Efficient Generation of Gaussian Varaiates Via Acceptance-Rejection Framework

A. Sajib
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Abstract

The Gaussian distribution is often considered to be the underlying distribution of many observed samples for modelling purposes, and hence simulation from the Gaussian density is required to verify the fitted model. Several methods, most importantly, Box-Muller method, inverse transformation method and acceptance-rejection method devised by Box and Muller1, Rao et al.7 and Sigman8 respectively, are available in the literature to generate samples from the Gaussian distribution. Among these methods, Box-Muller method is the most popular and widely used because of its easy implementation and high efficiency,which produces exact samples2. However, generalizing this method for generating non-standard multivariate Gaussian variates is not discovered yet. On the other hand, inverse transformation method uses numerical approximation to the CDF of Gaussian density which may not be desirable in some situations while performance of acceptance-rejection method depends on choosing efficient proposal density. In this paper, we introduce a more general technique by exploiting the idea invented by Wakefield9 under acceptance rejection framework to generate one dimensional Gaussian variates, in which we don’t require to choose any proposal density and it can be extended easily for non-standard multivariate Gaussian density. The proposed method is compared to the existing acceptance-rejection method (Sigman8 method), and we have shown both mathematically and empirically that the proposed method performs better than Sigman8 method as it has a higher acceptance rate (79.53 %) compared to Sigman (76.04 %) method. Dhaka Univ. J. Sci. 67(2): 123-130, 2019 (July)
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基于接受-拒绝框架的高斯变量高效生成
为了建模,高斯分布通常被认为是许多观测样本的潜在分布,因此需要从高斯密度进行模拟来验证拟合模型。文献中有几种方法可以从高斯分布中生成样本,最重要的是Box- muller法、逆变换法和接受-拒绝法,分别由Box和Muller1、Rao等人7和Sigman8提出。在这些方法中,Box-Muller法因其易于实现和效率高,产生的样品准确而最受欢迎和广泛使用2。然而,目前还没有发现将这种方法推广到非标准多元高斯变量的生成中。另一方面,逆变换方法采用数值逼近高斯密度的CDF,这在某些情况下可能不理想,而接受-拒绝方法的性能取决于选择有效的提议密度。在本文中,我们利用Wakefield9在接受拒绝框架下发明的思想,引入了一种更通用的技术来生成一维高斯变量,其中我们不需要选择任何提议密度,并且可以很容易地扩展到非标准多元高斯密度。将所提出的方法与现有的接受-拒绝方法(Sigman8方法)进行了比较,我们从数学和经验上都证明了所提出的方法优于Sigman8方法,其接受率(79.53%)高于Sigman8方法(76.04%)。达卡大学学报(自然科学版),67(2):123-130,2019 (7)
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