A Robust Two-Sample Test for Time Series data
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We develop a general framework for hypothesis testing with time series data. The problem is to distinguish between the mean functions of the underlying temporal processes of populations of times series, which are often irregularly sampled and measured with error. Such an observation pattern can result in substantial uncertainty about the underlying trajectory, quantifying it accurately is important to ensure robust tests. We propose a new test statistic that views each trajectory as a sample from a distribution on functions and considers the distributions themselves to encode the uncertainty between observations. We derive asymptotic null distributions and power functions for our test and put emphasis on computational considerations by giving an efficient kernel learning framework to prevent over-fitting in small samples and also showing how to scale our test to densely sampled time series. We conclude with performance evaluations on synthetic data and experiments on healthcare and climate change data.
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