nlreg (version 1.2-2.2)

Dvar: Differentiate the Variance Function of a Nonlinear Model

Description

Calculates the gradient and Hessian of the variance function of a nonlinear heteroscedastic model.

Usage

Dvar(nlregObj, hessian = TRUE)

Arguments

nlregObj

a nonlinear heteroscedastic model fit as obtained from a call to nlreg.

hessian

logical value indicating whether the Hessian should be computed. The default is TRUE.

Value

a function whose arguments are named according to the parameters of the nonlinear model nlregObj. When evaluated, it returns the value of the variance function along with attributes called gradient and hessian, the latter if requested. These are the gradient and Hessian of the variance function with respect to the model parameters.

Details

The variance function is differentiated with respect to the variance parameters specified in the varPar component of the nlregObj object and, if the variance function depends on them, with respect to the regression coefficients specified in the coef component. The returned function definition includes all parameters. When evaluated, it implicitly refers to the data to whom the nlreg object was fitted and which must be on the search list. The gradient and Hessian are calculated for each data point: the gradient attribute is a \(n\times p\) matrix, and the hessian attribute is a \(n\times p\times p\) array, where \(n\) and \(p\) are respectively the number of data points and the number of regression coefficients.

References

Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988) The New S Language: A Programming Environment for Data Analysis and Graphics. London: Chapman \& Hall. Section 9.6.

Brazzale, A. R. (2000) Practical Small-Sample Parametric Inference. Ph.D. Thesis N. 2230, Department of Mathematics, Swiss Federal Institute of Technology Lausanne.

See Also

Dmean, nlreg.object, deriv3, D

Examples

# NOT RUN {
library(boot)
data(calcium)
calcium.nl <- nlreg( cal ~ b0*(1-exp(-b1*time)), 
                     start = c(b0 = 4, b1 = 0.1), data = calcium )
Dvar( calcium.nl )
##function (b0, b1, logs)
##{
##    .expr1 <- exp(logs)
##    .value <- .expr1
##    .grad <- array(0, c(length(.value), 1), list(NULL, c("logs")))
##    .hessian <- array(0, c(length(.value), 1, 1), list(NULL,
##        c("logs"), c("logs")))
##    .grad[, "logs"] <- .expr1
##    .hessian[, "logs", "logs"] <- .expr1
##    attr(.value, "gradient") <- .grad
##    attr(.value, "hessian") <- .hessian
##    .value
##}
##
attach( calcium )
calcium.vd <- Dvar( calcium.nl )
param( calcium.nl )
##        b0         b1       logs
## 4.3093653  0.2084780 -1.2856765
##
attr( calcium.vd( 4.31, 0.208, -1.29 ), "gradient" )
##          logs
##[1,] 0.2752708
##
calcium.nl <- update( calcium.nl, weights = ~ ( 1+time^g )^2, 
                      start = c(b0 = 4, b1 = 0.1, g = 1))
Dvar( calcium.nl )
##function (b0, b1, g, logs) 
##{
##    .expr1 <- time^g
##    .expr2 <- 1 + .expr1
##    .expr4 <- exp(logs)
##    .expr5 <- .expr2^2 * .expr4
##    .expr6 <- log(time)
##    .expr7 <- .expr1 * .expr6
##    .expr10 <- 2 * (.expr7 * .expr2) * .expr4
##    .value <- .expr5
##    .grad <- array(0, c(length(.value), 2), list(NULL, c("g",
##        "logs")))
##    .hessian <- array(0, c(length(.value), 2, 2), list(NULL,
##        c("g", "logs"), c("g", "logs")))
##    .grad[, "g"] <- .expr10
##    .hessian[, "g", "g"] <- 2 * (.expr7 * .expr6 * .expr2 + .expr7 *
##        .expr7) * .expr4
##    .hessian[, "g", "logs"] <- .hessian[, "logs", "g"] <- .expr10
##    .grad[, "logs"] <- .expr5
##    .hessian[, "logs", "logs"] <- .expr5
##    attr(.value, "gradient") <- .grad
##    attr(.value, "hessian") <- .hessian
##    .value
##}
##
calcium.vd <- Dvar( calcium.nl )
param( calcium.nl )
##        b0         b1          g       logs 
## 4.3160408  0.2075937  0.3300134 -3.3447585
##
attr( calcium.vd(4.32, 0.208, 0.600, -2.66 ), "gradient" )
##                g      logs
## [1,] -0.11203422 0.1834220
## [2,] -0.11203422 0.1834220
## [3,] -0.11203422 0.1834220
## [4,]  0.09324687 0.3295266
## \dots
##
detach()
# }