The modeling and identification of time series data with a long memory are important in various fields. The streamflow discharge is one such example that can be reasonably described as an aggregated stochastic process of randomized affine processes where the probability measure, we call it reversion measure, for the randomization is not directly observable. Accurate identification of the reversion measure is critical because of its omnipresence in the aggregated stochastic process. However, the modeling accuracy is commonly limited by the available real-world data. One approach to this issue is to evaluate the upper and lower bounds of a statistic of interest subject to ambiguity of the reversion measure. Here, we use the Tsallis Value-at-Risk (TsVaR) as a convex risk measure to generalize the widely used entropic Value-at-Risk (EVaR) as a sharp statistical indicator. We demonstrate that the EVaR cannot be used for evaluating key statistics, such as mean and variance, of the streamflow discharge due to the blowup of some exponential integrand. In contrast, the TsVaR avoids this issue because it requires only the existence of some polynomial, not exponential moment. As a demonstration, we apply the semi-implicit gradient descent method to calculate the TsVaR and corresponding Radon-Nikodym derivative for time series data of actual streamflow discharges in mountainous river environments.

翻译：长记忆时间序列数据的建模与识别在各领域具有重要价值。河道径流量作为典型案例，可被合理描述为随机化仿射过程的聚合随机过程，其中用于随机化的概率测度（我们称其为逆转测度）无法直接观测。由于该测度在聚合随机过程中普遍存在，准确识别逆转测度至关重要。然而，建模精度通常受限于可获取的真实世界数据。针对这一问题，一种方法是在逆转测度不确定性的条件下，评估目标统计量的上下界。本文采用Tsallis风险价值（TsVaR）作为凸风险度量，对广泛使用的熵风险价值（EVaR）这一尖锐统计指标进行推广。研究表明，由于某些指数被积函数存在爆炸现象，EVaR无法用于评估河道径流量的关键统计量（如均值和方差）。相比之下，TsVaR仅需确保多项式矩（而非指数矩）的存在性，从而规避了该问题。为验证该方法，我们采用半隐式梯度下降法计算真实山地河流环境径流时间序列数据的TsVaR及其对应的Radon-Nikodym导数。