Joe Whittaker

Tracking and predicting a network traffic process.

This article outlines a partial description of a traffic monitoring and prediction system that is under development in the DRIVE DYNA project, concentrating on the underlying graphical representation of the state space model.

Recent advances and applications in the field of short-term traffic forecasting

Frequent road traffic congestion is now a global issue. One of the proposed solutions to this problem is dynamic traffic management (DTM): the management of traffic flows, vehicles and traffic demand based on data representing the current and near expected traffic situation. A key ingredient for DTM is accurate network-wide short-term traffic forecasts. This article gives a general overview of the state of the art together with some recent advances and applications derived from a number of field trials conducted as part of the DRIVE-II programme of the Commission of the European Communities. The article gives an introduction to DTM, and reviews the nature of traffic demand and supply and the traffic measurement process. The statistical methodology of short-term forecasts applied in transport is discussed and the articles in this issue are introduced. Mention is made of as yet unresolved problems. The article concludes that a great deal of work still remains to be done before the current methodology can consistently provide the desired level of accuracy needed for DTM. In the near future, more research will be needed and carried out, both with respect to methods already available, to methods available but not yet applied and perhaps to develop new methodology.

Sparse CCA using a Lasso with positivity constraints

Canonical correlation analysis (CCA) describes the relationship between two sets of variables by finding linear combinations of the variables with maximal correlation. A sparse version of CCA is proposed that reduces the chance of including unimportant variables in the canonical variates and thus improves their interpretation. A version of the Lasso algorithm incorporating positivity constraints is implemented in tandem with alternating least squares (ALS), to obtain sparse canonical variates. The proposed method is demonstrated on simulation studies and a data set from market basket analysis.

Quantile regression for modelling distributions of profit and loss

Quantile regression is applied in two retail credit risk assessment exercises exemplifying the power of the technique to account for the diverse distributions that arise in the financial service industry. The first application is to predict loss given default for secured loans, in particular retail mortgages. This is an asymmetric process since where the security (such as a property) value exceeds the loan balance the banks cannot retain the profit, whereas when the security does not cover the value of the defaulting loan then the bank realises a loss. In the light of this asymmetry it becomes apparent that estimating the low tail of the house value is much more relevant for estimating likely losses than estimates of the average value where in most cases no loss is realised. In our application quantile regression is used to estimate the distribution of property values realised on repossession that is then used to calculate loss given default estimates. An illustration is given for a mortgage portfolio from a European mortgage lender. A second application is to revenue modelling. While credit issuing organisations have access to large databases, they also build models to assess the likely effects of new strategies for which, by definition, there is no existing data. Certain strategies are aimed at increasing the revenue stream or decreasing the risk in specific market segments. Using a simple artificial revenue model, quantile regression is applied to elucidate the details of subsets of accounts, such as the least profitable, as predicted from their covariates. The application uses standard linear and kernel smoothed quantile regression.

Edge exclusion tests for graphical Gaussian models

Testing that an edge can be excluded from a graphical Gaussian model is an important step in model fitting and the form of the generalised likelihood ratio test statistic for this hypothesis is well known. Herein the modified profile likelihood test statistic for this hypothesis is obtained in closed form and is shown to be a function of the sample partial correlation. Related expressions are given for the Wald and the efficient score statistics. Asymptotic expansions of the exact distribution of this correlation coefficient under the hypothesis of conditional independence are used to compare the adequacy of the chi-squared approximation of these and Fisher’s Z statistics. While no statistic is uniformly best approximated, it is found that the coefficient of the O(n-1) term is invariant to the dimension of the multivariate Normal distribution in the case of the modified profile likelihood and Fisher’s Z but not for the other statistics. This underlines the importance of adjusting test statistics when there are large numbers of variables, and so nuisance parameters in the model.

Spectral preconditioning of Krylov spaces: Combining PLS and PC regression

In regularized regression the vectors that lie in Krylov and eigen subspaces, formed in PLS and PC regressions respectively, provide useful low dimensional approximations for the LS regression coefficient vector. By preconditioning the LS normal equations we provide a framework in which to combine these methods, and so exploit both of their respective advantages. The link between the proposed method to orthogonal signal correction and to cyclic subspace regression is made. The performance of the proposed solution in reducing the dimension of the regression problem, and the shrinkage properties of the resulting coefficient vector, are both examined.

A dynamic scorecard for monitoring baseline performance with application to tracking a mortgage portfolio.

A principled technique for monitoring the performance of a consumer credit scorecard through time is derived from Kalman filtering. Standard approaches sporadically compare certain characteristics of the new applicants with those predicted from the scorecard. The new approach systematically updates the scorecard combining new applicant information with the previous best estimate. The dynamically updated scorecard is tracked through time and compared to limits calculated by sequential simulation from the baseline scorecard. The observation equation of the Kalman filter is tailored to take the results of fitting local scorecards by logistic regression to batches of new clients that arrive in the current time interval. The states in the Kalman filter represent the true or underlying score for each attribute in the card: the parameters of the logistic regression. Their progress in time is modelled by a random walk and the filter provides the best estimate of the scores using past and present information. We illustrate the technique using a commercial mortgage portfolio and the results indicate significant emerging deficiencies in the baseline scorecard.

Glim Syntax and Simultaneous Tests for Graphical Log Linear Models

Within the class of hierarchical log linear models for contingency tables, conditional independence models have a special place and as they are well represented by their independence graphs they are known as graphical models. Standard GLIM model formulae are extended to give a simple representation of these models and rules are given to translate formulae between the standard and extended language. The deviances for all conditional independence models can be computed from the deviances of the elementary graphical models. These computations are employed in a simultaneous test procedure for selecting a parsimonious model from the class of all graphical models. This procedure is illustrated on a contingency table from the sociological literature classified by four response factors.

Standard errors for the parameters of graphical Gaussian models

The comparison of an estimated parameter to its standard error, the Wald test, is a well known procedure of classical statistics. Here we discuss its application to graphical Gaussian model selection. First we derive the Fisher information matrix and its inverse about the parameters of any graphical Gaussian model. Both the covariance matrix and its inverse are considered and a comparative analysis of the asymptotic behaviour of their maximum likelihood estimators (m.l.e.s) is carried out. Then we give an example of model selection based on the standard errors. The method is shown to produce almost identical inference to likelihood ratio methods in the example considered.

Correspondence analysis and association models constrained by a conditional independence graph

The manner in which the conditional independence graph of a multiway contingency table effects the fitting and interpretation of the Goodman association model (RC) and of correspondence analysis (CA) is considered.Estimation of the row and column scores is presented in this context by developing a unified framework that includes both models. Incorporation of the conditional independence constraints inherent in the graph may lead to equal or additive scores for the corresponding marginal tables, depending on the topology of the graph. An example of doubly additive scores in the analysis of a Burt subtable is given.