Details of Grant 

EPSRC Reference:	EP/D005221/1
Title:	Multiscale methods in statistics
Principal Investigator:	Nason, Professor G
Other Investigators:
Researcher Co-Investigators:
Project Partners:
Department:	Mathematics
Organisation:	University of Bristol
Scheme:	Standard Research (Pre-FEC)
Starts:	01 October 2005	Ends:	30 September 2008	Value (£):	168,395
EPSRC Research Topic Classifications:	Statistics & Appl. Probability
EPSRC Industrial Sector Classifications:	No relevance to Underpinning Sectors
Related Grants:
Panel History:
Summary on Grant Application Form
Many problems in science and engineering can be successfully addressed by analysing them at several scales simultaneously/multiscale analysis. Over the last 15 years a new technique called wavelet analysis has proved very successful for multiscale work in many real situations. E.g. the FBI fingerprint database and JPEG2000 image compression both use wavelet representations. Wavelets have several interesting and important properties: 1. They provide economical representations both for mathematically `nice' objects (like smooth functions) and also for many real-world objects (such as images that exhibit discontinuities). Thus wavelets are good for image compression; 2. Wavelets provide local information about an object at different scales and locations simultaneously; 3. Many wavelet algorithms are extremely fast and efficient, more efficient than other similar algorithms, such as the fast Fourier transform.Wavelets are very useful for statistical curve-fitting problems (regression) and the analysis of data through time (time series). In regression wavelets are provably more accurate than many existing methods and simultaneously faster and more efficient. Wavelets have provided new time series models that enable series analysis in terms of scale and, advanced representation models permit evolution of series characteristics over time (compared to stationary time series, where the characteristics are static over time which is unrealistic in many practical situations, such as financial data).However, for many statistical problems classical wavelets suffer since they are designed to represent objects that are defined on a regular grid. Many data are not regularly spaced: e.g. transport centres, population centres, environmental data such as wildlife reserves, epidemiological problems involving health or pollution to name but a few. Classical wavelets have difficulties in representing such data.Recently, a new wavelet generalization, called lifting, permits the advantages of wavelets to be extended to irregularly spaced data. Very recent statistical work has proposed using lifting for regression with irregularly spaced spatial data. We propose to improve on this early work in many ways but also greatly extend its usefulness. E.g. creating new techniques to estimate unknown quantities over a wider domain and not necessarily just on the sample collection points.We also propose to create and evaluate new stochastic models for irregularly spaced time series, spatial data and networks. Lifting gives us the opportunity to quantify local information about the composition of actual random processes (the local spectra and covariance structure), whilst naturally coping with irregularly spaced data. E.g. lifting can produce multiscale descriptions of networks which provide new ways to model quantities on networks (e.g. transport delays or volatility within communications networks). With these new techniques we shall be able to model and predict processes for which there are very few existing models.Finally, we propose multiscale approaches to some statistical bioinformatics problems. Molecules and proteins are not nice regularly spaced mathematical objects and so lifting methods are ideally suited for their multiscale modelling. An ambitious goal will be to provide a multiscale lifting model for large molecules which, after integrating other biochemical expertise, might enable rapid mathematical modelling of various interesting molecular scenarios: e.g. inter-molecular interactions.Overall our aims are to create and apply novel multiscale methods based on lifting to a variety of key statistical problems that realize the advantages of classical wavelet methods but for irregular data, spatial data and networks. The proposers are deeply involved in this new field and have long experience of multiscale methods in regression and time series and, as such, are uniquely placed to carry out the programme described here.
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