We study the probabilistic sampling of a random variable, in which the variable is sampled only if it falls outside a given set, which is called the silence set. This helps us to understand optimal event-based sampling for the special case of IID random processes, and also to understand the design of a sub-optimal scheme for other cases. We consider the design of this probabilistic sampling for a scalar, log-concave random variable, to minimize either the mean square estimation error, or the mean absolute estimation error. We show that the optimal silence interval: (i) is essentially unique, and (ii) is the limit of an iterative procedure of centering. Further we show through numerical experiments that super-level intervals seem to be remarkably near-optimal for mean square estimation. Finally we use the Gauss inequality for scalar unimodal densities, to show that probabilistic sampling gives a mean square distortion that is less than a third of the distortion incurred by periodic sampling, if the average sampling rate is between 0.3 and 0.9 samples per tick.
翻译:我们研究随机变量的概率抽样,在这种抽样中,该变量仅当它位于特定一组之外,即所谓的静默套件之外时才被抽样。这有助于我们理解ID随机过程的特殊情况的最佳事件抽样,并理解其他情况亚最佳方案的设计。我们考虑为一个标度、日志和随机变量设计这种概率抽样,以尽量减少平均平方估计误差,或平均绝对估计误差。我们表明,最佳静默间隔:(一)基本上是独特的,和(二)是迭接程序的核心界限。我们进一步通过数字实验表明,超水平间隔对于平均平方估计而言似乎非常接近最佳。最后,我们用标值不平等来计算标度的单模密度,以表明,如果平均取样率在每只0.3到0.9个标点之间,则概率抽样得出的平均平方形扭曲值不到定期抽样扭曲的三分之一。</s>