getDerivative: Combined Derivative Computation of beta's.
In paularaissa/distStatsClient: Distributed Multiple Imputations

Description Usage Arguments Details Value Dependencies Author(s) See Also

Computes the β for each interation.

1	getDerivative(formula, n.vars, family, learningrate, dif, datasources)

`formula`	a character that can be coerced to an object of class `formula`. It is a symbolic description of the model to be fitted.
`n.vars`	a numeric, the number of study variables.
`family`	a string character with the name of the error distribution and link function to be used in the analysis. If `family` is set to 'binomial' it defines the link function as logit, and likelihood as binomial. If `family` is set to 'poisson' it defines the link function as log, and likelihood as poisson.
`learningrate`	a numeric, controls how much we are adjusting the regression model.
`dif`	a numeric, controls the learning convergence.
`datasources`	a list of opal object(s) obtained after login in to opal servers; these objects hold also the data assign to R, as `data frame`, from opal datasources.

The straightforward way to compute the b coefficients is the Newton's method. Suppose that there is a valued function y = f(b). The problem is find the value b[k] such that f(b[k]) = 0. Starting with an initial value for b[0], the Taylor expansion of f can be done around b[0]:

f(b[0] + β) =~ f(b[0]) + f'(b[0]) * β

The f' is a matrix, a Jacobean if first derivative of f with respect to b. In this equation, setting the left side as zero, the β can be solved as

β[0] = -[f'(β[0])]^(-1) * f(b[0])

So the update of estimated for b is:

b[1] = b[0] + β[0]

and iterate until convergence.

Returns a list with the following components:

`call`	the model formula.
`coefficients`	a vector of linear regression coefficients.
`xtxw`	a data matrix, the Hessian matrix.
`xtyp`	a data matrix, that integrates the computation of derivatives.