UniformiteettiY

UniformiteettiY is a fictional mathematical construct used to assess how uniformly a dataset is distributed after applying a transformation Y. It is described in this article as a hypothetical tool within a theoretical framework called Uniformity Theory, a fictional branch intended for thought experiments and introductory demonstrations.

In basic terms, UniformiteettiY measures the spread of the transformed data Y(X) across a target space T. Let X = {x1, x2, ..., xn} be a finite sample from a domain D with an underlying measure µ, and let Y: D → T map into a bounded metric space T endowed with a probability measure ν. Partition T into m regions R1, R2, ..., Rm of equal ν-measure. For each region i, let ni denote the number of points in Y(X) that fall into Ri. The UniformiteettiY score for this partition is defined as a normalized Pearson chi-square statistic: U = 1 − [χ²(n1, ..., nm; p1, ..., pm) / χ²max], where pi = ν(Ri) and χ²max is a normalization constant chosen so that 0 ≤ U ≤ 1. A score of 1 indicates a perfect match to the target distribution ν, while lower scores reflect deviations from uniformity.

Properties of UniformiteettiY include nonnegativity, a range bounded by 0 and 1, and invariance under measure-preserving reparametrizations of the source domain. Different partitions or kernel-based refinements yield variant scores, allowing flexibility in applications.

Applications described in the hypothetical context include evaluating data pipelines, comparing the uniformity of randomized designs, and illustrating how transformations influence observed distributions in teaching settings. See also equidistribution, chi-square tests, and transformation invariance.