New challenges in time series analysis

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Time series are observations on a quantity or quantities, collected through time.They arise in many important areas of human endeavour, for examplefinance (daily closing values of the FTSE 100 index; the order book of a financialinstrument evolving over time), economics (interest rates set by a central bank;monthly changes to macroeconomic indicators; yield curves changing through time), engineering (speech signals), natural sciences (temperature, seismic signals) and neuroscience (brain activity measurements via EEG, fMRI or other techniques), to name but a few. Typical tasks faced by time series analysts include understanding the nature of and modelling the evolution of the time series, forecasting its future values, understanding how it impacts and is impacted by other factors, and classifying it to one of a number of categories. Solving these tasks adequatelycan have enormous positive impact on economy and society.Modern time series datasets often defy traditional statistical assumptions. In many contexts, time series data are massive in size and high-dimensional (e.g. in macroeconomic modelling, where many potential predictors are frequently included in models e.g. for GDP growth), non-normally-distributed (e.g. in finance where daily returns on many financial instruments show deviations from normality) and non-stationary, which means that their statistical properties such as the mean, variance or autocovariance change through time (e.g. in finance where co-dependence structure of markets is known to change in times of financial crises). Often, time series data arise as complex objects such as curves (e.g. yield curves). New theories and methods are needed to handle these new settings.The proposed research will break new ground in the analysis of non-stationary,high-dimensional and curve-valued time series. Although many of the problems we propose to tackle are motivated by financial applications, our solutions will be transferable to other fields. In particular, we will(i) re-define the way in which people think of non-stationarity. We will define (non-)stationarity to be a problem-dependent, rather than `fixed' property of time series, and propose new statistical model selection procedures in accordance with this new point of view. This will lead to the concept of (non-)stationarity beingput to much better use in solving practical problems (such as forecasting) thanit so far has been;(ii) propose new, problem-dependent dimensionality reduction procedures for time series which are both high-dimensional and non-stationary (dimensionality reductionis useful in practice as low-dimensional time series are much easier to handle). We hope that this problem-dependent approach will induce a completely new way of thinking of high-dimensional time series data and high-dimensional data in general;(iii) propose new methods for statistical model selection in high-dimensional time series regression problems, including the non-stationary setting. Our new methods will be useful in fields such as financial forecasting or statistical market research;(iv) investigate new methods for statistical model selection in high-dimensional time series (of, e.g., financial returns) in which the dependence structure changes in an abrupt fashion due to `shocks', e.g. macroeconomic announcements;(v) propose new multiscale time series models, specifically designed to solve a long-standing problem in finance of consistent modelling of financial returns on multiple time scales, e.g. intraday and interday;(vi) propose new ways of analysing time series of curves (e.g. yield curves) which can be non-stationary in a variety of ways.Overall, this is a comprehensive and ambitious research programme, which aims tooffer novel solutions to some of the most important questions in modern time seriesanalysis.