Invariance explains multiplicative and exponential skedactic functions

Abstract

In many situations, we have an (approximately) linear dependence between several quantities.(Formula presented.) The variance v=σ2 of the corresponding approximation error (Formula presented.) often depends on the values of the quantities x1,…,xn: v= v(x1,…,xn); the function describing this dependence is known as the skedactic function. Empirically, two classes of skedactic functions are most successful: multiplicative functions (Formula presented.) and exponential functions (Formula presented.).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.