Interpolating Location Data with Brownian Motion

TL;DR

This paper proposes using Brownian bridges to interpolate missing location data in travel surveys, providing a more realistic estimate of distance traveled than simple linear methods.

Contribution

It introduces a novel application of Brownian motion for interpolating gaps in location data, with explicit formulas for estimating travel distance.

Findings

Brownian bridge model accurately estimates travel distance.

Simulation results validate the method's effectiveness.

Provides explicit formulas for practical implementation.

Abstract

In 2018 the ''Onderweg in Nederland'' (translation: the Dutch Travel Survey) project by Statistics Netherlands was commenced, where participants were asked to track their location using an app, and check if the data is correct. An issue that occurs is so-called 'gaps' in the location data, where a whole sequence of data points is missing. The easiest way to fill such a gap is with a straight line, but this leads to systematic errors such as an underestimation of the distance travelled. A more realistic way to fill this gap is with a stochastic process. We use a Brownian bridge to model the movement of a traveller, as these have been used before successfully in ecological research. We find an explicit expression for the distance travelled in terms of some parameters that can be obtained from the data. To test whether this method gives an accurate estimation of the distance travelled, we simulate travel data using multiple different processes.