# Root Mean Square Curvature Calculation 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. package ## Details The DESCRIPTION file: tools:::Rd_package_DESCRIPTION("IPEC") tools:::Rd_package_indices("IPEC") ## Note We are deeply thankful to Paul Gilbert and Jinlong Zhang for their invaluable help during creating this package. We also thank Linli Deng, Kurt Hornik and Lin Wang for their statistical and 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 ## References Bates, D.M and Watts, D.G. (1988) **Nonlinear Regression Analysis and its Applications**. Wiley, New York. 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$-$309. tools:::Rd_expr_doi("10.1093/aesa/saw098") ## See Also `hessian` in package `numDeriv`, `jacobian` in package `numDeriv`, `rms.curv` in package `MASS` ## Examples ```r #### Example 1 ################################################################################## graphics.off() # The velocity of the reaction (counts/min^2) under different substrate concentrations # in parts per million (ppm) (Page 269 of Bates and Watts 1988) x1 <- c(0.02, 0.02, 0.06, 0.06, 0.11, 0.11, 0.22, 0.22, 0.56, 0.56, 1.10, 1.10) y1 <- c(76, 47, 97, 107, 123, 139, 159, 152, 191, 201, 207, 200) # Define the Michaelis-Menten model MM <- function(theta, x){ theta[1]*x / ( theta[2] + x ) } res0 <- fitIPEC( MM, x=x1, y=y1, ini.val=c(200, 0.05), xlim=c( 0, 1.5 ), ylim=c(0, 250), fig.opt=TRUE ) par1 <- res0$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-package