Comparing maximal mean values on different scales

TL;DR

This paper investigates the common assumption about how maximal mean values change with scale, revealing conditions under which the intuition holds or fails, with implications for analyzing GPS data and similar time series.

Contribution

It provides a rigorous analysis of the behavior of maximal mean values across scales, clarifying when the intuitive decrease with increasing interval length is valid.

Findings

Maximal mean values do not always decrease with increasing interval length.

Conditions are identified under which the intuitive behavior holds.

The results have implications for analyzing time series data like GPS speeds.

Abstract

When computing the average speed of a car over different time periods from given GPS data, it is conventional wisdom that the maximal average speed over all time intervals of fixed length decreases if the interval length increases. However, this intuition is wrong. We investigate this phenomenon and make rigorous in which sense this intuition is still true.