Invariance Explains Multiplicative and Exponential Skedactic Functions

Authors

Vladik Kreinovich, The University of Texas at El PasoFollow
Olga Kosheleva, The University of Texas at El PasoFollow
Hung T. Nguyen, New Mexico State University - Main CampusFollow
Songsak Sriboonchitta, Chiang Mai UniversityFollow

Publication Date

8-2015

Comments

Technical Report: UTEP-CS-15-63

Published in: Van Nam Huynh, Vladik Kreinovich, and Songsak Sriboonchitta (eds.), Causal Inference in Econometrics, Springer Verlag, Cham, Switzerland, 2016, pp. 119-131.

Abstract

In many situation, we have an (approximately) linear dependence between several quantities y = f(x₁, ..., x_n). The variance v of the corresponding approximation error often depends on the values of the quantities x₁, ..., x_n: v = v(x₁, ..., x_n); the function describing this dependence is known as the skedactic function. Empirically, two classes of skedactic functions are most successful: multiplicative functions v = c * |x₁|^γ₁ * ... * |x_n|^γ_n and exponential functions v = exp(α + γ₁ * x₁ + ... + γ_n * x_n). In this paper, we use natural invariance ideas to provide a possible theoretical explanation for this empirical success; we explain why in some situations multiplicative skedactic functions work better and in some exponential ones. We also come up with a general class of invariant skedactic function that includes both multiplicative and exponential functions as particular cases.