[3] **viXra:1304.0143 [pdf]**
*submitted on 2013-04-25 11:24:37*

**Authors:** Zhang Huiming

**Comments:** 6 Pages. In Chinese

In this paper, by using three kinds of ideas of probability theory, we proof the equivalence among three kinds of probability expressions in the problem of rational division of stakes by the method of mathematical analysis. In addition, different ideas of probability theory obtain the identity. Let one of the probability expressions be a function, we find the B-function is closely relate to the derivative the probability expression function. According to Beta distribution function, we proof that probability expression function in the problem of rational division is equal to the distribution function of Beta distribution.

**Category:** Statistics

[2] **viXra:1304.0055 [pdf]**
*submitted on 2013-04-11 17:24:10*

**Authors:** Li Charlie Xia

**Comments:** 56 Pages.

Local association analysis, such as local similarity analysis and local shape analysis, of biological time series data helps elucidate the varying dynamics of biological systems. However, their applications to large scale high-throughput data are limited by slow permutation procedures for statistical signicance evaluation. We developed a theoretical approach to approximate the statistical signicance of local similarity and local shape analysis based on the approximate tail distribution of the maximum partial sum of independent identically distributed (i.i.d) and Markovian random variables. Simulations show that the derived formula approximates the tail distribution reasonably well (starting at time points > 10 with no delay and > 20 with delay) and provides p-values comparable to those from permutations. The new approach enables ecient calculation of statistical signicance for pairwise local association analysis, making possible all-to-all association studies otherwise prohibitive. As a demonstration, local association analysis of human microbiome time series shows that core OTUs are highly synergetic and some of the associations are body-site specic across samples. The new approach is implemented in our eLSA package, which now provides pipelines for faster local similarity and shape analysis of time series data. The tool is freely available from eLSA's website:
http://meta.usc.edu/softs/lsa.

**Category:** Statistics

[1] **viXra:1304.0054 [pdf]**
*submitted on 2013-04-11 17:27:00*

**Authors:** Li Charlie Xia

**Comments:** 127 Pages.

Recent developments in experimental molecular techniques, such as microarray, next generation sequencing technologies, have led molecular biology into a high-throughput era with emergent omics research areas, including metagenomics and transcriptomics. Massive-size omics datasets generated and being generated from the experimental laboratories put new challenges to computational biologists to develop fast and accurate quantitative analysis tools. We have developed two statistical and algorithmic methods,
GRAMMy and eLSA, for metagenomics and microbial community time series analysis. GRAMMy provides a unied probabilistic framework for shotgun metagenomics, in which maximum likelihood method is employed to accurately compute Genome Relative Abundance of microbial communities using the Mixture Model theory (GRAMMy). We extended the Local Similarity Analysis technique (eLSA) to time series data with replicates, capturing statistically signicant local and potentially time-delayed associations. Both methods are validated through simulation studies and their capability to reveal new biology is also demonstrated through applications to real datasets. We implemented GRAMMy and eLSA as C++ extensions to Python, with both superior computational eciency and easy-to-integrate programming interfaces. GRAMMy and eLSA methods
will be increasingly useful tools as new omics researches accelerating their pace.http://meta.usc.edu/softs/lsa.

**Category:** Statistics