Stochastic Modeling and Applications

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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). **