Working Title: Novel Methods Toward Independent and Identically Distributed Test

描述：在概率论中，如果所有的随机变量都遵循相同的概率分布并且相互独立，那么这个随机变量的集合就是独立同分布的（IID）。由IID随机变量产生的观察假设被称为IID假设或随机性假设[5,7,10 -12]，它普遍存在，在统计学[5]、机器学习[14]、熵源估计[9]等领域经常作为基础。然而，在许多实际环境中，IID假设经常被违反[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。这个IID测试问题自1919年以来已经讨论了至少100年，目前仍在积极研究中[1,5,6,9 -12]。我们的项目目标是研究IID测试并开发新的测试方法。该项目属于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测试方法仅在某些情况下有用，而在许多其他情况下无用，即当IID假设被违反时，在许多非IID情况下不能拒绝IID的假设。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，如果不拒绝IID假设，就无法得出结论，因此经常受到批评[0]。因此，我们未来的工作之一是提出一种比假设检验更有信息量的检验方法。例如，该方法可以衡量观测值的IID属性强度，可能在0到1之间，接近0表示严重违反IID，接近1表示IID的高置信度。这一措施不仅表明是否拒绝IID假设作为假设检验，而且还显示了IID观察的强度，因此更有信息量，在实践中可能更有用。为了开发新的IID测试方法，至少有以下两种潜在的方法可以进行，值得更多的研究。使用(1)：边际似然比框架可以开发新的IID检验方法，其中边际似然反映了不同模型（两种模型都假设观测值IID和非IID）产生观测值的概率。随后，假设IID的模型的观测值产生概率与假设非IID的模型的观测值产生概率之比可以表示观测值的IID强度。(2)：排列，我们可以比较观测值和排列后的观测值。粗略地说，如果我们把排列的观测值看作是由一些（未知的）IID随机变量产生的，我们可以用一定的距离度量来度量观测值和排列观测值的距离。直观地说，距离越小，观测值与排列观测值越相似，因此观测值越有可能是IID。已经简要讨论了开发我们的方法的两种可能的方法，需要更多的研究。一旦完成，我们的方法可以对许多统计和机器学习模型有价值，这些模型可以对观察结果进行iid假设，我们的方法可以用来测试这个假设。与大多数现有方法相比，这些方法可能无法拒绝许多非IID病例，并且可能需要大量的观察值来拒绝，我们的IID检验方法在实践中具有价值，可以用更少的观察值来拒绝更多非IID病例下的IID假设，并产生更多信息丰富的检验结果。

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.