`R/nonlinearFitDist.R`

`nonlinearFitDist.Rd`

A wrapper to call functions `weibull3P`

,
`fitGammaDist`

, and `fitGGammaDist`

to operate on a
list of `GRanges-class`

objects.

```
nonlinearFitDist(
LR,
column = 9,
dist.name = "Weibull",
sample.size = 20,
location.par = FALSE,
absolute = FALSE,
min.scale = 1e-05,
min.mu = 0.01,
mu.rel.err = 0.89,
npoints = NULL,
model = "all",
maxiter = 1024,
tol = 1e-12,
ftol = 1e-12,
ptol = 1e-12,
minFactor = 10^-6,
num.cores = multicoreWorkers(),
tasks = 0L,
maxfev = 1e+05,
verbose = FALSE,
...
)
```

- LR
A list of

`GRanges-class`

objects with information divergence values in their meta-columns.- column
An integer number denoting the index of the GRanges column where the information divergence is given. Default column = 1

- dist.name
Name(s) of the distribution to fit. A single character string or character vector naming the distribution(s): 'Weibull' (default), gamma with three-parameter (Gamma3P), gamma with two-parameter (Gamma2P), generalized gamma with three-parameter ('GGamma3P') or four-parameter ('GGamma4P'), and Log-Normal (LogNorm).

- sample.size
size of the sample

- location.par
whether to consider the fitting to generalized gamma distribution (GGamma) including the location parameter, i.e., a GGamma with four parameters (GGamma4P).

- absolute
Logic (default, FALSE). Total variation (TV, the difference of methylation levels) is normally an output in the downstream MethylIT analysis. If 'absolute = TRUE', then TV is transformed into |TV|, which is an information divergence that can be fitted to Weibull or to Generalized Gamma distribution.

- min.scale
A number. The nonlinear fit of GGamma family distributions sometimes yields weird estimation of the

*scale*parameter with values < 1e-5. Based on our experience, scale parameter values lower than min.scale = 1e-5 are probably meaningless. The result of numerical errors, the nonlinear algorithm are not perfect.- min.mu, mu.rel.err
Numbers. Models with location parameter \(\mu\) sometimes reports \(\mu\) values close to zero, which generally corresponds to a non statistically significant model coefficient/parameter. This detail is not detected by the AIC or the cross-validations correlation coefficient. In this case the best model is the model without location parameter, where \(\mu = 0\). If \(\mu < min.mu\) and \(|E(\mu) - mean(idiv)|/mean(idiv) > mu.rel.err\), where \(E(\mu)\) is the estimated value and \(mean(idiv)\) is the mean observed value of the information divergence provided, then the issue is reported in the column named

*'model'*.- npoints
number of points to be used in the fit. Default is NULL.

- model
Optional. Only when dist.name = 'Weibull'. A selection of the distribution model, two-parameters and three-parameters Weibull model ('2P' and '3P'). Default is 'all' and the model with the best AIC criterion is reported. Alternatively, just use dist.name = 'Weibull2P' or dist.name = 'Weibull3P'.

- maxiter
positive integer. Termination occurs when the number of iterations reaches maxiter. Default value: 1024

- tol
A positive numeric value specifying the tolerance level for the relative offset convergence criterion. Default value: 1e-12,

- ftol
non-negative numeric. Termination occurs when both the actual and predicted relative reductions in the sum of squares are at most ftol. Therefore, ftol measures the relative error desired in the sum of squares. Default value: 1e-12.

- ptol
non-negative numeric. Termination occurs when the relative error between two consecutive iterates is at most ptol. Therefore, ptol measures the relative error desired in the approximate solution. Default value: 1e-12.

- minFactor
A positive numeric value specifying the minimum step-size factor allowed on any step in the iteration. The increment is calculated with a Gauss-Newton algorithm and successively halved until the residual sum of squares has been decreased or until the step-size factor has been reduced below this limit. Default value: 10^-6.

- num.cores
The number of cores to use, i.e. at most how many child processes will be run simultaneously (see

`bpmapply`

function from BiocParallel package).- tasks
integer. 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-class`

from BiocParallel package).- maxfev
integer; termination occurs when the number of calls to fn has reached maxfev. Note that nls.lm sets the value of maxfev to 100*(length(par)

if maxfev = integer(), where par is the list or vector of parameters to be optimized.

- verbose
If TRUE, prints the function log to stdout

- ...
Additional fitting parameters.

Model table with coeficients and goodness-of-fit results: Adj.R.Square, deviance, AIC, R.Cross.val, and rho, as well as, the coefficient covariance matrix.

The algorithm prepares the information divergence variable to try fitting Weibull or generalized gamma distribution model to the data. If Weibull distribution is selected (default: 'Weibull'), function 'Weibull2P' first attempts fitting to the two-parameter Weibull CDF (Weibull2P). If Weibull2P did not fit, then the algorithm will try to fit Weibull3P. The Levenberg-Marquardt algorithm implemented in R package 'minpack.lm' is used to perform the nonlinear fit. Cross-validations for the nonlinear regressions (R.Cross.val) are performed in each methylome as described in reference (1-2). In addition, Stein's formula for adjusted R squared (rho) is used as an estimator of the average cross-validation predictive power (2).

If 'GGamma3P' is selected the call to function `fitGGammaDist`

permits the fitting to the three-parameter GGamma CDF ('GGamma3P'). The fit
to the four-parameter GGamma ('GGamma4P') is also available. GGamma
distribution are fitted using a modification of Levenberg-Marquardt algorithm
implemented in function 'nls.lm' from the 'minpack.lm' R package. Notice that
the fit to GGamma distribution is computationally time consuming (see
?fitGGammaDist for additional information).

Notice that all the estimated parameters/coefficients must be positive. Negative parameter values would happen and in that case the model must be dicarded.

R. Sanchez and S. A. Mackenzie, “Information Thermodynamics of Cytosine DNA Methylation,” PLoS One, vol. 11, no. 3, p. e0150427, Mar. 2016.

Stevens JP. Applied Multivariate Statistics for the Social Sciences. Fifth Edit. Routledge Academic; 2009.

```
## Load a dataset with Hellinger Divergence of methylation levels on it.
data(HD)
## The nonlinear fit based on three-parameter GGamma distribution
nlms2 <- nonlinearFitDist(HD, npoints = 100, dist.name = 'GGamma3P',
verbose = FALSE)
## Weilbull distribution is a particular case of GGamma.
nlms <- nonlinearFitDist(HD, npoints = 100, verbose = FALSE)
## The goodness-of-fit indicators AIC suggests that the best fitted model
## is obtained with GGamma distribution (in this example).
res <- mapply(function(m1,m2) as.numeric(c(Weibull = m1$AIC[1],
GGamma = m2$AIC[1])),
nlms, nlms2)
rownames(res) <-c('Weibull', 'GGamma')
res
#> C1 C2 C3 T1 T2 T3
#> Weibull -965.8982 -516.0584 -954.5536 -1053.215 -735.8882 -1306.127
#> GGamma -42990.4444 -42340.3319 -42911.1631 -55836.638 -59430.0469 -60454.309
## However, the Cross-validations correlation coefficient is saying that
## the Weibull distribution would be a little better probability
## predictor.
res <- mapply(function(m1,m2) as.numeric(c(Weibull = m1$R.Cross.val[1],
GGamma = m2$R.Cross.val[1])),
nlms, nlms2)
rownames(res) <-c('Weibull', 'GGamma')
res
#> C1 C2 C3 T1 T2 T3
#> Weibull 0.9989671 0.9968109 0.9995595 0.9997388 0.9997335 0.9998669
#> GGamma 0.9993984 0.9995083 0.9993989 0.9994740 0.9997990 0.9996741
```