最优多谱估计

摘要：

我们表明，全逆协方差加权可以自然地纳入模态估计方法，使其最佳的CMB功率谱，双谱和三谱，以及在其他3D应用。模态估计方法非常高效，只需要求nmax × nmax协方差矩阵的逆，其中nmax是描述理论模型和影响它的噪声所需的有限数量的模态（与完整的lmax^2 × lmax^2逆协方差加权相反）。讨论了模态双谱估计的WMAP分辨率实现，并示出在lmax =500和nmax <= 50处产生局部和其他模型的最优约束。我们还讨论了进一步的改进，通过表征和预处理的协方差矩阵使用一个单一的双谱方差形状引起的噪声和掩模。我们描述了最佳模态估计一般适用于polyspectrum的任何顺序的形式主义，可以扩展到超越各向同性的情况下。我们使用双谱估计中的各向异性噪声的例子来显示各向同性模态基的完整性和减去线性估计项以获得最优性的必要性，去除方差中的交叉项。我们简要讨论了CMB功率谱和三谱，以及大尺度结构的应用。

英文摘要：

We show that full inverse covariance weighting can be naturally incorporated into modal estimation methods making them optimal for the CMB power spectrum, bispectrum and trispectrum, as well as in other 3D applications. Modal estimation methods are highly efficient requiring inversion of only an nmax x nmax covariance matrix, where nmax is the limited number of modes needed to describe both the theoretical model and the noise which affects it (in contrast to the full lmax^2 x lmax^2 inverse covariance weighting). A WMAP resolution implementation for modal bispectrum estimation is discussed and shown to yield optimal constraints for local and other models at lmax =500 and nmax <= 50. We also discuss further improvements by characterising and preconditioning the covariance matrix using a single bispectrum variance shape induced by the noise and mask. We describe optimal modal estimation in general terms applicable to polyspectra of any order in a formalism that can be extended beyond the isotropic case. We use the example of anisotropic noise in bispectrum estimation to show the completeness of the isotropic modal basis and the necessity of subtracting a linear estimator term for optimality, removing cross-terms in the variance. We briefly discuss applications to the CMB power spectrum and trispectrum, as well as large-scale structure.

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