A Bayesian Approach
to Retrospective Detection of Change-Points in
Road Surface Measurements
som för avläggande av filosofie doktorsexamen
vid Stockholms universitet
offentligen försvaras i
De Geerssalen, Geovetenskapshus, Stockholms Universitet
den 24 september 2001 kl 13.00
First-order autoregressive processes are
analysed for sudden changes in parameter value. In its most general form, a
multivariate vector of measurements is allowed, and no prior knowledge about
the involved parameters is required. Furthermore, no distributional assumptions
about the nature of the sudden change are made, and arbitrary prior distributions
over the space of all possible change-points are allowed. The change may be
in level, variance, or autocorrelation, or in some combinations of these.
Posterior probability distributions are derived for the location of a change-point conditional on the existence of such a change-point somewhere. The question whether or not there is a change present somewhere in the series is addressed in terms of posterior odds.
The for the posterior odds needed Bayes factors are only defined up to an arbitrary constant due to the use of improper prior distributions on most of the model parameters. This indeterminacy is resolved by the use of minimal imaginary training samples.
The emphasis is on analytical solutions based on an approximate version of the likelihood in order to allow for fast algorithms. Nevertheless, details for a Gibbs sampler are given based on the exact model, assuming that the unobserved initial conditions come from the stationary distributions of the involved processes. This sampler employs conditional conjugacy when possible. However, the full conditionals of the autoregressive coefficients are not of standard form and a slice sampler is implemented.
The motivating application is in the field of road maintenance, where pavement surfaces are frequently measured and the need arises to partition roads into parts, which can be considered homogeneous with respect to the measured characteristics. In Sweden, the international roughness index (IRI - a measure of a road's longitudinal unevenness) and rutting are the two measurements of foremost interest. The use of the developed theory is exemplified throughout by these measurement series, which are collected by so-called Laser-RST-vehicles. A well-motivated segmentation of a road is a prerequisite for a successful handling of the road sections in a pavement management system, which ultimately aims at efficiently providing road infrastructure of high quality.
In order to make this work accessible to others than professional statisticians, the basic concepts involved in the analysis are described in the introductory section.
Key words: Change-point detection, retrospective view, autoregressive processes, minimal imaginary training sample, road surface measurements, international roughness index, rutting, Laser-RST vehicles, road maintenance, pavement management.