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.
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()
.
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
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.
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.
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
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.
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