Adaptive Threshold Sampling and Estimation

Sampling is a fundamental problem in both computer science and statistics. A number of issues arise when designing a method based on sampling. These include statistical considerations such as constructing a good sampling design and ensuring there are good, tractable estimators for the quantities of interest as well as computational considerations such as designing fast algorithms for streaming data and ensuring the sample fits within memory constraints. Unfortunately, existing sampling methods are only able to address all of these issues in limited scenarios. We develop a framework that can be used to address these issues in a broad range of scenarios. In particular, it addresses the problem of drawing and using samples under some memory budget constraint. This problem can be challenging since the memory budget forces samples to be drawn non-independently and consequently, makes computation of resulting estimators difficult. At the core of the framework is the notion of a data adaptive thresholding scheme where the threshold effectively allows one to treat the non-independent sample as if it were drawn independently. We provide sufficient conditions for a thresholding scheme to allow this and provide ways to build and compose such schemes. Furthermore, we provide fast algorithms to efficiently sample under these thresholding schemes

Fast estimation of false alarm probabilities of STAP detectors - the AMF

This paper describes an attempt to harness the power of adaptive importance sampling techniques for estimating false alarm probabilities of detectors that use space-time adaptive processing. Fast simulation using these techniques have been notably successful in the study of conventional constant false alarm rate radar detectors, and in several other applications. The principal task here is to examine the viability of using importance sampling methods for STAP detection. Though a modest beginning, the adaptive matched filter detection algorithm is analysed successfully using fast simulation. Of the two biasing methods considered, one is implemented and shown to yield excellent results. The important problem of detector threshold determination is also addressed, with matching outcome. The work reported here serves to pave the way to development of more advanced estimation techniques that can facilitate design of powerful and robust detection algorithms designed to counter hostile and heterogeneous clutter environments

An Adaptive estimation scheme for reducing communications in a distributed control implementation

This paper examines the application of adaptive estimation and control techniques to reduce the amount of communication required between subsystems in a distributed control implementation. Rather than require a large amount of communications to broadcast the outputs or the states of each of the subsystem nodes to all of the other nodes at every sampling instant, local estimators in each subsystem are used to predict the state vectors for all of the other subsystems. These estimates are then used in the calculation of the controller outputs for each of the subsystems. Prior work in the literature has focused on static estimation schemes to achieve such reductions in communications. However, such schemes typically require very accurate models of the plant in order to maintain the desired reduction in communications. Poorly modeled dynamics or systems whose dynamics change slowly over time (due to aging of components, changes in plant parameters such as a robot picking up a heavy object, etc.) can cause a substantial increase in the amount of communications required to maintain the desired system performance. In order to avoid this, this paper presents an adaptive estimation and control scheme for each subsystem in the distributed implementation. The stability of the state estimators and the convergence of the state tracking errors to within a desired threshold is proven. The performance of the system using perfect communication at every sampling instant, using a static estimation scheme, and using the proposed adaptive estimation scheme are then compared in simulation

Nonparametric estimation of a renewal reward process from discrete data

We study the nonparametric estimation of the jump density of a renewal reward process from one discretely observed sample path over [0,T]. We consider the regime when the sampling rate goes to 0. The main difficulty is that a renewal reward process is not a Levy process: the increments are non stationary and dependent. We propose an adaptive wavelet threshold density estimator and study its performance for the Lp loss over Besov spaces. We achieve minimax rates of convergence for sampling rates that vanish with T at polynomial rate. In the same spirit as Buchmann and Gr\"ubel (2003) and Duval (2012), the estimation procedure is based on the inversion of the compounding operator. The inverse has no closed form expression and is approached with a fixed point technique.Comment: arXiv admin note: substantial text overlap with arXiv:1203.313

Improving Enrichment Strategies in Outcome-Dependent Sampling Designs and Adaptive Biomarker-Threshold Designs

In many study areas, such as epidemiological studies and clinical trials, both enrichment schemes and estimation methods have been investigated to achieve more efficient estimates of interest given fixed budget. In this dissertation, we study how to improve outcome-dependent-sampling (ODS) designs in epidemiological studies and adaptive enrichment designs in the ongoing randomized clinical trials, in the ways of both enrichment strategies and estimation methods. Bayesian method plays an important role in all topics by providing robust posterior or reasonable prediction based on all the available information. In Topic 1, we propose a new cost-effective sampling design, the extreme outcome dependent sampling (EODS) design and a Bayesian inference procedure, for studies with a continuous outcome. Compared to existing ODS designs, the new EODS design adopts a strategy to use the smallest and largest outcomes to identify supplemental samples. The EODS design provides an alternative way to make efficient use of limited resources, especially in multi-state studies, by targeting the more informative subjects for sampling. We develop a Bayesian Markov Chain Monte Carlo (MCMC) method for the EODS design. Our method can incorporate the information of all subjects, including those with unobserved exposure, and provide an unbiased and efficient estimator through posterior inference. Simulation results indicate that the newly proposed EODS scheme, coupled with the proposed MCMC estimator, is more efficient than the existing ODS designs and the simple random sampling design with the same sample size. In Topic 2, instead of sampling latter stage subjects based on the value of an outcome, we propose the likelihood dependent sampling (LDS) design, for studies with a continuous outcome. Compared to existing ODS designs, the LDS design selects supplemental samples from those subjects with the smallest estimated conditional likelihood values, and thus allows those subjects to directly impact the likelihood function. The proposed LDS design provides a new way to make efficient use of the limited resources for second stage sampling. Bayesian MCMC method is also used for inference of data from LDS design. Simulation results indicate that the newly proposed sampling scheme, coupled with the proposed MCMC estimator, is more efficient than existing ODS designs and the simple random sampling design. In Topic 3, we propose a new adaptive threshold detection and enrichment design -- Biomarker Enrichment and Adaptive Threshold (BEAT) design, in which the biomarker threshold is adaptively estimated and updated with a trade-off between the size of the positive population and the magnitude of the treatment effect of that population optimized. The enrichment of the patients is based on an enrollment criterion, which is determined flexibly through a user-specified parameter and the accuracy of the threshold to account for the uncertainty of the adaptive estimation. Early termination for futility is allowed based on the predictive probability of success for biomarker-positive patients. Valid test and estimation on treatment effects overall or in patient subgroups are also studied. Simulation results demonstrate that the proposed design has several advantages compared to existing designs, including more accurate estimation of the biomarker threshold and treatment effects and significant reduction in cost and time.Doctor of Philosoph