InitialParameters: Initial parameters estimates and comparison
In SGB: Simplicial Generalized Beta Regression

Description Usage Arguments Details Value References Examples

initpar.SGB computes an initial vector of parameters.
condshape2 computes the shape2 parameters by the same method as initpar.SGB, but from an arbitrary set of parameters (shape1,coefi) (e.g. the result of a SGB regression fit). These approximations are compared with the shape2 estimates.
compushape2 is internally called by initpar.SGB and condshape2. It computes shape2 parameters in function of shape1 and given regression parameters coefi.

1
2
3

initpar.SGB(d, u, V, weight = rep(1, dim(u)[1]), shape1 = 1, Mean2 = TRUE) 
condshape2(x,d,u,V)
compushape2(shape1, coefi, d, u, V)

`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
`V`	full rank transformation of log(parts) into log-ratios, matrix D \times (D-1)
`weight`	vector of length n; positive observation weights, default `rep(1,n)`. Should be scaled to sum to n.
`shape1`	positive number, overall shape parameter
`Mean2`	logical, if TRUE (default), the computed `shape2` parameters are each replaced by their average.
`coefi`	vector of regression coefficients of length (m+1)(D-1), resp. D-1* constants, then D-1 coef. of the 1st expl. variable,..., D-1 coef. of the m-th expl. variable
`x`	fitted SGB regression parameters, see `regSGB`.

The main function here is initpar.SGB. The initial value of shape1 must be specified by the user; by default, it takes the value 1. In the initial regression model, each column of log(u) % * % V is regressed by OLS on the columns of d. coefi is the vector of regression parameters, first the D-1 terms associated with the first explanatory variable in d, and so on similarily for each explanatory variable. The initial scale compositions are computed by back-transforming the predicted values to the simplex and used to compute the vector z=C[(u/scale)^{shape1}], where C[.] is the closure operation. Wicker et al. (2008), see also Ng et al. (2011) p.74-75, describe a procedure to find initial values for the shape parameters in a Dirichlet distribution. Their method is used on the (approximate) Dirichlet vector z.

initpar.SGB:
vector of length (1+ (D-1)*(m+1) + D) containing initial values for (shape1,coefi,shape2).
condshape2:
list with two components: 1. title and 2. data-frame with 2 columns: fitted shape2 and Wicker's approximation.

Wicker, N., J. Muller, R. K. R. Kalathur, and O. Poch (2008). A maximum likelihood approximation method for Dirichlet's parameter estimation. Computational Statistics & Data Analysis 52 (3), 1315-1322.

Kai Wang Ng, Guo-Liang Tian, Man-Lai Tang (2011). Dirichlet and Related Distributions: Theory, Methods and Applications. Wiley Series in Probability and Statistics.

## Explanatory variable
da <- data.frame(l.depth=log(arc[["depth"]]))  
damat <- as.matrix(da)
## Compositions
ua <- arc[,1:3]

## alr transforms
Va <- matrix(c(1,0,-1,0,1,-1),nrow=3)   
colnames(Va) <- c("alr1","alr2")
Va

## Initial values
initpar.SGB(damat,ua,Va)
initpar.SGB(damat,ua,Va,Mean2=FALSE)

## Conditional shape2 values; same as parameters computed with initpar
condshape2(initpar.SGB(damat,ua,Va,Mean2=FALSE),damat,ua,Va)

## Comparison with fitted parameters
oa <- regSGB(damat, as.matrix(ua), Va)
condshape2(oa[["par"]],damat,ua,Va)