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In modern contexts, some types of data are observed in high-resolution, essentially continuously in time. Such data units are best described as taking values in a space of functions. Subject units carrying the observations may have intrinsic relations among themselves, and are best described by the nodes of a large graph. It is often sensible to think that the underlying signals in these functional observations vary smoothly over the graph, in that neighboring nodes have similar underlying signals. This qualitative information allows borrowing of strength over neighboring nodes and consequently leads to more accurate inference. In this paper, we consider a model with Gaussian functional observations and adopt a Bayesian approach to smoothing over the nodes of the graph. We characterize the minimax rate of estimation in terms of the regularity of the signals and their variation across nodes quantified in terms of the graph Laplacian. We show that an appropriate prior constructed from the graph Laplacian can attain the minimax bound, while using a mixture prior, the minimax rate up to a logarithmic factor can be attained simultaneously for all possible values of functional and graphical smoothness. We also show that in the fixed smoothness setting, an optimal sized credible region has arbitrarily high frequentist coverage. A simulation experiment demonstrates that the method performs better than potential competing methods like the random forest. The method is also applied to a dataset on daily temperatures measured at several weather stations in the US state of North Carolina.
We propose modeling raw functional data as a mixture of a smooth function and a highdimensional factor component. The conventional approach to retrieving the smooth function from the raw data is through various smoothing techniques. However, the smoo
The issue of determining not only an adequate dose but also a dosing frequency of a drug arises frequently in Phase II clinical trials. This results in the comparison of models which have some parameters in common. Planning such studies based on Baye
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$ell_1$-penalized quantile regression is widely used for analyzing high-dimensional data with heterogeneity. It is now recognized that the $ell_1$-penalty introduces non-negligible estimation bias, while a proper use of concave regularization may lea
Functional data, with basic observational units being functions (e.g., curves, surfaces) varying over a continuum, are frequently encountered in various applications. While many statistical tools have been developed for functional data analysis, the