Stochastic Modeling and Applications

1. 非线性时间序列******近似共单调或协整的思想越来越重要。在过去的几十年里，金融市场，特别是股票市场和指数，大致以同样的方式运行，这对投资组合也有影响。******一个相关的问题也将研究线性时间序列的残差。具体来说，这些残差的样本均值不为0，但总体模型的均值为0。最好以样本均值为中心，而不是以已知的总体均值为中心。* * * * * * 2。随机捕食模型******我们考虑一个离散观察随机噪声捕食过程。目前的方法不能很好地估计初始值。在模拟研究中，从理想模型来看，拟合的ODE路径与观测数据的拟合非常差，拟合与观测数据的拟合要么过于分散，要么过于紧密。******我们将探索一种投影思想来改进估计的初始ODE点和ODE周期。* * * * * * 3。遥感和碳通量数据******这是一个应用统计学问题，基于与生态学家John Gamon教授（以前在阿尔伯塔大学，现在在内布拉斯加州大学）的一些早期讨论，是上述研究的一部分。一个方面是发现遥感数据在多大程度上可以预测地面碳通量测量。我们很自然地把这看作是一个回归问题。然而，每年都有不同的季节性影响，以及测量噪声。因此，我们考虑一个分层或多阶段模型。******方法包括注册，注册时间空间中的随机效应，然后随机扭曲函数将这些映射回观察到的时间空间。加上附加的太阳时间相关的附加噪声。作为一个副产品，这种方法将允许建立一个复杂的非线性模型（生长季节测量）和（未来）观测数据的预测区间。然后，它可以形成衡量随时间变化的基础。要研究的问题将包括对层次模型的诊断，以及测试一些模型假设的方法。******4。微阵列数据的空间方面******数据来自凯瑟琳·希尔教授的实验室。在遗传学中，有一个新的领域引起了人们的兴趣，因为现在人们认为突变的聚集是癌症的一个很好的指标（基于完整的DNA测序实验）。微阵列也便宜得多。***未来的工作是研究微阵列设计的效率，基于探针位置的一种数据库方法。如果我们可以访问一些完整的DNA序列数据，我们就可以使用它来测量各种阵列设计的簇检测效率，并且可以通过我们的遗传学和生物学联系以及拷贝数变异（CNV）到snp（突变）的调查来访问这些数据。**

1. Nonlinear Time Series******The idea of approximately co-monotonic or cointegrated is increasingly important. Over the past couple of decades financial markets, in particular stock markets and indices, move roughly in the same fashion, with it implications for portfolios. ******A related problem will also study residuals from linear time series. Specifically these residuals do not have sample mean 0, but the population model has known mean 0. It is better to centre at the sample mean instead of the known population mean of 0. ******2. Random Prey Predator Models******We consider a discretely observed random noise prey-predator process. Current methods do not estimate the initial value well. In a simulation study, from the ideal model, the fitted ODE path can fit the observed data very poorly, the fit being either far more spread or way too tight with respect to the observed data.******We will explore a projection idea to improve the estimated initial ODE point and the period of the ODE. ******3. Remote Sensing and Carbon Flux Data******This is an applied statistics question based on some earlier discussions with an ecologist Professor John Gamon (formerly at U Alberta, now at U Nebraska) and is part of the ABove study. One aspect is to find how well the remote sensing data can predict the ground carbon flux measurements. It is natural to view this as a regression problem. However there is a seasonal affect that varies each year, as well as measurement noise. We thus consider a hierarchical or multistage model.******The methods involve registration, the random effects in a registered time space, and then a random warping function to map these back to the observed time space. plus additive solar time dependent additive noise. As a by product this method will allow a complex nonlinear model of local productivity (growing season measurement) and prediction intervals of (future) observed data. It can then form the basis for measuring changes over time. Problems to be studied will include diagnostics for the hierarchical model, and methods of testing some of the model assumptions.******4. Spatial Aspects of Microarray Data******The data comes from the lab of Professor Kathleen Hill. In genetics there is a new area of interest since it is now thought that clustering of mutations is a good indicator of cancers (based on a full DNA sequencing experiment). Microarrays are a much cheaper too. ***Future work, for this proposal, is to study the efficiency of microarray designs, based on one of the data base methods for probe locations. If we can access some of the full DNA sequence data we can then use this to measure the cluster detection efficiency of various array designs, and can likely get access to this data through our genetics and biology contacts, as well as an investigation of copy number variants (CNV) to SNPs (mutations). **