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:模拟和识别具有长期记忆的时间序列数据在多个领域都很重要。流流流排放是一个可以被合理描述为随机随机松动过程的综合随机随机剖析过程,其概率测量,我们称之为回转测量,因为随机化无法直接观测。精确地识别回流测量至关重要,因为它在综合随机分析过程中具有无处不在性。然而,建模准确性通常受到现有真实世界数据的限制。这一问题的一个方法是评估受再转换测量模糊度影响的利益统计的上下限。在这里,我们使用Tsallis value-at-Risk(TsVaR)作为概率测量度测量度测量,因为随机回流测量值-Risk(EVaR)是一个精确的统计指标。我们证明EVaR不能用于评估关键统计数据,例如平均值和差异,因为流流流流流流流流流流的大小取决于某些指数性RVsal-Rgroism序列的振动。相比之下,TVa数据只能用于当前运行中的数据。