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.)
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