The Shannon entropy is a widely used summary statistic, for example, network traffic measurement, anomaly detection, neural computations, spike trains, etc. This study focuses on estimating Shannon entropy of data streams. It is known that Shannon entropy can be approximated by Reenyi entropy or Tsallis entropy, which are both functions of the p-th frequency moments and approach Shannon entropy as p->1. Compressed Counting (CC) is a new method for approximating the p-th frequency moments of data streams. Our contributions include: 1) We prove that Renyi entropy is (much) better than Tsallis entropy for approximating Shannon entropy. 2) We propose the optimal quantile estimator for CC, which considerably improves the previous estimators. 3) Our experiments demonstrate that CC is indeed highly effective approximating the moments and entropies. We also demonstrate the crucial importance of utilizing the variance-bias trade-off.
CC improves symmetric stable random projections @cite_4 @cite_9 uniformly for all @math as shown in Figure in Section . However, one can still considerably improve CC around @math , by developing better estimators, as in this study. In addition, no empirical studies on CC were reported.
@cite_9 @cite_4
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!# @math ) 4 ' " + @math 4 ) @ & A B C D E . * . ! F ' G H ' IJ K L * M & + *) N z'cC g) ' &  p : q :  r Htd|#t9 ,s tvuw x yFz |;w]tS 9 x |;   C n+ U <dU' 4 2 In this paper we introduce the Cognitive Compressive Sensing (CCS) problem, modeling a Cognitive Receiver (CR) that optimizes the @math projections of a @math dimensional vector dynamically, by optimizing the objective of correctly detecting the maximum number of idle entries, while updating each time its Bayesian beliefs on the future vector realizations. We formulate and study the CCS as a Restless Multi-Armed Bandit problem, generalizing the popular Cognitive Spectrum Sensing model, in which the CR can sense @math out of the @math sub-channels and propose a novel adaptive Finite Rate of Innovation (FRI) sampling method based on the CCS approach. While in general the optimum policy remains elusive, we provide sufficient conditions in which in the limit for large @math and @math the greedy policy is optimum. Numerical results corroborate our theoretical findings. 0. Introduction.- 1. Experimenting on nondeterministic machines.- 2. Synchronization.- 3. A case study in synchronization and proof techniques.- 4. Case studies in value-communication.- 5. Syntax and semantics of CCS.- 6. Communication trees (CTs) as a model of CCS.- 7. Observation equivalence and its properties.- 8. Some proofs about data structures.- 9. Translation into CCS.- 10. Determinancy and confluence.- 11. Conclusion. Abstract Background Chest compression (CC) quality is associated with rescuer posture and body weight. We designed a Kinect module-based real-time audiovisual feedback (AVF) device to investigate the relationship between rescuer posture, body weight, and CC quality. Methods A total of 100 healthcare professionals were enrolled as participants in this randomized trial. A Kinect-based sensor system was used to monitor the depth and rate of CC and provide further real-time feedback. All participants were asked to perform continuous CC on a manikin with and without feedback for 2 min individually in either a kneeling or standing position. Results A kneeling posture can provide higher rate of CC than a standing posture can (111.4 ± 22.6 per minute vs. 99.1 ± 18.9 per minute, p value = 0.005). Real-time AVF feedback can provide a better compression depth, rate, and effective compression ratio (6.16 ± 1.88 cm vs. 5.54 ± 1.89 cm, p value = 0.02; 103.2 ± 21.0 min vs. 96.7 ± 25.8 min, p value = 0.03; 62.6 ± 28.0 vs. 51.0 ± 33.2 , p value = 0.004). Regardless of the effect of real-time feedback, the CC depth correlated to the rescuers' body weight. Rescuers who weighed below 71 kg benefited from the Kinect module-based real-time AVF device in terms of improved CC quality. Conclusion The Kinect-based AVF device can significantly improve CC quality in manikin training in rescuers with their body weight