Evaluating the reliability of estimated phylogenetic trees is of critical importance in the field of molecular phylogenetics, and for other endeavors that depend on accurate phylogenetic reconstruction. The bootstrap method is a well-known computational approach to phylogenetic tree assessment, and more generally for assessing the reliability of statistical models. However, it is known to be biased under certain circumstances, calling into question the accuracy of the method. Several advanced bootstrap methods have been developed to achieve higher accuracy, one of which is the double bootstrap approach, but the computational burden of this method has precluded its application to practical problems of phylogenetic tree selection. We address this issue by proposing a simple method called the speedy double bootstrap, which circumvents the second-tier resampling step in the regular double bootstrap approach. We also develop an implementation of the regular double bootstrap for comparison with our speedy method. The speedy double bootstrap suffers no significant loss of accuracy compared with the regular double bootstrap, while performing calculations significantly more rapidly (at minimum around 371 times faster, based on analysis of mammalian mitochondrial amino acid sequences and 12S and 16S rRNA genes). Our method thus enables, for the first time, the practical application of the double bootstrap technique in the context of molecular phylogenetics. The approach can also be used more generally for model selection problems wherever the maximum likelihood criterion is used.
Based on the factors affecting sports performance, from a more comprehensive and broad perspective, after consulting the literature, 52 factors that affect the outcome of football matches are selected, including technology, tactics, physical fitness and referees’ penalties. By watching the video of the game, 52 influencing factors of 200 games and 400 teams were counted. The original data was statistically processed with correlation analysis and multiple linear regression analysis, and the statistics of the 26 European Cup games were substituted into the winning formula. To verify the scientific nature and objectivity of the formula, we aim to ascertain the core factors in the winning factors of a football game and the quantitative relationship between these factors and the result of the game, so as to provide a certain reference for football training, game analysis and scientific research. The technical and tactical ability of individuals and teams is the core competitive ability factor that affects the result of the game; from a single factor, 15 factor indicators have a significant impact on the result of a football match; on the whole, 10 factor indicators have a significant effect on the result of a football match. In addition, there is a certain quantitative relationship between these influencing factors and the results of the game; empirical evidence shows that the football game winning formula has a certain degree of science and objectivity.
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