nlsr (version 2023.8.31)

nlfb: nlfb: nonlinear least squares modeling by functions

Description

A simplified and hopefully robust alternative to finding the nonlinear least squares minimizer that causes 'formula' to give a minimal residual sum of squares.

Usage

nlfb(
  start,
  resfn,
  jacfn = NULL,
  trace = FALSE,
  lower = -Inf,
  upper = Inf,
  weights = NULL,
  data = NULL,
  ctrlcopy = FALSE,
  control = list(),
  ...
)

Value

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.

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

Author

J C Nash 2014-7-16 nashjc _at_ uottawa.ca

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

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)