Motivated by some cutting edge circular data such as from Smart Home technologies and roulette spins from online and casino, we construct some new rich classes of discrete distributions on the circle. We give four new general methods of construction, namely (i) maximum entropy, (ii) centered wrapping, (iii) marginalized and (iv) conditionalized methods. We motivate these methods on the line and then work on the circular case and provide some properties to gain insight into these constructions. We mainly focus on the last two methods (iii) and (iv) in the context of circular location families, as they are amenable to general methodology. We show that the marginalized and conditionalized discrete circular location families inherit important properties from their parent continuous families. In particular, for the von Mises and wrapped Cauchy as the parent distribution, we examine their properties including the maximum likelihood estimators, the hypothesis test for uniformity and give a test of serial independence. Using our discrete circular distributions, we demonstrate how to determine changepoint when the data arise in a sequence and how to fit mixtures of this distribution. Illustrative examples are given which triggered the work. For example, for roulette data, we test for uniformity (unbiasedness) , test for serial correlation, detect changepoint in streaming roulette-spins data, and fit mixtures. We analyse a smart home data using our mixtures. We examine the effect of ignoring discreteness of the underlying population, and discuss marginalized versus conditionalized approaches. We give various extensions of the families with skewness and kurtosis, to those supported on an irregular lattice, and discuss potential extension to general manifolds by showing a construction on the torus
翻译:以某些尖端循环数据为动力,比如Smart Home 技术和在线和赌场的游离性旋律等,我们根据一些来自在线和赌场的离散性循环数据,建造了一些新富的离散分布类别。我们给出了四种新的建设方法,即:(一) 最大通心球,(二) 中央包绕,(三) 边缘化和(四) 有条件的封闭性循环方法。我们在线上激励这些方法,然后在圆环中开展工作,并提供一些属性,以深入了解这些构造。我们主要侧重于在循环地点家庭的背景下,根据一般方法,构建一些新颖性方法(三)和(四),构建一些新颖的离散性循环性循环性循环性循环性分配。我们显示,处于边缘和有条件的离散性分散性分配家庭继承其父母连续家庭的重要属性。特别是,对于双向Mises和被包绕的圆形包装作为母体分布,我们检查其属性,包括最大可能性的估算、统一性的假设测试以及序列独立的测试。我们利用离心循环分布的分布,我们如何确定变化点,我们的数据在顺序和组合中,我们如何确定变化的顺序,我们如何调整。我们使用。我们用来分析那些变变的变变的变的变,我们用。