Invariance Explains Multiplicative and Exponential Skedactic Functions

Abstract

In many situations, we have an (approximately) linear dependence between several quantities: $y\approx c+\sum \limits _{i=1}^n a_i\cdot x_i$. The variance $v=\sigma ^2$ of the corresponding approximation error $\varepsilon =y-\left( c+\sum \limits _{i=1}^n a_i\cdot x_i\right)$ often depends on the values of the quantities $x_1,\ldots ,x_n$: $v=v(x_1,\ldots ,x_n)$; the function describing this dependence is known as the skedactic function. Empirically, two classes of skedactic functions are most successful: multiplicative functions $v=c\cdot \prod \limits _{i=1}^n |x_i|^{\gamma _i}$ and exponential functions $v=\exp \left( \alpha +\sum \limits _{i=1}^n \gamma _i\cdot x_i\right)$. 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.