Research

Our main areas of research within the current group are:

• time series analysis
• multivariate data analysis
• applications to market research
• search algorithms and stochastic global optimisation
• probabilistic number theory
• optimal experimental design
• stochastic processes and random fields with weak and strong dependence
• diffusion processes and PDE with random data
• anomalous diffusion
• Burgers and KPZ turbulence
• fractional ordinary and PDE, and statistical inference with higher-order information.

In focus

Time series analysis

In recent years a powerful technique of time series analysis has been developed and applied to many practical problems. This technique is based on the use of the Singular-value decomposition of the so-called trajectory matrix obtained from the initial time series by the method of delays. It is aimed at an expansion of the original time series into a sum of a small number of 'independent' and 'interpretable' components.

Also, the spatial analogies of the popular ARMA-type stochastic time series have been developed based on the fractional generalizations of the Laplacian with two fractal indices. These models describe important features of processes of anomalous diffusions such as strong dependence and/or intermittency.

Multivariate statistics

The objective is the development of a methodology of exploratory analysis of temporal-spatial data of complex structures with the final aim of construction of suitable parametric models.

The applications include various medical, biological, engineering and economical data. Several market research projects where the development of statistical models was a substantial part have taken place.

Stochastic global optimisation

Let ƒ be a function given on a d-dimensional compact set X and belonging to a suitable functional class F of multi extremal continuous functions.

We consider the problem of its minimization, that is approximation of a point x' such that ƒ(x')=min ƒ(x), using evaluations of ƒ at specially selected points.

Probabilistic methods in search and number theory

Several interesting features of the accuracy of diophantine approximations can be expressed in probabilistic terms.

Many diophantine approximation algorithms produce a sequence of sets F(n), indexed by n, of rational numbers p/q in [0,1]. Famous examples of F(n) are the Farey sequence, the collection of rationals p/q in [0,1] with q<=n, and the collection of all n-th continued fraction convergents.

Stochastic processes

New classes of stochastic processes with student distributions and various types of dependence structures have been introduced and studied. A particular motivation is the modelling of risk assets with strong dependence through fractal activity time.

The asymptotic theory of estimation of parameters of stochastic processes and random fields has been developed using higher-order information (that is, information on the higher-order cumulant spectra). This theory allows analysis of non-linear and non-Gaussian models with both short- and long-range dependence.

Burgers turbulence problem

Explicit analytical solutions of Burgers equation with quadratic potential have been derived and used to handle scaling laws results for the Burgers turbulence problem with quadratic potential and random initial conditions of Ornstein-Uhlenbeck type driven by Levy noise.

Results have considerable potential for stochastic modelling of observational series from a wide range of fields, such as turbulence or anomalous diffusion.

Topics in medical statistics

Several topics that have been associated with medical statistics presently researched in Cardiff include time-specific reference ranges, and errors in variables regression. Current research focuses on the search for a unified methodology and approach to the errors in variables problem.

Extreme Value Analysis

Extreme value analysis is a branch of probability and statistics that provides non-parametric procedures for extrapolation beyond the range of data (as good as possible and depending on the quality of data, knowing the limits is also an important issue). Its methods are usually relevant for institutions that are exposed to high risks, for instance, financial services and insurance companies or environmental engineering institutions.