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Concept
Parameter space
Summary
The parameter space is the space of possible parameter values that define a particular mathematical model, often a subset of finite-dimensional Euclidean space. Often the parameters are inputs of a function, in which case the technical term for the parameter space is domain of a function. 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.
In statistics, parameter spaces are particularly useful for describing parametric families of probability distributions. They also form the background for parameter estimation. In the case of extremum estimators for parametric models, a certain objective function is maximized or minimized over the parameter space. Theorems of existence and consistency of such estimators require some assumptions about the topology of the parameter space. For instance, compactness of the parameter space, together with continuity of the objective function, suffices for the existence of an extremum estimator.
A simple model of health deterioration after developing lung cancer could include the two parameters gender 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 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 the parameters are amplitude A > 0, angular frequency ω > 0, and phase φ ∈ S1. Thus the parameter space is
In complex dynamics, the parameter space is the complex plane C = { z = x + y i : x, y ∈ R }, where i2 = −1.
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