Linear Autonomous Compartmental Models as Continuous-Time Markov Chains: Transit-Time and Age Distributions
Linear Autonomous Compartmental Models as Continuous-Time Markov Chains: Transit-Time and Age Distributions
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DOI：
10.1007/s11004-017-9690-1
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发表时间：
2018-01-01
影响因子：
2.6
通讯作者：
Sierra, Carlos A.
中科院分区：
文献类型：
--
作者：
Metzler, Holger;Sierra, Carlos A.
线性区室模型通常用于不同的科学领域，特别是在模拟碳和其他生物地球化学元素的循环时。这些模型表示为线性自治分区系统，允许比较不同的模型结构和参数化。特别是，系统寿命和传输时间等度量是有用的模型诊断。然而，描述其概率分布的紧凑数学表达式仍有待推导。本文将开放线性自治分室系统理论转化为吸收连续时间马尔可夫链理论，得出所有开放线性自治分室系统的底层结构都是相型分布。这种概率分布将指数分布从其应用于单室系统推广到多室系统。此外，本文表明重要的系统诊断具有自然的概率对应物。例如，在稳定状态下，系统的渡越时间与相关马尔可夫链的吸收时间一致，而系统年龄和隔室年龄与适当更新过程的向后重现时间相对应。这些关系产生用于系统诊断的简单显式公式，该公式应用于稳定状态下的一种线性和一种非线性碳循环模型。人们发现简单系统的渡越时间和系统年龄密度的早期结果是相型概率密度函数的特殊情况。新的显式公式使得无需昂贵的长期模拟来获取和分析稳态下开放线性自治分区系统的年龄结构。
Linear compartmental models are commonly used in different areas of science, particularly in modeling the cycles of carbon and other biogeochemical elements. The representation of these models as linear autonomous compartmental systems allows different model structures and parameterizations to be compared. In particular, measures such as system age and transit time are useful model diagnostics. However, compact mathematical expressions describing their probability distributions remain to be derived. This paper transfers the theory of open linear autonomous compartmental systems to the theory of absorbing continuous-time Markov chains and concludes that the underlying structure of all open linear autonomous compartmental systems is the phase-type distribution. This probability distribution generalizes the exponential distribution from its application to one-compartment systems to multiple-compartment systems. Furthermore, this paper shows that important system diagnostics have natural probabilistic counterparts. For example, in steady state the system's transit time coincides with the absorption time of a related Markov chain, whereas the system age and compartment ages correspond with backward recurrence times of an appropriate renewal process. These relations yield simple explicit formulas for the system diagnostics that are applied to one linear and one nonlinear carbon-cycle model in steady state. Earlier results for transit-time and system-age densities of simple systems are found to be special cases of probability density functions of phase-type. The new explicit formulas make costly long-term simulations to obtain and analyze the age structure of open linear autonomous compartmental systems in steady state unnecessary.