values.

Alternatively, χ² can be thought of as an additional dimension. In this case, if we're optimizing 2 variables, variable space is still 2-dimensional, but the addition of χ² as a third dimension results in 3-dimensional "goodness-of-fit" landscapes where the best fit is represented by the lowest point in 3D space.

Examples

For complex quadratic mapping the parameter space is parameter plane ( c-plane), in which points ( complex numbers) are parameters of a complex quadratic function. In the parameter plane there is a Mandelbrot set.
Compare it with dynamical plane ( z-plane) which is a phase space for complex quadratic mapping. In dynamical planes one can find Julia and Fatou sets.

ee also

*Parametric equation
*Parametric surface
*data analysis
*Phase space