使用开放式数据科学工具计算Gamma差异分布的实用有效方法
A practical, effective calculation of gamma difference distributions with open data science tools
摘要：
目前，仍然没有正式接受和广泛验证的实现计算的伽玛差分布允许不相等的形状参数。我们探讨了不同形状参数的伽玛差分布的四种计算方法，分别是时间序列克里格法、基于最佳线性无偏预测的预测方法和线性混合模型。我们的数值研究的结果，重点是使用开放数据科学工具，表明我们在高性能Python中实现的开放工具（使用Numba）是指数级快速，高度准确和非常可靠的。它结合了数值反演的特征函数和梯形规则与双指数振荡变换（DE积分）。在双倍的53位精度下，我们的工具比基于CAS软件（商业Mathematica，开放SageMath）中Tricomi的$U(a, b, z)$函数的分析计算速度高出1.5-2个数量级。在科学数值计算工具的精度上，它超过了开放的SciPy、NumPy和商业MATLAB 5-10倍。我们的工具在混合特征函数中的潜在未来应用可能为基于多维统计、计量学中的测量不确定性分析以及金融数学和风险分析等领域的精确概率分布的快速数据分析开辟新的可能性。
英文摘要：
At present, there is still no officially accepted and extensively verified implementation of computing the gamma difference distribution allowing unequal shape parameters. We explore four computational ways of the gamma difference distribution with the different shape parameters resulting from time series kriging, a forecasting approach based on the best linear unbiased prediction, and linear mixed models. The results of our numerical study, with emphasis on using open data science tools, demonstrate that our open tool implemented in high-performance Python(with Numba) is exponentially fast, highly accurate, and very reliable. It combines numerical inversion of the characteristic function and the trapezoidal rule with the double exponential oscillatory transformation (DE quadrature). At the double 53-bit precision, our tool outperformed the speed of the analytical computation based on Tricomi's $U(a, b, z)$ function in CAS software (commercial Mathematica, open SageMath) by 1.5-2 orders. At the precision of scientific numerical computational tools, it exceeded open SciPy, NumPy, and commercial MATLAB 5-10 times. The potential future application of our tool for a mixture of characteristic functions could open new possibilities for fast data analysis based on exact probability distributions in areas like multidimensional statistics, measurement uncertainty analysis in metrology as well as in financial mathematics and risk analysis.
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