奇偶骨架首达动力学与双侧离散监测系统中的最优干预
Parity-Skeleton First-Passage Dynamics and Optimal Intervention in Two-Sided Discrete Monitoring Systems
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中文总结 AI 辅助
本文针对双侧离散监测系统，提出基于有偏随机游走和奇偶骨架的有限状态首达框架，推导老化性质与寿命公式，并利用更新成本准则制定最优预防性干预计划，通过数值实验验证其有效性。
中文摘要 AI 辅助
在离散监测系统中，当干预由偏离运行区间触发时，会出现双侧阈值交叉现象。本文基于具有对称吸收壁的有偏随机游走，建立了一个有限状态首达框架。为消除原始命中时间的奇偶间隙，我们通过偶数和奇数状态马尔可夫骨架分析了奇偶校正的寿命。相邻行似然比不等式验证了单调性条件，并确立了递增失效率和新优于旧的性质。这些老化性质产生了对数凹生存概率和几何持久性界限。推导了子随机矩阵公式，用于生存、风险、平均寿命、状态条件剩余使用寿命和有限时域交叉风险。随后，一个更新成本准则将首达分布转化为最优预防性干预计划。通过将观测增量的漂移和方差匹配到两点随机游走近似来获得模型参数，并针对非平稳情形提供了滚动窗口扩展。数值实验将精确计算与蒙特卡洛诊断、矩匹配逆高斯和威布尔基准、序列相关增量、校准扰动以及固定间隔策略进行了比较。一个合成的远程患者监测示例说明了如何将离散生理偏差映射为透明的风险评分和个性化的复查计划。该框架在离散随机动力学、剩余寿命预测和基于状态的干预之间提供了可审计的联系，同时将数学验证与临床验证分开。
英文摘要
Two-sided threshold crossings arise in discrete monitoring systems whenever intervention is triggered by departure from an operating band. This paper develops a finite-state first-passage framework based on a biased random walk with symmetric absorbing barriers. To remove parity gaps of the raw hitting time, we analyse a parity-corrected lifetime through even- and odd-state Markov skeletons. Adjacent-row likelihood-ratio inequalities verify the monotonicity condition and establish increasing failure rate and new-better-than-used properties. These ageing properties yield log-concave survival probabilities and geometric persistence bounds. Substochastic-matrix formulas are derived for survival, hazard, mean lifetime, state-conditioned remaining useful life, and finite-horizon crossing risk. A renewal-cost criterion then converts the first-passage distribution into an optimal preventive-intervention schedule. Model parameters are obtained by matching the drift and variance of observed increments to a two-point random-walk approximation, with a rolling-window extension for nonstationary regimes. Numerical experiments compare exact calculations with Monte Carlo diagnostics, moment-matched inverse-Gaussian and Weibull benchmarks, serially correlated increments, calibration perturbations, and fixed-interval policies. A synthetic remote-patient-monitoring example illustrates how discrete physiological deviations can be mapped to transparent risk scores and personalized review schedules. The framework provides an auditable link between discrete stochastic dynamics, remaining-lifetime prediction, and condition-based intervention, while separating mathematical validation from clinical validation.
