Everything about Parameter Space totally explained
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
. In this case the parameter space is the intervall
.
More precisely,
is a
parameter space of dimension
. 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've a "field" in our 10-dimensional space. Following this "field" downwards leads us to the "location" in parameter space with the lowest χ², for example 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 parameter space is parameter plane (c-plane), which
points (
complex numbers) are
parameters of
complex quadratic function. In parameter plane there's
Mandelbrot set.
Compare it with dynamical plane (z-plane) which is a phase space for complex quadratic mapping. In dynamical plane one can find
Julia and
Fatou sets.
Further Information
Get more info on 'Parameter Space'.
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