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

该奖学金建议在时间序列分析和回归方面进行研究。时间序列分析涉及通过时间记录的数据。时间序列出现在各种对社会非常重要的领域，如医学(记录生命体征)、经济和金融(GDP或股价)、环境(空气污染)、能源(国家电力需求)和交通(交通流量)，仅举几例。一种常见的显示时间序列的方法是通过时间图来显示时间序列，它通过时间连续地绘制序列的值，从而能够很容易地观察到主要特征，如趋势或季节性影响。总的来说，社会需要确保对序列进行适当的收集和记录，建立适当的模型，以了解它们的行为，并经常通过预测来估计它们的未来价值(预测)。许多现实世界的分析假设，序列来自静态模型，这允许序列的值在每个时间段发生变化，但基本统计数据不会改变(例如，固定的股价每小时都会变化，但总体水平或平均值保持不变)。越来越清楚的是，静态模型不适用于许多真实的序列。例如，股票价格统计确实会因政治动荡或自然灾害等突发事件而发生变化，有时是例外的，而非平稳模型往往是合适和有用的替代。这个项目旨在开发非平稳技术，重点放在能源和经济应用上。例如，能源公司对非平稳模型感兴趣，因为放松管制和日益多样化的能源导致许多以前稳定的数据集变得不那么平稳，更不可预测。这个项目将创建新的非平稳模型，旨在更现实、更灵活，并导致更好的建模、预测和因此更好的决策。非平稳模型还可以揭示不可能用于平稳模型的任务，例如确定序列是否被足够频繁地采样。我们还将研究非平稳函数模型，其中每个观测值不是单个数字，而是一条完整的曲线，例如全国一天的用电量记录。回归涉及不同变量之间关系的建模，在现实世界中被广泛使用。许多重要的回归方法都假定数据具有常方差和钟形分布。许多真实数据不是这样的，但运算，如取每个观测值的平方根，可以使数据至少近似地满足那些常方差/钟形曲线假设。最近，一种新的、有前途的、非常不同的多尺度类被称为Haar-Fisz变换。新的类对于计数数据工作得非常好，并且已经显示出一些相关的理论属性，例如模仿众所周知的对数。该项目将研究这一新类别耐人寻味的理论基础，并开发进一步的方法来清除噪声信号，例如，从天文或微光安全图像中去除噪声。此外，我们还将研究同时使用多个尺度(多尺度)的不规则数据处理技术的回归。第一代多尺度方法在JPEG图像压缩等方面非常有价值，但不容易适应不规则情况。本项目致力于研究第二代多尺度方法，适用于不规则数据。例如，为了更好地估计和控制网络上的信息(例如识别和缓解传输网络上的延迟)或不规则间隔的系统(例如识别与几种复杂疾病(如癌症)有关的基因组区域)。

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.)