This article aims to investigate the impact of noise on parameter fitting for an Ornstein-Uhlenbeck process, focusing on the effects of multiplicative and thermal noise on the accuracy of signal separation. To address these issues, we propose algorithms and methods that can effectively distinguish between thermal and multiplicative noise and improve the precision of parameter estimation for optimal data analysis. Specifically, we explore the impact of both multiplicative and thermal noise on the obfuscation of the actual signal and propose methods to resolve them. First, we present an algorithm that can effectively separate thermal noise with comparable performance to Hamilton Monte Carlo (HMC) but with significantly improved speed. We then analyze multiplicative noise and demonstrate that HMC is insufficient for isolating thermal and multiplicative noise. However, we show that, with additional knowledge of the ratio between thermal and multiplicative noise, we can accurately distinguish between the two types of noise when provided with a sufficiently large sampling rate or an amplitude of multiplicative noise smaller than thermal noise. Thus, we demonstrate the mechanism underlying an otherwise counterintuitive phenomenon: when multiplicative noise dominates the noise spectrum, one can successfully estimate the parameters for such systems after adding additional white noise to shift the noise balance.

翻译：本文旨在研究噪声对Ornstein-Uhlenbeck过程参数拟合的影响，重点关注乘性噪声和热噪声对信号分离精度的影响。为解决这些问题，我们提出了能够有效区分热噪声与乘性噪声并提高参数估计精度的算法与方法，以优化数据分析。具体而言，我们探讨了乘性噪声和热噪声对实际信号混淆的影响，并提出了相应的解决方法。首先，我们提出一种算法，能够有效分离热噪声，其性能与汉密尔顿蒙特卡洛（HMC）方法相当，但速度显著提升。随后，我们分析了乘性噪声，并证明HMC方法不足以分离热噪声与乘性噪声。然而，我们发现，在已知热噪声与乘性噪声比值的前提下，当采样率足够大或乘性噪声幅度小于热噪声时，我们能够准确区分这两类噪声。因此，我们揭示了一个看似反直觉现象的内在机制：当乘性噪声主导噪声谱时，通过添加额外的白噪声来改变噪声平衡，可以成功估计此类系统的参数。