# nlfb: nonlinear least squares modeling by functions A simplified and hopefully robust alternative to finding the nonlinear least squares minimizer that causes 'formula' to give a minimal residual sum of squares. ```r nlfb( start, resfn, jacfn = NULL, trace = FALSE, lower = -Inf, upper = Inf, weights = NULL, data = NULL, ctrlcopy = FALSE, control = list(), ... ) ``` ## Arguments - `start`: a numeric vector with all elements present e.g., start=c(b1=200, b2=50, b3=0.3) The start vector for this `nlfb`, unlike `nlxb`, does not need to be named. - `resfn`: A function that evaluates the residual vector for computing the elements of the sum of squares function at the set of parameters `start`. Where this function is created by actions on a formula or expression in `nlxb`, this residual vector will be created by evaluation of the 'model - data', rather than the conventional 'data - model' approach. The sum of squares is the same. - `jacfn`: A function that evaluates the Jacobian of the sum of squares function, that is, the matrix of partial derivatives of the residuals with respect to each of the parameters. If NULL (default), uses an approximation. The Jacobian MUST be returned as the attribute "gradient" of this function, allowing `jacfn` to have the same name and be the same code block as `resfn`, which may permit some efficiencies of computation. - `trace`: TRUE for console output during execution - `lower`: a vector of lower bounds on the parameters. If a single number, this will be applied to all. Default `-Inf`. - `upper`: a vector of upper bounds on the parameters. If a single number, this will be applied to all parameters. Default `Inf`. - `weights`: A vector of fixed weights or a function producing one. See the Details below. - `data`: a data frame of variables used by resfn and jacfn to compute the required residuals and Jacobian. - `ctrlcopy`: If TRUE use control supplied as is. This avoids reprocessing controls. - `control`: a list of control parameters. See nlsr.control(). - `...`: additional data needed to evaluate the modeling functions ## Returns A list of the following items: - **coefficients**: A named vector giving the parameter values at the supposed solution. - **ssquares**: The sum of squared residuals at this set of parameters. - **resid**: The weighted residual vector at the returned parameters. - **jacobian**: The jacobian matrix (partial derivatives of residuals w.r.t. the parameters) at the returned parameters. - **feval**: The number of residual evaluations (sum of squares computations) used. - **jeval**: The number of Jacobian evaluations used. - **weights0**: The weights argument as specified. - **weights**: The weights vector at the final fit. ## Details nlfb is particularly intended to allow for the resolution of very ill-conditioned or else near zero-residual problems for which the regular nls() function is ill-suited. This variant uses a qr solution without forming the sum of squares and cross products t(J) Neither this function nor `nlxb` have provision for parameter scaling (as in the `parscale` control of `optim` and package `optimx`). This would be more tedious than difficult to introduce, but does not seem to be a priority feature to add. The `weights` argument can be a vector of fixed weights, in which case the objective function that will be minimized is the sum of squares where each residual is multiplied by the square root of the corresponding weight. Default `NULL` implies unit weights. `weights` may alternatively be a function with header `function(parms, resids)` to compute such a vector. ## Examples ```r library(nlsr) # Scaled Hobbs problem shobbs.res <- function(x){ # scaled Hobbs weeds problem -- residual # This variant uses looping if(length(x) != 3) stop("shobbs.res -- parameter vector n!=3") y <- c(5.308, 7.24, 9.638, 12.866, 17.069, 23.192, 31.443, 38.558, 50.156, 62.948, 75.995, 91.972) tt <- 1:12 res <- 100.0*x[1]/(1+x[2]*10.*exp(-0.1*x[3]*tt)) - y } shobbs.jac <- function(x) { # scaled Hobbs weeds problem -- Jacobian jj <- matrix(0.0, 12, 3) tt <- 1:12 yy <- exp(-0.1*x[3]*tt) zz <- 100.0/(1+10.*x[2]*yy) jj[tt,1] <- zz jj[tt,2] <- -0.1*x[1]*zz*zz*yy jj[tt,3] <- 0.01*x[1]*zz*zz*yy*x[2]*tt attr(jj, "gradient") <- jj jj } st <- c(b1=2, b2=1, b3=1) # a default starting vector (named!) # Default controls, standard Nash-Marquardt algorithm anlf0 <- nlfb(start=st, resfn=shobbs.res, jacfn=shobbs.jac, trace=TRUE, control=list(prtlvl=1)) anlf0 # Hartley with step reduction factor of .2 anlf0h <- nlfb(start=st, resfn=shobbs.res, jacfn=shobbs.jac, trace=TRUE, control=list(prtlvl=1, lamda=0, laminc=1.0, lamdec=1.0, phi=0, stepredn=0.2)) anlf0h anlf1bm <- nlfb(start=st, resfn=shobbs.res, jacfn=shobbs.jac, lower=c(2,0,0), upper=c(2,6,3), trace=TRUE, control=list(prtlvl=1)) anlf1bm cat("backtrack using stepredn=0.2\n") anlf1bmbt <- nlfb(start=st, resfn=shobbs.res, jacfn=shobbs.jac, lower=c(2,0,0), upper=c(2,6,3), trace=TRUE, control=list(stepredn=0.2, prtlvl=1)) anlf1bmbt ## Short output pshort(anlf1bm) anlf2bm <- nlfb(start=st, resfn=shobbs.res, jacfn=shobbs.jac, lower=c(2,0,0), upper=c(2,6,9), trace=TRUE, control=list(prtlvl=1)) anlf2bm cat("backtrack using stepredn=0.2\n") anlf2bmbt <- nlfb(start=st, resfn=shobbs.res, jacfn=shobbs.jac, lower=c(2,0,0), upper=c(2,6,9), trace=TRUE, control=list(stepredn=0.2, prtlvl=1)) anlf2bmbt ## Short output pshort(anlf2bm) ``` ## Author(s) J C Nash 2014-7-16 nashjc _at_ uottawa.ca

nlfb function