We present a novel method for averaging a sequence of histogram states visited by a Metropolis-Hastings Markov chain whose stationary distribution is the posterior distribution over a dense space of tree-based histograms. The computational efficiency of our posterior mean histogram estimate relies on a statistical data-structure that is sufficient for nonparametric density estimation of massive, multidimensional metric data. This data-structure is formalized as statistical regular paving (SRP). A regular paving (RP) is a binary tree obtained by selectively bisecting boxes along their first widest side. SRP augments RP by mutably caching the recursively computable sufficient statistics of the data. The base Markov chain used to propose moves for the Metropolis-Hastings chain is a random walk that data-adaptively prunes and grows the SRP histogram tree. We use a prior distribution based on Catalan numbers and detect convergence heuristically. The performance of our posterior mean SRP histogram is empirically assessed for large sample sizes simulated from several multivariate distributions that belong to the space of SRP histograms.
We present a data-adaptive multivariate histogram estimator of an unknown density f based on n independent samples from it. Such histograms are based on binary trees called regular pavings (RPs). RPs represent a computationally convenient class of simple functions that remain closed under addition and scalar multiplication. Unlike other density estimation methods, including various regularization and Bayesian methods based on the likelihood, the minimum distance estimate (MDE) is guaranteed to be within an L 1 distance bound from f for a given n, no matter what the underlying f happens to be, and is thus said to have universal performance guarantees (Devroye and Lugosi, Combinatorial methods in density estimation. Springer, New York, 2001). Using a form of tree matrix arithmetic with RPs, we obtain the first generic constructions of an MDE, prove that it has universal performance guarantees and demonstrate its performance with simulated and real-world data. Our main contribution is a constructive implementation of an MDE histogram that can handle large multivariate data bursts using a tree-based partition that is computationally conducive to subsequent statistical operations.
A variety of tasks conducted by aviation system decision makers and researchers requires analyzing aircraft trajectory data. Datasets containing high frequency aircraft position information collected over large geographic areas and long periods of time are too large to store in the primary memory of personal computers. This paper introduces the use of statistical regular pavings as data structures capable of summarizing very large aircraft trajectory datasets. Recursively computable statistics can be stored for variable-sized regions of airspace. The regions themselves can be created automatically to reflect the varying density of aircraft observations, dedicating more computational resources and providing more detailed information in areas with more air traffic. In particular, statistical regular pavings are able to very quickly aggregate or separate data with different characteristics so that data describing individual aircraft or collected using different technologies (reflecting different levels of precision) can be stored separately and yet also very quickly combined using standard arithmetic operations.
Background: The intrinsic motivation behind the "need to complete" is more influential than external incentives. We introduced a novel progress-bar tool to motivate the completion of programs designed to treat stimulant and cannabis use disorders. We further examined the effectiveness of the progress bar's scoring approach in forecasting consistently negative urine tests.Methods: This study's participants included 568 patients with stimulant, amphetaminetype, and cannabis use disorders who were undergoing 12-month mandatory treatment programs at Taichung Veterans General Hospital in Taiwan. Patients were given scores of 1, -1, or 0 depending on whether they received negative, positive, or missing urinalysis reports, respectively. The autonomic progress bar generated weekly score totals. At the group level, score i donated scores from all patients for a given week (i denoted the week). Score i was standardized to adjusted score i . We then conducted Autoregressive Integrated Moving Average (ARIMA) Model of time-series analyses for the adjusted score i .Results: A total of 312 patients maintained treatment progress over the 12-month program. The autonomic score calculator totaled the shared achievements of these patients. The coefficients of the lag variables for mean (p), lag variables for residual error term (q), and number of orders for ensuring stationary (d) were estimated at p = 3, d = 4, and q = 7 for the first half of the treatment program, and were estimated at p = 2, d = 2, and q = 3 for the second half. Both models were stationary and tested as fit for prediction (p < 0.05). Sharply raised adjusted scores were predicted during the high-demand treatment phase.Discussion: This study's novel progress-bar tool effectively motivated treatment completion. It was also effective in forecasting continually negative urine tests. The tool's free open-source code makes it easy to implement among many substancetreatment services.
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