A new numerical method to solve the downdating problem (and variants thereof), namely removing the effect of some observations from the generalized least squares (GLS) estimator of the general linear model (GLM) after it has been estimated, is extensively investigated. It is verified that the solution of the downdated least squares problem can be obtained from the estimation of an equivalent GLM, where the original model is updated with the imaginary deleted observations. This updated GLM has a non positive definite dispersion matrix which comprises complex covariance values and it is proved herein to yield the same normal equations as the downdated model. Additionally, the problem of deleting observations from the seemingly unrelated regressions model is addressed, demonstrating the direct applicability of this method to other multivariate linear models. The algorithms which implement the novel downdating method utilize efficiently the previous computations from the estimation of the original model. As a result, the computational cost is significantly reduced. This shows the great usability potential of the downdating method in computationally intensive problems. The downdating algorithms have been applied to real and synthetic data to illustrate their efficiency.
The ability to automatically identify industry sector coverage in articles on legal developments, or any kind of news articles for that matter, can bring plentiful of benefits both to the readers and the content creators themselves. By having articles tagged based on industry coverage, readers from all around the world would be able to get to legal news that are specific to their region and professional industry. Simultaneously, writers would benefit from understanding which industries potentially lack coverage or which industries readers are currently mostly interested in and thus, they would focus their writing efforts towards more inclusive and relevant legal news coverage. In this paper, a Machine Learning-powered industry analysis approach which combined Natural Language Processing (NLP) with Statistical and Machine Learning (ML)
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