Characterization of Phase-type Distributions and their Applications

相型(PH)分布由Marcel Neuts于1975年提出，并在科学和工程中得到了广泛的应用。PH分布是有用的，因为i)它们可以在非负支持下近似概率分布；ii)它们适于建立数学模型；iii)它们导致了随机分析的算法方法。PH分布适用于随机系统的随机建模和统计分析，如排队模型、排队网络、可靠性模型、库存模型、供应链、风险/保险模型等。随着PH分布应用的扩大，对随机建模的数据拟合精度和计算速度提出了更高的要求和标准。研究者和实践者经常遇到的两个问题是a)PH分布类型的选择和b)利用PH分布选择合适的近似方法。受到它们的应用以及矩阵分析方法和机器学习领域最新进展的启发，我们计划探索关于PH分布的三个问题，以解决精度和速度问题。目标1(Obj1)：MMPP(马尔可夫调制泊松过程)型相型分布的特征。我们计划证明所有MMPP型PH分布的平方变异系数(SCV)大于或等于1，并利用随机比较方法和鞅来研究进一步的性质，例如，矩的界。所得结果可用于选择合适的PH分布和马尔可夫调制泊松过程，从而提高数据拟合的精度和速度。目标2(Obj2)：用连续PH分布逼近离散概率分布。我们计划使用Erlang化方法来构造连续的PH分布来近似离散的概率分布。我们期望基于Erlangation的近似不仅具有与原始分布相同的均值，而且确保方差、SCV和分布函数接近，从而提高数据拟合的精度。目标3(Obj3)：有限支持PH分布的特征。我们计划研究有限支撑相型分布在一定条件下(例如，在给定均值下)的矩、方差和SCV的范围，这可能会提高参数估计的精度和速度。该研究一方面推进了PH分布理论的发展，另一方面也使研究者和实践者有可能有效地使用PH分布和马尔可夫调制泊松过程。这项研究为HQP提供了训练数据拟合、参数估计和随机建模的机会，这些都在机器学习和数据分析中发挥着重要作用。

Phase-type (PH) distributions were introduced by Marcel Neuts in 1975 and have been widely used in science and engineering. PH-distributions are useful since i) they can approximate probability distributions with nonnegative support; ii) they are amenable for building mathematical models; and iii) they lead to an algorithmic approach for stochastic analysis. PH-distributions are suitable for stochastic modeling and statistical analysis of stochastic systems such as queueing models, queueing networks, reliability models, inventory models, supply chains, risk/insurance models, etc.  As the application of PH-distributions expands, there come higher requirements and standards on the accuracy in data fitting and the speed in computation in stochastic modeling. Two problems that are commonly encountered by researcher and practitioners are a) the selection of the proper type of PH-distributions, and b) the selection of proper approximation methods using PH-distributions. Motivated by their applications and inspired by recent progress in the area of matrix-analytic methods and machine learning, we plan to explore three issues on PH-distributions to address the accuracy and speed problems. Objective 1 (Obj1): Characterization of the MMPP (Markov modulated Poisson process) type phase-type distributions. We plan to prove that the squared coefficient of variations (SCV) of all MMPP type PH-distributions are greater than or equal to one, and to investigate further properties, e.g., bounds on the moments, by using stochastic comparison methods and martingales. The results are useful for selecting the proper PH-distributions and Markov modulated Poisson processes and, consequently, improving the accuracy and speed of data fitting. Objective 2 (Obj2): Approximation of discrete probability distributions by continuous PH-distributions. We plan to use the Erlangization method to construct continuous PH-distributions to approximate discrete probability distributions. We expect that the Erlangization based approximation not only has the same mean as the original distribution, but also ensures that the variance, SCV, and distribution function are close, which may improve the accuracy in data fitting. Objective 3 (Obj3): Characterization of finite support PH-distributions. We plan to investigate the ranges of the moments, variances, and SCVs of finite support phase-type distributions under certain conditions (e.g., with a given mean), which may improve the accuracy and speed in parameter estimation. The proposed research, on one hand, advances the theory of PH-distributions, and, on the other hand, makes it possible for researchers and practitioners to use PH-distributions and Markov modulated Poisson processes effectively and efficiently. The proposed research offers opportunities to train HQP for data fitting, parameter estimation, and stochastic modeling, which play an important role in machine learning and data analytics.