# Parameter space

In statistics, a parameter space is the space of all possible combinations of values for all the different parameters contained in a particular mathematical model. If data generating processes of a particular model are collected in the set ${\displaystyle {\mathcal {M}}}$, then a parameter-defining mapping ${\displaystyle \theta }$ is understood as a submersion acting from ${\displaystyle {\mathcal {M}}}$ to a parameter space ${\displaystyle \Theta }$; that is, a typical data generating process ${\displaystyle \mu _{0}\in {\mathcal {M}}}$ is associated with a point ${\displaystyle \theta _{0}\equiv \theta (\mu _{0})}$. The pair ${\displaystyle ({\mathcal {M}},\theta )}$ is referred to as the parametrized model.[1] Usually, the parameter space ${\displaystyle \Theta }$ is a subset of the finite-dimensional Euclidean space ${\displaystyle \mathbb {R} ^{k}}$, where ${\displaystyle k}$ is a positive integer that gives the dimension. In low dimensions, 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 behavior in the model.

Oftentimes additional restrictions are imposed on the parameter space to establish the existence of certain estimators. For instance, compactness of the parameter space guarantees the existence of extremum estimators.[2]

## Examples

• A simple model of health deterioration after developing lung cancer could include the two parameters gender[3] 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 ${\displaystyle x_{n+1}=rx_{n}(1-x_{n})}$ has one parameter, r, which can take any positive value. The parameter space is therefore positive real 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 ${\displaystyle y(t)=A\cdot \sin(\omega t+\phi ),}$ the parameters are amplitude A > 0, angular frequency ω > 0, and phase φ ∈ S1. Thus the parameter space is ${\displaystyle R^{+}\times R^{+}\times S^{1}.}$
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".[4]:165

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.[4]:168

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