# Derivative Calculation Function Calculates the Jacobian and Hessian matrices of model parameters at a number or a vector `z`. ```r 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 number or 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 number or a vector `z`, which represents a value of a single independent variable or a combination of different values of multiple independent variables. Note: `z` actually corresponds to a combination observation of $x$ rather than all $n$ observations. If there is only a preditor, `z` is a numerical value; there are several predictors, then `z` is a vector corresponding to one combination observation of those predictors. ## Returns - **Jacobian**: The Jacobian matrix of parameters at `z` - **Hessian**: The Hessian matrix of parameters at `z` ## 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. ## 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 . ## See Also `biasIPEC`, `skewIPEC`, `curvIPEC`, `parinfo`, `hessian` in package `numDeriv`, `jacobian` in package `numDeriv` ## Examples ```r #### 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 ################################################################################################# #### 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 ) } 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$par Resul1 <- derivIPEC( isom.fun, theta=par8, z=X[1, ], method="Richardson", method.args=list(eps=1e-4, d=0.11, zero.tol=sqrt(.Machine$double.eps/7e-7), r=6, v=2) ) Resul1 ################################################################################################## ```

derivIPEC function