Parametrization

Parameterization (or parametrization; parameterisation in British English) is the process of defining or deciding the parameters - usually of some model - that are salient to the question being asked of that model.

Context-dependent meaning

If, for example, the model is of a wind turbine with a particular interest in the efficiency of power generation, then the parameters of interest will probably include the number, length and pitch of the blades.

In the context of mathematics and physics, parameterization may instead imply the identification of a complete set of effective coordinates or degrees of freedom of the system, process, or model, i.e. without regard to their utility in some design. Parameterization of a line, surface or volume, for example, implies identification of a set of coordinates (a chart) that allows one to uniquely identify any point (on the line, surface, or volume) with an (ordered) list of numbers.

Non-uniqueness

Parameterizations are not generally unique. The ordinary 3-volume can be parameterized (or 'coordinatized') equally efficiently with Cartesian coordinates - generally denoted (x,y,z) - or with cylindrical polar coordinates - generally denoted (ρ,φ,z), or with spherical (r,φ,θ) or other coordinate systems. Similarly, the color space of human trichromatic color vision can be parameterized in terms of the three colors red, green and blue, RGB, or equally well with Cyan, Magenta and Yellow, CMYK.

Dimensionality

Generally, the minimum number of parameters required to describe a model or geometric object is equal to its dimension, and the scope of the parameters - within their allowed ranges - is called the "parameter space". Though a good set of parameters permits identification of every point in the parameter space, it may be that, for a given parameterization, different parameter values can refer to the same 'physical' point. Such mappings are surjective but not injective. An example is the pair of cylindrical polar coordinates (ρ,φ,z) and (ρ,φ + 2π,z).

Parameterization invariance

As indicated above, there is arbitrariness in the choice of parameters of a given model, geometric object, etc. Often, one wishes to determine intrinsic properties of an object that do not depend on this arbitrariness, which are therefore independent of any particular choice of parameters. This is particularly the case in physics, wherein parameterization invariance (or 're-parameterization invariance') is a guiding principle in the search for physically acceptable theories (particularly in General relativity).

For example, whilst the location of a fixed point on some (curved) line may be given by different numbers depending on how the line is parameterized, nonetheless the "length" of the line between two such fixed points will be independent of the choice of parameterization, even though it might have been computed using algebra specific to one or other particular coordinate system.

Parameterization invariance implies that either the dimensionality or the volume of the parameter space is larger than that which is necessary to describe the physics in question. (Such may arise in circumstances of Scale invariance, for example).

Related links

Parameterization may refer to:

* A mathematical concept related to coordinate system
**Parametric equation
**Parametric surface
**Position vector
**Surjective function
**Vector-valued function
* The representation of processes in general circulation models and numerical weather prediction models: Parametrization (climate)