258,104 research outputs found
Perturbative approach to covariance matrix of the matter power spectrum
We evaluate the covariance matrix of the matter power spectrum using
perturbation theory up to dominant terms at 1-loop order and compare it to
numerical simulations. We decompose the covariance matrix into the disconnected
(Gaussian) part, trispectrum from the modes outside the survey (beat coupling
or super-sample variance), and trispectrum from the modes inside the survey,
and show how the different components contribute to the overall covariance
matrix. We find the agreement with the simulations is at a 10\% level up to . We show that all the connected components are
dominated by the large-scale modes (), regardless of
the value of the wavevectors of the covariance matrix, suggesting
that one must be careful in applying the jackknife or bootstrap methods to the
covariance matrix. We perform an eigenmode decomposition of the connected part
of the covariance matrix, showing that at higher it is dominated by a
single eigenmode. The full covariance matrix can be approximated as the
disconnected part only, with the connected part being treated as an external
nuisance parameter with a known scale dependence, and a known prior on its
variance for a given survey volume. Finally, we provide a prescription for how
to evaluate the covariance matrix from small box simulations without the need
to simulate large volumes.Comment: 22 pages, 17 figures, 1 tabl
A cautionary note on robust covariance plug-in methods
Many multivariate statistical methods rely heavily on the sample covariance
matrix. It is well known though that the sample covariance matrix is highly
non-robust. One popular alternative approach for "robustifying" the
multivariate method is to simply replace the role of the covariance matrix with
some robust scatter matrix. The aim of this paper is to point out that in some
situations certain properties of the covariance matrix are needed for the
corresponding robust "plug-in" method to be a valid approach, and that not all
scatter matrices necessarily possess these important properties. In particular,
the following three multivariate methods are discussed in this paper:
independent components analysis, observational regression and graphical
modeling. For each case, it is shown that using a symmetrized robust scatter
matrix in place of the covariance matrix results in a proper robust
multivariate method.Comment: 24 pages, 7 figure
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