A Bayesian Approach
to Retrospective Detection of Change-Points in
Road Surface Measurements
Akademisk avhandling
som för avläggande av filosofie doktorsexamen
vid Stockholms universitet
offentligen försvaras i
De Geerssalen, Geovetenskapshus, Stockholms Universitet
den 24 september 2001 kl 13.00
av
Fridtjof Thomas
fil mag
Abstract
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.
ISBN 91-7265-307-8