Abstracts of research reports - 1996

This page contains abstracts of Research Reports published by the Department of Statistical Science at University College London during 1996. Usually a compressed postscript file can be downloaded by clicking the download link next to the research report number. Help on viewing these files is available at this link

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A. P. Dawid : Conditional Independence

Date: February 1996

This article has appeared in the Wiley Encyclopedia of Statistical Sciences (Update, Volume 2 (1998), pp. 146-155). It gives a brief overview of fundamental properties and applications of conditional independence.

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Richard E. Chandler: The second-order spectral analysis of spatial-temporal rainfall models

Date: February 1996.

Point-process based models provide a convenient and flexible framework within which to study rainfall in space and time. Models for rainfall at a single site have been extensively discussed in the literature; one of the main problems is a lack of adequate inferential procedures. This is largely due to the presence of complex dependencies in both data and models. A procedure which overcomes these difficulties to some extent involves the use of a `spectral likelihood' function (the Whittle Criterion) where the sample Fourier coefficients rather than the observations themselves are regarded as data; to use this method one needs to know the mean and second-order spectral density of the model to be fitted. This paper presents a multidimensional version of the method for use in the study of spatial-temporal rainfall fields. A review of the necessary multidimensional Fourier theory is given which motivates the use of the Whittle Criterion as an objective function. A general class of point process models for spatial-temporal rainfall is proposed and first- and second-order properties derived for this class. Some preliminary analysis of data from a radar station in the South-West of England is presented with an emphasis on computational issues.

Keywords: Fast Fourier transform, parameter estimation, periodogram, rainfall modelling, spatial processes, spectral analysis, Whittle criterion.

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Kostas Skouras and A. Philip Dawid: On Efficient Probability Forecasting Systems

Date: March 1996

In this paper we study the asymptotic behaviour of probability forecasting systems, and discuss their usefulness as inferential tools for statistical problems such as model verification and selection. Our theoretical setting is the prequential (predictive sequential) framework proposed by Dawid (Journal of the Royal Statistical Society, Series A,, vol. 147 (1984), pp.278-292). We study especially the notion of prequential efficiency of a forecasting system and present some new results. We focus on plug-in (estimative) forecasting systems where the forecast distribution is generated by replacing the unknown parameter with an estimate.

A revised version of this paper has appeared in Biometrika (1999).

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K. Skouras and A.P. Dawid: On Efficient Point Prediction Systems

Date: March 1996

Assume that a forecaster observes a sequence of random variables and issues predictions according to a point prediction system i.e. a rule which at every time t issues a point prediction for the next observation at time t+1. We introduce the concept of efficiency of a point prediction system for the case that the joint distribution of the sequence of observations is known to belong to a parametric family and performance is assessed by the long-run sum of squared prediction errors. Independence is not a requirement. Under weak conditions the class of efficient point prediction systems is non-empty and any two efficient point prediction systems will in a certain strong sense make asymptotically identical predictions for the infinite future. We discuss the efficiency of point prediction systems based on Bayesian predictive means and on plugging-in parameter estimates. The results are applied to probability forecasting and stochastic regression.

Keywords: Point Prediction Systems, Efficiency, Optimal Predictors, Time Series, Prequential Inference.

A revised version of this paper has appeared in the Journal of the Royal Statistical Society (1998, Series B, No. 60, pp.765-780).

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R.J. Brooks and M. Stone: Efficient Cross-Validation of Reduced Rank Regression and JCR0

Date: March 1996

This paper proves that the procedure "JCR0" as defined by Brooks and Stone (1994) as a special case of joint continuum regression is equivalent to a generalised form of reduced rank regression. It then describes a procedure for "leave-me-out" cross-validatory assessment of the rank constraint in reduced rank regression which does not require recalculation of regressions coefficients on omission of each observation. This method is found to simplify cross-validatory assesssment of JCR0.

Keywords: Cross-validation; Joint continuum regression; Multivariate predictand; Reduced rank regression.

162 Downlaod

M. Hazelton: Optimal rates for local bandwidth selection

Date: March 1996.

The problem of bandwidth selection for kernel density estimation at a point is considered. Asymptotic lower bounds are established for the relative rate of convergence of data-driven bandwidth selectors to their optimal values. It is noted that some existing methods of local bandwidth selection, using high order kernel functions, attain these rates. Nevertheless, a simulation study indicates that improved performance predicted by asymptotic theory may not occur in practice for sample sizes as large as 10^4 or 10^5. The paper finishes with a comparison of local and global bandwidth selection, observing that in some sense the local problem is the more difficult.

Keywords: Bandwidth selector, kernel density estimation, mean squared error, relative rate of convergence

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Hilde M. Herbots: The dependence of measures of subpopulation differentiation on population structure and mutation rate. Part I: The symmetric case

Date: March 1996.

Exact and approximate analytical values of Wright's FST, obtained by coalescent techniques, are studied for several symmetric models of population structure. The dependence of FST on the parameters of population structure and on the neutral mutation rate is examined and compared between the different population structures considered. It is shown that FST depends strongly on the mutation rate when gene flow is restricted to essentially one dimension and the number of subpopulations is large. The accuracy of Slatkin's approximation for FST in terms of mean coalescence times of pairs of genes and that of a first order approximation for FST are investigated and criteria for accuracy are proposed. An appendix lists the analytical values of FST, its approximations and related quantities under the various models of population structure studied in this paper.

Hilde M. Herbots:The dependence of measures of subpopulation differentiation on population structure and mutation rate. Part II: The asymmetric case

Date: March 1996.

The effect of asymmetries in population structure (for example unequal subpopulation sizes) on the value of Wright's FST is examined. Exact and approximate analytical FST values (obtained by coalescent techiques) are studied for three models of population structure which have less symmetry than the traditional island and stepping-stone models. The evidence suggests that asymmetries in the population structure can have an important impact on the FST value when groups of subpopulations are relatively isolated from each other, when the overall level of gene flow is low, and, if FST is calculated with equal weights on all subpopulations, at very high mutation rates. If the subpopulations are weighted by their relative sizes, unqeual subpopulations sizes can strongly affect FST at small mutation rates, while little effect on FST is seen at very high mutation rates. Asymmetries can both decrease or increase the dependence of FST on the mutation rate. The accuracy of two suggested approximations for the exact FST value is considered. An appendix lists the exact values of FST and related quantities under the models of population structure studied in this paper.

Hilde M. Herbots:The structured coalescent

Date: March 1996.

The genealogy of a sample from a subdivided population is, under reasonable assumptions about reproduction and migrations, well described by the structured coalescent, which is the extension coalescent, the distribution of the time since the most recent common ancestor of a pair of individuals from the same subpopulation in a symmetric island model of population structure is compared to that of two individuals from a panmictic population of the same total size. For a simple model for reproduction and migration, a rigorous proof is given that, measuring time in diffusion time-scale, the ancestral process converges (weakly) to the structured coalescent, as the subpopulation sizes tend to infinity.

Hilde M. Herbots, Martin B. Richards, Peter Forster and Bryan C. Sykes: Site 73 in hypervariable region II of the human mitochondrial genome and the origin of European populations

Date: March 1996.

The majority of human mitochondrial DNA sequence data published to date are confined to hypervariable region I in the control region. By contrast, this paper focusses on a nucleotide site in hypervariable region II, a site which is shown to be of considerable interest when studying the origins of the British "white Caucasian" population. Unlike most non-European populations whose mtDNA sequences have been studied in the literature, the British "white Caucasian" population has a high level of variation at site 73 (following the site numbering by Anderson et al. 1981). This variation appears to have its origin largely in a mutation from guanine to adenine at that site with an estimated minimum age between 15,000 and 25,000 years. The data of Piercy et at.(1993) suggest that roughly half of the British "white Caucasian" population are descended from a common maternal ancestor who carried this mutation at site 73. Incorporating site 73 into a phylogenetic analysis otherwise identical to that of Richards et al. (1996) greatly clarifies the structure of the British "white Caucasian" population, resolving much more clearly the five major European mtDNA clusters identified by Richards et al. We suggest that the lineages carrying an A at site 73, together with some other lineages, perhaps have their origins in a small founder population which expanded after the last glacial maximum about 20,000 years ago. Adenine at site 73 would thus have been common amongst the hunter-gatherers who inhabited Europe before the spread of agriculture from the Middle East. We conclude that, in addition to region I sequences, site 73 is worth determining in studies of Caucasian populations.

A. P. Dawid and P. Sebastiani: Coherent Criteria for Optimal Experimental Design

Date: March 1996

An approach to the choice of experiment to aid prediction is proposed in a Bayes decision-theoretic framework. Estimation is included as a special case. We characterize coherent design criteria which are functions of the dispersion matrix and their associated decision problems. Connexions are established with the classical approach to optimal design theory for the normal linear model thus making its optimization techniques available for suitable problems in this new setting. Implications of the theory for non-normal non-linear models are also considered.

A revised and retitled version of this paper has appeared in Ann. Statist., No. 27 (1999), pp.65-87.

P.J. Brown, T. Fearn and M.S. Haque: Discrimination with many variables

Date: April 1996.

No abstract available.

T. Fearn: Detecting adulteratism of Basmati rice by the analysis of mixtures

Date: April 1996.

No abstract available.

Download 170

A. P. Dawid and I.W. Evett: Using a graphical method to take account of dependencies when evaluating complicated patterns of evidence

Date: April 1996

The forensic scientist often faces the task of interpreting patterns of evidence which involve many variables. Combining different items of evidence within a complex framework of circumstances requires logical powers of reasoning and this can be assisted by formal methods. We discuss one such method which, as has already been pointed out by Aitken and Gammerman (J. Forensic Science Society, vol.29, 1989), offers considerable potential for creating probabilistic expert systems to assist in evidence interpretation. In particular, we show how the method, which is based on a directed acyclic graph, enables dependencies between different aspects of the evidence to be considered. The discussion is based on an imaginary case example.

A revised and retitled version has been published as "Using a graphical method to assist the evaluation of complicated patterns of evidence". Journal of Forensic Sciences, No. 42 (1997), pp.226-231.

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171 Download

A.P. Dawid and V.G. Vovk: Prequential Probability: Principles and Properties

Date: May 1996

Note for users in more westerly longitudes: another postscript version of this paper is available for download here (CASBS, Stanford).

Forecaster has to predict, sequentially, a string of uncertain quantities (X1, X2, ... ), whose values are determined and revealed one by one, by Nature. Various criteria may be proposed to assess Forecaster's empirical performance. The Weak Prequential Principle requires that such a criterion should depend on the forecaster's behaviour or strategy only through the actual forecasts issued. A wide variety of appealing criteria is shown to respect this Principle. We further show that many such criteria also obey the Strong Prequential Principle, which requires that, when both Nature and Forecaster make their choices in accordance with a common joint distribution P for (X1, X2, ... ), certain stochastic properties, underlying and justifying the criterion and inferences based on it, hold regardless of the detailed specification of P.

In order to understand further this compliant behaviour, we introduce the prequential framework, a game-theoretic basis for Probability Theory in which it is impossible to violate the Prequential Principles, and we describe its connexions with classical Probability Theory. In this framework, in order to show that some criterion for assessing Forecaster's empirical performance is a valid one, we have to exhibit a winning strategy for a third player, Statistician, in a certain perfect-information game. We demonstrate that many performance criteria can be formulated and are valid in this framework and, therefore, satisfy both Prequential Principles.

Keywords: Farthingale, forecasting, goodness of fit, limit theorems of probability theory, martingale, perfect-information game, strong prequential principle, weak prequential principle.

This paper has appeared in Bernoulli, No. 5 (1999), pp.125-162.

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A.P.Dawid, M.Stone and J.V.Zidek: Critique of E.T.Jaynes's "Paradoxes of Probability Theory"

Date: September 1996

In Chapter 15 "Paradoxes of Probability Theory" of his draft book "Probability Theory: the Logic of Science" (obtainable via Internet from here), E.T. Jaynes (1994) has made numerous criticisms of our work on problems arising from the use of improper prior distributions in Bayesian Statistics. Specifically he has attacked our analyses of the "Marginalization Paradox" (Dawid Stone and Zidek 1973) and of "Strong Inconsistency" (Stone 1976). In this article we respond to these criticisms and attempt to clarify further the nature of the underlying arguments.

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Richard E. Chandler: A note on analytical solutions to the Whittle likelihood equation

Date: October 1996

Estimation of the parameter vector of a stochastic process model may be carried out using an approximate `spectral likelihood' method, providing the model mean and spectral density are known. We derive analytical solutions to the estimating equations for parameters which contribute to the spectral density via some multiplicative function; this considerably speeds up the numerical solution of the estimating equations for the remaining parameters. In particular, if there are two such parameters then the Whittle estimates reproduce the mean of the data.

Keywords: Whittle estimates, parameter estimation, spectral analysis.

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D.R. Cox and V.S. Isham: Stochastic spatial-temporal models for rain

Date:October 1996

The emphasis of this paper is on stochastic models for spatial rainfall fields and their evolution in time: the development of models, their fitting to empirical data, and the assessment of the adequacy of the fitted models. However, the paper starts with a brief discussion of these issues in the context of more general stochastic models and their application. The paper is particularly concerned with models that aim to represent only the main observable features of physical procecesses (using relatively few parameters, interpretable in terms of the process) rather than attempting to incorporate detailed underlying physical laws. The models for rainfall described here are based on Poisson cluster processes and involve random moving clusters of elliptical rain cells.

Keywords: regression, Bayesian inference, nonconjugate prior, indeterminism.

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B.Q. Fang and A.P. Dawid: Nonconjugate Bayesian regression on many variables

Date: November 1996

In this paper the problem of normal linear regression on arbitrarily many variables is considered. It is shown that a nonconjugate prior implies indeterminism for the Bayes predictor, in contrast to determinism induced by the use of the usual conjugate priors. A comparison is made between the Bayes predictor and a sample-based least squares predictor which fits the data perfectly.

AMS 1991 subject classification: 62A15, 62J05.

Keywords and phrases: regression, Bayesian inference, nonconjugate prior, indeterminism.

This paper has appeared in J. Statist. Plann. Inf. (2002)

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[This page last modified 19 December, 2008]