Ata (Additional file 1). Since H3K27me3 and H3K9me3 modifications are associated with gene silencing, the

Ata (Additional file 1). Since H3K27me3 and H3K9me3 modifications are associated with gene silencing, the heavy tail with high occupancy values can be associated with lowly expressed genes and the low occupancy counts with highly expressed genes. Using gene expression increased the performance of the algorithm (Additional file 1), both for the single GS-9620 chemical information sample analysis and for the sample comparison.Bivariate hidden Markov modelNormal marginal CDFs. To obtain a CDF for the origia nal random variables X and Y with marginal CDFs Fx and b we use again the probability integral transformation to Fy a b obtain the uniform variables ux = Fx (x) and uy = Fy (y). For a more detailed introduction to copula theory we refer the reader to [47]. Now putting it all together, we used a Gaussian copula to define the bivariate cumulative distribution function of each component F((x, y), a,b ) asa a C a,b Fx (x), Fy (y) = P (X x, Y y)= wherea,b-a,ba Fx (x) ,-b Fy (y),zx , zy = zx , zy =zxzy- -a,bzx , zy dzx dzy ,histoneHMM is primarily designed to compare two ChIPseq samples, say A and B. For each individual ChIP-seq sample, we partition the genome into m equally sized bins (1000 bp by default). Let xi and yi be the read counts for the ith bin for sample A and B, respectively. Further we define the indicator variable a = 0 if sample A is unmodified and a = 1 if it is modified. Similarly the indicator variable b is defined for sample B. We denote the parameters of the univariate mixture of sample A as A and that of sample B as B . The probability of the random pair (xi , yi ) is given by a bivariate count distribution with four mixing components, corresponding to the situations where both samples are unmodified (a = 0, b = 0), both samples are modified (a = 1, b = 1), only sample A is modified (a = 1, b = 0) or only sample B is modified (a = 0, b = 1). We write this four component mixture as11 2 x y 1 – 2 ?exp -1 2(1 – 2 )2 2 zy 2zx zy zx + 2- 2 x y x ya,bP((x, y)| ) =a=0 b=a,b f ((x, y), a,b ),(3)where a,b are the mixing weights and a,b are the component density parameters for each component j, corresponding to a pair a, b. Calculating the bivariate components f ((x, y), a,b ) is challenging as bivariate (or multivariate) count distributions are difficult to work with and often do not exist in closed form. Copula theory offers an elegant PubMed ID:https://www.ncbi.nlm.nih.gov/pubmed/25447644 way to obtain multivariate distributions once the marginals are known [46]. A copula C = C(u1 , u2 , . . . , up ) = P(U1 u1 , U2 u2 , . . . , Up up ) is a multivariate cumulative density function (CDF) defined over the p-dimensional unit cube C :[ 0, 1]p [ 0, 1], where each Ui Unif(0, 1). For two random variables Zx , Zy with joint CDF G and marginal CDFs Gx , Gy the probability integral transformation can be used to obtain a copula C(ux , uy ) = G(G-1 (ux ), G-1 (uy )). Here we used a Gaussian copula, x y such that G is the CDF of the multivariate Normal distribution and Gx , Gy are the corresponding univariateis the bivariate Gaussian CDF with zero mean and covariance matrix corresponding to , x , y . -1 is the inverse a of the univariate standard normal CDF and Fx = P(X x f (x, A,a ) is the CDF for the marginal distribux) = 0 b tion of component a of sample A, and Fy = P(Y y) = y 0 f (y, B,b ) for the component b of B, respectively (Eq. 1, Eq. 2). The covariance matrix a,b between the transformed a b variables -1 (Fx (x)) and -1 (Fy (y)) is computed as follows: first we called each region modified or unmodified i.