用Fokker-Planck方程模拟太阳黑子数分布
Modeling the Sunspot Number Distribution with a Fokker-Planck Equation
摘要：
太阳黑子的数量表现出很大的短时间尺度（每日每月）的变化，除了较长的时间尺度的变化，由于太阳周期。一个正式的统计框架，估计和预测随机性的太阳黑子数的确定性（包括混乱）模型的太阳周期。Fokker-Planck方法通过一个驱动函数假定了太阳黑子数在一个太阳周期内的长期或长期变化。然后，该模型描述了在这个驱动函数之上观察到的太阳黑子数量的随机性。我们考虑一个简单的谐波选择的驱动程序功能，但该方法是通用的，可以很容易地扩展到包括其他驱动程序，占基本的物理过程和/或经验的太阳黑子数的功能。该框架在太阳活动最大值和最小值期间是一致的，并且不需要任何参数限制来确保非负的太阳黑子数。模型参数估计使用统计最优技术。该模型同意定性和定量与每月的太阳黑子数据，即使与简单的代表性的太阳周期。这个框架应该是特别有用的太阳周期预报，是现有的建模技术的补充。提出了福克-普朗克方程的解析近似，这是类似于欧拉近似，它允许有效的极大似然估计的大数据集和/或当使用难以评估的驱动程序功能。
英文摘要：
Sunspot numbers exhibit large short-timescale (daily-monthly) variation in addition to longer timescale variation due to solar cycles. A formal statistical framework is presented for estimating and forecasting randomness in sunspot numbers on top of deterministic (including chaotic) models for solar cycles. The Fokker-Planck approach formulated assumes a specified long-term or secular variation in sunspot number over an underlying solar cycle via a driver function. The model then describes the observed randomness in sunspot number on top of this driver function. We consider a simple harmonic choice for the driver function, but the approach is general and can easily be extended to include other drivers which account for underlying physical processes and/or empirical features of the sunspot numbers. The framework is consistent during both solar maximum and minimum, and requires no parameter restrictions to ensure non-negative sunspot numbers. Model parameters are estimated using statistically optimal techniques. The model agrees both qualitatively and quantitatively with monthly sunspot data even with the simplistic representation of the periodic solar cycle. This framework should be particularly useful for solar cycle forecasters and is complementary to existing modeling techniques. An analytic approximation for the Fokker-Planck equation is presented, which is analogous to the Euler approximation, which which allows for efficient maximum likelihood estimation of large data sets and/or when using difficult to evaluate driver functions.
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