powermodified

Powermodified is a term used in mathematical and applied contexts to describe transformations that modify a function or data by applying a power operation. It is not a standard formal term in most mathematical literatures, but it is used informally to describe two common variants: input-power modification and output-power modification.

In input-power modification, for a function f defined on a domain D and a real exponent p > 0, the powermodified function is f_p_in(x) = f(x^p), with the domain adjusted to include only those x for which x^p lies in D. This form changes the input scale and can alter properties such as domain, range, and monotonicity via the chain rule: the derivative is f_p_in'(x) = p x^{p-1} f'(x^p). In output-power modification, the powermodified function is f_p_out(x) = [f(x)]^p, defined where f(x) is defined and, if p is not an integer, nonnegative. Its derivative is f_p_out'(x) = p [f(x)]^{p-1} f'(x).

Examples illustrate the idea: f_p_in for f(x) = x yields x^p; f_p_out for f(x) = x also yields x^p; f_p_in for f(x) = e^x gives e^{x^p}. Applications span data transformation and statistical modeling, such as variance stabilization and scale adjustment, as well as signal processing and feature engineering in machine learning. Cautions include domain restrictions, sign issues for non-integer powers, and preserving or altering monotonicity.

Related concepts include power transformations, Box-Cox and Yeo-Johnson transformations, and other monotone feature engineering techniques that reshape distributions or scales.