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

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This fellowship proposes research in time series analysis and regression. Time seriesanalysis is concerned with data recorded through time. Time series occur in a varietyof 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 needsto ensure that series are properly collected and recorded, modelled appropriately,to gain an understanding of their behaviour, and often predicted to estimate theirfuture values (forecasting).Much real world analysis assumes that series arise from stationary models, whichpermit the values of the series to change at each time, but the underlying statisticsdo 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 thatstationary models are not appropriate for many real series. For example, share pricestatistics do change, sometimes exceptionally, due to sudden events such as politicalupheaval or natural disasters, and often nonstationary models are appropriate anduseful alternatives.This project intends to develop nonstationary techniques with a focus on energy andeconomics applications. For example, energy companies are interested in nonstationarymodels because deregulation and increasingly diverse energy sources have causedmany previously stable data sets to become less stationary and more unpredictable.This project will create new nonstationary models intended to be more realistic, flexible andlead to better modelling, forecasting and consequently better decision-making.Nonstationary models can also shed light on tasks that are infeasible for stationary onessuch as ascertaining whether a series has been sampled frequently enough. We will alsoresearch nonstationary functional models, where each observation is not a single numberbut an entire curve, such as national electricity consumption recorded across a day.Regression is concerned with the modelling of relationships between different variablesand is used extensively in the real world. Many important regression methods assumethat data have constant variance and a `bell-curve' distribution. Much real data are notlike that, but operations, such as taking each observation's square root, can make thedata 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 somefascinating theoretical properties, such as mimicking the well-known logarithm. This projectwill investigate the intriguing theoretical underpinnings of this new class as well as developfurther methods for cleaning up noisy signals, for example, removing noise from astronomicalor low-light security images. Additionally, we will investigate regression for irregular datausing techniques that make use of multiple scales simultaneously (multiscale).First generation multiscale methods, highly valued for purposes such as image compressionin JPEG, are not easily adapted to irregular situations. This project seeks to investigatesecond generation multiscale methods, suitable for irregular data. For example, to betterestimate and control information on networks (such as identify and mitigate delays ontransport networks) or irregularly-spaced systems (such as identify regions of the genomethat are implicated in several complex diseases such as cancer.)