Nfwaarden

Nfwaarden is a theoretical concept used in discrete mathematics and data analysis to describe the set of values produced when a base value function is passed through a normalization function. The term is Dutch in origin, with waarden meaning values and Nf indicating the normalization step. The concept helps compare datasets after applying a common scaling or transformation.

Definition: Let X be a finite set, f: X -> R a value function, and N: R -> R a nondecreasing normalization function. The Nfwaarden of X under (f, N) is the multiset {N(f(x)) : x in X}. If N is the identity function, the Nfwaarden reduce to the values of f. If f is injective, the Nfwaarden are distinct unless N compresses values.

Properties: The ordering of Nfwaarden reflects the order of f modulo the monotonicity of N. They can be normalized further to form a probability distribution by dividing by the sum of all elements. The multiset nature means repeated values are allowed, which is important for assessing distributional features.

Applications: Nfwaarden are used in data preprocessing to compare distributions after scaling, in teaching about value transformations, and in analyses of algorithmic fairness where normalized value distributions must be reconciled across datasets.

Example: X = {a, b, c}, f(a)=2, f(b)=5, f(c)=7; N(y)=log(1+y). Then Nfwaarden are {log 3, log 6, log 8}.

See also: normalization, transformation, value function, multiset.