Theoretical insight into diagnosing observation error correlations using observation-minus-background and observation-minus-analysis statistics

To improve the quantity and impact of observations used in data assimilation it is necessary to take into account the full, potentially correlated, observation error statistics. A number of methods for estimating correlated observation errors exist, but a popular method is a diagnostic that makes use of statistical averages of observation-minus-background and observation-minus-analysis residuals. The accuracy of the results it yields is unknown as the diagnostic is sensitive to the difference between the exact background and exact observation error covariances and those that are chosen for use within the assimilation. It has often been stated in the literature that the results using this diagnostic are only valid when the background and observation error correlation length scales are well separated. Here we develop new theory relating to the diagnostic. For observations on a 1D periodic domain we are able to the show the effect of changes in the assumed error statistics used in the assimilation on the estimated observation error covariance matrix. We also provide bounds for the estimated observation error variance and eigenvalues of the estimated observation error correlation matrix. We demonstrate that it is still possible to obtain useful results from the diagnostic when the background and observation error length scales are similar. In general, our results suggest that when correlated observation errors are treated as uncorrelated in the assimilation, the diagnostic will underestimate the correlation length scale. We support our theoretical results with simple illustrative examples. These results have potential use for interpreting the derived covariances estimated using an operational system

Bayesian Statistics as a New Tool for Spectral Analysis: I. Application for the Determination of Basic Parameters of Massive Stars

Spectral analysis is a powerful tool to investigate stellar properties and it has been widely used for decades now. However, the methods considered to perform this kind of analysis are mostly based on iteration among a few diagnostic lines to determine the stellar parameters. While these methods are often simple and fast, they can lead to errors and large uncertainties due to the required assumptions. Here we present a method based on Bayesian statistics to find simultaneously the best combination of effective temperature, surface gravity, projected rotational velocity, and microturbulence velocity, using all the available spectral lines. Different tests are discussed to demonstrate the strength of our method, which we apply to 54 mid-resolution spectra of field and cluster B stars obtained at the Observatoire du Mont-M\'egantic. We compare our results with those found in the literature. Differences are seen which are well explained by the different methods used. We conclude that the B-star microturbulence velocities are often underestimated. We also confirm the trend that B stars in clusters are on average faster rotators than field B stars.Comment: 31 pages, 22 figure

On the insufficiency of arbitrarily precise covariance matrices: non-Gaussian weak lensing likelihoods

We investigate whether a Gaussian likelihood, as routinely assumed in the analysis of cosmological data, is supported by simulated survey data. We define test statistics, based on a novel method that first destroys Gaussian correlations in a dataset, and then measures the non-Gaussian correlations that remain. This procedure flags pairs of datapoints which depend on each other in a non-Gaussian fashion, and thereby identifies where the assumption of a Gaussian likelihood breaks down. Using this diagnostic, we find that non-Gaussian correlations in the CFHTLenS cosmic shear correlation functions are significant. With a simple exclusion of the most contaminated datapoints, the posterior for $s_8$ is shifted without broadening, but we find no significant reduction in the tension with $s_8$ derived from Planck Cosmic Microwave Background data. However, we also show that the one-point distributions of the correlation statistics are noticeably skewed, such that sound weak lensing data sets are intrinsically likely to lead to a systematically low lensing amplitude being inferred. The detected non-Gaussianities get larger with increasing angular scale such that for future wide-angle surveys such as Euclid or LSST, with their very small statistical errors, the large-scale modes are expected to be increasingly affected. The shifts in posteriors may then not be negligible and we recommend that these diagnostic tests be run as part of future analyses.Comment: Replacement to match accepted MNRAS versio

Diagnosing horizontal and inter-channel observation error correlations for SEVIRI observations using observation-minus-background and observation-minus-analysis statistics

It has been common practice in data assimilation to treat observation errors as uncorrelated; however, meteorological centres are beginning to use correlated inter-channel observation errors in their operational assimilation systems. In this work, we are the first to characterise inter-channel and spatial error correlations for Spinning Enhanced Visible and Infrared Imager (SEVIRI) observations that are assimilated into the Met Office high-resolution model. The errors are calculated using a diagnostic that calculates statistical averages of observation-minus-background and observation-minus-analysis residuals. This diagnostic is sensitive to the background and observation error statistics used in the assimilation, although, with careful interpretation of the results, it can still provide useful information. We find that the diagnosed SEVIRI error variances are as low as one-tenth of those currently used in the operational system. The water vapour channels have significantly correlated inter-channel errors, as do the surface channels. The surface channels have larger observation error variances and inter-channel correlations in coastal areas of the domain; this is the result of assimilating mixed pixel (land-sea) observations. The horizontal observation error correlations range between 30 km and 80 km, which is larger than the operational thinning distance of 24 km. We also find that estimates from the diagnostics are unaffected by biased observations, provided that the observation-minus-background and observation-minus-analysis residual means are subtracted

Ensemble prediction for nowcasting with a convection-permitting model - II: forecast error statistics

A 24-member ensemble of 1-h high-resolution forecasts over the Southern United Kingdom is used to study short-range forecast error statistics. The initial conditions are found from perturbations from an ensemble transform Kalman filter. Forecasts from this system are assumed to lie within the bounds of forecast error of an operational forecast system. Although noisy, this system is capable of producing physically reasonable statistics which are analysed and compared to statistics implied from a variational assimilation system. The variances for temperature errors for instance show structures that reflect convective activity. Some variables, notably potential temperature and specific humidity perturbations, have autocorrelation functions that deviate from 3-D isotropy at the convective-scale (horizontal scales less than 10 km). Other variables, notably the velocity potential for horizontal divergence perturbations, maintain 3-D isotropy at all scales. Geostrophic and hydrostatic balances are studied by examining correlations between terms in the divergence and vertical momentum equations respectively. Both balances are found to decay as the horizontal scale decreases. It is estimated that geostrophic balance becomes less important at scales smaller than 75 km, and hydrostatic balance becomes less important at scales smaller than 35 km, although more work is required to validate these findings. The implications of these results for high-resolution data assimilation are discussed

A mixed effect model for bivariate meta-analysis of diagnostic test accuracy studies using a copula representation of the random effects distribution

Diagnostic test accuracy studies typically report the number of true positives, false positives, true negatives and false negatives. There usually exists a negative association between the number of true positives and true negatives, because studies that adopt less stringent criterion for declaring a test positive invoke higher sensitivities and lower specificities. A generalized linear mixed model (GLMM) is currently recommended to synthesize diagnostic test accuracy studies. We propose a copula mixed model for bivariate meta-analysis of diagnostic test accuracy studies. Our general model includes the GLMM as a special case and can also operate on the original scale of sensitivity and specificity. Summary receiver operating characteristic curves are deduced for the proposed model through quantile regression techniques and different characterizations of the bivariate random effects distribution. Our general methodology is demonstrated with an extensive simulation study and illustrated by re-analysing the data of two published meta-analyses. Our study suggests that there can be an improvement on GLMM in fit to data and makes the argument for moving to copula random effects models. Our modelling framework is implemented in the package CopulaREMADA within the open source statistical environment R

Is there evidence for dark energy evolution?

Recently, Sahni, Shafielo o & Starobinsky (2014) combined two independent measurements of $H(z)$ from BAO data with the value of the Hubble constant $H_0 = H(z=0)$, in order to test the cosmological constant hypothesis by means of an improved version of the $Om$ diagnostic. Their result indicated a considerable tension between observations and predictions of the $\Lambda$CDM model. However, such strong conclusion was based only on three measurements of $H(z)$. This motivated us to repeat similar work on a larger sample. By using a comprehensive data set of 29 $H(z)$, we find that discrepancy indeed exists. Even though the value of $\Omega_{m,0} h^2$ inferred from $Omh^2$ diagnostic depends on the way one chooses to make a summary statistics (weighted mean or the median), the persisting discrepancy supports the claims of Sahni, Shafielo o & Starobinsky (2014) that $\Lambda$CDM model may not be the best description of our Universe.Comment: 8 pages, 2 figures. Accepted for publication in the ApJ