| skewIPEC {IPEC} | R Documentation |
Skewness Calculation Function
Description
Calculates the skewness in the estimates of the parameters of a given model.
Usage
skewIPEC( expr, theta, x, y, tol = sqrt(.Machine$double.eps), method = "Richardson",
method.args = list(eps = 1e-04, d = 0.11,
zero.tol = sqrt(.Machine$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 |
x |
A vector or matrix of observations of independent variable(s) |
y |
A vector of observations of response variable |
tol |
The tolerance for detecting linear dependencies in the columns of a matrix for calculating its inverse.
See the input argument of |
method |
It is the same as the input argument of |
method.args |
It is the same as the input argument of |
side |
It is the same as the input argument of |
Details
The defined model should have two input arguments: a parameter vector and an independent variable vector or matrix,
e.g. myfun <- function(P, x){...}, where P represents the parameter vector and x represents the independent variable vector or matrix.
Let |g_{1i}| be a measure of the skewness of the estimate of the i-th parameter. If |g_{1i}| < 0.1,
the estimator \hat \theta_i of parameter \theta_i is very close-to-linear in behavior;
if 0.1 \le |g_{1i}| < 0.25, the estimator is reasonably close-to-linear; if |g_{1i}| \ge 0.25,
the skewness is very apparent; if |g_{1i}| > 1, the estimator is considerably nonlinear in behavior (Pages 27-28 in Ratkowsky 1990).
Value
skewness |
The skewness |
Note
The current function can be applicable to nonlinear models with multiple independent variables.
Author(s)
Peijian Shi pjshi@njfu.edu.cn, Peter M. Ridland p.ridland@unimelb.edu.au, David A. Ratkowsky d.ratkowsky@utas.edu.au, Yang Li yangli@fau.edu.
References
Hougaard, P. (1985) The appropriateness of the asymptotic distribution in a nonlinear regression model in relation to curvature.
J. R. Statist. Soc., Ser. B 47, 103-114.
Ratkowsky, D.A. (1990) Handbook of Nonlinear Regression Models, Marcel Dekker, New York.
See Also
derivIPEC, hessian
in package numDeriv, jacobian in package numDeriv
Examples
#### Example 1 #################################################################################
# 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 (MM) model
MM <- function(theta, x){
theta[1]*x / ( theta[2] + x )
}
par1 <- c(212.68490865, 0.06412421)
res5 <- skewIPEC( MM, theta=par1, x=x1, y=y1, tol= 1e-20 )
res5
#################################################################################################
#### Example 2 ##################################################################################
# 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 )
# 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)
# To calculate bias
resu5 <- skewIPEC( myfun, theta=par2, x=x2, y=y2, tol= 1e-20 )
resu5
#################################################################################################
#### 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 first case of Mitscherlich equation
MitA <- function(P1, x){
P1[3] + P1[2]*exp(P1[1]*x)
}
# Define the second case of Mitscherlich equation
MitB <- function(P2, x){
log( P2[3] ) + exp(P2[2] + P2[1]*x)
}
# Define the third case of Mitscherlich equation
MitC <- function(P3, x, x1=1, x2=13){
theta1 <- P3[1]
beta2 <- P3[2]
beta3 <- P3[3]
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.val3 <- c(-0.1, 2.5, 1)
r0 <- fitIPEC( MitA, x=x4, y=y4, ini.val=ini.val3, xlim=NULL, ylim=NULL,
fig.opt=TRUE, control=list(trace=FALSE, reltol=1e-20, maxit=50000) )
parA <- r0$par
parA
r5 <- skewIPEC(MitA, theta=parA, x=x4, y=y4, tol=1e-20)
r5
ini.val4 <- c(exp(-0.1), log(2.5), 1)
R0 <- fitIPEC( MitB, x=x4, y=y4, ini.val=ini.val3, xlim=NULL, ylim=NULL,
fig.opt=TRUE, control=list(trace=FALSE, reltol=1e-20, maxit=50000) )
parB <- R0$par
parB
R5 <- skewIPEC( MitB, theta=parB, x=x4, y=y4, tol=1e-20 )
R5
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
RES5 <- skewIPEC( MitC, theta=parC, x=x4, y=y4, tol=1e-20 )
RES5
#################################################################################################
#### 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.
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 )
}
par8 <- c(35.92831619, 0.07084811, 0.03772270, 0.16718384)
cons5 <- skewIPEC( isom.fun, theta=par8, x=X, y=Y, tol= 1e-20 )
cons5
#################################################################################################