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This is joint work with Deborah Burr.

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I will present an overview of a coherent and unified body of statistical methods that are based on the SLEX (smooth localized complex exponentials) library. The SLEX are time-localized Fourier waveforms with multi-resolution support. The SLEX library provides a systematic and efficient way of extracting transient spectral and cross-spectral features. In addition, the SLEX methods are able to handle massive data sets because they utilize computationally efficient algorithms. As a matter of practical importance, the SLEX methods give results that are easy to understand because they are time-dependent analogues of the classical Fourier methods for stationary signals. Finally, under the SLEX models, we develop theoretical results of consistency for spectral estimation and classification.

The SLEX methods will be illustrated using biological and physical
data sets, namely, brain waves, fMRI time series, a speech signal
and seismic waves recorded during earthquake and explosion events.
This talk will conclude with open and challenging problems in signal
analysis and the potential of SLEX in addressing these.

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(i) We have developed a new method of evaluating numerical models in air quality using monitoring data. For numerical models in air quality, it is important to evaluate the numerical model output in the space-time context and especially to check how the numerical model follows the dynamics of the real process. We suggest that by comparing certain space-time correlations from observations with those from numerical model output, we can achieve this goal. I will demonstrate how our method is applied to a numerical model called CMAQ for sulfate levels over North America.

(ii) For space-time processes on global or large scales, it is critical to use models that respect the Earth's spherical shape. However, there has been almost no research in this regard. We have developed a new class of space-time covariance functions on the sphere crossed with time from a sum of independent processes, where each process is obtained by applying a first-order differential operator to a fully symmetric process on sphere crossed with time. The resulting covariance functions can produce various types of space-time interactions and give different covariance structures along different latitudes. Our approach yields explicit expressions for the covariance functions and can also be applied to other spatial domains such as flat surfaces. I will show the fitted result of our new covariance functions to observed sulfate levels.

Finally, I will describe how we build a space-time model that combines
numerical model output and observations to build a space-time map of air
pollution levels. The information on space-time covariance structure
obtained from numerical model evaluation procedure is useful for building
the space-time model. Moreover, the space-time covariance functions on
spheres play a critical role here because of large spatial domain.

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We propose the elastic net, a new regularization and variable selection method. Real world data and a simulation study show that the elastic net often outperforms the lasso, while enjoying a similar sparsity of representation. In addition, the elastic net encourages a grouping effect, where strongly correlated predictors tend to be in or out of the model together. The elastic net is particularly useful when the number of predictors is much bigger that the number of samples. We have implemented an algorithm called LARS-EN for efficiently computing the entire elastic net regularization path, much like the LARS algorithm does for the lasso. In this talk, I will also describe some interesting applications of the elastic net in other statistical areas such as the sparse principal component analysis and the margin-based kernel classifier.

This is joint work with Trevor Hastie.

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One approach I'll describe extends results by Beran and Dumbgen (1998) to wavelet bases and weighted L2 loss. I will demonstrate these methods through an analysis of the Cosmic Microwave Background spectrum, which is used by cosmologists to understand the physics of the early universe.

While it is possible to construct estimators in nonparametric regression that adapt to the unknown smoothness of the function, constructing adpative confidence sets is not always possible. I'll discuss this issue and describe an approach to nonparametric inference, which I call confidence catalogs, in which the end product is a mapping from assumptions to confidence sets.

This is joint work with Larry Wasserman.

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