Fractal Scaling of Population Counts Over Time Spans

TL;DR

This paper demonstrates that population counts over time follow a fractal scaling law when lifetimes are Pareto distributed, enabling better early estimates of reach in digital advertising.

Contribution

It introduces a fractal scaling model for population counts based on Pareto lifetime distributions and proposes a method to combine counts using the $L^p$-norm for improved estimates.

Findings

Counts scale as time to the power of r in Pareto distributions

The $L^p$-norm with p=1/r approximates combined population counts

Fractal scaling aids early reach estimation in digital advertising

Abstract

Attributes which are infrequently expressed in a population can require weeks or months of counting to reach statistical significance. But replacement in a stable population increases long-term counts to a degree determined by the probability distribution of lifetimes. If the lifetimes are in a Pareto distribution with shape factor $1-r$ between 0 and 1, then the expected counts for a stable population are proportional to time raised to the $r$ power. Thus $r$ is the fractal dimension of counts versus time for this population. Furthermore, the counts from a series of consecutive measurement intervals can be combined using the $L^p$-norm where $p=1/r$ to approximate the population count over the combined time span. Data from digital advertising support these assertions and find that fractal scaling is useful for early estimates of reach, and that the largest reachable fraction of an audience over a long time span is about $1-r$.