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Ignoring correlation in uncertainty and sensitivity analysis in life cycle assessment : what is the risk? Groen, E.A. ; Heijungs, R.  \ 2017
Environmental Impact Assessment Review 62 (2017).  ISSN 01959255  p. 98  109. correlation  covariance matrix  global sensitivity analysis  Uncertainty propagation
Life cycle assessment (LCA) is an established tool to quantify the environmental impact of a product. A good assessment of uncertainty is important for making wellinformed decisions in comparative LCA, as well as for correctly prioritising data collection efforts. Under or overestimation of output uncertainty (e.g. output variance) will lead to incorrect decisions in such matters. The presence of correlations between input parameters during uncertainty propagation, can increase or decrease the the output variance. However, most LCA studies that include uncertainty analysis, ignore correlations between input parameters during uncertainty propagation, which may lead to incorrect conclusions. Two approaches to include correlations between input parameters during uncertainty propagation and global sensitivity analysis were studied: an analytical approach and a sampling approach. The use of both approaches is illustrated for an artificial case study of electricity production. Results demonstrate that both approaches yield approximately the same output variance and sensitivity indices for this specific case study. Furthermore, we demonstrate that the analytical approach can be used to quantify the risk of ignoring correlations between input parameters during uncertainty propagation in LCA. We demonstrate that: (1) we can predict if including correlations among input parameters in uncertainty propagation will increase or decrease output variance; (2) we can quantify the risk of ignoring correlations on the output variance and the global sensitivity indices. Moreover, this procedure requires only little data. 

Considering Horn’s Parallel Analysis from a Random Matrix Theory Point of View Saccenti, Edoardo ; Timmerman, Marieke E.  \ 2017
Psychometrika 82 (2017)1.  ISSN 00333123  p. 186  209. common factor analysis  covariance matrix  number of common factors  number of principal components  principal component analysis
Horn’s parallel analysis is a widely used method for assessing the number of principal components and common factors. We discuss the theoretical foundations of parallel analysis for principal components based on a covariance matrix by making use of arguments from random matrix theory. In particular, we show that (i) for the first component, parallel analysis is an inferential method equivalent to the Tracy–Widom test, (ii) its use to test highorder eigenvalues is equivalent to the use of the joint distribution of the eigenvalues, and thus should be discouraged, and (iii) a formal test for higherorder components can be obtained based on a Tracy–Widom approximation. We illustrate the performance of the two testing procedures using simulated data generated under both a principal component model and a common factors model. For the principal component model, the Tracy–Widom test performs consistently in all conditions, while parallel analysis shows unpredictable behavior for higherorder components. For the common factor model, including major and minor factors, both procedures are heuristic approaches, with variable performance. We conclude that the Tracy–Widom procedure is preferred over parallel analysis for statistically testing the number of principal components based on a covariance matrix. 