傅立叶光谱学:贝叶斯方法

S. Schmuck, J. Svensson
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引用次数: 2

摘要

傅立叶光谱领域中应用的标准分析技术的概念对基本方面的处理不够。例如,要推断的光谱受到以下因素的影响:噪声对干涉数据的贡献、与傅里叶系数在一定阶以上相关联的未探测空间域、奈奎斯特假设通常没有给出的光谱极限、以及手头问题的附加参数(如零径差)。为了考虑这些基本原理,介绍了一种基于贝叶斯定理的概率方法,该方法利用多元正态分布。对于实例应用,我们用一个先验协方差表示的布朗桥的高斯过程来模拟谱。光谱本身由若干线性映射到数据域的参数表示。这些线性参数的后验是解析得到的,这些参数的边缘是微不足道的。这允许对涉及的非线性参数(如零路径差位置和光谱限制)和超参数(如高斯过程的缩放)的后验进行直接调查。对于线性问题,这可以解释为奥卡姆剃刀原理的实现。
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Fourier Spectroscopy: A Bayesian Way
The concepts of standard analysis techniques applied in the field of Fourier spectroscopy treat fundamental aspects insufficiently. For example, the spectra to be inferred are influenced by the noise contribution to the interferometric data, by nonprobed spatial domains which are linked to Fourier coefficients above a certain order, by the spectral limits which are in general not given by the Nyquist assumptions, and by additional parameters of the problem at hand like the zero-path difference. To consider these fundamentals, a probabilistic approach based on Bayes’ theorem is introduced which exploits multivariate normal distributions. For the example application, we model the spectra by the Gaussian process of a Brownian bridge stated by a prior covariance. The spectra themselves are represented by a number of parameters which map linearly to the data domain. The posterior for these linear parameters is analytically obtained, and the marginalisation over these parameters is trivial. This allows the straightforward investigation of the posterior for the involved nonlinear parameters, like the zero-path difference location and the spectral limits, and hyperparameters, like the scaling of the Gaussian process. With respect to the linear problem, this can be interpreted as an implementation of Ockham’s razor principle.
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