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Collinearity a review of methods to deal with it and a simulation study evaluating their performance
(2013)
Collinearity refers to the non independence of predictor variables, usually in a regression-type analysis. It is a common feature of any descriptive ecological data set and can be a problem for parameter estimation because it inflates the variance of regression parameters and hence potentially leads to the wrong identification of relevant predictors in a statistical model. Collinearity is a severe problem when a model is trained on data from one region or time, and predicted to another with a different or unknown structure of collinearity. To demonstrate the reach of the problem of collinearity in ecology, we show how relationships among predictors differ between biomes, change over spatial scales and through time. Across disciplines, different approaches to addressing collinearity problems have been developed, ranging from clustering of predictors, threshold-based pre-selection, through latent variable methods, to shrinkage and regularisation. Using simulated data with five predictor-response relationships of increasing complexity and eight levels of collinearity we compared ways to address collinearity with standard multiple regression and machine-learning approaches. We assessed the performance of each approach by testing its impact on prediction to new data. In the extreme, we tested whether the methods were able to identify the true underlying relationship in a training dataset with strong collinearity by evaluating its performance on a test dataset without any collinearity. We found that methods specifically designed for collinearity, such as latent variable methods and tree based models, did not outperform the traditional GLM and threshold-based pre-selection. Our results highlight the value of GLM in combination with penalised methods (particularly ridge) and threshold-based pre-selection when omitted variables are considered in the final interpretation. However, all approaches tested yielded degraded predictions under change in collinearity structure and the folk lore'-thresholds of correlation coefficients between predictor variables of |r| >0.7 was an appropriate indicator for when collinearity begins to severely distort model estimation and subsequent prediction. The use of ecological understanding of the system in pre-analysis variable selection and the choice of the least sensitive statistical approaches reduce the problems of collinearity, but cannot ultimately solve them.
We present an analytical formula for the asymptotic relative entropy of entanglement for Werner states of arbitrary dimensionality. We then demonstrate its validity using methods from convex optimization. This is the first case in which the value of a subadditive entanglement measure has been obtained in the asymptotic limit. This formula also gives the sharpest known upper bound on the distillable entanglement of these states.
Context. The Milky Way is surrounded by large amounts of diffuse gaseous matter that connects the stellar body of our Galaxy with its large-scale Local Group (LG) environment. Aims. To characterize the absorption properties of this circumgalactic medium (CGM) and its relation to the LG we present the so-far largest survey of metal absorption in Galactic high-velocity clouds (HVCs) using archival ultraviolet (UV) spectra of extragalactic background sources. The UV data are obtained with the Cosmic Origins Spectrograph (COS) onboard the Hubble Space Telescope (HST) and are supplemented by 21 cm radio observations of neutral hydrogen. Methods. Along 270 sightlines we measure metal absorption in the lines of Si II, Si III, C II, and C IV and associated H I 21 cm emission in HVCs in the velocity range vertical bar v(LSR)vertical bar = 100-500 km s(-1). With this unprecedented large HVC sample we were able to improve the statistics on HVC covering fractions, ionization conditions, small-scale structure, CGM mass, and inflow rate. For the first time, we determine robustly the angular two point correlation function of the high-velocity absorbers, systematically analyze antipodal sightlines on the celestial sphere, and compare the HVC absorption characteristics with that of damped Lyman alpha absorbers (DLAs) and constrained cosmological simulations of the LG (CLUES project).