Time and frequency domain statistical methods for high-frequency time series

Elayouty, Amira Sherif Mohamed (2017) Time and frequency domain statistical methods for high-frequency time series. PhD thesis, University of Glasgow.

Abstract

Advances in sensor technology enable environmental monitoring programmes to record and store measurements at high-temporal resolution over long time periods. These large volumes of high-frequency data promote an increasingly comprehensive picture of many environmental processes that would not have been accessible in the past with monthly, fortnightly or even daily sampling. However, benefiting from these increasing amounts of high-frequency data presents various challenges in terms of data processing and statistical modeling using standard methods and software tools. These challenges are attributed to the large volumes of data, the persistent and long memory serial correlation in the data, the signal to noise ratio, and the complex and time-varying dynamics and inter-relationships between the different drivers of the process at different timescales.

This thesis aims at using and developing a variety of statistical methods in both the time and frequency domains to effectively explore and analyze high-frequency time series data as well as to reduce their dimensionality, with specific application to a 3 year hydrological time series. Firstly, the thesis investigates the statistical challenges of exploring, modeling and analyzing these large volumes of high-frequency time series. Thereafter, it uses and develops more advanced statistical techniques to: (i) better visualize and identify the different modes of variability and common patterns in such data, and (ii) provide a more adequate dimension reduction representation to the data, which takes into account the persistent serial dependence structure and non-stationarity in the series. Throughout the thesis, a 15-minute resolution time series of excess partial pressure of carbon dioxide (EpCO2) obtained for a small catchment in the River Dee in Scotland has been used as an illustrative data set. Understanding the bio-geochemical and hydrological drivers of EpCO 2 is very important to the assessment of the global carbon budget.

Specifically, Chapters 1 and 2 present a range of advanced statistical approaches in both the time and frequency domains, including wavelet analysis and additive models, to visualize and explore temporal variations and relationships between variables for the River Dee data across the different timescales to investigate the statistical challenges posed by such data. In Chapter 3, a functional data analysis approach is employed to identify the common daily patterns of EpCO2 by means of functional principal component analysis and functional cluster analysis. The techniques used in this chapter assume independent functional data. However, in numerous applications, functional observations are serially correlated over time, e.g. where each curve represents a segment of the whole time interval. In this situation, ignoring the temporal dependence may result in an inappropriate dimension reduction of the data and inefficient inference procedures.

Subsequently, the dynamic functional principal components, recently developed by Hor
mann et al. (2014), are considered in Chapter 4 to account for the temporal correlation using a frequency domain approach. A specific contribution of this thesis is the extension of the methodology of dynamic functional principal components to temporally dependent functional data estimated using any type of basis functions, not only orthogonal basis functions. Based on the scores of the proposed general version of dynamic functional principal components, a novel clustering approach is proposed and used to cluster the daily curves of EpCO2 taking into account the dependence structure in the data.

The dynamic functional principal components depend in their construction on the assumption of second-order stationarity, which is not a realistic assumption in most environmental applications. Therefore, in Chapter 5, a second specific contribution of this thesis is the development of a time-varying dynamic functional principal components which allows the components to vary smoothly over time. The performance of these smooth dynamic functional principal components is evaluated empirically using the EpCO2 data and using a simulation study. The simulation study compares the performance of smooth and original dynamic functional principal components under both stationary and non-stationary conditions. The smooth dynamic functional principal components have shown considerable improvement in representing non-stationary dependent functional data in smaller dimensions.

Using a bootstrap inference procedure, the smooth dynamic functional principal components have been subsequently employed to investigate whether or not the spectral density and covariance structure of the functional time series under study change over time. To account for the possible changes in the covariance structure, a clustering approach based on the proposed smooth dynamic functional principal components is suggested and the results of application are discussed.

Finally, Chapter 6 provides a summary of the work presented within this thesis, discusses the limitations and implications and proposes areas for future research.

• Item Type: Thesis (PhD)
• Qualification Level: Doctoral
• Keywords: High-frequency time series, excess partial pressure of carbon dioxide, functional data analysis, functional principal components, non-stationarity, smoothing, clustering, time domain, frequency domain.
• Subjects: G Geography. Anthropology. Recreation > GE Environmental Sciences; H Social Sciences > HA Statistics
• Colleges/Schools: College of Science and Engineering > School of Mathematics and Statistics > Statistics
• Supervisor's Name: Scott, Prof. Marian, Miller, Dr. Claire and Susan, Prof. Waldron
• Date of Award: 2017
• Depositing User: Miss Amira Sherif Mohamed Elayouty
• Unique ID: glathesis:2017-8061
• Copyright: Copyright of this thesis is held by the author.
• Date Deposited: 05 Apr 2017 12:46
• Last Modified: 17 May 2017 07:45
• URI: https://theses.gla.ac.uk/id/eprint/8061