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Register a new userJackknife resampling technique on mocks: an alternative method for covariance matrix estimation
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Added by
St\\'ephanie Escoffier
Publication date
2016
fields
Physics
and research's language is
English
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We present a fast and robust alternative method to compute covariance matrix in case of cosmology studies. Our method is based on the jackknife resampling applied on simulation mock catalogues. Using a set of 600 BOSS DR11 mock catalogues as a reference, we find that the jackknife technique gives a similar galaxy clustering covariance matrix estimate by requiring a smaller number of mocks. A comparison of convergence rates show that $sim$7 times fewer simulations are needed to get a similar accuracy on variance. We expect this technique to be applied in any analysis where the number of available N-body simulations is low.
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We present correction terms that allow delete-one Jackknife and Bootstrap methods to be used to recover unbiased estimates of the data covariance matrix of the two-point correlation function $xileft(mathbf{r}right)$. We demonstrate the accuracy and precision of this new method using a large set of 1000 QUIJOTE simulations that each cover a comoving volume of $1rm{left[h^{-1}Gpcright]^3}$. The corrected resampling techniques accurately recover the correct amplitude and structure of the data covariance matrix as represented by its principal components. Our corrections for the internal resampling methods are shown to be robust against the intrinsic clustering of the cosmological tracers both in real- and redshift space using two snapshots at $z=0$ and $z=1$ that mimic two samples with significantly different clustering. We also analyse two different slicing of the simulation volume into $n_{rm sv}=64$ or $125$ sub-samples and show that the main impact of different $n_{rm sv}$ is on the structure of the covariance matrix due to the limited number of independent internal realisations that can be made given a fixed $n_{rm sv}$.
Cosmological large-scale structure analyses based on two-point correlation functions often assume a Gaussian likelihood function with a fixed covariance matrix. We study the impact on cosmological parameter estimation of ignoring the parameter dependence of this covariance matrix, focusing on the particular case of joint weak-lensing and galaxy clustering analyses. Using a Fisher matrix formalism (calibrated against exact likelihood evaluation in particular simple cases), we quantify the effect of using a parameter dependent covariance matrix on both the bias and variance of the parameters. We confirm that the approximation of a parameter-independent covariance matrix is exceptionally good in all realistic scenarios. The information content in the covariance matrix (in comparison with the two point functions themselves) does not change with the fractional sky coverage. Therefore the increase in information due to the parameter dependent covariance matrix becomes negligible as the number of modes increases. Even for surveys covering less than $1%$ of the sky, this effect only causes a bias of up to ${cal O}(10%)$ of the statistical uncertainties, with a misestimation of the parameter uncertainties at the same level or lower. The effect will only be smaller with future large-area surveys. Thus for most analyses the effect of a parameter-dependent covariance matrix can be ignored both in terms of the accuracy and precision of the recovered cosmological constraints.
We propose a novel pilot structure for covariance matrix estimation in massive multiple-input multiple-output (MIMO) systems in which each user transmits two pilot sequences, with the second pilot sequence multiplied by a random phase-shift. The covariance matrix of a particular user is obtained by computing the sample cross-correlation of the channel estimates obtained from the two pilot sequences. This approach relaxes the requirement that all the users transmit their uplink pilots over the same set of symbols. We derive expressions for the achievable rate and the mean-squared error of the covariance matrix estimate when the proposed method is used with staggered pilots. The performance of the proposed method is compared with existing methods through simulations.
We give an analytical interpretation of how subsample-based internal covariance estimators lead to biased estimates of the covariance, due to underestimating the super-sample covariance (SSC). This includes the jackknife and bootstrap methods as estimators for the full survey area, and subsampling as an estimator of the covariance of subsamples. The limitations of the jackknife covariance have been previously presented in the literature because it is effectively a rescaling of the covariance of the subsample area. However we point out that subsampling is also biased, but for a different reason: the subsamples are not independent, and the corresponding lack of power results in SSC underprediction. We develop the formalism in the case of cluster counts that allows the bias of each covariance estimator to be exactly predicted. We find significant effects for a small-scale area or when a low number of subsamples is used, with auto-redshift biases ranging from 0.4% to 15% for subsampling and from 5% to 75% for jackknife covariance estimates. The cross-redshift covariance is even more affected; biases range from 8% to 25% for subsampling and from 50% to 90% for jackknife. Owing to the redshift evolution of the probe, the covariances cannot be debiased by a simple rescaling factor, and an exact debiasing has the same requirements as the full SSC prediction. These results thus disfavour the use of internal covariance estimators on data itself or a single simulation, leaving analytical prediction and simulations suites as possible SSC predictors.
This paper proposed a low-complexity antenna layout-aware (ALA) covariance matrix estimation method. In the estimation process, antenna layout is assumed known at the estimator. Using this information, the estimator finds antenna pairs with statistically equivalent covariance values and sets their covariance values to the average of covariance values of all these antenna pairs. ALA for both uniform linear array (ULA) and uniform planar array (UPA) is discussed. This method takes the benefit that covariance matrices do not have full degrees of freedom. Then, the proposed ALA covariance matrix method is applied to a multi-cell network. Simulations have demonstrated that the proposed method can provide better performance than the widely used viaQ method, with respect to mean square errors and downlink spectral efficiencies.
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