Modelling survival functions and their critical points

这项研究的主要部分解决了对两个生物变量之间的周期性相互作用进行建模的问题。我们用Bart等人提出的具有马尔可夫结构的中心二维高斯过程来模拟这两个变量之间的相互作用。(2005)。我们开发了适当的置信域来估计交互作用的周期和强度，并对该模型进行了调整，以处理有缺失观测的数据。我们还讨论了周期的可辨识性与实验设计的关系。我们应用这个模型来识别酵母中周期性相互作用的基因对。然后，该方法被应用于对急性胰腺炎(AP)疾病进展的周期性进行建模，AP是一种频繁且具有潜在致命性的全身性炎症性疾病，疾病进展由两个相反的因素描述，包括假定的“保护性”和“破坏性”变量。特别是，我们发现AP数据中的前两个主成分可以被解释为“保护性”和“破坏性”变量，并有明确的临床解释，估计有缺失观测的观察数据的疾病进展周期，并使最佳医疗干预的时机和机会合理化。然后，我们将该模型重新表述为潜在增长模型，该模型可以更容易地处理非规则间隔的观测数据和缺失数据，从而能够使用更广泛的数据集。研究的其余部分涉及开发指数衰减周期函数的推论，特别是当周期函数的最大值和最小值的衰减率存在不对称时。这一推断是在模拟数据的基础上发展起来的，然后应用于从小鼠脑片获得的昼夜节律基因表达数据。

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.