Locally Stationary Time Series and Multiscale Methods for Statistics (LuSTruM)

This fellowship proposes research in time series analysis and regression. Time series
analysis is concerned with data recorded through time. Time series occur in a variety
of areas of great importance to society such as medicine (recording of vital signs),
economics and finance (GDP or share prices), the environment (air pollution),
energy (national electricity demand), and transportation (traffic flow), to name but a few.
A common way of displaying time series, often seen in the media, is via the time plot,
which plots the series' values consecutively through time enabling major features,
such as trend or seasonal effects, to be readily observed. Collectively, society needs
to ensure that series are properly collected and recorded, modelled appropriately,
to gain an understanding of their behaviour, and often predicted to estimate their
future values (forecasting).

Much real world analysis assumes that series arise from stationary models, which
permit the values of the series to change at each time, but the underlying statistics
do not change (for example, a stationary share price changes from hour to hour,
but the overall level, or mean, stays constant). It is becoming increasingly clear that
stationary models are not appropriate for many real series. For example, share price
statistics do change, sometimes exceptionally, due to sudden events such as political
upheaval or natural disasters, and often nonstationary models are appropriate and
useful alternatives.

This project intends to develop nonstationary techniques with a focus on energy and
economics applications. For example, energy companies are interested in nonstationary
models because deregulation and increasingly diverse energy sources have caused
many previously stable data sets to become less stationary and more unpredictable.

This project will create new nonstationary models intended to be more realistic, flexible and
lead to better modelling, forecasting and consequently better decision-making.
Nonstationary models can also shed light on tasks that are infeasible for stationary ones
such as ascertaining whether a series has been sampled frequently enough. We will also
research nonstationary functional models, where each observation is not a single number
but an entire curve, such as national electricity consumption recorded across a day.

Regression is concerned with the modelling of relationships between different variables
and is used extensively in the real world. Many important regression methods assume
that data have constant variance and a `bell-curve' distribution. Much real data are not
like that, but operations, such as taking each observation's square root, can make the
data fulfil those constant variance/`bell curve' assumptions, at least approximately.
Recently, a new, promising, very different, multiscale class, called the Haar-Fisz transform,
was developed. The new class works extremely well for count data and has shown some
fascinating theoretical properties, such as mimicking the well-known logarithm. This project
will investigate the intriguing theoretical underpinnings of this new class as well as develop
further methods for cleaning up noisy signals, for example, removing noise from astronomical
or low-light security images. Additionally, we will investigate regression for irregular data
using techniques that make use of multiple scales simultaneously (multiscale).
First generation multiscale methods, highly valued for purposes such as image compression
in JPEG, are not easily adapted to irregular situations. This project seeks to investigate
second generation multiscale methods, suitable for irregular data. For example, to better
estimate and control information on networks (such as identify and mitigate delays on
transport networks) or irregularly-spaced systems (such as identify regions of the genome
that are implicated in several complex diseases such as cancer.)

The beneficiaries of the proposed research fall into two groups: direct and indirect.

The direct beneficiaries consist of Shell Research, the Bank of England and Prof. Ina Hoeschele and the wider medical team at the Virginia Bioinformatics Institute (VBI) and the Wake Forest School of Medicine: these groups are named and their interests are explained in the proposal and pathways to impact document. I intend to work closely with these beneficiaries and provide added value as described in the proposal in each case. For example, the VBI would value being able to use the "best-in-class" signal extraction benefits of wavelets (which they already use) but directly on irregularly spaced data via lifting (which we will work on); the Bank of England is interested in new approaches to financial stability and, in particular, the benefits of local stationarity for economic modelling in terms of more accurate forecasts or attempting to understand the sampling scheme that gives maximum information to such public bodies with least cost; Shell, like many other energy companies, are interested in many aspects of nonstationarity particularly the multivariate situation in a world where massively high-dimensional data collection is a reality. Specific aspects of the project, of interest to particular partners, is described in the Case for Support.

Another important group of indirect beneficiaries are the energy companies that I am already working with on a separate EPSRC grant (LETS) funded by the Mathematics and Energy programme. These energy companies are EDF (Paris), Vattenfall and GL Garrad Hassan. The current proposal and the Maths/Energy proposal are supportive and complementary in the sense that the latter has a narrower focus on energy-related issues, and also a small amount of my time. However, it is likely that new developments in the current proposal will also be of interest to these energy partners (for example, continuous time nonstationary processes, new regression methods) and we would seek to continue any productive LETS interactions as part of this proposal after LETS finishes in March 2015. On the (statistical) economics side we are building relationships with Antonis Michis at the Central Bank of Cyprus who has already worked on applications of multiscale methods to economics.

Another key group of beneficiaries will be the RAs themselves and, in particular, those organisations that later employ them or work with them. I have a strong record of developing excellent people and helping them start productive academic and non-academic careers. Many of these people are themselves further creating and innovating in cognate statistical areas and this is a powerful way of initiating impact.

Another constituency are those that will take ideas and, particularly, software developed by the proposed research. Much of what we develop is generic and can be used in many application areas where time series or regression problems occur. It is difficult to quantify precisely the number and identity of such groups who might use our work. However, from our experience with earlier software, such as WaveThresh, it is clear that the distribution and use of such free software is very wide, both in terms of research publications that use such software and in direct communication with users in banking (Bank of America, Reserve Bank of New Zealand), government (Centers for Disease Control, NIMR-MRC, NIST, Los Alamos National Lab) and industry (Detica, Merck, AC Nielsen, Plant Automation Services, Siemens, Unilever) and a large number in academic institutions.

Overall, we intend our impact to be specific (with the partners mentioned) and generic/widespread as a result of our publications and software. Through these routes we intend to have a positive impact on national/global economic competitiveness, increasing the efficiency of analysis, forecasting and hence policy and enhancing health through our bioinformatics work.