IPEC-package: Root Mean Square Curvature Calculation
In IPEC: Root Mean Square Curvature Calculation
IPEC-package R Documentation
Root Mean Square Curvature Calculation
Description
Calculates the RMS intrinsic and parameter-effects curvatures of a nonlinear regression model.
The curvatures are global measures of assessing whether a model/data set combination
is close-to-linear or not. See Bates and Watts (1980) and Ratkowsky and Reddy (2017) for details.
Details
The DESCRIPTION file:
This package was not yet installed at build time.
Index: This package was not yet installed at build time.
Note
We are deeply thankful to Drs. Paul Gilbert and Jinlong Zhang for their invaluable help
during creating this package. We also thank Dr. Kurt Hornik for his technical guidance in
updating the package.
Author(s)
Peijian Shi [aut, cre], Peter Ridland [aut], David A. Ratkowsky [aut], Yang Li [aut]
Maintainer: Peijian Shi <pjshi@njfu.edu.cn>
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.
Ratkowsky, D.A. & Reddy, G.V.P. (2017) Empirical model with excellent statistical properties for describing temperature-dependent
developmental rates of insects and mites. Ann. Entomol. Soc. Am. 110, 302-309. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1093/aesa/saw098")}
See Also
hessian in package numDeriv, jacobian
in package numDeriv, rms.curv in package MASS
Examples
#### Example 1 ##################################################################################
graphics.off()
# The velocity of the reaction (counts/min^2) under different substrate concentrations
# in parts per million (ppm) (Page 269 of Bates and Watts 1988)
x1 <- c(0.02, 0.02, 0.06, 0.06, 0.11, 0.11, 0.22, 0.22, 0.56, 0.56, 1.10, 1.10)
y1 <- c(76, 47, 97, 107, 123, 139, 159, 152, 191, 201, 207, 200)
# Define the Michaelis-Menten model
MM <- function(theta, x){
theta[1]*x / ( theta[2] + x )
}
res0 <- fitIPEC( MM, x=x1, y=y1, ini.val=c(200, 0.05),
xlim=c( 0, 1.5 ), ylim=c(0, 250), fig.opt=TRUE )
par1 <- res0$par
par1
res1 <- derivIPEC( MM, theta=par1, z=x1[1], method="Richardson",
method.args=list(eps=1e-4, d=0.11,
zero.tol=sqrt(.Machine$double.eps/7e-7), r=6, v=2) )
res1
# To calculate curvatures
res2 <- curvIPEC( MM, theta=par1, x=x1, y=y1, alpha=0.05, method="Richardson",
method.args=list(eps=1e-4, d=0.11,
zero.tol=sqrt(.Machine$double.eps/7e-7), r=6, v=2) )
res2
# To calculate bias
res3 <- biasIPEC(MM, theta=par1, x=x1, y=y1, tol= 1e-20)
res3
set.seed(123)
res4 <- bootIPEC( MM, x=x1, y=y1, ini.val=par1,
control=list(reltol=1e-20, maxit=40000),
nboot=2000, CI=0.95, fig.opt=TRUE )
res4
set.seed(NULL)
# To calculate skewness
res5 <- skewIPEC(MM, theta=par1, x=x1, y=y1, tol= 1e-20)
res5
#################################################################################################
#### Example 2 ##################################################################################
graphics.off()
# Development data of female pupae of cotton bollworm (Wu et al. 2009)
# References:
# Ratkowsky, D.A. and Reddy, G.V.P. (2017) Empirical model with excellent statistical
# properties for describing temperature-dependent developmental rates of insects
# and mites. Ann. Entomol. Soc. Am. 110, 302-309.
# Wu, K., Gong, P. and Ruan, Y. (2009) Estimating developmental rates of
# Helicoverpa armigera (Lepidoptera: Noctuidae) pupae at constant and
# alternating temperature by nonlinear models. Acta Entomol. Sin. 52, 640-650.
# 'x2' is the vector of temperature (in degrees Celsius)
# 'D2' is the vector of developmental duration (in d)
# 'y2' is the vector of the square root of developmental rate (in 1/d)
x2 <- seq(15, 37, by=1)
D2 <- c(41.24,37.16,32.47,26.22,22.71,19.01,16.79,15.63,14.27,12.48,
11.3,10.56,9.69,9.14,8.24,8.02,7.43,7.27,7.35,7.49,7.63,7.9,10.03)
y2 <- 1/D2
y2 <- sqrt( y2 )
ini.val1 <- c(0.14, 30, 10, 40)
# 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
xlab1 <- expression( paste("Temperature (", degree, "C)", sep="" ) )
ylab1 <- expression( paste("Developmental rate"^{1/2}, " (", d^{"-1"}, ")", sep="") )
resu0 <- fitIPEC( myfun, x=x2, y=y2, ini.val=ini.val1, xlim=NULL, ylim=NULL,
xlab=xlab1, ylab=ylab1, fig.opt=TRUE,
control=list(trace=FALSE, reltol=1e-20, maxit=50000) )
par2 <- resu0$par
par2
resu1 <- derivIPEC( myfun, theta=par2, z=x2[1], method="Richardson",
method.args=list(eps=1e-4, d=0.11,
zero.tol=sqrt(.Machine$double.eps/7e-7), r=6, v=2) )
resu1
# To calculate curvatures
resu2 <- curvIPEC( myfun, theta=par2, x=x2, y=y2, alpha=0.05, method="Richardson",
method.args=list(eps=1e-4, d=0.11,
zero.tol=sqrt(.Machine$double.eps/7e-7), r=6, v=2) )
resu2
# To calculate bias
resu3 <- biasIPEC(myfun, theta=par2, x=x2, y=y2, tol= 1e-20)
resu3
set.seed(123)
resu4 <- bootIPEC( myfun, x=x2, y=y2, ini.val=ini.val1,
nboot=2000, CI=0.95, fig.opt=TRUE )
resu4
set.seed(NULL)
# To calculate skewness
resu5 <- skewIPEC(myfun, theta=par2, x=x2, y=y2, tol= 1e-20)
resu5
#################################################################################################
#### Example 3 ##################################################################################
graphics.off()
# Height growth data of four species of bamboo (Gramineae: Bambusoideae)
# Reference(s):
# Shi, P., Fan, M., Ratkowsky, D.A., Huang, J., Wu, H., Chen, L., Fang, S. and
# Zhang, C. (2017) Comparison of two ontogenetic growth equations for animals and plants.
# Ecol. Model. 349, 1-10.
data(shoots)
# Choose a species
# 1: Phyllostachys iridescens; 2: Phyllostachys mannii;
# 3: Pleioblastus maculatus; 4: Sinobambusa tootsik.
# 'x3' is the vector of the investigation times from a specific starting time of growth
# 'y3' is the vector of the aboveground height values of bamboo shoots at 'x3'
ind <- 4
x3 <- shoots$x[shoots$Code == ind]
y3 <- shoots$y[shoots$Code == ind]
# Define the beta sigmoid model (bsm)
bsm <- function(P, x){
P <- cbind(P)
if(length(P) !=4 ) {stop("The number of parameters should be 4!")}
ropt <- P[1]
topt <- P[2]
tmin <- P[3]
tmax <- P[4]
tailor.fun <- function(x){
x[x < tmin] <- tmin
x[x > tmax] <- tmax
return(x)
}
x <- tailor.fun(x)
ropt*(x-tmin)*(x-2*tmax+topt)/(topt+tmin-2*tmax)*(
(x-tmin)/(topt-tmin) )^((topt-tmin)/(tmax-topt))
}
# Define the simplified beta sigmoid model (simp.bsm)
simp.bsm <- function(P, x, tmin=0){
P <- cbind(P)
ropt <- P[1]
topt <- P[2]
tmax <- P[3]
tailor.fun <- function(x){
x[x < tmin] <- tmin
x[x > tmax] <- tmax
return(x)
}
x <- tailor.fun(x)
ropt*(x-tmin)*(x-2*tmax+topt)/(topt+tmin-2*tmax)*
((x-tmin)/(topt-tmin))^((topt-tmin)/(tmax-topt))
}
# For the original beta sigmoid model
ini.val2 <- c(40, 30, 5, 50)
xlab2 <- "Time (d)"
ylab2 <- "Height (cm)"
re0 <- fitIPEC( bsm, x=x3, y=y3, ini.val=ini.val2, xlim=NULL, ylim=NULL,
xlab=xlab2, ylab=ylab2, fig.opt=TRUE,
control=list(trace=FALSE, reltol=1e-20, maxit=50000) )
par3 <- re0$par
par3
re1 <- derivIPEC( bsm, theta=par3, x3[15], method="Richardson",
method.args=list(eps=1e-4, d=0.11, zero.tol=
sqrt(.Machine$double.eps/7e-7), r=6, v=2) )
re1
re2 <- curvIPEC( bsm, theta=par3, x=x3, y=y3, alpha=0.05, method="Richardson",
method.args=list(eps=1e-4, d=0.11, zero.tol=
sqrt(.Machine$double.eps/7e-7), r=6, v=2) )
re2
re3 <- biasIPEC( bsm, theta=par3, x=x3, y=y3, tol= 1e-20 )
re3
re4 <- bootIPEC( bsm, x=x3, y=y3, ini.val=ini.val2,
control=list(trace=FALSE, reltol=1e-20, maxit=50000),
nboot=2000, CI=0.95, fig.opt=TRUE, fold=3.5 )
re4
re5 <- skewIPEC( bsm, theta=par3, x=x3, y=y3, tol= 1e-20 )
re5
# For the simplified beta sigmoid model
# (in comparison with the original beta sigmoid model)
ini.val7 <- c(40, 30, 50)
RESU0 <- fitIPEC( simp.bsm, x=x3, y=y3, ini.val=ini.val7,
xlim=NULL, ylim=NULL, xlab=xlab2, ylab=ylab2,
fig.opt=TRUE, control=list(trace=FALSE, reltol=1e-20, maxit=50000) )
par7 <- RESU0$par
par7
RESU1 <- derivIPEC( simp.bsm, theta=par7, x3[15], method="Richardson",
method.args=list(eps=1e-4, d=0.11,
zero.tol=sqrt(.Machine$double.eps/7e-7), r=6, v=2) )
RESU1
RESU2 <- curvIPEC( simp.bsm, theta=par7, x=x3, y=y3, alpha=0.05, method="Richardson",
method.args=list(eps=1e-4, d=0.11,
zero.tol=sqrt(.Machine$double.eps/7e-7), r=6, v=2) )
RESU2
RESU3 <- biasIPEC( simp.bsm, theta=par7, x=x3, y=y3, tol= 1e-20 )
RESU3
set.seed(123)
RESU4 <- bootIPEC( simp.bsm, x=x3, y=y3, ini.val=ini.val7,
control=list(trace=FALSE, reltol=1e-20, maxit=50000),
nboot=2000, CI=0.95, fig.opt=TRUE, fold=3.5 )
RESU4
set.seed(NULL)
RESU5 <- skewIPEC( simp.bsm, theta=par7, x=x3, y=y3, tol= 1e-20 )
RESU5
##################################################################################################
#### Example 4 ###################################################################################
# Data on biochemical oxygen demand (BOD; Marske 1967)
# References:
# Pages 56, 255 and 271 in Bates and Watts (1988)
# Carr, N.L. (1960) Kinetics of catalytic isomerization of n-pentane. Ind. Eng. Chem.
# 52, 391-396.
graphics.off()
data(isom)
Y <- isom[,1]
X <- isom[,2:4]
# There are three independent variables saved in matrix 'X' and one response variable (Y)
# The first column of 'X' is the vector of partial pressure of hydrogen
# The second column of 'X' is the vector of partial pressure of n-pentane
# The third column of 'X' is the vector of partial pressure of isopentane
# Y is the vector of experimental reaction rate (in 1/hr)
isom.fun <- function(theta, x){
x1 <- x[,1]
x2 <- x[,2]
x3 <- x[,3]
theta1 <- theta[1]
theta2 <- theta[2]
theta3 <- theta[3]
theta4 <- theta[4]
theta1*theta3*(x2-x3/1.632) / ( 1 + theta2*x1 + theta3*x2 + theta4*x3 )
}
ini.val8 <- c(35, 0.1, 0.05, 0.2)
cons1 <- fitIPEC( isom.fun, x=X, y=Y, ini.val=ini.val8, control=list(
trace=FALSE, reltol=1e-20, maxit=50000) )
par8 <- cons1$par
cons2 <- curvIPEC( isom.fun, theta=par8, x=X, y=Y, alpha=0.05, method="Richardson",
method.args=list(eps=1e-4, d=0.11,
zero.tol=sqrt(.Machine$double.eps/7e-7), r=6, v=2))
cons2
cons3 <- biasIPEC( isom.fun, theta=par8, x=X, y=Y, tol= 1e-20 )
cons3
set.seed(123)
cons4 <- bootIPEC( isom.fun, x=X, y=Y, ini.val=ini.val8,
control=list(trace=FALSE, reltol=1e-20, maxit=50000),
nboot=2000, CI=0.95, fig.opt=TRUE, fold=10000 )
cons4
set.seed(NULL)
cons5 <- skewIPEC( isom.fun, theta=par8, x=X, y=Y, tol= 1e-20 )
cons5
##################################################################################################
IPEC documentation built on Nov. 2, 2023, 6:14 p.m.
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| IPEC-package | R Documentation |
Root Mean Square Curvature Calculation
Description
Calculates the RMS intrinsic and parameter-effects curvatures of a nonlinear regression model. The curvatures are global measures of assessing whether a model/data set combination is close-to-linear or not. See Bates and Watts (1980) and Ratkowsky and Reddy (2017) for details.
Details
The DESCRIPTION file:
This package was not yet installed at build time.
Index: This package was not yet installed at build time.
Note
We are deeply thankful to Drs. Paul Gilbert and Jinlong Zhang for their invaluable help during creating this package. We also thank Dr. Kurt Hornik for his technical guidance in updating the package.
Author(s)
Peijian Shi [aut, cre], Peter Ridland [aut], David A. Ratkowsky [aut], Yang Li [aut]
Maintainer: Peijian Shi <pjshi@njfu.edu.cn>
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.
Ratkowsky, D.A. & Reddy, G.V.P. (2017) Empirical model with excellent statistical properties for describing temperature-dependent
developmental rates of insects and mites. Ann. Entomol. Soc. Am. 110, 302-309. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1093/aesa/saw098")}
See Also
hessian in package numDeriv, jacobian
in package numDeriv, rms.curv in package MASS
Examples
#### Example 1 ##################################################################################
graphics.off()
# The velocity of the reaction (counts/min^2) under different substrate concentrations
# in parts per million (ppm) (Page 269 of Bates and Watts 1988)
x1 <- c(0.02, 0.02, 0.06, 0.06, 0.11, 0.11, 0.22, 0.22, 0.56, 0.56, 1.10, 1.10)
y1 <- c(76, 47, 97, 107, 123, 139, 159, 152, 191, 201, 207, 200)
# Define the Michaelis-Menten model
MM <- function(theta, x){
theta[1]*x / ( theta[2] + x )
}
res0 <- fitIPEC( MM, x=x1, y=y1, ini.val=c(200, 0.05),
xlim=c( 0, 1.5 ), ylim=c(0, 250), fig.opt=TRUE )
par1 <- res0$par
par1
res1 <- derivIPEC( MM, theta=par1, z=x1[1], method="Richardson",
method.args=list(eps=1e-4, d=0.11,
zero.tol=sqrt(.Machine$double.eps/7e-7), r=6, v=2) )
res1
# To calculate curvatures
res2 <- curvIPEC( MM, theta=par1, x=x1, y=y1, alpha=0.05, method="Richardson",
method.args=list(eps=1e-4, d=0.11,
zero.tol=sqrt(.Machine$double.eps/7e-7), r=6, v=2) )
res2
# To calculate bias
res3 <- biasIPEC(MM, theta=par1, x=x1, y=y1, tol= 1e-20)
res3
set.seed(123)
res4 <- bootIPEC( MM, x=x1, y=y1, ini.val=par1,
control=list(reltol=1e-20, maxit=40000),
nboot=2000, CI=0.95, fig.opt=TRUE )
res4
set.seed(NULL)
# To calculate skewness
res5 <- skewIPEC(MM, theta=par1, x=x1, y=y1, tol= 1e-20)
res5
#################################################################################################
#### Example 2 ##################################################################################
graphics.off()
# Development data of female pupae of cotton bollworm (Wu et al. 2009)
# References:
# Ratkowsky, D.A. and Reddy, G.V.P. (2017) Empirical model with excellent statistical
# properties for describing temperature-dependent developmental rates of insects
# and mites. Ann. Entomol. Soc. Am. 110, 302-309.
# Wu, K., Gong, P. and Ruan, Y. (2009) Estimating developmental rates of
# Helicoverpa armigera (Lepidoptera: Noctuidae) pupae at constant and
# alternating temperature by nonlinear models. Acta Entomol. Sin. 52, 640-650.
# 'x2' is the vector of temperature (in degrees Celsius)
# 'D2' is the vector of developmental duration (in d)
# 'y2' is the vector of the square root of developmental rate (in 1/d)
x2 <- seq(15, 37, by=1)
D2 <- c(41.24,37.16,32.47,26.22,22.71,19.01,16.79,15.63,14.27,12.48,
11.3,10.56,9.69,9.14,8.24,8.02,7.43,7.27,7.35,7.49,7.63,7.9,10.03)
y2 <- 1/D2
y2 <- sqrt( y2 )
ini.val1 <- c(0.14, 30, 10, 40)
# 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
xlab1 <- expression( paste("Temperature (", degree, "C)", sep="" ) )
ylab1 <- expression( paste("Developmental rate"^{1/2}, " (", d^{"-1"}, ")", sep="") )
resu0 <- fitIPEC( myfun, x=x2, y=y2, ini.val=ini.val1, xlim=NULL, ylim=NULL,
xlab=xlab1, ylab=ylab1, fig.opt=TRUE,
control=list(trace=FALSE, reltol=1e-20, maxit=50000) )
par2 <- resu0$par
par2
resu1 <- derivIPEC( myfun, theta=par2, z=x2[1], method="Richardson",
method.args=list(eps=1e-4, d=0.11,
zero.tol=sqrt(.Machine$double.eps/7e-7), r=6, v=2) )
resu1
# To calculate curvatures
resu2 <- curvIPEC( myfun, theta=par2, x=x2, y=y2, alpha=0.05, method="Richardson",
method.args=list(eps=1e-4, d=0.11,
zero.tol=sqrt(.Machine$double.eps/7e-7), r=6, v=2) )
resu2
# To calculate bias
resu3 <- biasIPEC(myfun, theta=par2, x=x2, y=y2, tol= 1e-20)
resu3
set.seed(123)
resu4 <- bootIPEC( myfun, x=x2, y=y2, ini.val=ini.val1,
nboot=2000, CI=0.95, fig.opt=TRUE )
resu4
set.seed(NULL)
# To calculate skewness
resu5 <- skewIPEC(myfun, theta=par2, x=x2, y=y2, tol= 1e-20)
resu5
#################################################################################################
#### Example 3 ##################################################################################
graphics.off()
# Height growth data of four species of bamboo (Gramineae: Bambusoideae)
# Reference(s):
# Shi, P., Fan, M., Ratkowsky, D.A., Huang, J., Wu, H., Chen, L., Fang, S. and
# Zhang, C. (2017) Comparison of two ontogenetic growth equations for animals and plants.
# Ecol. Model. 349, 1-10.
data(shoots)
# Choose a species
# 1: Phyllostachys iridescens; 2: Phyllostachys mannii;
# 3: Pleioblastus maculatus; 4: Sinobambusa tootsik.
# 'x3' is the vector of the investigation times from a specific starting time of growth
# 'y3' is the vector of the aboveground height values of bamboo shoots at 'x3'
ind <- 4
x3 <- shoots$x[shoots$Code == ind]
y3 <- shoots$y[shoots$Code == ind]
# Define the beta sigmoid model (bsm)
bsm <- function(P, x){
P <- cbind(P)
if(length(P) !=4 ) {stop("The number of parameters should be 4!")}
ropt <- P[1]
topt <- P[2]
tmin <- P[3]
tmax <- P[4]
tailor.fun <- function(x){
x[x < tmin] <- tmin
x[x > tmax] <- tmax
return(x)
}
x <- tailor.fun(x)
ropt*(x-tmin)*(x-2*tmax+topt)/(topt+tmin-2*tmax)*(
(x-tmin)/(topt-tmin) )^((topt-tmin)/(tmax-topt))
}
# Define the simplified beta sigmoid model (simp.bsm)
simp.bsm <- function(P, x, tmin=0){
P <- cbind(P)
ropt <- P[1]
topt <- P[2]
tmax <- P[3]
tailor.fun <- function(x){
x[x < tmin] <- tmin
x[x > tmax] <- tmax
return(x)
}
x <- tailor.fun(x)
ropt*(x-tmin)*(x-2*tmax+topt)/(topt+tmin-2*tmax)*
((x-tmin)/(topt-tmin))^((topt-tmin)/(tmax-topt))
}
# For the original beta sigmoid model
ini.val2 <- c(40, 30, 5, 50)
xlab2 <- "Time (d)"
ylab2 <- "Height (cm)"
re0 <- fitIPEC( bsm, x=x3, y=y3, ini.val=ini.val2, xlim=NULL, ylim=NULL,
xlab=xlab2, ylab=ylab2, fig.opt=TRUE,
control=list(trace=FALSE, reltol=1e-20, maxit=50000) )
par3 <- re0$par
par3
re1 <- derivIPEC( bsm, theta=par3, x3[15], method="Richardson",
method.args=list(eps=1e-4, d=0.11, zero.tol=
sqrt(.Machine$double.eps/7e-7), r=6, v=2) )
re1
re2 <- curvIPEC( bsm, theta=par3, x=x3, y=y3, alpha=0.05, method="Richardson",
method.args=list(eps=1e-4, d=0.11, zero.tol=
sqrt(.Machine$double.eps/7e-7), r=6, v=2) )
re2
re3 <- biasIPEC( bsm, theta=par3, x=x3, y=y3, tol= 1e-20 )
re3
re4 <- bootIPEC( bsm, x=x3, y=y3, ini.val=ini.val2,
control=list(trace=FALSE, reltol=1e-20, maxit=50000),
nboot=2000, CI=0.95, fig.opt=TRUE, fold=3.5 )
re4
re5 <- skewIPEC( bsm, theta=par3, x=x3, y=y3, tol= 1e-20 )
re5
# For the simplified beta sigmoid model
# (in comparison with the original beta sigmoid model)
ini.val7 <- c(40, 30, 50)
RESU0 <- fitIPEC( simp.bsm, x=x3, y=y3, ini.val=ini.val7,
xlim=NULL, ylim=NULL, xlab=xlab2, ylab=ylab2,
fig.opt=TRUE, control=list(trace=FALSE, reltol=1e-20, maxit=50000) )
par7 <- RESU0$par
par7
RESU1 <- derivIPEC( simp.bsm, theta=par7, x3[15], method="Richardson",
method.args=list(eps=1e-4, d=0.11,
zero.tol=sqrt(.Machine$double.eps/7e-7), r=6, v=2) )
RESU1
RESU2 <- curvIPEC( simp.bsm, theta=par7, x=x3, y=y3, alpha=0.05, method="Richardson",
method.args=list(eps=1e-4, d=0.11,
zero.tol=sqrt(.Machine$double.eps/7e-7), r=6, v=2) )
RESU2
RESU3 <- biasIPEC( simp.bsm, theta=par7, x=x3, y=y3, tol= 1e-20 )
RESU3
set.seed(123)
RESU4 <- bootIPEC( simp.bsm, x=x3, y=y3, ini.val=ini.val7,
control=list(trace=FALSE, reltol=1e-20, maxit=50000),
nboot=2000, CI=0.95, fig.opt=TRUE, fold=3.5 )
RESU4
set.seed(NULL)
RESU5 <- skewIPEC( simp.bsm, theta=par7, x=x3, y=y3, tol= 1e-20 )
RESU5
##################################################################################################
#### Example 4 ###################################################################################
# Data on biochemical oxygen demand (BOD; Marske 1967)
# References:
# Pages 56, 255 and 271 in Bates and Watts (1988)
# Carr, N.L. (1960) Kinetics of catalytic isomerization of n-pentane. Ind. Eng. Chem.
# 52, 391-396.
graphics.off()
data(isom)
Y <- isom[,1]
X <- isom[,2:4]
# There are three independent variables saved in matrix 'X' and one response variable (Y)
# The first column of 'X' is the vector of partial pressure of hydrogen
# The second column of 'X' is the vector of partial pressure of n-pentane
# The third column of 'X' is the vector of partial pressure of isopentane
# Y is the vector of experimental reaction rate (in 1/hr)
isom.fun <- function(theta, x){
x1 <- x[,1]
x2 <- x[,2]
x3 <- x[,3]
theta1 <- theta[1]
theta2 <- theta[2]
theta3 <- theta[3]
theta4 <- theta[4]
theta1*theta3*(x2-x3/1.632) / ( 1 + theta2*x1 + theta3*x2 + theta4*x3 )
}
ini.val8 <- c(35, 0.1, 0.05, 0.2)
cons1 <- fitIPEC( isom.fun, x=X, y=Y, ini.val=ini.val8, control=list(
trace=FALSE, reltol=1e-20, maxit=50000) )
par8 <- cons1$par
cons2 <- curvIPEC( isom.fun, theta=par8, x=X, y=Y, alpha=0.05, method="Richardson",
method.args=list(eps=1e-4, d=0.11,
zero.tol=sqrt(.Machine$double.eps/7e-7), r=6, v=2))
cons2
cons3 <- biasIPEC( isom.fun, theta=par8, x=X, y=Y, tol= 1e-20 )
cons3
set.seed(123)
cons4 <- bootIPEC( isom.fun, x=X, y=Y, ini.val=ini.val8,
control=list(trace=FALSE, reltol=1e-20, maxit=50000),
nboot=2000, CI=0.95, fig.opt=TRUE, fold=10000 )
cons4
set.seed(NULL)
cons5 <- skewIPEC( isom.fun, theta=par8, x=X, y=Y, tol= 1e-20 )
cons5
##################################################################################################
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