Martin L. Hazelton

Plug-in bandwidth matrices for bivariate kernel density estimation

We consider bandwidth matrix selection for bivariate kernel density estimators. The majority of work in this area has been directed towards selection of diagonal bandwidth matrices, but full bandwidth matrices can give markedly better performance for some types of target density. Our methodological contribution has been to develop a new version of the plug-in selector for full bandwidth matrices. Our approach has the advantage, in comparison to existing full bandwidth matrix plug-in techniques, that it will always produce a finite bandwidth matrix. Furthermore, it requires computation of significantly fewer pilot bandwidths. Numerical studies indicate that the performance of our bandwidth selector is best when implemented with two pilot estimation stages and applied to sphered data. In this case our methodology performs at least as well as any competing method considered, while being simpler to implement than its competitors.

The dynamics and equilibria of day-to-day assignment models

Traffic network modelling is a field that has developed over a number of decades, largely from the economics of predicting equilibria across route travel choices, in consideration of the congestion levels on those routes. More recently, there has been a growing influence from the psychological and social science fields, leading to a greater interest in understanding behavioural mechanisms that underlie such travel choice decisions. The purpose of the present paper is to describe mathematical models which aim to reflect day-to-day dynamic adjustments in route choice behaviour in response to previous travel experiences. Particularly, the aim is to set these approaches in a common framework with the conventional economic equilibrium models. Starting from the analysis of economic equilibria under perturbations, the presentation moves onto deterministic dynamical system models and stochastic processes. Simple illustrative examples are used to introduce the modelling approaches. It is argued that while such dynamical approaches have appeal, in terms of the range of adaptive behavioural processes that can be incorporated, their estimation may not be trivial. In particular, the obvious solution technique (namely, explicit simulation of the dynamics) can lead to a rather complex problem of interpretation for the model-user, and that more “analytical” approximation techniques may be a better way forward.

Estimation of origin-destination matrices from link flows on uncongested networks

Given link flow data from an uncongested network over a number of time periods, the problem of estimating the origin-destination (O-D) traffic intensities is considered. A statistical model of the transport system with Poisson distributed O-D flows is developed, in which the variation of route choice proportions is represented. The distribution theory of the model is discussed, parameter identifiability investigated, and the full likelihood function derived. This function proves somewhat too cumbersome for practical use but two feasible estimation procedures are established, both based upon maximization of multivariate normal approximations to the likelihood. Although these methods can operate using link flow data alone, incorporation of prior information into the inferential process is also detailed. The basic statistical model is then extended to encompass measurement error in the link flow data and modified methods of parameter estimation are investigated. The paper finishes with a numerical study of the proposed estimation procedures and discussion of some suggested avenues for future research.

SOME COMMENTS ON ORIGIN-DESTINATION MATRIX ESTIMATION

Estimation of origin-destination (O-D) matrices from link count data is considered. This problem is challenging because the number of parameters to be estimated is typically larger than the number of network links. As a result, it is (usually) impossible to identify a unique optimal estimate of the O-D matrix from mean link traffic counts. However, information from the covariance matrix of link count data collected over a sequence of days can relieve this problem of indeterminacy. This fact is illustrated through a simple example. The use of second-order statistical properties of the data in O-D matrix estimation is then explored, and a class of estimators proposed. Practical problems of model mis-specification are discussed and some avenues for future research outlined.

INFERENCE FOR ORIGIN-DESTINATION MATRICES: ESTIMATION, PREDICTION AND RECONSTRUCTION

Abstract This paper concerns inference about an origin–destination (O–D) matrix from a single observation on a set of network link flows. Two problems in this area have received attention; first, the problem of reconstructing the actual number of O–D trips, and second, estimation of mean O–D trip rates. Little distinction has been drawn between these in the literature; indeed, it has sometimes been implicitly suggested that their a posteriori most probable solutions are identical. We show that this is not the case. An example is provided to demonstrate that the dissimilarity between the solutions to the two problems is potentially unbounded. The relative merits of reconstructed versus estimated mean trip count vectors are discussed with particular reference to their use as predictors of future traffic flows.

Adaptive kernel estimation of spatial relative risk.

Kernel smoothing is routinely used for the estimation of relative risk based on point locations of disease cases and sampled controls over a geographical region. Typically, fixed-bandwidth kernel estimation has been employed, despite the widely recognized problems experienced with this methodology when the underlying densities exhibit the type of spatial inhomogeneity frequently seen in geographical epidemiology. A more intuitive approach is to utilize a spatially adaptive, variable smoothing parameter. In this paper, we examine the properties of the adaptive kernel estimator by both asymptotic analysis and a simulation study, finding advantages over the fixed kernel approach in both the cases. We also look at practical issues with implementation of the adaptive relative risk estimator (including bandwidth choice and boundary correction), and develop a computationally inexpensive method for generating tolerance contours to highlight areas of significantly elevated risk.

Value of retinal vein pulsation characteristics in predicting increased optic disc excavation

Background: Retinal vein pulsation is often absent in glaucoma, but can be induced by applying a graded ophthalmodynamometric force (ODF) to the eye, which is elevated in glaucoma. Aim: To assess whether ODF has a predictive value in determining glaucoma progression. Methods: 75 patients with glaucoma and suspected glaucoma were examined prospectively in 1996, and then re-examined at a mean of 82 months later. All subjects had intraocular pressure, visual fields, stereo optic disc photography and ODF measured on their initial visit. When venous pulsation was spontaneous, the ODF was said to be 0 g. At re-examination, central corneal thickness and blood pressure were also measured. Initial and subsequent optic disc photographs were compared and graded into those that had increased excavation and those that had remained stable. The relationship between increased excavation (recorded as a binary response) and the measured variables was modelled using a multiple mixed effects logistic regression. Results: ODF at the initial visit was strongly predictive of increased excavation (p = 0.004, odds ratio 1.16/g, range 0–60 g), with greater predictive value in women than in men (p = 0.004). Visual field mean deviation was predictive of increased excavation (p = 0.044), as was optic nerve haemorrhage in association with older age (p = 0.038). Central corneal thickness was not significantly predictive of increased excavation (p = 0.074) after having adjusted for other variables. Conclusion: ODF measurement seems to be strongly predictive of the patient’s risk for increased optic disc excavation. This suggests that ODF measurement may have predictive value in assessing the likelihood of glaucoma progression.

Inference based on kernel estimates of the relative risk function in geographical epidemiology.

Kernel smoothing is a popular approach to estimating relative risk surfaces from data on the locations of cases and controls in geographical epidemiology. The interpretation of such surfaces is facilitated by plotting of tolerance contours which highlight areas where the risk is sufficiently high to reject the null hypothesis of unit relative risk. Previously it has been recommended that these tolerance intervals be calculated using Monte Carlo randomization tests. We examine a computationally cheap alternative whereby the tolerance intervals are derived from asymptotic theory. We also examine the performance of global tests of hetereogeneous risk employing statistics based on kernel risk surfaces, paying particular attention to the choice of smoothing parameters on test power.

Estimating Vehicle Speed from Traffic Count and Occupancy Data

Automatic vehicle detectors are now common on road systems across the world. Many of these detectors are based on single inductive loops, from which data on traffic volumes (i.e. vehicle counts) and occupancy (i.e. proportion of time during which the loop is occupied) are available for 20 or 30 second observational periods. However, for the purposes of traffic management it is frequently useful to have data on (mean) vehicle speeds, but this is not directly available from single loop detectors. While detector occupancy is related in a simple fashion to vehicle speed and length, the latter variable is not measured on the vehicles that pass.In this paper a new method for speed estimation from traffic count and occupancy data is proposed. By assuming a simple random walk model for successive vehicle speeds an MCMC approach to speed estimation can be applied, in which missing vehicle lengths are sampled from an exogenous data set. Unlike earlier estimation methods, measurement error in occupancy data is explicitly modelled. The proposed methodology is applied to traffic flow data from Interstate 5 near Seattle, during a weekday morning. The efficacy of the estimation scheme is examined by comparing the estimates with independently collected vehicle speed data. The results are encouraging.

Boundary kernels for adaptive density estimators on regions with irregular boundaries

In some applications of kernel density estimation the data may have a highly non-uniform distribution and be confined to a compact region. Standard fixed bandwidth density estimates can struggle to cope with the spatially variable smoothing requirements, and will be subject to excessive bias at the boundary of the region. While adaptive kernel estimators can address the first of these issues, the study of boundary kernel methods has been restricted to the fixed bandwidth context. We propose a new linear boundary kernel which reduces the asymptotic order of the bias of an adaptive density estimator at the boundary, and is simple to implement even on an irregular boundary. The properties of this adaptive boundary kernel are examined theoretically. In particular, we demonstrate that the asymptotic performance of the density estimator is maintained when the adaptive bandwidth is defined in terms of a pilot estimate rather than the true underlying density. We examine the performance for finite sample sizes numerically through analysis of simulated and real data sets.