Working Title: Novel Methods Toward Independent and Identically Distributed Test

产品说明：在概率论中，如果所有随机变量都遵循相同的概率分布并且相互独立，则随机变量的集合是独立同分布的（IID）。IID随机变量产生的观测假设称为IID假设或随机性假设[5，7，10-12]，它是普遍存在的，经常作为统计学[5]，机器学习[14]，熵源估计[9]等的基础。然而，在许多实际环境中，IID假设通常被违反[3，4]，13]因此，检验假设的正确性是有价值的，称为IID检验或随机性检验。自1919年以来，IID检验问题已经讨论了至少100年[8]，并且仍在积极研究[1，5，6，9-12]。本项目的主要目的是研究IID检验方法，并开发新的IID检验方法，本项目属于EPSRC统计和应用概率研究领域。更具体地说，我们提出新的IID测试方法的动机和目标至少有以下两个方面。首先，我们发现，大多数现有的IID测试方法是有用的，只有在某些情况下，可以是无用的，在许多其他情况下，即不能拒绝IID的假设在许多非IID的情况下，当IID假设被违反。因此，开发一种能够在更多非IID情况下拒绝IID假设的测试方法将是有价值的。其次，几乎所有的IID检验方法都是基于假设检验框架的，这经常被批评为如果不拒绝IID假设就无法得出结论[2]。因此，我们未来的工作之一是提出一个比假设检验更具信息量的检验方法。例如，该方法可以测量观测的IID属性的强度，可能在0和1之间，接近0表示严重违反IID，接近1表示IID的高置信度。该度量不仅指示是否拒绝IID假设作为假设检验，而且还显示了观测的IID的强度，因此信息量更大，在实践中可能更有用。为了开发新的IID测试方法，至少可以进行以下两种潜在的方法，值得更多的研究。有可能使用（1）：边际似然比框架开发新的IID检验方法，其中比较反映不同模型（两个模型假设的观测值IID和非IID）生成观测值的概率的边际似然。随后，假设IID的模型和假设非IID的模型的观测生成概率的比率可以指示观测的IID的强度。(2)：permutation，我们可以比较观察和置换观察。粗略地说，如果我们把置换观测看作是由一些（未知的）IID随机变量产生的，我们可以使用一定的距离测量来测量观测和置换观测的距离。直觉上，距离越小，观测值与置换观测值越相似，因此观测值越有可能是IID。已经简要讨论了发展我们的方法的两种潜在方法，需要更多的研究。一旦完成，我们的方法对于许多统计和机器学习模型都是有价值的，这些模型对观测值进行了IID假设，我们的方法可以用来测试这个假设。与大多数现有的方法相比，这可能无法拒绝许多非IID的情况下，可能需要大量的观察拒绝，我们的方法的IID测试可能是有价值的，在实践中拒绝IID假设下，更多的非IID的情况下，使用更少的观察，并产生更多的信息测试结果。

Description: In probability theory, a collection of random variables is independent and identically distributed (IID) if all random variables follow the same probability distribution and are mutually independent. The assumption viewingobservations generated by IID random variables is called IID assumption or randomness assumption [5, 7, 10-12], which is ubiquitous and often serves as the foundation in statistics [5], machine learning [14], entropy source estimations[9] and so on. Nevertheless, the IID assumption is commonly violated in many practical environments [3, 4, 13] and it is therefore valuable to test the correctness of the assumption, called IID test or randomness test.This IID test problem has been discussed for at least 100 years since 1919 [8] and is still being actively researched [1, 5, 6, 9-12]. Our aims for the project is to investigate IID test and develop novel methods of it. This project fallswithin the EPSRC statistics and applied probability research area. More specifically, our motivations and objectives of proposing novel IID test methods are at least the following two-fold. First, we find that most existingIID test methods are useful only under certain cases and can be useless in many others, i.e. cannot reject the assumptions of IID under many non-IID cases when the IID assumption is violated. Therefore, it would be valuable todevelop a test method being able to reject the IID assumption under more non-IID cases. Secondly, nearly all IID test methods are based on hypothesis test framework, which is often criticized as no conclusion can be drawn if the IIDassumption is not rejected [2]. One of our future work is thus to propose a more informative test method than hypothesis test. For example, this method could be a measure of the strength of the IID property of observations, potentiallybetween 0 and 1 with closing to 0 indicating severe violation of IID and closing to 1 high confidence of IID. This measure not only indicate whether to reject the IID assumption as hypothesis test but also shows the strength of the IID ofobservations and therefore is more informative and could potentially be more useful in practice. For our aims of developing novel IID test methods, at least the following two potential methodologies could be carried out and deserve more research. It would be possible to develop new IID test methods using (1): the marginal like-1lihood ratio framework, in which marginal likelihoods reflecting the probability of different models (both models assumed observations IID and non-IID) generating observations are compared. Subsequently, the ratios of the probability of generating of observations of the models assuming IID and that of models assuming non-IID can indicate the strength of the IID of observations. (2):permutation, where we can compare the observations and permuted observations. Roughly speaking, if we regard the permuted observations as generated by some (unknown) IID random variables, we can use certain distance measurements to measure the distances of observations and permuted observations. Intuitively, the smaller the distance is, the more similar the observations are to the permuted observations and therefore the more likely the observations are IID. Two potential methodologies for developing our methods have been briefly discussed and needed more research. Once completed, our methods can be valuable for many statistic and machine learning models which make the IIDassumption of observations and our methods can be used to test this assumption. Compared with most existing methods, which may not be able to reject many non-IID cases and could potentially require a large number of observations toreject, our methods of IID test could be valuable in practice by rejecting IID assumptions under more non-IID cases using fewer observations and produce more informative test results.