通过Bhattacharyya距离量化异方差性
Quantifying Heteroskedasticity via Bhattacharyya Distance
摘要：
异方差性是描述时间序列数据集中误差项的不同方差的统计异常。数据中异方差的存在对预测模型提出了严峻的挑战，许多统计检验在异方差存在下无效。数据的异方差性影响预测变量与结果之间的关系，从而导致假设检验中的假阳性和假阴性决策。迄今为止，研究异方差性的现有方法采用了在时间序列中容纳异方差性的策略，并认为它是不可避免的噪声源。在这些现有的方法中，两个预测模型准备正常和异方差的情况下，统计测试是为了确定数据是否是异方差的。这项正在进行的研究引入了异方差性的量化测量。所提出的度量背后的想法是异方差时间序列具有均匀分布的局部方差的事实。所提出的测量是通过使用线性时不变滤波器计算局部方差来获得的。然后推导出计算出的局部方差的密度函数，并与使用统计发散测量的理论最终异方差的均匀分布进行比较。在合成数据集上的结果表明，在异方差时间序列中，所提出的度量和局部估计的方差数之间具有很强的相关性。
英文摘要：
Heteroskedasticity is a statistical anomaly that describes differing variances of error terms in a time series dataset. The presence of heteroskedasticity in data imposes serious challenges for forecasting models and many statistical tests are not valid in the presence of heteroskedasticity. Heteroskedasticity of the data affects the relation between the predictor variable and the outcome, which leads to false positive and false negative decisions in the hypothesis testing. Available approaches to study heteroskedasticity thus far adopt the strategy of accommodating heteroskedasticity in the time series and consider it an inevitable source of noise. In these existing approaches, two forecasting models are prepared for normal and heteroskedastic scenarios and a statistical test is to determine whether or not the data is heteroskedastic. This work-in-progress research introduces a quantifying measurement for heteroskedasticity. The idea behind the proposed metric is the fact that a heteroskedastic time series features a uniformly distributed local variances. The proposed measurement is obtained by calculating the local variances using linear time invariant filters. A probability density function of the calculated local variances is then derived and compared to a uniform distribution of theoretical ultimate heteroskedasticity using statistical divergence measurements. The results demonstrated on synthetic datasets shows a strong correlation between the proposed metric and number of variances locally estimated in a heteroskedastic time series.
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