Modelling and analysis of Time Series

Scientific advances arise increasingly from analysing large amounts of data collected through organised experiments. Processes that evolve through time frequently occur in a variety of fields, from energy (wave heights/wind speeds at certain points in an ocean) to medicine (heart rate signals for newborns) or biology (circadian plant rhythms) and transportation (average delay per station over a railway network). The modelling and analysis of time series, to the point of predicting their future behaviour, form a vital area of statistics.

Currently, time series are often assumed to be observed at regular intervals and to have a stationary behaviour: that is, their underlying statistical properties are constant over time. Such a key statistical property is the spectrum which explains the power of oscillation at different frequencies. For stationary series the spectrum is constant over time, even though the process itself is not. Unfortunately, in reality many important time series are not stationary and drawing inference for nonstationary time series as if they were stationary can have grave consequences. A further complexity in real data analysis arises from irregular sampling- wave heights or railway station loads change over time and measuring devices often cannot transmit regularly spaced data, while networks have a spatial distribution. Alongside nonstationarity and sampling irregularity, replicates are typically available for experimental data.

The key to improved modelling (and subsequent data analysis tasks such as forecasting, clustering, etc) lies in developing models that closely reflect the process characteristics. Such modelling can yield well-behaved reliable estimators for both the time-varying spectrum and covariance. In turn, for the examples given, this may result in better forecasting of ocean storms for offshore oil drilling stations, predicting neonatal distress for treatment administration, predicting plant response to light or thermal entrainment, anticipating railway station overload in order to facilitate remedial actions.

The situation of stationary processes observed at regularly spaced locations has been studied extensively and several approaches exist for departures from stationarity within regular sampling. Historically due to the lack of appropriate tools, when stationary processes are irregularly sampled or have missing data, the research literature is far less developed, and even less so for irregularly sampled nonstationary time series. The availability of replicates, while increasing the signal-to-noise ratio, also calls for different models to embed this supplementary information, rather than recourse to (e.g. spectral) estimate averaging.

This research proposal aims to establish new time-scale data representations, whose essential ingredients are second generation wavelets (constructed via the lifting scheme) and the locally stationary time series to deal with sampling irregularity and process nonstationarity, respectively.
The overarching aim is to develop and disseminate novel models which incorporate nonstationary process behaviour and replicate information while naturally coping with irregular sampling.

In order to achieve this goal, this project will
Develop novel methods for the modelling and estimation of multivariate nonstationary time series that feature replicates and sampling missingness.
Develop novel methods for the modelling and estimation of nonstationary time series that evolve over complex structures such as networks, and possibly feature replicates.
Disseminate and support the application of these methods via freely accessible R software.