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Monte Carlo Markov Chain Algorithm Convergence
Published in Richard M. Golden, Statistical Machine Learning, 2020
Let p* be an M-dimensional probability vector specifying the stationary distribution of an M-state Markov chain. Assume M is very large (e.g., M = 210000). Explain how the MCMC Strong Law of Large Numbers can be used to approximately evaluate the expectation ∑k=1Mgkpk∗ where pk* is the kth element of p* and g1, …, gM are known constants.
Population genetics and Markov chains
Published in Henry C. Tuckwell, Elementary Applications of Probability Theory, 2018
Definition Let P be the transition matrix of a temporally homogeneous Markovchain. If there exists a probability vector p̂ such that p̂ = p̂P, then p̂ is called a stationary distribution for the Markov chain.
Distribution-free multivariate time-series monitoring with analytically determined control limits
Published in International Journal of Production Research, 2023
Di Liu, Heeseon Kim, Seong-Hee Kim, Taeheung Kim, Dongki Lee, Yao Xie
Note that we do not assume a multivariate normal marginal distribution but assume a general stationary distribution F for when the process is in control. Further, and represent the in-control mean vector and an out-of-control mean vector, respectively. We assume the marginal distribution family F and lag-ℓ covariance matrices for do not change but the mean vector is shifted to when a monitored process is out of control. Covariance matrices for are assumed to be positive definite. Our goal is to develop a monitoring procedure that raises an alarm as soon as possible after a shift in the mean occurs.
Real-Time Monitoring of High-Dimensional Functional Data Streams via Spatio-Temporal Smooth Sparse Decomposition
Published in Technometrics, 2018
Hao Yan, Kamran Paynabar, Jianjun Shi
The proof is shown in Appendix C. Note that in (9), the temporal structure of μt is modeled by the weighted average of the previous estimation and the current estimation of , which is a recursive equation similar to the monitoring statistic of the EWMA control chart. Therefore, for a stationary process, can help average the noise over time, which leads to a stationary distribution with a much smaller variance than the original data. However, different from the EWMA control chart, we use (9) to estimate the true dynamic trend in dynamic processes. The spatial structure of μt is captured by applying the projection matrix Hs. However, (or equivalently ) is unknown and should also be estimated. To efficiently solve for θa, t, we first show that the loss function is equivalent to a weighted lasso formulation, which can be solved via an accelerated proximal gradient algorithm.
Equilibrium and social optimality in queues with service rate and customers’ joining decisions
Published in Quality Technology & Quantitative Management, 2022
Ruiling Tian, Siping Su, Zhe George Zhang
Denote the stationary probability by , then is the stationary distribution of the process. Clearly, these stationary probabilities satisfy the following flow-balance equations: