The processes which govern the function of biological organisms are inherently
dynamic and studying their behaviour over time is critical for gaining insight
into their underlying mechanisms. They are also incredibly complex with tens
of thousands of interacting variables comprising their state. In recent years,
the development of high-throughput assaying technologies such as microarrays
and nuclear magnetic resonance spectroscopy have revolutionised the fields of
genomics and metabolomics respectively with their ability to quickly and easily
interrogate these states at a single moment in time. When these assaying
technologies are used to collect measurements repeatedly on the same biological
unit, such as a human patient, laboratory rat or cell line, then the temporal
behaviour of the system can begin to emerge. Furthermore, when several of
these units are studied simultaneously then the experiment is said to be biologically replicated and such data sets permit the inference of systemic behaviour
in the population as a whole.
The time series data sets arising from these replicated `omics experiments
possess unique characteristics that make for challenging statistical analysis.
They are very short (3-10 time points is typical), heterogeneous, noisy, frequently
irregularly sampled and often have missing observations, in addition
to being very highly dimensional. To overcome some of these difficulties, researchers
in the field of genomics have turned to functional data analysis, which
has proven to be successful in modelling unreplicated data sets. Replicated
data sets, however, have received far less attention, due to the complexity introduced
by the extremely small sample sizes and multiple levels of variation - the between-variable and the between-replicate. Furthermore, despite the
remarkable similarities between genomics and metabolomics time series data
sets, these methods have been far less successful at establishing themselves in
the latter field. In this thesis we present a general statistical framework for the analysis
of replicated, high-dimensional biological time series data sets. Supported by
three case studies, we develop novel models and algorithms for tackling the
unique challenges that each data set presents. We show how these fitted models
can be used in dimensionality reduction, summarising the thousands of
observed time series into a small number of representative temporal profiles
that are eminently biologically interpretable. We introduce a novel moderated
functional t -statistic that can be used for detecting variables that differ significantly between two biological groups, leveraging the high dimensionality of the
data in order to increase power. In all instances detailed simulation studies
are used to demonstrate that the methods outperform existing state-of-the-art
approaches. With practical data analysis in mind, careful consideration is
given to the implementation of the methods in software that is computationally
efficient, with parallel programming exploited wherever possible. In most
instances, the methods have resulted in novel biological findings when applied
to real data, and represent, as far as we are aware, the first application of such
functional data analysis models to metabolomics time series experiments.