We aim to analyze the behaviour of a finite-time stochastic system, whose model is not available, in the context of more rare and harmful outcomes. Standard estimators are not effective in making predictions about such outcomes due to their rarity. Instead, we use Extreme Value Theory (EVT), the theory of the long-term behaviour of normalized maxima of random variables. We quantify risk using the upper-semideviation $\rho(Y) = E(\max\{Y - \mu,0\})$ of an integrable random variable $Y$ with mean $\mu = E(Y)$. $\rho(Y)$ is the risk-aware part of the common mean-upper-semideviation functional $\mu + \lambda \rho(Y)$ with $\lambda \in [0,1]$. To assess more rare and harmful outcomes, we propose an EVT-based estimator for $\rho(Y)$ in a given fraction of the worst cases. We show that our estimator enjoys a closed-form representation in terms of the popular conditional value-at-risk functional. In experiments, we illustrate the extrapolation power of our estimator using a small number of i.i.d. samples ($<$50). Our approach is useful for estimating the risk of finite-time systems when models are inaccessible and data collection is expensive. The numerical complexity does not grow with the size of the state space.
翻译:我们的目标是在更罕见和有害的结果中分析一个没有模型的有限时间随机系统的行为。 标准估计值由于这些结果的罕见性, 无法有效预测这些结果。 相反, 我们使用极值理论(EVT), 即随机变量标准化最大值的长期行为理论(Y) 。 我们用高缩降 $\rho(Y) = E( max ⁇ Y- mu,0 ⁇ ) 来量化风险。 为了评估更罕见和有害的结果, 我们建议使用基于 EVT 的估测器, 以美元为随机随机可变值, 平均$\ mu = E(Y) $。 $\rho(Y) 美元(Y) 是普通平均降值理论(EV) 的一部分 。 使用我们最坏的精确度模型, 我们的功能性变异性数据 显示我们货币变异性变异性变异性模型 。 我们的变异性变异性模型 显示我们货币变异性变异性模型 值 。 我们的变异性变异性模型 显示我们货币变异性变异性变异性变变变变变变变的 。