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A nonparametric model is a model that has no built-in assumption of a particular type of relation (such as a linear, square, or exponential relationship or a common statistical distribution) that might suggest some underlying cause and effect. The term nonparametric might mislead because the model relationship ultimately will have parameters, often many, but not just a few pre-specified parameters to discover and plug into a particular formula. The term is used when the parameterized formula is not intended to represent the underlying mathematics (and physics), but merely to produce the same curve. An example such formula could a polynomial of high enough degree to allow many different simpler function "shapes" to be approximated. The same tests as used for parameterizing a parametric model can be used to evaluate the a match. My above description presumes an equation of one independent variable and one dependent variable, but the principles apply to higher-dimensional models as well. There exist some formulae able to produce a wide variety of curves, surfaces, etc., that are commonly used for nonparametric models. As always, it is helpful if the evaluation of the fit also yields boundaries on adjustments that are likely to improve the fit.
Statistical work using such models is termed nonparametric statistics, e.g., producing distributions without presuming some particular template for the distribution function (e.g., not assuming it is Gaussian).
One area that uses such models is machine learning (ML). The techniques are useful in other areas where underlying equations are not known or easily guessed.