Geophysical and other natural processes often exhibit non-stationary covariances and this feature is important to take into account for statistical models that attempt to emulate the physical process. A convolution-based model is used to represent non-stationary Gaussian processes that allows for variation in the correlation range and variance of the process across space. Application of this model has two steps: windowed estimates of the covariance function under the assumption of local stationary and encoding the local estimates into a single spatial process model that allows for efficient simulation. Specifically we give evidence to show that non-stationary covariance functions based on the Matèrn family can be reproduced by the Lat-ticeKrig model, a flexible, multi-resolution representation of Gaussian processes. We propose to fit locally stationary models based on the Matèrn covariance and then assemble these estimates into a single, global LatticeKrig model. One advantage of the LatticeKrig model is that it is efficient for simulating non-stationary fields even at 10 5 locations. This work is motivated by the interest in emulating spatial fields derived from numerical model simulations such as Earth system models. We successfully apply these ideas to emulate fields that describe the uncertainty in the pattern scaling of mean summer (JJA)
We extend the work of Robinson and Turner to use hypothesis testing with persistence homology to test for measurable differences in shape between point clouds from three or more groups. Using samples of point clouds from three distinct groups, we conduct a large-scale simulation study to validate our proposed extension. We consider various combinations of groups, samples sizes and measurement errors in the simulation study, providing for each combination the percentage of p-values below an alpha-level of 0.05. Additionally, we apply our method to a Cardiotocography data set and find statistically significant evidence of measurable differences in shape between normal, suspect and pathologic health status groups.2010 Mathematics Subject Classification. 55N35, 62H15.
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