This function prepares the data for the estimation of information divergences and works as a wrapper calling the functions that compute selected information divergences of methylation levels. In the downstream analysis, the probability distribution of a given information divergence is used in Methyl-IT as the null hypothesis of the noise distribution, which permits, in a further signal detection step, the discrimination of the methylation regulatory signal from the background noise.

For the current version, two information divergences of methylation levels are computed by default: 1) Hellinger divergence (H) and 2) the total variation distance (TVD). In the context of methylation analysis TVD corresponds to the absolute difference of methylation levels. Here, although the variable reported is the total variation (TV), the variable actually used for the downstream analysis is TVD. Once a differentially methylated position (DMP) is identified in the downstream analysis, TV is the standard indicator of whether the cytosine position is hyper- or hypo-methylated.

The option to compute the J-information divergence (JD) is currently provided. The motivation to introduce this divergence is given in the help of function estimateJDiv.

estimateDivergence(
ref,
indiv,
Bayesian = FALSE,
columns = NULL,
min.coverage = 4,
min.meth = 4,
min.umeth = 0,
min.sitecov = 4,
high.coverage = NULL,
percentile = 0.999,
JD = FALSE,
num.cores = 1L,
meth.level = FALSE,
logbase = 2,
verbose = TRUE,
...
)

## Arguments

ref The GRanges object of the reference individual that will be used in the estimation of the information divergence. A list of GRanges objects from the individuals that will be used in the estimation of the information divergence. logical(1). Whether to perform the estimations based on posterior estimations of methylation levels. Vector of one or two integer numbers denoting the indexes of the columns where the methylated and unmethylated read counts are found or, if meth.level = TRUE, the columns corresponding to the methylation levels. If columns = NULL and meth.level = FALSE, then columns = c(1,2) is assumed. If columns = NULL and meth.level = TRUE, then columns = 1 is assumed. An integer or an integer vector of length 2. Cytosine sites where the coverage in both samples, 'x' and 'y', are less than 'min.coverage' are discarded. The cytosine site is preserved, however, if the coverage is greater than 'min.coverage' in at least one sample. If 'min.coverage' is an integer vector, then the corresponding min coverage is applied to each sample. An integer or an integer vector of length 2. Cytosine sites where the numbers of read counts of methylated cytosine in both samples, '1' and '2', are less than 'min.meth' are discarded. If 'min.meth' is an integer vector, then the corresponding min number of reads is applied to each sample. Default is min.meth = 4. An integer or an integer vector of length 2 ($$min.umeth = c(min.umeth1, min.umeth2)$$). Min number of reads to consider cytosine position. Specifically cytosine positions where (uC <= min.umeth) & (mC > 0) & (mC <= min.meth) hold will be removed, where mC and uC stand for the numbers of methylated and unmethylated reads. Default is min.umeth = 0. An integer. The minimum total coverage. Only sites where the total coverage (cov1 + cov2) is greater than 'min.sitecov' are considered for downstream analysis, where cov1 and cov2 are the coverages for samples 1 and 2, respectively. An integer for read counts. Cytosine sites having higher coverage than this are discarded. Threshold to remove the outliers from each file and all files stacked. Logic (Default:FALSE). Option on whether to add a column with values of J-information divergence (see estimateJDiv). It is only compute if JD = TRUE and meth.level = FALSE. The number of cores to use, i.e. at most how many child processes will be run simultaneously (see 'bplapply' function from BiocParallel package). integer(1). The number of tasks per job. value must be a scalar integer >= 0L. In this documentation a job is defined as a single call to a function, such as bplapply, bpmapply etc. A task is the division of the X argument into chunks. When tasks == 0 (default), X is divided as evenly as possible over the number of workers (see MulticoreParam from BiocParallel package). logical(1) Whether methylation levels are given in place of counts. Default is FALSE. Logarithm base used to compute the JD (if JD = TRUE). Logarithm base 2 is used as default (bit unit). Use logbase = $$exp(1)$$ for natural logarithm. if TRUE, prints the function log to stdout Additional parameters for 'uniqueGRanges' function.

## Value

An object from 'infDiv' class with the four columns of counts, the information divergence, and additional columns:

1) A matrix:

The original matrix of methylated $$c_{ij}$$ and unmethylated $$t_{ij}$$ read counts from control $$j=1$$ and treatment $$j=2$$ samples at positions $$i$$.

2) 'p1' and 'p2':

methylation levels for control and treatment, respectively. If 'meth.level = FALSE' and 'Bayesian = TRUE' (recommended), 'p1' and 'p2' are estimated following the Bayesian approach described in reference (1).

3) 'bay.TV':

total variation TV = p2 - p1

4) 'TV':

total variation based on simple counts: $$TV=c_1/(c_1+t_1)-c_2/(c_2+t_2)$$, where $$c_i$$ and $$t_i$$ denote methylated and unmethylated read counts, respectively.

5) Hellinger divergence, 'hdiv':

If Bayesian = TRUE, the results are based on the posterior estimations of methylation levels. if meth.level = FALSE', then 'hdiv' is computed as given in reference (2), otherwise as: $$hdiv = (sqrt(p_1) - sqrt(p_2))^2 + (sqrt(1 -p_1) - sqrt(1 - p_2))^2$$

## Details

If read counts are provided, then Hellinger divergence is computed as given in the first formula from Theorem 1 from reference (1). In the present case:

$$H = 2 (n_1 + 1) (n_2 + 1)*((sqrt(p_1) - sqrt(p_2))^2 + (sqrt(1-p_2) - sqrt(1-p_2))^2)/(n_1 + n_2 + 2)$$

where $$n_1$$ and $$n_2$$ are the coverage for the control and treatment, respectively. Notice that each row from the matrix of counts correspond to a single cytosine position and has four values corresponding to 'mC1' and 'uC1' (control), and mC2' and 'uC2' for treatment.

According with the above equation, to estimate Hellinger divergence, not only the methylation levels are considered in the estimation of H, but also the control and treatment coverage at each given cytosine site. At this point, it is worthy to do mention that if the reference sample is derived with function poolFromGRlist using the 'sum' of read counts to compute a methylation pool, then 'min.coverage' parameter value must be used to prevent an over estimation of the divergence for low coverage cytosines sites. For example, if a reference sample is derived as the methylation pool of read count sum from 3 individuals and we want to consider only methylation sites with minimum coverage of 4, then we can set min.coverage = c(12, 4), where the number 12 (3 x 4) is the minimum coverage requested for the each cytosine site in the reference sample.

If the methylation levels are provided in place of counts, then the Hellinger divergence is computed as: $$H = (sqrt(p_1) - sqrt(p_2))^2 + (sqrt(1 - p_1) - sqrt(1 - p_2))^2$$

This formula assumes that the probability vectors derived from the methylation levels $$p_i = c(p_{i1}, 1 - p_{i2}$$) (see estimateHellingerDiv are an unbiased estimation of the expected one. The function applies a pairwise filtering after building a single GRanges from the two GRanges objects. Experimentally available cytosine sites are paired using the function 'uniqueGRanges'.

It is important to observe that several filtering conditions are provided to select biological meaningful cytosine positions, which prevent to carry experimental errors in the downstream analyses. By filtering the read count we try to remove bad quality data, which would be in the edge of the experimental error originated by the BS-seq sequencing. It is user responsibility to check whether cytosine positions used in the analysis are biological meaningful. For example, a cytosine position with counts mC1 = 10 and uC1 = 20 in the 'ref' sample and mC2 = 1 & uC2 = 0 in an 'indv' sample will lead to methylation levels p1 = 0.333 and p2 = 1, respectively, and TV = p2 - p1 = 0.667, which apparently indicates a hypermethylated site. However, there are not enough reads supporting p2 = 1. A Bayesian estimation of TV will reveal that this site would be, in fact, hypomethylated. So, the best practice will be the removing of sites like that. This particular case is removed under the default settings: min.coverage = 4, min.meth = 4, and min.umeth = 0 (see example for function uniqueGRfilterByCov, called by 'estimateDivergence').

## References

1. Sanchez R, Yang X, Maher T, Mackenzie S. Discrimination of DNA Methylation Signal from Background Variation for Clinical Diagnostics. Int. J. Mol Sci, 2019, 20:5343.

2. Basu A., Mandal A., Pardo L. Hypothesis testing for two discrete populations based on the Hellinger distance. Stat. Probab. Lett. 2010, 80: 206-214.

estimateBayesianDivergence

## Examples

## The read count data are created
num.samples <- 250
x <- data.frame(chr = 'chr1', start = 1:num.samples,
end = 1:num.samples,strand = '*',
mC = rnbinom(size = num.samples, mu = 4, n = 500),
uC = rnbinom(size = num.samples, mu = 4, n = 500))

y <- data.frame(chr = 'chr1', start = 1:num.samples, end = 1:num.samples,
strand = '*', mC = rnbinom(size = num.samples,
mu = 4, n = 500),
uC = rnbinom(size = num.samples, mu = 4, n = 500))

x <- makeGRangesFromDataFrame(x, keep.extra.columns = TRUE)
y <- makeGRangesFromDataFrame(y, keep.extra.columns = TRUE)
hd <- estimateDivergence(ref = x, indiv = list(y), JD = TRUE,
verbose = FALSE)[[1]]

## Keep in mind that Hellinger and J divergences are, in general,
## correlated
cor.test(x = as.numeric(hd$hdiv), y = as.numeric(hd$jdiv),
method = 'kendall')#>
#> 	Kendall's rank correlation tau
#>
#> data:  as.numeric(hd$hdiv) and as.numeric(hd$jdiv)
#> z = 18.941, p-value < 2.2e-16
#> alternative hypothesis: true tau is not equal to 0
#> sample estimates:
#>       tau
#> 0.8926499
#>