InequalityConstr: Inequality constraints and jacobian
In SGB: Simplicial Generalized Beta Regression

Description Usage Arguments Details Value Examples

Setting of inequality constraints on shape parameters.
hin.SGB sets inequality constraints on the shape parameters in a SGB regression.
hin.SGB.jac defines the corresponding Jacobian.

1 2	hin.SGB(x, d, u, bound, ...) hin.SGB.jac(x, d, u, ...)

`x`	vector of parameters (`shape1`, `coefi`, `shape2`) where `shape1` is the overall shape, `coefi` is the vector of regression coefficients (see `initpar.SGB`) and `shape2` the vector of D Dirichlet shape parameters.
`d`	data matrix of explanatory variables (without constant vector) (N \times m); N: sample size, m: number of auxiliary variables.
`u`	data matrix of compositions (independent variables) (N \times D); D: number of parts.
`bound`	the estimates of shapes are constrained by `shape1*shape2[i] > bound, i=1,...,D.` By default `bound = 2.1`.
`...`	not used.

These functions are invoked internally by regSGB with bound specified by the user.
shape1 is constrained to be larger than 0.1, in order to avoid numerical problems and shape2 must be positive.
Moments of ratios of parts only exist up to bound. Thus bound = 2.1 guarantees the existence of variances of ratios of parts.

hin.SGB : vector of length D+1 with the current value of c(shape1-0.1,shape1*shape2-bound). It should be non-negative at convergence of the regression algorithm.
hin.SGB.jac : corresponding jacobian matrix of dimensions (D+1) \times length(x).

## Parameter vector for a 3 parts composition with one explanatory variable (+ intercept):
x <- c(1,3.2,0.04,0.05,6,7:9)
bound <- 2.1
u <- t(c(0.1,0.5,0.4))  # only used to compute the number of parts.
hin.SGB(x, d, u, bound)
# = c(shape1-0.1, shape1*shape2-bound,shape2)
# all must be positive.