Wavelet Analysis of Nonstationary Circadian Time Series

Rhythmic data are ubiquitous in the life sciences, with biologists needing reliable statistical tools for the analysis of such data. When these signals display rhythmic yet nonstationary behaviour, common in many biological systems, the established methodologies are often misleading. Chapter 2 develops and tests a new method for clustering nonstationary rhythmic biological data. The method combines locally stationary wavelet time series modelling with functional principal components analysis and thus extracts time—scale patterns useful for identifying common characteristics. We demonstrate the advantages of our methodology over alternative approaches by means of a simulation study and for real circadian data applications. Motivated by three complementary applications in circadian biology, Chapter 3 develops new reliable statistical tests to identify whether a particular experimental treatment has caused a significant change in a rhythmic signal that displays nonstationary characteristics. As circadian behaviour is best understood in the spectral domain, we develop novel hypothesis testing procedures in the (wavelet) spectral domain, which facilitate the identification of three specific types of spectral difference. We demonstrate the advantages of our methodology over alternative approaches by means of a comprehensive simulation study and for real data applications, involving both plant and animal signals. Chapter 4 investigates the effect of industrial and agricultural pollutants on the plant circadian clock. We examine the impact of exposure to a comprehensive range of environmentally relevant pollutants by utilising the methodologies developed in Chapters 2 and 3. Our findings indicate that many of the tested chemicals have an effect on the plant circadian clock, most of which would have remained undetected by classical methods overlooking nonstationarity. The results of Chapter 4 demonstrate the additional insight gained by using the appropriate methodologies, as developed in Chapters 2 and 3, and also have important implications for understanding environmental ramifications associated with soil pollution

Wavelet spectral testing : application to nonstationary circadian rhythms

Rhythmic data are ubiquitous in the life sciences. Biologists need reliable statistical tests to identify whether a particular experimental treatment has caused a significant change in a rhythmic signal. When these signals display nonstationary behaviour, as is common in many biological systems, the established methodologies may be misleading. Therefore, there is a real need for new methodology that enables the formal comparison of nonstationary processes. As circadian behaviour is best understood in the spectral domain, here we develop novel hypothesis testing procedures in the (wavelet) spectral domain, embedding replicate information when available. The data are modelled as realisations of locally stationary wavelet processes, allowing us to define and rigorously estimate their evolutionary wavelet spectra. Motivated by three complementary applications in circadian biology, our new methodology allows the identification of three specific types of spectral difference. We demonstrate the advantages of our methodology over alternative approaches, by means of a comprehensive simulation study and real data applications, using both published and newly generated circadian datasets. In contrast to the current standard methodologies, our method successfully identifies differences within the motivating circadian datasets, and facilitates wider ranging analyses of rhythmic biological data in general

Automatic Locally Stationary Time Series Forecasting with application to predicting U.K. Gross Value Added Time Series under sudden shocks caused by the COVID pandemic

Accurate forecasting of the U.K. gross value added (GVA) is fundamental for measuring the growth of the U.K. economy. A common nonstationarity in GVA data, such as the ABML series, is its increase in variance over time due to inflation. Transformed or inflation-adjusted series can still be challenging for classical stationarity-assuming forecasters. We adopt a different approach that works directly with the GVA series by advancing recent forecasting methods for locally stationary time series. Our approach results in more accurate and reliable forecasts, and continues to work well even when the ABML series becomes highly variable during the COVID pandemic.Comment: 21 pages, 4 figure

A wavelet approach to modelling the evolutionary dynamics across ordered replicate time series

Experimental time series data collected across a sequence of ordered replicates often crop up in many fields, from neuroscience to circadian biology. In practice, it is natural to observe variability across time in the dynamics of the underlying process within a single replicate and wavelets are essential in analysing nonstationary behaviour. Additionally, signals generated within an experiment may also exhibit evolution across replicates even for identical stimuli. We propose the Replicate-Evolving Locally Stationary Wavelet process (REv-LSW) which gives a stochastic wavelet representation of the replicate time series. REv-LSW yields a natural desired time- and replicate-localisation of the process dynamics, capturing nonstationary behaviour both within and across replicates, while accounting for between-replicate correlation. Firstly, we rigorously develop the associated wavelet spectral estimation framework along with its asymptotic properties for the particular case that replicates are uncorrelated. Next, we crucially develop the framework to allow for dependence between replicates. By means of thorough simulation studies, we demonstrate the theoretical estimator properties hold in practice. Finally, it is unreasonable to make the typical assumption that all replicates stem from the same process if a replicate spectral evolution exists. Thus, we propose two novel tests that assess whether a significant replicate-effect is manifest across the replicate time series. Our modelling framework uses wavelet multiscale constructions that mitigate against the potential nonstationarities, across both times and replicates. Thorough simulation studies prove both tests to be flexible tools and allow the analyst to accordingly tune their subsequent analysis. Throughout this thesis, our work is motivated by an investigation into the evolutionary dynamics of brain processes during an associative learning experiment. The neuroscience data analysis illustrates the utility of our proposed methodologies and demonstrates the wider experimental data analysis achievable that is also of benefit to other experimental fields, e.g. circadian biology, and not just the neurosciences