Parameter space
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In generative art people talk about parameter space as the set of possible parameters 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 <math>p</math>. In this case the parameter space is the intervall <math>[0,1]</math>.
More precisely, <math>\Theta</math> is a parameter space of dimension <math>p\in\mathbb{N}^*</math> if there exists a <math>p</math>-dimensional vector space <math>E</math> such that <math>\Theta\subseteq E</math>. <math>p</math> is called number of parameters.
For example, <math>\mathbb{R}\times\mathbb{R}^+</math> is a parameter space because it is included in <math>\mathbb{R}^2</math>. 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 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.
