Modelling survival functions and their critical points

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The main part of this research addresses the problem of modelling a periodic interaction between two biological variables. We model the interaction of the two variables by a centred two-dimensional Gaussian process with Markovian structure, as suggested by Bart et al. (2005). We develop appropriate confidence regions for estimating periodicity and the strength of the interaction and we adapt the model to deal with data with missing observations. We also discuss how identifiability of the periodicity is related to the experimental design. We apply this model to identify pairs of genes in yeast that interact periodically. The approach is then applied to model periodicity in the disease progression of acute pancre- atitis (AP), a frequent and potentially fatal systemic inflammatory disease, where the disease progression is described by two opposing factors, comprising of putative 'protective' and 'damaging' variables. In particular, we show that the first two principal components in the AP data can be interpreted as the 'protective' and 'damaging' variables and have a clear clinical interpretation, estimate periodicity of the disease progression for observed data with missing observations and rationalise the timing and opportunities for optimal medical intervention. We then reformulate this model into a latent growth model, which can more easily handle non- regularly spaced observations and missing data, enabling the use of a wider range of data sets. The remaining part of the research involves developing inference for exponentially decaying periodic functions, in particular when there is asymmetry in the decay rates of the maxima and minima of the periodic function. The inference is developed on simulated data and then applied to circadian gene expression data obtained from murine brain slices.