Specifying and Using Fixed Parameters in Nonlinear Estimation

In nlsr: Functions for Nonlinear Least Squares Solutions - Updated 2022

Motivation

In finding optimal parameters in nonlinear optimization and nonlinear least squares problems, we frequently wish to fix one or more parameters while allowing the rest to be adjusted to explore or optimize an objective function.

This vignette discusses some ideas about specifying the fixed parameters. A lot of the material is drawn from Nash J C (2014) Nonlinear parameter optimization using R tools Chichester UK: Wiley, in particular chapters 11 and 12. There is, however, additional material concerning ways to manage extensible models, as well as some update to the package nlsr. The algorithm has been marginally altered to allow for different sub-variants to be used, and mechanisms for specifying parameter constraints and providing Jacobian approximations have been changed.

Background

Here are some of the ways fixed parameters may be specified in R packages.

Function nlxb() in package nlsr has argument masked:

| Character vector of quoted parameter names. These parameters will NOT be | altered by the algorithm.

This approach has a simplicity that is attractive, but introduces an extra argument to calling sequences. (This approach was previously in defunct package nlmrt.)

Simlarly, function nlfb() in nlsr has argument maskidx:

| Vector of indices of the parameters to be masked. These parameters will NOT | be altered by the algorithm. Note that the mechanism here is different from
| that in nlxb which uses the names of the parameters.

From Rvmmin and Rcgmin in package optimx the argument bdmsk:

| An indicator vector, having 1 for each parameter that is "free" or | unconstrained, and 0 for any parameter that is fixed or MASKED for the | duration of the optimization.

Note that the function bmchk() in package optimx contains a much more extensive examination of the bounds on parameters. In particular, it considers the issues of inadmissible bounds (lower > upper), when to convert a pair of bounds where upper["parameter"] - lower["parameter"] < tol to a fixed or masked parameter (maskadded) and whether parameters outside of bounds should be moved to the nearest bound (parchanged). It may be useful to use inadmissible to refer to situations where a lower bound is higher than an upper bound and infeasible where a parameter value, especially in a given starting vector, is outside the bounds.

Further in package optimx, the function optimr() can call many different "optimizers" (actually function minimization methods that may include bounds and possibly masks). These may be specified by setting the lower and upper bounds equal for the parameters to be fixed. This seems a simple method for specifying masks, but does pose some issues. For example, what happens when the upper bound is only very slightly greater than the lower bound? Also should we stop or declare an error if starting values are NOT on the fixed value?

Of these methods, my preference is now to use the last one -- setting lower and upper bounds equal, and furthermore requiring the starting value of such a parameter to this fixed value, otherwise declaring an error. The approach does not add any special argument for masking, and is relatively obvious to novice users. However, such users may be tempted to put in narrow bounds rather than explicit equalities, and this could have deleterious consequences.

In the revision to package nlsr, package nlsr, I have stopped using masked in nlxb() and maskidx in nlfb() (though the latter is a returned value). This is because I feel the use of equal lower and upper bounds is a better approach. Moreover, though it is not documented, it appears to "mostly work" for the base R function nls() with the algorithm="port" option and with minpack.lm::nlsLM().

Internal structures

bdmsk is the internal structure used in Rcgmin, Rvmmin and nlfb to handle bounds constraints as well as masks. There is one element of bdmsk for each parameter, and in Rcgmin and Rvmmin, this is used on input to specify parameter i as fixed or masked by setting bdmsk[i] <- 0. Free parameters have their bdmsk element 1, but during optimization in the presence of bounds, we can set other values. The full set is as follows

• 1 for a free or unconstrained parameter
• 0 for a masked or fixed parameter
• -0.5 for a parameter that is out of bounds high (> upper bound)
• -1 for a parameter at its upper bound
• -3 for a parameter at its lower bound
• -3.5 for a parameter that is out of bounds low (< lower bound)

Not all these possibilities will be used by all methods that use bdmsk.

The -1 and -3 are historical, and arose in the development of BASIC codes for @jnmws87 (This is now available for free download from archive.org. (https://archive.org/details/NLPE87plus). In particular, adding 2 to the bdmsk element gives 1 for an upper bound and -1 for a lower bound, simplifying the expression to decide if an optimization trial step will move away from a bound.

Proposed algorithmic approaches

Because masks (fixed parameters) reduce the dimensionality of the optimization problem, we can consider modifying the problem to the lower dimension space. This is Duncan Murdoch's suggestion, using

• fn0(par0) to be the initial user function of the full dimension parameter vector par0
• fn1(par1) to be the reduced or internal functin of the reduced dimension vector par1
• par1 <- forward(par0)
• par0 <- inverse(par1)

The major advantage of this approach is explicit dimension reduction. The main disadvantage is the effort of transformation at every step of an optimization.

An alternative is to use the bdmsk vector to mask the optimization search or adjustment vector, including gradients and (approximate) Hessian or Jacobian matrices. A 0 element of bdmsk "multiplies" any adjustment. The principal difficulty is to ensure we do not essentially divide by zero in applying any inverse Hessian. This approach avoids forward, inverse and fn1. However, it may hide the reduction in dimension, and caution is necessary in using the function and its derived gradient, Hessian and derived information.

Examples of use

For optimx

require(optimx)
sq<-function(x){
   nn<-length(x)
   yy<-1:nn
   f<-sum((yy-x)^2)
   f
}
sq.g <- function(x){
   nn<-length(x)
   yy<-1:nn
   gg<- 2*(x - yy)
}
xx <- c(.3, 4)
uncans <- Rvmmin(xx, sq, sq.g)
proptimr(uncans)
mybm <- c(0,1) # fix parameter 1
cans <- Rvmmin(xx, sq, sq.g, bdmsk=mybm)
proptimr(cans)
require(nlsr)
weed <- 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)
ii <- 1:12
wdf <- data.frame(weed, ii)
weedux <- nlxb(weed~b1/(1+b2*exp(-b3*ii)), start=c(b1=200, b2=50, b3=0.3)) 
weedux
weedcx <- nlxb(weed~b1/(1+b2*exp(-b3*ii)), start=c(b1=200, b2=50, b3=0.3), masked=c("b1")) 
weedcx
rfn <- function(bvec, weed=weed, ii=ii){
  res <- rep(NA, length(ii))
  for (i in ii){
    res[i]<- bvec[1]/(1+bvec[2]*exp(-bvec[3]*i))-weed[i]
  }
  res
}
weeduf <- nlfb(start=c(200, 50, 0.3),resfn=rfn,weed=weed, ii=ii,
               control=list(japprox="jacentral"))
weeduf
weedcf <- nlfb(start=c(200, 50, 0.3),resfn=rfn,weed=weed, ii=ii, lower=c(200, 0, 0),
               upper=c(200, 100,100), control=list(japprox="jacentral"))
weedcf

An extensible bell-curve model

Package nlraa has a selfStart model SSbell (@ArchMiguez2013) of which the formula is

$$ y \approx ymax * exp(a (x - xc)^2 + b(x-xc)^3) $$

This is essentially the Gaussian bell curve with an additional cubic element in the exponential function. If we fix $b = 0$, then we have the usual Gaussian, and we can use the standard deviation $sigma$ of the variable $x$ with $xc$ equal to its mean and our parameters will be given approximately by

$$ ymax = max(y)$$ $$ a = -0.5/sigma^2$$ $$ xc = mean(y) $$

We illustrate this in the following example.

References

nlsr documentation built on Sept. 8, 2023, 5:48 p.m.