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. Firstly, we present an algorithm that can effectively separate thermal noise with comparable performance to Hamilton Monte Carlo (HMC) but with significantly improved speed. Subsequently, we 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. This finding results in a situation that initially seems counterintuitive. When multiplicative noise dominates the noise spectrum, we 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奥恩斯坦-乌伦贝克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。针对这些问题，我们提出了能够有效区分热噪声与乘性噪声的算法和方法，从而提升参数估计的精度以实现最优数据分析。具体而言，我们探究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策略。首先，我们给出一种算法，能以显著提升的速度有效分离热噪声，其性能与哈密顿蒙特卡洛（HMC）方法相当。随后，我们分析乘性噪声并证明，仅凭HMC无法区分热噪声与乘性噪声。然而，研究表明，若已知热噪声与乘性噪声的强度比，且在足够大的采样率或乘性噪声振幅小于热噪声的条件下，可准确区分两类噪声。这一发现导致了一种看似违反直觉的情形：当乘性噪声主导噪声谱时，通过额外引入白噪声以改变噪声平衡，仍能成功估计此类系统的参数。