Regression with Time Series Regressors
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The projects to be investigated are motivated by the following two different problems. It is well known that sea ice in the Arctic is receding. Changes in climate are thought to be a contributing factor. Therefore, it is important to understand if, when, and how daily temperatures in the Arctic region are impacting sea ice. The second motivation comes from neuroscience. Is it possible to predict the decision a person makes based on biometric and neurophysiological responses, such as eye dilation observed over time or an EEG? The objectives in both examples are very different, however they are bound by a common theme; to understand how data observed over time (usually called a time series) affects an outcome of interest. These problems fall under the canopy of regression (a broad topic in statistics), which is a widely researched area in statistics. However, what distinguishes these problems from the classical regression framework is that the regressors have a well-defined structure, which is rarely exploited in most classical regression techniques. By modelling the time series, methods will be developed that exploit the structure of the time series. This will facilitate estimation in models that otherwise would not be possible. The approach will be used to identify salient periods in the time series that have the greatest impact on the outcome and can be used to physically interpret the data.In recent years, there has been a growing number of data sets, from a wide spectrum of applications ranging from the neurosciences to the geosciences, where an outcome is observed together with a time series that is believed to influence the outcome. Despite the clear need in applications, there exists surprisingly few results that exploit the properties of a time series in the prediction of outcomes. This project will bridge this gap by developing regression methods that utilize the fact that the regressors are a time series or are spatially dependent. To achieve these aims, many new statistical methods will be developed. In signal processing, deconvolution methods are often used to estimate the parameters of a two-sided linear filter. This is because the deconvolution is computationally very fast to implement. However, there has been very little exploration on the use of deconvolution methods within the framework of estimating regression parameters. This project will develop deconvolution techniques for (i) linear regression models, and (ii) generalized linear models and when the regressors are stationary and locally stationary. The focus will be on the realistic situation where the time series or spatial data is far larger than the number of responses. Besides the computational simplicity of deconvolution, by isolating the Fourier transform of the regression coefficients, diagnostic tools to understand the nature of the underlying regression coefficients will be developed. For example, the methods can tell whether the coefficients are smooth, contain periodicities, or are sparse. The project will develop inferential methods for parameter estimators that allow for uncertainty quantification, construction of confidence intervals, and tests for linear dependence. Included is a new technique for estimating the variance of the regression coefficients.This award reflects NSF''s statutory mission and has been deemed worthy of support through evaluation using the Foundation''s intellectual merit and broader impacts review criteria.