天文数据线性回归中测量误差的几个问题
Some Aspects of Measurement Error in Linear Regression of Astronomical Data
摘要：
我描述了一种贝叶斯方法来解释天文数据线性回归中的测量误差。该方法允许异方差和可能相关的测量误差，以及回归关系中的内在分散。该方法是基于推导出的测量数据的似然函数，我专注于独立变量的内在分布时，可以近似使用混合高斯的情况下。我将该方法推广到多个自变量、未检测和选择效应（例如，Malmquist偏差）。描述了一个吉布斯采样器，用于模拟随机抽取的参数的概率分布，给定的观测数据。我使用模拟比较的方法与其他常见的估计。仿真结果表明，高斯混合模型优于其他常见的估计，并可以有效地给回归参数的约束，即使当测量误差占主导地位的观察到的散射，源检测分数低，或独立变量的内在分布不是高斯的混合。最后，我使用这种方法来拟合X射线光谱斜率作为爱丁顿比的函数，使用39个z < 0.8的射电宁静类星体的样本。我证实了其他作者所看到的无线电宁静类星体的X射线光谱斜率和爱丁顿比之间的相关性，其中X射线光谱斜率软化的爱丁顿比增加。
英文摘要：
I describe a Bayesian method to account for measurement errors in linear regression of astronomical data. The method allows for heteroscedastic and possibly correlated measurement errors, and intrinsic scatter in the regression relationship. The method is based on deriving a likelihood function for the measured data, and I focus on the case when the intrinsic distribution of the independent variables can be approximated using a mixture of Gaussians. I generalize the method to incorporate multiple independent variables, non-detections, and selection effects (e.g., Malmquist bias). A Gibbs sampler is described for simulating random draws from the probability distribution of the parameters, given the observed data. I use simulation to compare the method with other common estimators. The simulations illustrate that the Gaussian mixture model outperforms other common estimators and can effectively give constraints on the regression parameters, even when the measurement errors dominate the observed scatter, source detection fraction is low, or the intrinsic distribution of the independent variables is not a mixture of Gaussians. I conclude by using this method to fit the X-ray spectral slope as a function of Eddington ratio using a sample of 39 z < 0.8 radio-quiet quasars. I confirm the correlation seen by other authors between the radio-quiet quasar X-ray spectral slope and the Eddington ratio, where the X-ray spectral slope softens as the Eddington ratio increases.
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