PMC:4992282 / 17153-19513 JSONTXT

{"project":"2_test","denotations":[{"id":"27542753-21385052-8217011","span":{"begin":95,"end":96},"obj":"21385052"},{"id":"27542753-21385052-8217012","span":{"begin":657,"end":658},"obj":"21385052"}],"text":"Expectation-Maximization (EM) step\nWe follow the same Expectation-Maximization approach as in [8] to identify the mixture of Poisson distributions that best fits the observed k-mer frequencies. Briefly, each Poisson distribution of parameter λi models the probability of certain k-mers coming from species (or group of species with similar abundance level) of relative abundance λi. If S denotes the set of species, then the probability of k-mer kj with n(kj) total counts coming from species (or group of species) Si of relative abundance λi and genome length gi (or total genome length in the case of a group of species of similar abundance) is given by [8]:\\documentclass[12pt]{minimal} \t\t\t\t\\usepackage{amsmath} \t\t\t\t\\usepackage{wasysym} \t\t\t\t\\usepackage{amsfonts} \t\t\t\t\\usepackage{amssymb} \t\t\t\t\\usepackage{amsbsy} \t\t\t\t\\usepackage{mathrsfs} \t\t\t\t\\usepackage{upgreek} \t\t\t\t\\setlength{\\oddsidemargin}{-69pt} \t\t\t\t\\begin{document}$$ P\\left({k}_j\\in {S}_i\\left|n\\left({k}_j\\right)\\right.\\right)=\\frac{{\\mathsf{g}}_i}{{\\displaystyle {\\sum}_{m=1}^{\\left|S\\right|}{\\mathsf{g}}_m}{\\left(\\frac{\\lambda_m}{\\lambda_i}\\right)}^{n\\left({k}_j\\right)}e\\left({\\lambda}_i-{\\lambda}_m\\right)} $$\\end{document}Pkj∈Sinkj=gi∑m=1Sgmλmλinkjeλi−λm\nThe EM process starts by defining n distributions with random initial parameters and iteratively updates parameters until convergence or until a maximum number of iterations is reached. The parameters of the Poisson distributions are updated iteratively according to:\\documentclass[12pt]{minimal} \t\t\t\t\\usepackage{amsmath} \t\t\t\t\\usepackage{wasysym} \t\t\t\t\\usepackage{amsfonts} \t\t\t\t\\usepackage{amssymb} \t\t\t\t\\usepackage{amsbsy} \t\t\t\t\\usepackage{mathrsfs} \t\t\t\t\\usepackage{upgreek} \t\t\t\t\\setlength{\\oddsidemargin}{-69pt} \t\t\t\t\\begin{document}$$ {\\mathit{\\mathsf{g}}}_i={\\displaystyle \\sum_{j=1}^KP\\left({k}_j\\in {S}_i\\left|n\\left({k}_j\\right)\\right.\\right)} $$\\end{document}gi=∑j=1KPkj∈Sinkj\\documentclass[12pt]{minimal} \t\t\t\t\\usepackage{amsmath} \t\t\t\t\\usepackage{wasysym} \t\t\t\t\\usepackage{amsfonts} \t\t\t\t\\usepackage{amssymb} \t\t\t\t\\usepackage{amsbsy} \t\t\t\t\\usepackage{mathrsfs} \t\t\t\t\\usepackage{upgreek} \t\t\t\t\\setlength{\\oddsidemargin}{-69pt} \t\t\t\t\\begin{document}$$ {\\lambda}_i=\\frac{{\\displaystyle {\\sum}_{j=1}^Kn\\left({k}_j\\right)P\\left({k}_j\\in {S}_i\\left|n\\left({k}_j\\right)\\right.\\right)}}{{\\mathit{\\mathsf{g}}}_i} $$\\end{document}λi=∑j=1KnkjPkj∈Sinkjgi"}