Parameter space

In science, a parameter space is the set of all possible combinations of values for all the different parameters contained in a particular mathematical model. 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 behaviour in the model.

Often the parameters are inputs of a function, in which case the technical term for the parameter space is domain of a function.

Parameter spaces are particularly useful for describing families of probability distributions that depend on parameters. More generally in science, the term parameter space is used to describe experimental variables. For example, the concept has been used in the science of soccer in the article "Parameter space for successful soccer kicks." In the study, "Success rates are determined through the use of four-dimensional parameter space volumes."

In the context of statistics, parameter spaces form the background for parameter estimation. As Ross describes in his book:

The idea of intentionally truncating the parameter space has also been advanced elsewhere.

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

The requirement for higher dimensions is illustrated by Plücker's line geometry. Struik writes

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