derivIPEC: Derivative Calculation Function

In IPEC: Root Mean Square Curvature Calculation

Derivative Calculation Function

Description

Calculates the Jacobian and Hessian matrices of model parameters at a vector z.

Usage

derivIPEC(expr, theta, z, method = "Richardson", 
          method.args = list(eps = 1e-04, d = 0.11, 
          zero.tol = sqrt(.Machine$double.eps/7e-07), r = 6, v = 2, 
          show.details = FALSE), side = NULL)

Arguments

expr	A given parametric model
theta	A vector of parameters of the model
z	A vector where the derivatives are calculated
method	It is the same as the input argument of method of the hessian function in package numDeriv
method.args	It is the same as the input argument of method.args of the hessian function in package numDeriv
side	It is the same as the input argument of side of the jacobian function in package numDeriv

Details

The Hessian and Jacobian matrices are calculated at a vector z, which represents a value of a single independent variable or a combination of different values of multiple independent variables.

Value

Jacobian	The Jacobian matrix of parameters at z
Hessian	The Hessian matrix of parameters at z

Author(s)

Peijian Shi [e-mail], Peter M. Ridland [e-mail], David A. Ratkowsky [e-mail], Yang Li [e-mail].

References

Bates, D.M and Watts, D.G. (1988) Nonlinear Regression Analysis and its Applications. Wiley, New York. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1002/9780470316757")}

Ratkowsky, D.A. (1983) Nonlinear Regression Modeling: A Unified Practical Approach. Marcel Dekker, New York.

Ratkowsky, D.A. (1990) Handbook of Nonlinear Regression Models, Marcel Dekker, New York.

See Also

biasIPEC, skewIPEC, curvIPEC, hessian in package numDeriv, jacobian in package numDeriv

Examples

#### Example 1 #####################################################################################
# Define the Michaelis-Menten model
MM <- function(theta, x){
    theta[1]*x / ( theta[2] + x )    
}

par1 <- c(212.68490865, 0.06412421)
res1 <- derivIPEC(MM, theta=par1, z=0.02, method="Richardson",
            method.args=list(eps=1e-4, d=0.11, 
            zero.tol=sqrt(.Machine$double.eps/7e-7), r=6, v=2))
res1
####################################################################################################


#### Example 2 #####################################################################################
# Define the square root function of the Lobry-Rosso-Flandrois (LRF) model
sqrt.LRF <- function(P, x){
  ropt <- P[1]
  Topt <- P[2]
  Tmin <- P[3]
  Tmax <- P[4]
  fun0 <- function(z){
    z[z < Tmin] <- Tmin
    z[z > Tmax] <- Tmax
    return(z)
  }
  x <- fun0(x)
  if (Tmin >= Tmax | ropt <= 0 | Topt <= Tmin | Topt >= Tmax) 
    temp <- Inf
  if (Tmax > Tmin & ropt > 0 & Topt > Tmin & Topt < Tmax){
    temp <- sqrt( ropt*(x-Tmax)*(x-Tmin)^2/((Topt-Tmin)*((Topt-Tmin
      )*(x-Topt)-(Topt-Tmax)*(Topt+Tmin-2*x))) )  
  }
  return( temp )
}

myfun <- sqrt.LRF
par2  <- c(0.1382926, 33.4575663, 5.5841244, 38.8282021)
resu1 <- derivIPEC( myfun, theta=par2, z=15, method="Richardson", 
            method.args=list(eps=1e-4, d=0.11, 
            zero.tol=sqrt(.Machine$double.eps/7e-7), r=6, v=2) )
resu1
####################################################################################################


#### Example 3 #####################################################################################
# Weight of cut grass data (Pattinson 1981)
# References:
#   Clarke, G.P.Y. (1987) Approximate confidence limits for a parameter function in nonlinear 
#       regression. J. Am. Stat. Assoc. 82, 221-230.
#   Gebremariam, B. (2014) Is nonlinear regression throwing you a curve? 
#       New diagnostic and inference tools in the NLIN Procedure. Paper SAS384-2014.
#       http://support.sas.com/resources/papers/proceedings14/SAS384-2014.pdf
#   Pattinson, N.B. (1981) Dry Matter Intake: An Estimate of the Animal
#       Response to Herbage on Offer. unpublished M.Sc. thesis, University
#       of Natal, Pietermaritzburg, South Africa, Department of Grassland Science.

# 'x4' is the vector of weeks after commencement of grazing in a pasture
# 'y4' is the vector of weight of cut grass from 10 randomly sited quadrants

x4 <- 1:13
y4 <- c(3.183, 3.059, 2.871, 2.622, 2.541, 2.184, 2.110, 2.075, 2.018, 1.903, 1.770, 1.762, 1.550)

# Define the third case of Mitscherlich equation
MitC <- function(P3, x){
    theta1 <- P3[1]
    beta2  <- P3[2]
    beta3  <- P3[3]
    x1     <- 1
    x2     <- 13
    theta2 <- (beta3 - beta2)/(exp(theta1*x2)-exp(theta1*x1))
    theta3 <- beta2/(1-exp(theta1*(x1-x2))) - beta3/(exp(theta1*(x2-x1))-1)
    theta3 + theta2*exp(theta1*x)
}

ini.val6 <- c(-0.15, 2.52, 1.09)
RES0     <- fitIPEC( MitC, x=x4, y=y4, ini.val=ini.val6, xlim=NULL, ylim=NULL,  
                     fig.opt=TRUE, control=list(trace=FALSE, reltol=1e-20, maxit=50000) )
parC     <- RES0$par
parC
RES1     <- derivIPEC( MitC, theta=parC, z=2, method="Richardson", 
                       method.args=list(eps=1e-4, d=0.11, 
                       zero.tol=sqrt(.Machine$double.eps/7e-7), r=6, v=2) )
RES1
#################################################################################################

IPEC documentation built on Nov. 2, 2023, 6:14 p.m.