Fuzzy transform and least-squares approximation: Analogies, differences, and generalizations

This paper investigates the relations between the least-squares approximation techniques and the Fuzzy Transform. Assuming that the function f : R → R underlying a discrete data set D := {(xi, f(xi))}i=1 has been computed with interpolating or least-squares constraints, we prove that the Discrete Fuzzy Transform of the sets {f(xi)}i=1 and {f(xi)}i=1 is the same. This result shows that the Discrete Fuzzy Transform is invariant with respect to the interpolating and least-squares approximation of D. Additionally, the Fuzzy Transform of f outside P is approximated by simply resampling the continuous map f at a set of points of R\P. Using numerical linear algebra, we also derive new properties (e.g., stability to noise, additivity with respect to P) and characterizations (e.g., radial and dual membership maps) of the Discrete Fuzzy Transform. Finally, we define the geometryand confidence-driven Discrete Fuzzy Transform, which take into account the intrinsic geometry of the input data and the confidence weights associated to the f -values or the points of P.