EPSRC Project Summary: New Methods for Network Time Series Analysis

具有显式或隐式网络结构的多变量时间序列数据的快速增加导致了对用于预测或网络结构推断的网络时间序列模型的高度兴趣。这些模型已被用于预测从流行病学到气象学再到社交媒体网络等各种研究领域的时间序列。例如，一个给定地区的风速可能取决于同一地点过去的观测结果及其地理相邻地区的观测结果，具有不同的滞后和效应大小。准确的风速模型可能是一个有用的工具，用于决定新的风力涡轮机的位置，当它是不符合成本效益的收集数据在所有候选位置的长时间。最近开发的广义网络自回归（GNAR）模型提供了一个灵活的和高度简约的方法来建模这样的数据，通过允许依赖于一个自回归组件和邻居跨多个协变量网络的建模系列。我的研究旨在扩展GNAR建模框架，并在多个领域开发与网络时间序列分析有关的新方法。一个扩展将涉及在没有任何网络先验的情况下开发GNAR网络结构推断的新算法，允许将所有多变量时间序列数据集作为网络时间序列处理。这将建立在现有的研究贝叶斯网络的结构推理。第二，GNAR模型结构可以扩展，以纳入特定节点的外源时间序列回归因子，这将导致更好的预测和有用的推论时，提供信息的解释变量。第三，我的研究将尝试将网络时间序列模型推广到张量值时间序列。例如，在流行病学领域，这将允许对网络时间序列进行简约建模，（或网络中的节点）拥有多个时间序列，代表病例数、气象条件和其他与疾病传播相关的因素。最后，我将研究深度学习在大网络时间序列数据集上的应用，通过使用初始网络提升预处理步骤对数据集进行去趋势和空间去相关。特别令人感兴趣的是“混合”深度学习架构的扩展，例如最近开发的高斯过程长短期记忆（GP-LSTM）模型。GP-LSTM使用递归神经网络来嵌入高斯过程的内核矩阵，并以高度可扩展的方式执行推理。除了在时间序列预测任务中实现最先进的性能外，GP-LSTM还允许直接估计传统“不透明”神经网络预测中的不确定性。此外，据我所知，在机器学习文献中还没有研究过使用网络提升方案将数据输入到这种深度学习模型中，我希望这些领域的研究将为网络时间序列分析领域做出新的贡献，即提供方法学工具，使用利用网络结构的高度简约模型来预测多变量时间序列。本项目属于EPSRC统计和应用概率研究领域的福尔斯。_

The rapidly increasing availability of multivariate time series data with explicit or implicit network structure have resulted in a heightened interest in network time series models for the purposes of forecasting or network structure inference. Such models have been used to forecast time series across a diverse array of research areas, from epidemiology to meteorology to social media networks. For example, the wind speed in a given area may depend on both past observations in the same location and those of its geographic neighbours, with varying lags and effect sizes. An accurate wind speed model may be a useful tool for deciding on the locations of new wind turbines, when it is not cost effective to collect the data at all candidate locations for long periods of time. The recently-developed generalised network autoregressive (GNAR) model provides both a flexible and highly parsimonious approach to the modelling of such data, by allowing dependence of the modelled series on an autoregressive component and neighbours across multiple covariate networks. My research aims to extend the GNAR modelling framework and develop new methods pertaining to network time series analysis in several areas. One extension would involve the development of novel algorithms for GNAR network structure inference in the absence of any network priors, allowing the treatment of all multivariate time series data sets as network time series. This would build on existing research for structural inference of Bayesian networks. Secondly, the GNAR model structure may be extended to incorporate node-specific exogenous time series regressors, which should lead to better forecasts and useful inferences when informative explanatory variables are available. Thirdly, my research will attempt to generalise network time series models to tensor-valued time series. For example, in the area of epidemiology, this would allow the parsimonious modelling of network time series where each location (or node in the network) possesses multiple time series representing case numbers, meteorological conditions and other factors relevant to disease transmission.Finally, I will examine the applications of deep learning to big network time series data sets, by using an initial network lifting preprocessing step to detrend and spatially decorrelate the data set. Of particular interest are extensions of `hybrid' deep learning architectures, such as the recently-developed Gaussian Process Long Short Term Memory (GP-LSTM) model. GP-LSTM uses a recurrent neural network to embed the kernel matrix of a Gaussian process and perform inference in a highly scalable fashion. As well as achieving state-of-the-art performance in time series forecasting tasks, the GP-LSTM allows for the straightforward estimation of the uncertainty in predictions of traditionally `opaque' neural networks. Furthermore, to my knowledge, the use of a network lifting scheme for feeding data into such deep learning models has not yet been examined in the machine learning literature.It is my hope that research in these areas will present novel contributions to the field of network time series analysis, that is to provide methodological tools to forecast multivariate time series using highly parsimonious models that exploit network structure. This project falls within the EPSRC Statistics and Applied Probability research area.________________________________________