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Functions for Nonlinear Least Squares Solutions - Updated 2022

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nlsr/nlsr Background, Development, and Discussion

nlsr/nlsr Background, Development, and Discussion
In nlsr: Functions for Nonlinear Least Squares Solutions - Updated 2022 

rm(list=ls()) # clear workspace for each major section

\pagebreak

Overview and objectives

This extended vignette is to explain and to (partially) record the development and testing of nlsr, an R package to try to bring the R function nls() up to date and to add capabilities for the extension of the symbolic and automatic derivative tools in R. As of 2022, it also records the upgrade to nlsr.

A particular goal in nlsr (and previously nlsr) is to attempt, wherever possible, to use analytic or automatic derivatives. The function nls() generally uses a rather weak forward derivative approximation, though a central difference approximation is available if one uses advanced options.

A second objective is to use a Marquardt stabilization of the Gauss-Newton equations to avoid the commonly encountered "singular gradient" failure of nls(). This refers to the loss of rank of the Jacobian at the parameters for evaluation. The particular stabilization also incorporates a very simple trick to avoid very small diagonal elements of the Jacobian inner product, though in the present implementations, this is accomplished indirectly. See the section below Implementation of method

Two other objectives have arisen in 2022 in the move to a new nlsr:

In preparing the nlsr package there was a sub-goal to unify, or at least render compatible, various packages in R for the estimation or analysis of problems amenable to nonlinear least squares solution. This was expanded in 2021 with a Google Summer of Code initiative Improvements to nls() in which Arkajyoti Bhattacharjee was the contributor. Unfortunately, while we made some progress, we were NOT able to overcome some of the densely entangled code sufficiently to make more than limited improvements.

A large part of the work for the nlsr package family -- particularly the parts concerning derivatives and R language structures -- was initially carried out by Duncan Murdoch in 2014. Without his input, much of the capability of the package would not exist, even though there was an earlier 2012 package nlmrt (@jnnlmrt2012). That package was clumsily written, but did show the possibilities for automatically providing analytic Jacobian information to a nonlinear regression package.

The nlsr package and this vignette are works in progress, and assistance and examples are welcome. Note that there is similar work on symbolic derivatives in the package Deriv (Andrew Clausen and Serguei Sokol (2015) Symbolic Differentiation, version 2.0, https://cran.r-project.org/package=Deriv), and making the current work "play nicely" with that package is desirable. There are also aspects of nls() in base R and the package minpack.lm (@minpacklm12) which could be better aligned. Much more on nls() in particular is in process with Arkajyoti Bhattacharjee in mid 2022.

As a mechanism for highlighting issues that remain to be resolved, I have put double question marks (??) where I believe attention is needed in this document.

0. Issues remaining to address and TODOS

Issues are related to concerns about one or more of the nonlinear modeling tools as revealed by tests and examples used in this vignette. I have labeled some of these as MINOR, and regard these as issues that could be fixed easily.

nls() uses different models with different algorithm choices

In Section 7. under Partially Linear Models, nls() uses different model forms according to the choice of algorithm in some cases where the model can be partially linear. I believe there should at least be a WARNING about this possibility, though preferably I would like to see this treated as an error in the code. This is discussed in more detail in the paper "Refactoring nls()" (@JNABRefactoringNLS22).

MINOR -- Bounds specification and warnings for nlsLM and nls.lm

nls(), nlsr::nlxb and nlsr::nlfb allow a single value to be given for the lower and upper bounds. This single value is expanded to a vector of length equal to the number of parameters, but the minpack.lm routines are more fussy.

Also, while nls() and the nlsr functions warn of initial starting parameters that violate specified bounds, the minpack.lm routines do not. This is easily fixable with a warning().

These are illustrated below.

# Bounds Test nlsbtsimple.R (see BT.RES in Nash and Walker-Smith (1987))
rm(list=ls())
bt.res<-function(x){ x }
bt.jac<-function(x){nn <- length(x); JJ <- diag(nn); attr(JJ, "gradient") <- JJ;  JJ}
n <- 4
x<-rep(0,n) ; lower<-rep(NA,n); upper<-lower # to get arrays set
for (i in 1:n) { lower[i]<-1.0*(i-1)*(n-1)/n; upper[i]<-1.0*i*(n+1)/n }
x <-0.5*(lower+upper) # start on mean
xnames<-as.character(1:n) # name our parameters just in case
for (i in 1:n){ xnames[i]<-paste("p",xnames[i],sep='') }
names(x) <- xnames
require(minpack.lm)
require(nlsr)
bsnlf0<-nlfb(start=x, resfn=bt.res, jacfn=bt.jac) # unconstrained
bsnlf0
bsnlm0<-nls.lm(par=x, fn=bt.res, jac=bt.jac) # unconstrained
bsnlm0
bsnlf1<-nlfb(start=x, resfn=bt.res, jacfn=bt.jac, lower=lower, upper=upper)
bsnlf1
bsnlm1<-nls.lm(par=x, fn=bt.res, jac=bt.jac, lower=lower, upper=upper)
bsnlm1

# single value bounds
bsnlf2<-nlfb(start=x, resfn=bt.res, jacfn=bt.jac, lower=0.25, upper=4)
bsnlf2
# nls.lm will NOT expand single value bounds
bsnlm2<-try(nls.lm(par=x, fn=bt.res, jac=bt.jac, lower=0.25, upper=4))

For example (from the file nlsbtsimple.R):

MINOR -- nlsLM and nls.lm do not warn of out of bounds initial parameters

x<-rep(0,n) # resetting to this puts x out of bounds
cat("lower:"); print(lower)
cat("upper:"); print(upper)
names(x) <- xnames # to ensure names set
obnlm<-nls.lm(par=x, fn=bt.res, jac=bt.jac, lower=lower, upper=upper)
obnlm

For nonlinear regression with minpack.lm::nlsLM there is also no warning. The example using file Hobbsbdformula1.R in the vignette R: Examples of different nonlinear least squares calculations (file NLS-Examples.Rmd) shows this. However, in this case, the program actually gets a good answer, despite the failure to warn. In a start from all 1's, which is feasible, a poor answer is obtained, as noted in the next section.

Bounded minimization with minpack.lm

While minpack.lm mentions lower and upper bounds in the manual pages, there are no examples in the package (that I could find) of their application. For the file nlsbtsimple.R, we get acceptable answers. For the Hobbs problem, from a start of all 1's, both nlsLM and nls.lm converge to sums of squares higher than nlxb, nlfb or nls() using the "port" algorithm (when the last is able to get started). Moreover, in one case for the scaled Hobbs problem, the two minpack.lm functions give different results. I have not yet been able to understand how these routines apply bounds constraints. There is mention in \url{https://lmfit-py.readthedocs.io/en/latest/bounds.html} that the original MINPACK did NOT cater for bounds constraints on parameters, and that MINUIT, which uses the MINPACK ideas, used a transformation of the parameters to accomplish this. (Hans Werner Borchers uses the same idea in the the code dfoptim::nmkb, which he calls a transfinite transformation.)

Comments:

1) the transfinite idea is useful in that it may be applicable quite generally. If a wrapper for unconstrained minimizers could be devised, it would enlarge the capability of a number of R tools.

2) The Hobbs problem, even when scaled, may be unscaled again by such a transformation of the parameters.

I have seen many cases where methods that are generally reliable give an occasional unsatisfactory result. However, it would be useful to know the precise reasons why or where minpack.lm routines are obtaining these results, since it may be possible to either fix the issues or else provide some warnings or diagnostics, which hopefully would be of wider application.

TODOS

1. Summary of capabilities and functions in nlsr

Throughout this exposition, we have chosen to set trace variables to FALSE to reduce the volume of output. Changing such values to TRUE will expand output.

We will illustrate many of the capabilities with the Hobbs weed problem (@jncnm79, page 121). This can be set up in a scaled or unscaled form.

## Use the Hobbs Weed problem
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)
tt <- 1:12
weeddf <- data.frame(y=weed, tt=tt)
st <- c(b1=1, b2=1, b3=1) # a default starting vector (named!)
## Unscaled model
wmodu <- y ~ b1/(1+b2*exp(-b3*tt))
## Scaled model
wmods <- y ~ 100*b1/(1+10*b2*exp(-0.1*b3*tt))
## We can provide the residual and Jacobian as functions
# Unscaled Hobbs problem
hobbs.res  <-  function(x){ # scaled Hobbs weeds problem -- residual
  # This variant uses looping
  if(length(x) != 3) stop("hobbs.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  <-  x[1]/(1+x[2]*exp(-x[3]*tt)) - y
}

hobbs.jac  <-  function(x) { # scaled Hobbs weeds problem -- Jacobian
  jj  <-  matrix(0.0, 12, 3)
  tt  <-  1:12
  yy  <-  exp(-x[3]*tt)
  zz  <-  1.0/(1+x[2]*yy)
  jj[tt,1]   <-   zz
  jj[tt,2]   <-   -x[1]*zz*zz*yy
  jj[tt,3]   <-   x[1]*zz*zz*yy*x[2]*tt
  attr(jj, "gradient") <- jj
  jj
}
# 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
}

nlxb

This is the likely the function most called by users from nlsr.

require(nlsr)
# Use Hobbs scaled formula model
anlxb1  <-  try(nlxb(wmods, start=st, data=weeddf))
print(anlxb1)
# summary(anlxb1) # for different output
# Test without starting parameters
anlxb1nostart  <-  try(nlxb(wmodu, data=weeddf))
print(anlxb1nostart)

Here we see that the Jacobian at the solution is essentially of rank 1, even though there are 3 coefficients. It is therefore not surprising that nls(), which does not benefit from the Levenberg-Marquardt stabilization in solving the nonlinear least squares program fails for this problem. See the subsection below for wrapnlsr. Note that the singular values of the Jacobian computed via a numerical approximation are more extreme (i.e., nearly singular) than those determined by analytic derivatives in the preceding solution anlxb1. For this example, there is a clear advantage to the analytic derivatives.

While there is an almost liturgical adherence to setting up models where the dependent (or predicted) variable is on the left hand side (LHS) of the model formula and the independent (or predictor) variable(s) on the right hand side (RHS), this is
NOT an actual requirement in most cases, though there are some situations such as the minpack.lm function wfct that do assume this structure. Here is an example of solving the Hobbs unscaled problem using a 1-sided model formula.

# One-sided unscaled Hobbs weed model formula
wmodu1  <-  ~ b1/(1+b2*exp(-b3*tt)) - y
anlxb11  <-  try(nlxb(wmodu1, start=st, data=weeddf))
print(anlxb11)

model2rjfun, model2ssgrfun, modelexpr

# st  <-  c(b1=1, b2=1, b3=1)
wrj <- model2rjfun(wmods, st, data=weeddf)
wrj
weedux <- nlxb(wmods, start=st, data=weeddf) 
print(weedux)

wss <- model2ssgrfun(wmods, st, data=weeddf)
print(wss)

# We can get expressions used to calculate these as follows:
wexpr.rj <- modelexpr(wrj) 
print(wexpr.rj)

wexpr.ss <- modelexpr(wss) 
print(wexpr.ss)

nlfb -- minimize nonlinear least squares residual functions

require(nlsr)

cat("try nlfb\n")
st  <-  c(b1=1, b2=1, b3=1)
ans1 <- nlfb(st, shobbs.res, shobbs.jac, trace=FALSE)
summary(ans1)

## No jacobian function -- use internal approximation
ans1n <- nlfb(st, shobbs.res, trace=FALSE, control=list(japprox="jafwd", watch=FALSE)) # NO jacfn -- tell it fwd approx
print(ans1n)
## difference
coef(ans1)-coef(ans1n)

coef.nlsr

coef(anlxb1) # this is solution of scaled problem from unit start

print.nlsr

### From examples above
print(weedux)
print(ans1)

summary.nlsr

### From examples above
summary(weedux)
summary(ans1)

wrapnlsr

A particular purpose of this function is to create the nls-style model object for a problem when the solution has been obtained by nlsr::nlxb. minpack.lm::nlsLM creates a structure that is parallel to that from nls().

## The following attempt at Hobbs unscaled with nls() fails!
rm(list=ls()) # clear before starting
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)
tt <- 1:12
weeddf <- data.frame(tt, weed)
st <- c(b1=1, b2=1, b3=1) # a default starting vector (named!)
wmodu<- weed ~ b1/(1 + b2 * exp(- b3 * tt))
anls1u  <-  try(nls(wmodu, start=st, trace=FALSE, data=weeddf)) # fails
## But we succeed by calling nlxb first.
library(nlsr)
anlxb1un  <-  try(nlxb(wmodu, start=st, trace=FALSE, data=weeddf))
print(anlxb1un)
st2 <- coef(anlxb1un) # We try to start nls from solution using nlxb
anls2u  <-  try(nls(wmodu, start=st2, trace=FALSE, data=weeddf))
print(anls2u)
## Or we can simply call wrapnlsr FROM THE ORIGINAL start
anls2a  <-  try(wrapnlsr(wmodu, start=st, trace=FALSE, data=weeddf))
summary(anls2a)
# # For comparison could run nlsLM
# require(minpack.lm)
# anlsLM1u  <-  try(nlsLM(wmodu, start=st, trace=FALSE, data=weeddf))
# print(anlsLM1u)

resgr, resss

## Use shobbs example -- Scaled Hobbs problem
shobbs.res  <-  function(x){ # scaled Hobbs weeds problem -- residual
  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
}
RG <- resgr(st, shobbs.res, shobbs.jac)
RG
SS <- resss(st, shobbs.res)
SS

nlsDeriv, codeDeriv, fnDeriv, newDeriv

These functions are mainly for test and development use.

newDeriv() # a call with no arguments returns a list of available functions
           # for which derivatives are currently defined
newDeriv(sin(x)) # a call with a function that is in the list of available derivatives
                 # returns the derivative expression for that function
nlsDeriv(~ sin(x+y), "x") # partial derivative of this function w.r.t. "x"

## CAUTION !! ##
newDeriv(joe(x)) # but an undefined function returns NULL
newDeriv(joe(x), deriv=log(x^2)) # We can define derivatives, though joe() is meaningless.
nlsDeriv(~ joe(x+z), "x")

# Some examples of usage
f <- function(x) x^2
newDeriv(f(x), 2*x*D(x))
nlsDeriv(~ f(abs(x)), "x")
nlsDeriv(~ x^2, "x")
nlsDeriv(~ (abs(x)^2), "x")

# derivatives of distribution functions
nlsDeriv(~ pnorm(x, sd=2, log = TRUE), "x")
# get evaluation code from a formula
codeDeriv(~ pnorm(x, sd = sd, log = TRUE), "x")
# wrap it in a function call
fnDeriv(~ pnorm(x, sd = sd, log = TRUE), "x")
f <- fnDeriv(~ pnorm(x, sd = sd, log = TRUE), "x", args = alist(x =, sd = 2))
f
f(1)
100*(f(1.01) - f(1))  # Should be close to the gradient
                      # The attached gradient attribute (from f(1.01)) is
                      # meaningless after the subtraction.

nlsSimplify and related functions

The related tools are: newSimplification, sysSimplifications, isFALSE, isZERO, isONE, isMINUSONE, isCALL, findSubexprs, sysDerivs.

nlsSimplify simplifies expressions according to rules specified by newSimplification.

findSubexprs finds common subexpressions in an expression vector so that duplicate computation can be avoided.

## nlsSimplify
nlsSimplify(quote(a + 0))
nlsSimplify(quote(exp(1)), verbose = TRUE)

nlsSimplify(quote(sqrt(a + b)))  # standard rule
## sysSimplifications
# creates a new environment whose parent is emptyenv()  Why?
str(sysSimplifications)
myrules <- new.env(parent = sysSimplifications)
## newSimplification
newSimplification(sqrt(a), TRUE, a^0.5, simpEnv = myrules)
nlsSimplify(quote(sqrt(a + b)), simpEnv = myrules)

## isFALSE
print(isFALSE(1==2))
print(isFALSE(2==2))

## isZERO
print(isZERO(0))
x <- -0
print(isZERO(x))
x <- 0
print(isZERO(x))
print(isZERO(~(-1)))
print(isZERO("-1"))
print(isZERO(expression(-1)))


## isONE
print(isONE(1))
x <- 1
print(isONE(x))
print(isONE(~(1)))
print(isONE("1"))
print(isONE(expression(1)))


## isMINUSONE
print(isMINUSONE(-1))
x <- -1
print(isMINUSONE(x))
print(isMINUSONE(~(-1)))
print(isMINUSONE("-1"))
print(isMINUSONE(expression(-1)))

## isCALL  ?? don't have good understanding of this
x <- -1
print(isCALL(x,"isMINUSONE"))
print(isCALL(x, quote(isMINUSONE)))

## findSubexprs
findSubexprs(expression(x^2, x-y, y^2-x^2))

## sysDerivs
# creates a new environment whose parent is emptyenv()  Why? 
# Where are derivative definitions are stored?
str(sysDerivs)

2. Analytic versus approximate Jacobians

A key goal of package nlsr was to be able to use analytic or symbolic derivatives for the Jacobian in nonlinear least squares computations, in particular when the model is specified as a formula or expression. In this nlsr::nlxb() has been quite successsful. It should be pointed out that the principal advantage of using analytic derivatives is that we get a more assured measure of the "flatness" of the sum of squares surface at the termination point. My experience is that there is no particular gain in the speed in getting to that termination point.

A disadvantage of the approach is that specifying a "formula" that includes an R function will (usually) fail. nls(), because it defaults to using a derivative approximation, can accept formulas that include R functions, and indeed, one of the examples in the manual page nls.Rd is of this type. In package nlsr we have introduced the possibility of using Jacobian approximations via the control element japprox which is discussed in several places below.

A second goal of including approximations for the Jacobian is to be able to specify or control the approximation. nls(), as shown in Appendix A, has a rather complicated code involving both R and C to compute a forward difference approximation to the Jacobian. In this computation, each parameter is adjusted by its absolute value times the square root of the machine precision (double). Call this the ndstep. So the parameter delta is the absolute value of the parameter times approximately 1.5e-8. If the parameter is zero, then the delta is ndstep. In nlsr::jafwd(), I use a delta of ndstep times the absolute value of the parameter PLUS the ndstep. This avoids the check for a zero parameter.

It is known (https://en.wikipedia.org/wiki/Finite_difference) that the forward (and backward) difference approximations are not ideal. Central differences and higher approximations such as those found in the CRAN package numDeriv are better, and it is desirable to be able to specify that these be used.

Specifying approximations to nlfb

We first look at the nlfb() function that uses R functions for the residual and Jacobian. Borrowing from the mechanism used in optimx::optimr(), we can invoke an approximation by putting the name of an appropriate R function in quotation marks. In nlsr there are four functions jafwd(), jaback, jacentral, and jand for the forward, backward, central and numDeriv (default) approximations. Moreover, the ndstep can be set in the control() list in the nlfb() call. Its default value is 1e-7. An open question is whether to change to the value sqrt(.Machine$double.eps) to more closely match nls(). 

## Test with functional spec. of problem
## Default call WITH jacobian function
ans1 <- nlfb(st, resfn=shobbs.res, jacfn=shobbs.jac)
ans1
## No jacobian function -- and no japprox control setting
ans1n <- try(nlfb(st, shobbs.res)) # NO jacfn
ans1n
## Force jafwd approximation
ans1nf <- nlfb(st, shobbs.res, control=list(japprox="jafwd")) # NO jacfn, but specify fwd approx
ans1nf
## Alternative specification
ans1nfa <- nlfb(st, shobbs.res, jacfn="jafwd") # NO jacfn, but specify fwd approx in jacfn
ans1nfa
## Coeff differences from analytic: 
ans1nf$coefficients-ans1$coefficients
## Force jacentral approximation
ans1nc <- nlfb(st, shobbs.res, control=list(japprox="jacentral")) # NO jacfn
ans1nc
## Coeff differences from analytic:
ans1nc$coefficients-ans1$coefficients
## Force jaback approximation
ans1nb <- nlfb(st, shobbs.res, control=list(japprox="jaback")) # NO jacfn
ans1nb
## Coeff differences from analytic:
ans1nb$coefficients-ans1$coefficients
## Force jand approximation
ans1nn <- nlfb(st, shobbs.res, control=list(japprox="jand"), trace=FALSE) # NO jacfn
ans1nn
## Coeff differences from analytic:
ans1nc$coefficients-ans1$coefficients

Specifying Jacobian approximations to nlxb

Since nlxb() provides the model as a formula or expression, we need to tell it how to get an approximate Jacobian. First, we can specify WHICH approximation to use by putting the name of the jacobian approximating function in the control list element japprox in quotation marks.

The default mechanism for using nlxb() is to create a function trjfn (by calling model2rjfun() ). To make our work a lot easier, trjfn is used as BOTH the residual and Jacobian function by copying the created Jacobian matrix into the "gradient" attribute of the returned object from trjfn.

NOTE: This requirement that the Jacobian matrix be returned in the "gradient" attribute of the returned object of the jacfn specified in the call to nlfb() is one that users should be careful to observe.

When we specify a Jacobian approximation (via control$japprox) to nlxb(), the call to model2rjfun is made with jacobian=FALSE and the appropriate function for the approximation is supplied in the subsequent call the nlfb() to minimize the sum of squared objective. The jacobian parameter in the call to model2rjfun() defaults to TRUE.

## rm(list=ls()) # clear workspace for each major section
## Use the Hobbs Weed problem
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)
tt <- 1:12
weeddf <- data.frame(y=weed, tt=tt)
st1 <- c(b1=1, b2=1, b3=1) # a default starting vector (named!)
wmods <- y ~ 100*b1/(1+10*b2*exp(-0.1*b3*tt)) ## Scaled model
print(wmods)
anlxb1  <-  try(nlxb(wmods, start=st1, data=weeddf))
print(anlxb1)

anlxb1fwd  <-  (nlxb(wmods, start=st1, data=weeddf,
    control=list(japprox="jafwd")))
print(anlxb1fwd)

## fwd - analytic
print(anlxb1fwd$coefficients - anlxb1$coefficients)

anlxb1bak  <- (nlxb(wmods, start=st1, data=weeddf,
    control=list(japprox="jaback")))
print(anlxb1bak)
## back - analytic
print(anlxb1bak$coefficients - anlxb1$coefficients)

anlxb1cen  <- (nlxb(wmods, start=st1, data=weeddf,
    control=list(japprox="jacentral")))
print(anlxb1cen)
## central - analytic
print(anlxb1cen$coefficients - anlxb1$coefficients)

anlxb1nd  <-  (nlxb(wmods, start=st1, data=weeddf,
    control=list(japprox="jand")))
print(anlxb1nd)
## numDeriv - analytic
print(anlxb1nd$coefficients - anlxb1$coefficients)

3. Weighted nonlinear regression

While the standard approach to nonlinear regression is to minimize the sum of squared residuals, there are frequently good reasons to weight each residual to scale them according to their importance. That is, we wish to favour those residuals believed to be more certain. Until 2020, nlsr did not provide for weights that could alter with the parameters of the model. That is, the weights were pre-specified, at least in nlxb. The resfn and jacfn of nlfb could, of course be specified with almost any loss function that results in a sum of squared residuals.

With the addition of the possibility of forcing the use of an approximation to the Jacobian (Section 2 above), formulas that allow the inclusion of functions (i.e., subprograms) are possible, and these may include weighting.

The following examples illustrate how this works.

Static weights

## weighted nonlinear regression using inverse squared variance of the response
require(minpack.lm)
Treated <- Puromycin[Puromycin$state == "treated", ]
# We want the variance of each "group" of the rate variable 
# for which the conc variable is the same. We first find
# this variance by group using the tapply() function.
var.Treated <- tapply(Treated$rate, Treated$conc, var)
var.Treated <- rep(var.Treated, each = 2)
Pur.wt1nls <- nls(rate ~ (Vm * conc)/(K + conc), data = Treated, 
               start = list(Vm = 200, K = 0.1), weights = 1/var.Treated^2)
Pur.wt1nlm <- nlsLM(rate ~ (Vm * conc)/(K + conc), data = Treated, 
               start = list(Vm = 200, K = 0.1), weights = 1/var.Treated^2)
Pur.wt1nlx <- nlxb(rate ~ (Vm * conc)/(K + conc), data = Treated, 
               start = list(Vm = 200, K = 0.1), weights = 1/var.Treated^2)
rnls <- summary(Pur.wt1nls)$residuals
ssrnls<-as.numeric(crossprod(rnls))
rnlm <- summary(Pur.wt1nlm)$residuals
ssrnlm<-as.numeric(crossprod(rnlm))
rnlx <- Pur.wt1nlx$resid
ssrnlx<-as.numeric(crossprod(rnlx))
cat("Compare nls and nlsLM: ", all.equal(coef(Pur.wt1nls), coef(Pur.wt1nlm)),"\n")
cat("Compare nls and nlsLM: ", all.equal(coef(Pur.wt1nls), coef(Pur.wt1nlx)),"\n")
cat("Sumsquares nls - nlsLM: ", ssrnls-ssrnlm,"\n")
cat("Sumsquares nls - nlxb: ", ssrnls-ssrnlx,"\n")

Dynamic weights via the wfct() function

minpack.lm provides an interesting function that allows us to access current values of various quantities associated with our model. There are five possibilities, of which two are static weightings -- the name of the response or the predictor variable. (It seems wfct assumes only one such variable.) The other three possibilities are the current values of the "fitted" model, the residuals as specified by "resid", or the "error", which is the square root of the variance of the response variable. The last possibility requires repetitions of the independent or predictor variable.

Concerns: I have found that the structure of wfct means that examples using it in calls to nlsLM, nls or nlxb with fail if they are accessed via source() or if the call is surrounded by a print() or try(). This remains to be sorted out.

Note that nlxb cannot use fitted or resid in the wfct function to specify weights. nls() generates such functions as part of the returned object. nlsr does have predict.nlsr() that could be used to generate fits and by extension, residuals. What is possible??

## minpack.lm::wfct
### Examples from 'nls' doc where wfct used ###
## Weighting by inverse of response 1/y_i:
wtt1nlm<-nlsLM(rate ~ Vm * conc/(K + conc), data = Treated,
              start = c(Vm = 200, K = 0.05), weights = wfct(1/rate))
print(wtt1nlm)

wtt1nls<-nls(rate ~ Vm * conc/(K + conc), data = Treated,
          start = c(Vm = 200, K = 0.05), weights = wfct(1/rate))
print(wtt1nls)

wtt1nlx<-nlxb(rate ~ Vm * conc/(K + conc), data = Treated,
          start = c(Vm = 200, K = 0.05), weights = wfct(1/rate))
print(wtt1nlx)

## Weighting by square root of predictor \sqrt{x_i}:
# ?? why does try() not work
wtt2nlm<-nlsLM(rate ~ Vm * conc/(K + conc), data = Treated,
       start = c(Vm = 200, K = 0.05), weights = wfct(sqrt(conc)))
print(wtt2nlm)

wtt2nls<-nls(rate ~ Vm * conc/(K + conc), data = Treated,
       start = c(Vm = 200, K = 0.05), weights = wfct(sqrt(conc)))
print(wtt2nls)

wtt2nlx<-nlxb(rate ~ Vm * conc/(K + conc), data = Treated,
       start = c(Vm = 200, K = 0.05), weights = wfct(sqrt(conc)))
print(wtt2nlx)

## Weighting by inverse square of fitted values 1/\hat{y_i}^2:
wtt3nlm<-nlsLM(rate ~ Vm * conc/(K + conc), data = Treated,
       start = c(Vm = 200, K = 0.05), weights = wfct(1/fitted^2))
print(wtt3nlm)

wtt3nls<-nls(rate ~ Vm * conc/(K + conc), data = Treated,
       start = c(Vm = 200, K = 0.05), weights = wfct(1/fitted^2))
print(wtt3nls)

# These don't work -- there is no fitted() function available in nlsr
# wtt3nlx<-try(nlxb(rate ~ Vm * conc/(K + conc), data = Treated,
#        start = c(Vm = 200, K = 0.05), weights = wfct(1/fitted^2)))
##  (nlxb(rate ~ Vm * conc/(K + conc), data = Treated,
##        start = c(Vm = 200, K = 0.05), weights = wfct(1/(resid+rate)^2)))
## (wrapnlsr(rate ~ Vm * conc/(K + conc), data = Treated,
##        start = c(Vm = 200, K = 0.05), weights = wfct(1/(resid+rate)^2)))

## Weighting by inverse variance 1/\sigma{y_i}^2:
wtt4nlm<-nlsLM(rate ~ Vm * conc/(K + conc), data = Treated,
       start = c(Vm = 200, K = 0.05), weights = wfct(1/error^2))
print(wtt4nlm)

wtt4nls<-nls(rate ~ Vm * conc/(K + conc), data = Treated,
       start = c(Vm = 200, K = 0.05), weights = wfct(1/error^2))
print(wtt4nls)

wtt4nlx<-nlxb(rate ~ Vm * conc/(K + conc), data = Treated,
       start = c(Vm = 200, K = 0.05), weights = wfct(1/error^2))
print(wtt4nlx)

wtt5nls<-nls(rate ~ Vm * conc/(K + conc), data = Treated,
              start = c(Vm = 200, K = 0.05), weights = wfct(1/resid^2))
print(wtt5nls)
wtt5nlm<-nlsLM(rate ~ Vm * conc/(K + conc), data = Treated,
              start = c(Vm = 200, K = 0.05), weights = wfct(1/resid^2))
print(wtt5nlm)
## Won't work! No resid function available from nlxb.
## wtt5nlx<-nlxb(rate ~ Vm * conc/(K + conc), data = Treated,
##            start = c(Vm = 200, K = 0.05), weights = wfct(1/resid^2))
## print(wtt5nlx)

Weights built into a one-sided model function

In all three formula-based nonlinear model estimators (nls, nlsLM, nlxb), the "formula" can be set so there is no so-called Left Hand Side (LHS). 

st<-c(b1=1, b2=1, b3=1)
frm <- ~ b1/(1+b2*exp(-b3*tt))-y
nolhs <- nlxb(formula=frm, data=weeddf, start=st)
nolhs

Since we can set up problems in this way, we could, in some cases, build weights into the expression on the right hand side. For nlxb, attempts to develop the analytic Jacobian will generally fail if the formula includes an R function call. This can be overcome by specifying a Jacobian approximation in the control list element japprox. 

## weighted nonlinear regression using 1-sided model functions
Treated <- Puromycin[Puromycin$state == "treated", ]
weighted.MM <- function(resp, conc, Vm, K)
{
  ## Purpose: exactly as white book p. 451 -- RHS for nls()
  ##  Weighted version of Michaelis-Menten model
  ## ----------------------------------------------------------
  ## Arguments: 'y', 'x' and the two parameters (see book)
  ## ----------------------------------------------------------
  ## Author: Martin Maechler, Date: 23 Mar 2001

  pred <- (Vm * conc)/(K + conc)
  (resp - pred) / sqrt(pred)
}

Pur.wtMMnls <- nls( ~ weighted.MM(rate, conc, Vm, K), data = Treated,
               start = list(Vm = 200, K = 0.1))
print(Pur.wtMMnls)

Pur.wtMMnlm <- nlsLM( ~ weighted.MM(rate, conc, Vm, K), data = Treated,
                    start = list(Vm = 200, K = 0.1))
print(Pur.wtMMnlm)

Pur.wtMMnlx <- nlxb( ~ weighted.MM(rate, conc, Vm, K), data = Treated,
                    start = list(Vm = 200, K = 0.1), control=list(japprox="jafwd"))
print(Pur.wtMMnlx)

# Another approach
## Passing arguments using a list that can not be coerced to a data.frame
lisTreat <- with(Treated, list(conc1 = conc[1], conc.1 = conc[-1], rate = rate))
weighted.MM1 <- function(resp, conc1, conc.1, Vm, K){
  conc <- c(conc1, conc.1)
  pred <- (Vm * conc)/(K + conc)
  (resp - pred) / sqrt(pred)
}
Pur.wtMM1nls <- nls( ~ weighted.MM1(rate, conc1, conc.1, Vm, K),
                data = lisTreat, start = list(Vm = 200, K = 0.1))
print(Pur.wtMM1nls)
# Should we put in comparison of coeffs / sumsquares??
# stopifnot(all.equal(coef(Pur.wt), coef(Pur.wt1)))
Pur.wtMM1nlm <- nlsLM( ~ weighted.MM1(rate, conc1, conc.1, Vm, K),
                     data = lisTreat, start = list(Vm = 200, K = 0.1))
print(Pur.wtMM1nlm)
Pur.wtMM1nlx <- nlxb( ~ weighted.MM1(rate, conc1, conc.1, Vm, K),
                     data = lisTreat, start = list(Vm = 200, K = 0.1), control=list(japprox="jafwd"))
print(Pur.wtMM1nlx)
# yet another approach
## Chambers and Hastie (1992) Statistical Models in S  (p. 537):
## If the value of the right side [of formula] has an attribute called
## 'gradient' this should be a matrix with the number of rows equal
## to the length of the response and one column for each parameter.
weighted.MM.grad <- function(resp, conc1, conc.1, Vm, K) {
  conc <- c(conc1, conc.1)
  K.conc <- K+conc
  dy.dV <- conc/K.conc
  dy.dK <- -Vm*dy.dV/K.conc
  pred <- Vm*dy.dV
  pred.5 <- sqrt(pred)
  dev <- (resp - pred) / pred.5
  Ddev <- -0.5*(resp+pred)/(pred.5*pred)
  attr(dev, "gradient") <- Ddev * cbind(Vm = dy.dV, K = dy.dK)
  dev
}
Pur.wtMM.gradnls <- nls( ~ weighted.MM.grad(rate, conc1, conc.1, Vm, K),
                    data = lisTreat, start = list(Vm = 200, K = 0.1))
print(Pur.wtMM.gradnls)
Pur.wtMM.gradnlm <- nlsLM( ~ weighted.MM.grad(rate, conc1, conc.1, Vm, K),
                         data = lisTreat, start = list(Vm = 200, K = 0.1))
print(Pur.wtMM.gradnlm)
Pur.wtMM.gradnlx <- nlxb( ~ weighted.MM.grad(rate, conc1, conc.1, Vm, K),
                         data = lisTreat, start = list(Vm = 200, K = 0.1),
                         control=list(japprox="jafwd"))
print(Pur.wtMM.gradnlx)
#3 To display the coefficients for comparison:
## rbind(coef(Pur.wtMMnls), coef(Pur.wtMM1nls), coef(Pur.wtMM.gradnls))
## In this example, there seems no advantage to providing the gradient.
## In other cases, there might be.

4. Relative offset and other convergence criteria

Here we will mostly look at the relative offset convergence criterion (ROCC) that was proposed by @BatesWatts81. The primary merit of this test is that it it compares the estimated progress towards a minimum sum of squares with the current value. When this is very small, the method terminates. Both nls() and nlsr try to use this criterion for terminating the process of minimizing the objective, though there are minor differences of detail. nlsr also includes the option of turning off the ROCC with the control parameter rofftest=FALSE. nlsr::nlfb() also has a small sum of squares test: if the sum of squares is less than the fourth power of the machine precision, .Machine$double.eps, the iteration will terminate unless control smallsstest=FALSE. In some extreme problems this may be necessary, and they would likely run until limits on maximum number of evaluations of the sum of squares (control femax) or Jacobian (control jemax) are exceeded. In other words, we are dealing with problems at the edge of computability.

There is a weakness in the ROCC, in that it does not work well with problems that have a zero sum of squares as the solution, so called zero-residual problems. In October 2020, I suggested a patch to the stats::nls() function to avoid a zero-divide in the ROCC of nls(), using ideas, but not the exact implementation, already in nlsr::nlfb(). Previously, the documentation of nls() said:

Warning

The default settings of nls generally fail on small-residual problems.

The nls function uses a relative-offset convergence criterion that compares the numerical imprecision at the current parameter estimates to the residual sum-of-squares. To avoid a zero-divide in computing the test value, a positive constant convTestAdd should be added to the sum-of-squares. This quantity is set via nls.control(), as in the example below.

Overview of the ROCC test

The ROCC test uses the calculated reduction in the sum of squared residuals in the linearized sub-problem for the latest iteration and compares this to the current sum of squared residuals. The comparison is made by division of the reduction in the sum of squares to the current sum of squares. Clearly if the denominator is zero, we have the awkwardness of zero divided by zero. This is overcome if we add a positive value scaleOffset to the denominator. A default value of 0.0 for scaleOffset preserves the legacy behaviour of nls().

While I have used ROCC as if there is a single definition, nls() and nlsr::nffb() use somewhat different formulas and scalings. However, I believe they result in essentially similar outcomes once the scaleOffset safeguard is applied.

Some implementation ideas for the ROCC

Below in Section 5 we will see that the essential iteration in either a Gauss-Newton or Marquardt method finds a least squares solution to the linearized problem

$$ A \delta = b $$

where $A$ is either the Jacobian or a modification thereof for a Marquardt statbilization, and $b$ the appropriate negative of the residuals, possibly augmented with zeros according to the needs of the stabilization.

Clearly we can compute the current sum of squared as the cross product $b^T b$, since the zeros do not alter this value. It turns out the QR decomposition gives us a quite convenient computation for the reduction in this value by the current linearized sub-problem. This happens to be the sum of squares of the elements of the vector

$$ r_{proj} = Q' b $$

where $Q'$ is the transpose of the orthogonal decomposition matrix $Q$ from the decomposition.

Let us illustrate this with a linear least squares problem. Note that this problem uses an explicit column of 1s. Most "regression" calculations assume a constant term which is included automatically. Furthermore, a linear model with just a single constant minimizes the sum of squares when that constant is the mean of the dependent variable, that is the mean of column $b$. This sum of squares is generally called the "Total Sum of Squares" and any linear model with more than a constant will have a smaller some of squares. Here, and in most nonlinear models, we do not have that guarantee. Hence we will talk of the "overall" sum of squares for want of a better term, and this is simply the sum of squares of the elements of $b$. We will judge our progress by this number.

First we create some data for the $A$ and $b$.

# QRsolveEx.R -- example of solving least squares with QR
# J C Nash 2020-12-06
# Enter matrix from Compact Numerical Methods for Computers, Ex04
text <- c(563, 262, 461, 221, 1, 305,
658, 291, 473, 222, 1, 342,
676, 294, 513, 221, 1, 331,
749, 302, 516, 218, 1, 339,
834, 320, 540, 217, 1, 354,
973, 350, 596, 218, 1, 369,
1079, 386, 650, 218, 1, 378,
1151, 401, 676, 225, 1, 368,
1324, 446, 769, 228, 1, 405,
1499, 492, 870, 230, 1, 438,
1690, 510, 907, 237, 1, 438,
1735, 534, 932, 235, 1, 451,
1778, 559, 956, 236, 1, 485)
mm<-matrix(text, ncol=6, byrow=TRUE) # byrow is critical!
A <- mm[,1:5] # select the matrix
A # display
b <- mm[,6] # rhs
b
oss<-as.numeric(crossprod(b))
cat("Overall sumsquares of b =",oss,"\n")
cat("(n-1)*variance=",(length(b)-1)*var(b),"= tss = ",as.numeric(crossprod(b-mean(b))),"\n")
Aqr<-qr(A) # compute the QR decomposition of A
QAqr<-qr.Q(Aqr) # extract the Q matrix of this decomposition
RAqr<-qr.R(Aqr) # This is how to get the R matrix
XAqr<-qr.X(Aqr) # And this is the reconstruction of A from the decomposition
cat("max reconstruction error = ", max(abs(XAqr-A)),"\n")
cat("check the orthogonality Q' * Q: ", max(abs(t(QAqr)%*%QAqr-diag(rep(1,dim(A)[2])))),"\n")
cat("note failure of orthogonality of Q * Q': ",max(abs(QAqr%*%t(QAqr)-diag(rep(1,dim(A)[1])))),"\n")
Absol<-qr.solve(A,b, tol=1000*.Machine$double.eps) # solve the linear ls problem
Absol # and display it 
Abres<-qr.resid(Aqr,b)# and get the residual
rss<-as.numeric(crossprod(Abres))
cat("residual sumsquares=",rss,"\n")
Qtb<-as.vector(t(QAqr)%*%b) # Q' * b -- turns out to be reduction in sumsquares
ssQtb<-as.numeric(crossprod(Qtb))
cat("oss-ssQtb = overall sumsquares minus sumsquares Q' * b = ",oss-ssQtb,"\n")
cat("Thus reduction in sumsquares is ", ssQtb,"\n")
Qtr<-as.vector(t(QAqr)%*%Abres) # projection of residuals onto range space of A
Qtr

We thus have a quite convenient and available method to compute a relative offset test criterion if we solve our Gauss-Newton or Marquardt sub-problem with a QR decomposition.

Other convergence and termination tests

In order to catch runaway computations where some numerical imprecision causes convergence tests to fail, methods generally check the number of Jacobian or sum of squares (i.e., residual) calculations. nlsr does this with the femax and jemax elements of the control list, for which the defaults are 10000 and 5000 respectively, which is possibly overly loose. minpack.lm::nlsLM uses control list elements maxfev and maxiter (default 50) which I believe parallel femax and jemax. The default value of maxfev is 100 times the (number of parameters to estimate + 1). nls() appears to have only control element maxiter for which the default is 50 also.

While nls() and nlsr use the ROCC, minpack.lm uses a combination of tests:

5. Implementation of nonlinear least squares methods

This section is a review of approaches to solving the nonlinear least squares problem that underlies nonlinear regression modelling. In particular, we look at using a QR decomposition for the Levenberg-Marquardt stabilization of the solution of the Gauss-Newton equations.

Gauss-Newton variants

Nonlinear least squares methods are mostly founded on some or other variant of the Gauss-Newton algorithm. The function we wish to minimize is the sum of squares of the (nonlinear) residuals $r(x)$ where there are $m$ observations (elements of $r$) and $n$ parameters $x$. Hence the function is

$$ f(x) = \sum_i({r_i}^2)$$

Newton's method starts with an original set of parameters $x[0]$. At a given iteraion, which could be the first, we want to solve

$$ x[k+1] = x[k] - H^{-1} g$$

where $H$ is the Hessian and $g$ is the gradient at $x[k]$. We can rewrite this as a solution, at each iteration, of

$$ H \delta = -g$$

with

$$ x[k+1] = x[k] + \delta$$

For the particular sum of squares, the gradient is

$$ g(x) = 2 * r[k]$$

and

$$ H(x) = 2 ( J' J + \sum_i(r_i * Z_i))$$

where $J$ is the Jacobian (first derivatives of $r$ w.r.t. $x$) and $Z_i$ is the tensor of second derivatives of $r_i$ w.r.t. $x$). Note that $J'$ is the transpose of $J$.

The primary simplification of the Gauss-Newton method is to assume that the second term above is negligible. As there is a common factor of 2 on each side of the Newton iteration after the simplification of the Hessian, the Gauss-Newton iteration equation is

$$ (J' J) \delta = - J' r$$

This iteration frequently fails. The approximation of the Hessian by the Jacobian inner-product is one reason, but there is also the possibility that the sum of squares function is not "quadratic" enough that the unit step reduces it. @Hartley61 introduced a line search along delta. @Marquardt1963 suggested replacing $J' J$ with $(J' J + \lambda * D)$ where $D$ is a diagonal matrix intended to partially approximate the omitted portion of the Hessian. Choices suggested by Marquardt were $D = I$ (a unit matrix) or $D$ = (diagonal part of $J' J$). The former approach, when $\lambda$ is large enough that the iteration is essentially

$$ \delta = - g / \lambda $$

gives a version of the steepest descents algorithm. Using the diagonal of $J' J$, we have a scaled version of this (see \url{https://en.wikipedia.org/wiki/Levenberg-Marquardt_algorithm}; @Levenberg1944 predated Marquardt, but the latter seems to have done the practical work that brought the approach to general attention.)

Choices

Both nlsr::nlxb() and minpack.lm::nlsLM use a Levenberg-Marquardt stabilization of the iteration described above, with nlsr using the modification involving the $\phi$ control parameter. The complexities of the code in minpack.lm are such that I have relied largely on the documentation to judge how the iteration is accomplished. nls() uses a straightforward Hartley variant of the Gauss-Newton iteration, but rather than form the sum of squares and cross-products, uses a QR decomposition of the matrix $J$ that has been found by a forward difference approximation. The line search used by nls() is a simple back-tracking search using a step reduction factor of 0.5 as the default stepsize reduction.

@jncnm79 and @jnmws87 solve

$$ (J^T J + \lambda D_x) \delta = - J^T r $$

by the Cholesky decomposition. In this $D_x = (D + \phi * I)$ as described above and $\lambda$ is a number of modest size initially. Clearly for $\lambda = 0$ we have a Gauss-Newton method. Typically, the sum of squares of the residuals calculated at the "new" set of parameters is used as a criterion for keeping those parameter values. If so, the size of $\lambda$ is reduced. If not, we increase the size of $\lambda$ and compute a new $\delta$. Note that a new $J$, the expensive step in each iteration, is NOT required in this latter case.

In 2022, a modification to use $D_y = (\psi * D + \phi * I)$ was introduced, though the matrix equations are solved via a QR decomposition approach. Within the code, control parameters psi, phi and stepredn were introduced so that a variety of Gauss-Newton, Hartley, or Marquardt approaches are available by simple control modifications. Experience so far suggests that a Levenberg-Marquardt stabilization is much more reliable than the Gauss-Newton-Hartley choices, but that different selections of psi and phi perform rather similarly. As for Gauss-Newton methods, the details of how to start, adjust and terminate the iteration lead to many variants, increased by these different possibilities for specifying $D$. See @jncnm79.

Using matrix decompositions

In @jncnm79, the iteration equation was solved as stated. However, this involves forming the sum of squares and cross products of $J$, a process that loses some numerical precision. A better way to solve the linear equations is to apply the QR decomposition to the matrix $J$ itself. However, we still need to incorporate the $\lambda * I$ or $\lambda * D$ adjustments. This is done by adding rows to $J$ that are the square roots of the "pieces". We add 1 row for each diagonal element of $I$ and each diagonal element of $D$. Note that we also need to add a zero to the residual vector (the right-hand side) for each diagonal element.

Various authors (including the present one) have suggested different strategies for modification of $\lambda$. In the present approach, we reduce the $lambda$ parameter before solution. The initial value is the control element lamda (note the mis-spelling), which has default 1e-4. If the resulting sum of squares is not reduced, $lambda$ is increased, otherwise we move to the next iteration. My current opinion is that a "quick" increase, say a factor of 10, and a "slow" decrease, say a factor of 0.4, work quite well. However, it is worth checking that lamda has not got too small or underflowed before applying the increase factor. On the other hand, it is useful to be able to set lamda = 0 along with zero increase and decrease factors laminc and lamdec so a Hartley method can be evaluated with the program(s) by setting the control stepredn. Regularly, the current code nlfb() uses the line

if (lamda<1000*.Machine$double.eps) lamda<-1000*.Machine$double.eps

to ensure we get an increase. However, these possibilities are really for those of us playing to improve algorithms and not for practitioner use.

The Levenberg-Marquardt adjustment to the Gauss-Newton approach is the second major improvement of nlsr (and also its predecessor nlmrt and the package minpack-lm) over nls().

We could implement the methods using the equations above. However, the accumulation of inner products in $J^T J$ occasions some numerical error, and it is generally both safer and more efficient to use matrix decompositions. In particular, if we form the QR decomposition of $J$

$$ Q R = J$$

where Q is an orthonormal matrix and R is Right or Upper triangular, we can easily solve

$$ R \delta = Q^T r$$ for which the solution is also the solution of the Gauss-Newton equations. But how do we get the Marquardt stabilization?

If we augment $J$ with a square matrix $sqrt(\lambda D)$ whose diagonal elements are the square roots of $\lambda$ times the diagonal elements of $D$, and augment the vector $r$ with $n$ zeros where $n$ is the column dimension of $J$ and $D$, we achieve our goal.

Typically we can use $D = 1_n$ (the identity of order $n$), but @Marquardt1963 showed that using the diagonal elements of $J^T J$ for $D$ results in a useful implicit scaling of the parameters. @jn77ima pointed out that on computers with limited arithmetic (which now are rare since the IEEE 754 standard appeared in 1985), underflow might cause a problem of very small elements in $D$ and proposed adding $\phi 1_n$ to the diagonals of $J^T J$ before multiplying by $\lambda$ in the Marquardt stabilization. This avoids some awkward cases with very little extra overhead. It is accomplished with the QR approach by appending $sqrt(\phi * \lambda)$ times a unit matrix $I_n$ to the matrix already augmented matrix. We must also append a further $n$ zeros to the augmented $r$.

6. Fixed parameters

Motivation for fixed parameters

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

Background for fixed parameters

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

Note that the function bmchk() in package optimx contains a much more extensive examination of the bounds on parameters. In particular, it considers such issues as:

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 is outside the bounds.

From optimx the function optimr() can call many different "optimizers" (actually function minimization methods that may include bounds and possibly masks). Masks could be specified by setting the lower and upper bounds equal for the parameters to be fixed. While this may seem to be a simple method for specifying masks, there can be computational as well as conceptual difficulties. 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 choices, my current preference is to use the last one -- setting lower and upper bounds equal for fixed parameters, and furthermore setting the starting value to this fixed value, and 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.

Internal structures to specify fixed parameters

bdmsk is the internal structure used in Rcgmin and Rvmmin 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 which is now available for free download at https://archive.org/details/NLPE87plus. In particular, adding 2 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 approaches to fix parameters

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

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 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 of fixed parameters

Using Rvmmin, we easily minimize the function

$$ sq(x) = \sum_i{(i - x_i)^2} $$ a simple quadratic. The unconstrained solution has the function zero when the parameters are the sequence 1 to the number of parameters. When we fix the first parameter at 0.3, the smallest we can make the function is 0.49, i.e., $(1 - 0.3)^2$. In both cases the convergence indicator, 2, means that the gradient at the suggested solution was very small.

require(optimx)
sq<-function(x){
   nn<-length(x)
   yy<-1:nn
   f<-sum((yy-x)^2)
#   cat("Fv=",f," at ")
#   print(x)
   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)
uncans
mybm <- c(0,1) # fix parameter 1
cans <- Rvmmin(xx, sq, sq.g, bdmsk=mybm)
cans

Using the same example for nlsr::nlfb(), we get the following.

## rm(list=ls()) # clear workspace??
library(nlsr)
sqres<-function(x){
   nn<-length(x)
   yy<-1:nn
   res <- (yy-x)
}
sqjac <- function(x){
   nn<-length(x)
   yy<-1:nn
   JJ <- matrix(-1, nrow=nn, ncol=nn)
   attr(JJ, "gradient") <- JJ ## !! Critical statement
   JJ
}
xx <- c(.3, 4) # repeat for completeness
anlfbu<-nlfb(xx, sqres, sqjac)
anlfbu
anlfbc<-nlfb(xx, sqres, sqjac, lower=c(xx[1], -Inf), upper=c(xx[1], Inf), trace=FALSE)
anlfbc
# Following will warn All parameters are masked
anlfbe<-try(nlfb(xx, sqres, sqjac, lower=c(xx[1], xx[2]), upper=c(xx[1],xx[2])))

It is difficult to run this example with nlsr::nlxb(), as we need to provide a formula that spans the "data". Note that the unconstrained problem gives a warning when we try to compute a p-value for an essentially perfect minimum of the sum of squares.

?? Note that Dec 28 version does NOT work, but Dec 03 version does. Issue with changes between 03 and 28, probably in how "formula" is treated. -- see stats::model.frame -- need to have a version that handles "masked"

nn<-length(xx) # Also length(yy)
if (nn != 2) stop("This example has nn=2 only!")
yy<-1:2
v1 <- c(1,0); v2 <- c(0,1)
anlxbu <- nlxb(yy~v1*p1+v2*p2, start=c(p1=0.3, p2=4) )
anlxbu  
anlxbc <- nlxb(yy~v1*p1+v2*p2, start=c(p1=0.3, p2=4), masked=c("p1") )
anlxbc

We can also use a different example that better illustrates nlsr::nlxb(). 

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
#- Old mechanism of 'nlsr' NO longer works
#- weedcx <- nlxb(weed~b1/(1+b2*exp(-b3*ii)), start=c(b1=200, b2=50, b3=0.3), masked=c("b1")) 
weedcx <- nlxb(weed~b1/(1+b2*exp(-b3*ii)), start=c(b1=200, b2=50, b3=0.3), 
               lower=c(b1=200, b2=0, b3=0), upper=c(b1=200, b2=100, b3=40)) 
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
#- maskidx method to specify masks no longer works
#- weedcf <- nlfb(start=c(200, 50, 0.3),resfn=rfn,weed=weed, ii=ii, 
#-                maskidx=c(1), control=list(japprox="jacentral"))
weedcf <- nlfb(start=c(200, 50, 0.3),resfn=rfn,weed=weed, ii=ii, lower=c(200, 0, 0), 
               upper=c(200, 100, 40), control=list(japprox="jacentral"))
weedcf

7. Capabilities added to nlsr in the 2022 version

In the review of nonlinear modelling tools starting in December 2020, several aspects of nlsr were modified.

Numerical approximations to Jacobians

While a defining aspect of nlsr is the ability to develop analytic Jacobian functions for nlfb to use from the formula used in the call to nlxb, there are many models for which analytic derivatives are either impossible or, more commonly, simply not available through the table of derivatives in the nlsr package. Thus it is important to be able to specify an approximation.

This has been documented above in "2. Analytic versus approximate Jacobians", with examples. Note that the formula can use R functions that are not in the derivatives table if we specify a Jacobian approximation.

Self start models

nls() and nlsLM() allow for self-starting models, but they are not explicitly part of nlsr::nlxb(). nls() does not need a start argument if the formula contains as its right-hand side (rhs) the name of a selfStart function that is available in the current search path.

Such selfStart functions often also include code to compute the Jacobian, though this does not seem to be a requirement. We have also noted that some of the selfStart models -- particularly those in the base-R file ./src/library/stats/R/zzModels.R -- may simply use a crude start and a different algoithm, such as the 'plinear' option. Thus they are not really developing an approximate start, but conducting an alternative solution.

While I contemplated using the "no start argument" approach to selfStart models, the mechanism chosen to use their capabilities is to invoke the getInitial() function to set the starting parameter vector. Moreover, by setting control element japprox to 'SSJac', we use the Jacobian code available in the selfStart function.

Here is an example using the Michaelis-Menten model in the stats package.

cat("=== SSmicmen ===\n")
dat <- PurTrt <- Puromycin[ Puromycin$state == "treated", ]
frm <- rate ~ SSmicmen(conc, Vm, K)
strt<-getInitial(frm, data = dat)
print(strt)
fm1x <- nlxb(frm, data = dat, start=strt,
             trace=FALSE, control=list(japprox="SSJac"))
fm1x

We see very quick convergence using the approximation generated by the SSmicmen code.

nls(), in the absence of a selfStart model and with no start specified, puts all parameters equal to 1. Since there are situations such as the Lanczos multiple exponential model, i.e.,

y ~ a * exp(-b*x) + c * exp(-d*x) + e * exp(-f*x)

where the three terms would be equivalent with parameters a through f all at 1, a minor modification to set the parameters so they are all different seems more appropriate. While this could be automated, I prefer to insist that the user take responsibility for an initial guess in these cases.

Models with partially linear parameters

nls() offers the choice of solving a problem with partially linear parameters with a version of the Golub-Pereyra VARPRO algorithm. The choice is specified in by algorithm="plinear" in the call. (Note that algorithm="port" is also possible and replaces the default Gauss-Newton method with the nl2sol code from the Bell Port library (@PORTlib, @nl2solDGW81). However, "port" does not provide for partially linear parameters, though it does allow bounds constraints on them.)

Unfortunately, at least in my opinion, when the algorithm is changed to "plinear", the user must also change the formula that specifies the model to be fitted! This is illustrated with one of the examples from the manual page of nls(). Note how asking for the plinear option introduces another parameter into the model. We can set up the computation with this parameter explicitly included when using the default algorithm, which I believe is a more transparent choice. However, the conditional linearity (VARPRO) algorithm is generally more reliable in getting the optimal parameters in difficult cases. Ideally, the details of use of the "plinear" approach would be hidden from view. Accomplishing that remains an open issue.

## Comment in example is "## using conditional linearity"
DNase1 <- subset(DNase, Run == 1) # data
## using nls with plinear
# Using a formula that explicitly includes the asymptote. (Default algorithm.)
fm2DNase1orig <- nls(density ~ Asym/(1 + exp((xmid - log(conc))/scal)),
                 data = DNase1,
                 start = list(Asym = 10, xmid = 0, scal = 1))
summary(fm2DNase1orig)
# Using conditional linearity. Note the linear paraemter does not appear explicitly.
#  And we must change the starting vector.
fm2DNase1 <- nls(density ~ 1/(1 + exp((xmid - log(conc))/scal)),
                 data = DNase1,
                 start = list(xmid = 0, scal = 1),
                 algorithm = "plinear")
summary(fm2DNase1)
# How to run with "port" algorithm -- NOT RUN
# fm2DNase1port <- nls(density ~ Asym/(1 + exp((xmid - log(conc))/scal)),
#                  data = DNase1,
#                  start = list(Asym = 10, xmid = 0, scal = 1),
#                  algorithm = "port")
# summary(fm2DNase1port)

# require(minpack.lm) # Does NOT offer VARPRO. First example is WRONG. NOT RUN.
# fm2DNase1mA <- nlsLM(density ~ 1/(1 + exp((xmid - log(conc))/scal)),
#                  data = DNase1,
#                  start = list(xmid = 0, scal = 1))
# summary(fm2DNase1mA)
# fm2DNase1mB <- nlsLM(density ~ Asym/(1 + exp((xmid - log(conc))/scal)),
#                  data = DNase1,
#                  start = list(Asym=10, xmid = 0, scal = 1))
# summary(fm2DNase1mB)
# 
# # This gets wrong answer. NOT RUN
# fm2DNase1xA <- nlxb(density ~ 1/(1 + exp((xmid - log(conc))/scal)),
#                  data = DNase1,
#                  start = list(xmid = 0, scal = 1))
# print(fm2DNase1xA)
# Original formula with nlsr::nlxb().
fm2DNase1xB <- nlxb(density ~ Asym/(1 + exp((xmid - log(conc))/scal)),
                 data = DNase1,
                 start = list(Asym=10, xmid = 0, scal = 1))
print(fm2DNase1xB)

There are, however, problems where being able to exploit the partial linearity is an advantage. It would be very nice to be able to use the capability WITHOUT having to adjust the model specification.

8. Capabilities still missing from nlsr

This section last updated 2021-02-19.

Automatic differentiation of functional models

This is the natural extension of Multi-line expressions. The authors would welcome collaboration with someone who has expertise in this area. Some progress has been made with the autodiffr package of Changcheng Li (see https://github.com/Non-Contradiction/autodiffr).

At the time of writing, functional models require that nlxb be called with the control japprox set to an available approximating function, as discussed above.

Indexed parameters

There is an example in nls.Rd where indexed parameters are used. That is, parameters can be given a subscript e.g., b[2]. As far as I can determine, this facility is not documented, and neither minpack.lm::nlsLM nor nlsr::nlxb can do this.

Here is the example in the nls() documentation.

## The muscle dataset in MASS is from an experiment on muscle
## contraction on 21 animals.  The observed variables are Strip
## (identifier of muscle), Conc (Cacl concentration) and Length
## (resulting length of muscle section).
utils::data(muscle, package = "MASS")

## The non linear model considered is
##       Length = alpha + beta*exp(-Conc/theta) + error
## where theta is constant but alpha and beta may vary with Strip.

with(muscle, table(Strip)) # 2, 3 or 4 obs per strip

## We first use the plinear algorithm to fit an overall model,
## ignoring that alpha and beta might vary with Strip.

musc.1 <- nls(Length ~ cbind(1, exp(-Conc/th)), muscle,
              start = list(th = 1), algorithm = "plinear")
summary(musc.1)

## Then we use nls' indexing feature for parameters in non-linear
## models to use the conventional algorithm to fit a model in which
## alpha and beta vary with Strip.  The starting values are provided
## by the previously fitted model.
## Note that with indexed parameters, the starting values must be
## given in a list (with names):
b <- coef(musc.1)
musc.2 <- nls(Length ~ a[Strip] + b[Strip]*exp(-Conc/th), muscle,
              start = list(a = rep(b[2], 21), b = rep(b[3], 21), th = b[1]))
## IGNORE_RDIFF_BEGIN
summary(musc.2)
## IGNORE_RDIFF_END

The structure of the call is interesting in that start is a list and the elements of each part are NOT equal in length, as can bee seen from 

istart = list(a = rep(b[2], 21), b = rep(b[3], 21), th = b[1])
str(istart)

9. Nonlinear equations and other non-modelling problems

Solution of sets of nonlinear equations is generally NOT a problem that is commonly required for statisticians or data analysts. My experience is that the occasions where it does arise are when workers try to solve the first order conditions for optimality of a function, rather than try to optimize the function. If this function is a sum of squares, then we have a nonlinear least squares problem, and generally such problems are best approached by methods of the type discussed in this article, in particular codes nlfb and nls.lm.

Conversely, since our problem is, using the notation already established, equivalent to residuals equal to zero, namely,

r(x) = 0

the solution of a nonlinear least squares problem for which the final sum of squares is zero provides a solution to the nonlinear equations. In my experience this is a valid approach to the nonlinear equations problem, especially if there is concern that a solution may not exist. However, there are methods for nonlinear equations, some of which (e.g., @p-nleqslv) are available in R packages, and they may be more appropriate. On the other hand, if the nonlinear least squares tools are familiar, it may be more human-efficient to use them, at least as a first try.

\pagebreak

Appendix A: Providing exogenous data

These examples show dotargs do NOT work for any of nlsr, nls, or minpack.lm. Use of a dataframe or local (calling) environment objects does work in all.

# NOTE: This is OLD material and not consistent in usage with rest of vignette
library(knitr)
# try different ways of supplying data to R nls stuff
ydata <- 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)
ttdata <- seq_along(ydata) # for testing
mydata <- data.frame(y = ydata, tt = ttdata)
hobsc <- y ~ 100*b1/(1 + 10*b2 * exp(-0.1 * b3 * tt))
ste <- c(b1 = 2, b2 = 5, b3 = 3)

# let's try finding the variables
findmainenv <- function(formula, prm) {
  vn <- all.vars(formula)
  pnames <- names(prm)
  ppos <- match(pnames, vn)
  datvar <- vn[-ppos]
  cat("Data variables:")
  print(datvar)
  cat("Are the variables present in the current working environment?\n")
  for (i in seq_along(datvar)){
    cat(datvar[[i]]," : present=",exists(datvar[[i]]),"\n")
  }
}

findmainenv(hobsc, ste)
y <- ydata
tt <- ttdata
findmainenv(hobsc, ste)
rm(y)
rm(tt)

# ===============================


# let's try finding the variables in dotargs
finddotargs <- function(formula, prm, ...) {
  dots <- list(...)
  cat("dots:")
  print(dots)
  cat("names in dots:")
  dtn <- names(dots)
  print(dtn)
  vn <- all.vars(formula)
  pnames <- names(prm)
  ppos <- match(pnames, vn)
  datvar <- vn[-ppos]
  cat("Data variables:")
  print(datvar)
  cat("Are the variables present in the dot args?\n")
  for (i in seq_along(datvar)){
    dname <- datvar[[i]]
    cat(dname," : present=",(dname %in% dtn),"\n")
  }
}

finddotargs(hobsc, ste, y=ydata, tt=ttdata)
# ===============================

y <- ydata
tt <- ttdata
tryq <- try(nlsquiet <- nls(formula=hobsc, start=ste))
if (class(tryq) != "try-error") {print(nlsquiet)} else {cat("try-error\n")}
#- OK
rm(y); rm(tt)

## this will fail
tdots1<-try(nlsdots <- nls(formula=hobsc, start=ste, y=ydata, tt=ttdata))
# But ...
y <- ydata # put data in globalenv
tt <- ttdata
tdots2<-try(nlsdots <- nls(formula=hobsc, start=ste))
tdots2
rm(y); rm(tt)
## but ...
mydata<-data.frame(y=ydata, tt=ttdata)
tframe <- try(nlsframe <- nls(formula=hobsc, start=ste, data=mydata))
if (class(tframe) != "try-error") {print(nlsframe)} else {cat("try-error\n")}
#- OK

y <- ydata
tt <- ttdata
tquiet <- try(nlsrquiet <- nlxb(formula=hobsc, start=ste))
if ( class(tquiet) != "try-error") {print(nlsrquiet)}
#- OK
rm(y); rm(tt)
test <- try(nlsrdots <- nlxb(formula=hobsc, start=ste, y=ydata, tt=ttdata))
#- Note -- does NOT work -- do we need to specify the present env. in nlfb for y, tt??
if (class(test) != "try-error") { print(nlsrdots) } else {cat("Try error\n") }
test2 <- try(nlsframe <- nls(formula=hobsc, start=ste, data=mydata))
if (class(test) != "try-error") {print(nlsframe) } else {cat("Try error\n") }
#- OK


library(minpack.lm)
y <- ydata
tt <- ttdata
nlsLMquiet <- nlsLM(formula=hobsc, start=ste)
print(nlsLMquiet)
#- OK
rm(y)
rm(tt)
## Dotargs
## Following fails
## tdots <- try(nlsLMdots <- nlsLM(formula=hobsc, start=ste, y=ydata, tt=ttdata))
## but ... 
tdots <- try(nlsLMdots <- nlsLM(formula=hobsc, start=ste, data=mydata))
if (class(tdots) != "try-error") { print(nlsLMdots) } else {cat("try-error\n") }
#-  Note -- does NOT work
## dataframe
tframe <- try(nlsLMframe <- nlsLM(formula=hobsc, start=ste, data=mydata) )
if (class(tdots) != "try-error") {print(nlsLMframe)} else {cat("try-error\n") }
#- does not work

## detach("package:nlsr", unload=TRUE)
## Uses nlmrt here for comparison
## library(nlmrt)
## txq <- try( nlxbquiet <- nlxb(formula=hobsc, start=ste))
## if (class(txq) != "try-error") {print(nlxbquiet)} else { cat("try-error\n")}
#- Note -- does NOT work
## txdots <- try( nlxbdots <- nlxb(formula=hobsc, start=ste, y=y, tt=tt) )
## if (class(txdots) != "try-error") {print(nlxbdots)} else {cat("try-error\n")}
#- Note -- does NOT work
## dataframe
## nlxbframe <- nlxb(formula=hobsc, start=ste, data=mydata)
## print(nlxbframe)
#- OK

\pagebreak

Appendix B: Derivative approximation in nls()

This Appendix could benefit from some examples.

From nls.R

numericDeriv <- function(expr, theta, rho = parent.frame(), dir=1.0)
{
    dir <- rep_len(dir, length(theta))
    val <- .Call(C_numeric_deriv, expr, theta, rho, dir)
    valDim <- dim(val)
    if (!is.null(valDim)) {
        if (valDim[length(valDim)] == 1)
            valDim <- valDim[-length(valDim)]
        if(length(valDim) > 1L)
            dim(attr(val, "gradient")) <- c(valDim,
                                            dim(attr(val, "gradient"))[-1L])
    }
    val
}

From nls.c

/*
 *  call to numeric_deriv from R -
 *  .Call("numeric_deriv", expr, theta, rho)
 *  Returns: ans
 */
SEXP
numeric_deriv(SEXP expr, SEXP theta, SEXP rho, SEXP dir)
{
    SEXP ans, gradient, pars;
    double eps = sqrt(DOUBLE_EPS), *rDir;
    int start, i, j, k, lengthTheta = 0;

    if(!isString(theta))
    error(_("'theta' should be of type character"));
    if (isNull(rho)) {
    error(_("use of NULL environment is defunct"));
    rho = R_BaseEnv;
    } else
    if(!isEnvironment(rho))
        error(_("'rho' should be an environment"));
    PROTECT(dir = coerceVector(dir, REALSXP));
    if(TYPEOF(dir) != REALSXP || LENGTH(dir) != LENGTH(theta))
    error(_("'dir' is not a numeric vector of the correct length"));
    rDir = REAL(dir);

    PROTECT(pars = allocVector(VECSXP, LENGTH(theta)));

    PROTECT(ans = duplicate(eval(expr, rho)));

    if(!isReal(ans)) {
    SEXP temp = coerceVector(ans, REALSXP);
    UNPROTECT(1);
    PROTECT(ans = temp);
    }
    for(i = 0; i < LENGTH(ans); i++) {
    if (!R_FINITE(REAL(ans)[i]))
        error(_("Missing value or an infinity produced when evaluating the model"));
    }
    const void *vmax = vmaxget();
    for(i = 0; i < LENGTH(theta); i++) {
    const char *name = translateChar(STRING_ELT(theta, i));
    SEXP s_name = install(name);
    SEXP temp = findVar(s_name, rho);
    if(isInteger(temp))
        error(_("variable '%s' is integer, not numeric"), name);
    if(!isReal(temp))
        error(_("variable '%s' is not numeric"), name);
    if (MAYBE_SHARED(temp)) /* We'll be modifying the variable, so need to make sure it's unique PR#15849 */
        defineVar(s_name, temp = duplicate(temp), rho);
    MARK_NOT_MUTABLE(temp);
    SET_VECTOR_ELT(pars, i, temp);
    lengthTheta += LENGTH(VECTOR_ELT(pars, i));
    }
    vmaxset(vmax);
    PROTECT(gradient = allocMatrix(REALSXP, LENGTH(ans), lengthTheta));

    for(i = 0, start = 0; i < LENGTH(theta); i++) {
    for(j = 0; j < LENGTH(VECTOR_ELT(pars, i)); j++, start += LENGTH(ans)) {
        SEXP ans_del;
        double origPar, xx, delta;

        origPar = REAL(VECTOR_ELT(pars, i))[j];
        xx = fabs(origPar);
        delta = (xx == 0) ? eps : xx*eps;
        REAL(VECTOR_ELT(pars, i))[j] += rDir[i] * delta;
        PROTECT(ans_del = eval(expr, rho));
        if(!isReal(ans_del)) ans_del = coerceVector(ans_del, REALSXP);
        UNPROTECT(1);
        for(k = 0; k < LENGTH(ans); k++) {
        if (!R_FINITE(REAL(ans_del)[k]))
            error(_("Missing value or an infinity produced when evaluating the model"));
        REAL(gradient)[start + k] =
            rDir[i] * (REAL(ans_del)[k] - REAL(ans)[k])/delta;
        }
        REAL(VECTOR_ELT(pars, i))[j] = origPar;
    }
    }
    setAttrib(ans, install("gradient"), gradient);
    UNPROTECT(4);
    return ans;
}

nlsr::numericDerivR.R

This is a replacement for the nls() function numericDeriv() that is coded all in R.

Let us compute the Jacobian for the Hobbs problem with this function and its nls() original. I find the mechanism for these functions awkward.

library(nlsr) # so we have numericDerivR code
# Data for Hobbs problem
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) # for testing
tt  <-  seq_along(weed) # for testing
# A simple starting vector -- must have named parameters for nlxb, nls, wrapnlsr.
st  <-  c(b1=1, b2=1, b3=1)
wmodu  <-   weed ~ b1/(1+b2*exp(-b3*tt))
weeddf  <-  data.frame(weed=weed, tt=tt)
weedenv <- list2env(weeddf)
weedenv$b1 <- st[[1]]
weedenv$b2 <- st[[2]]
weedenv$b3 <- st[[3]]
rexpr<-call("-",wmodu[[3]], wmodu[[2]])
r0<-eval(rexpr, weedenv)
cat("Sumsquares at 1,1,1 is ",sum(r0^2),"\n")## Another way
expr <- wmodu
rho <- weedenv
rexpr<-call("-",wmodu[[3]], wmodu[[2]])
res0<-eval(rexpr, rho) # the base residuals
res0
## Try the numericDeriv option
theta<-names(st)
nDnls<-numericDeriv(rexpr, theta, weedenv)
nDnls
nDnlsR<-numericDerivR(rexpr, theta, weedenv)
nDnlsR

\pagebreak

Appendix C: A comparison of nlsr::nlxb with nls and minpack::nlsLM

R has several tools for estimating nonlinear models and minimizing sums of squares functions. Sometimes we talk of nonlinear regression and at other times of minimizing a sum of squares function. Many workers conflate these two tasks. In this appendix, some of the differences between the tools available in R for these two computational tasks are highlighted. In particular, we compare the tools from the package nlsr (@nlsr2019manual), particularly function nlxb() with those from base-R nls() and the nlsLM function of package minpack.lm (@minpacklm12). We also compare how nlsr:nlfb() and minpack.lm:nls.lm allow a sum of squares function to be minimized.

Principal differences

The main differences in the tools relate to the following features:

Derivative information

As detailed above, nlsr::nlxb() attempts to use symbolic and algorithmic tools to obtain the derivatives of the model expression that are needed for the Jacobian matrix that is used in creating a linearized sub-problem at each iteration of an attempted solution of the minimization of the sum of squared residuals. As discussed in the section "Analytic versus approximate Jacobians" and using the code in Appendix B, nls() and minpack.lm::nlsLM() use a very simple forward-difference approximation for the partial derivatives for the Jacobian.

Forward difference approximations are less accurate than central differences, and both are subject to numerical error when the modelling function is "flat", so that there is a large amount of digit cancellation in the subtraction necessary to compute the derivative approximation.

minpack.lm::nlsLM uses the same derivatives as far as I can determine. The loss of information compared to the analytic or algorithmic derivatives of nlsr::nlxb() is important in that it can lead to Jacobian matrices that are computationally singular, where nls() will stop with "singular gradient". (It is actually the Jacobian which is singular here, and I will stay with that terminology.) minpack.lm::nlsLM() may fail to get started if the initial Jacobian is singular, but is less susceptible in general, as described in the sub-section on Marquardt stabilization which follows.

Consequences of different derivative computations

While readers might expect that the precise derivative information of nlsr::nlxb() would mean a faster solution, this is quite often not the case. Approximate derivatives may allow faster approach to the solution by "ironing out" wrinkles in the function surface. In my opinion, the main advantage of precise derivative information is in testing that we actually have arrived at a solution.

There are even some cases where the approximation may be helpful, though users may not realize the potential danger. Thanks to Karl Schilling for an example of modelling with the function

a * (x ^ b)

where x is our data and we wish to estimate a and b. Now the partial derivative of this function w.r.t. b is 

partialderiv <- D(expression(a * (x ^ b)),"b")
print(partialderiv)

The danger here is that we may have data values x = 0, in which case the derivative is not defined, though the model can still be evaluated. Thus nlsr::nlxb() will not compute a solution, while nls() and minpack.lm::nlsLM() will generally proceed. A workaround is to provide a very small value instead of zero for the data, though I find this inelegant. Another approach is to drop the offending element of the data, though this risks altering the model estimated. A proper treatment might be to develop the limit of the derivative as the data value goes to zero, but finding general software that can detect and deal with this is a large project.

Timing comparisons

Let us compare timings on the (scaled) Hobbs weed problem. 

require(microbenchmark)
## nls on Hobbs scaled model
wmods  <-   weed ~ 100*b1/(1+10*b2*exp(-0.1*b3*tt))
stx<-c(b1=2, b2=5, b3=3)
tnls<-microbenchmark((anls<-nls(wmods, start=stx, data=weeddf)), unit="us")
tnls
## nlsr::nlfb() on Hobbs scaled model
tnlxb<-microbenchmark((anlxb<-nlsr::nlxb(wmods, start=stx, data=weeddf)), unit="us")
tnlxb
## minpack.lm::nlsLM() on Hobbs scaled model
tnlsLM<-microbenchmark((anlsLM<-minpack.lm::nlsLM(start=stx, formula=wmods, data=weeddf)), 
                       unit="us")
tnlsLM

Programmatic modelling functions

A consequence of the symbolic derivative approach in nlsr::nlxb() is that it cannot be applied to a modelling expression that includes an R function i.e., sub-program.

This limitation could be overcome using appropriate automatic differentiation code (to provide derivative computations based on transformation of the modelling function's programmatic form). The present work-around is to use numerical approximation by specifying the control element japprox.

Functional expression of residuals and Jacobian

require(microbenchmark)
## nlsr::nlfb() on Hobbs scaled
tnlfb<-microbenchmark((anlfb<-nlsr::nlfb(start=st1, resfn=shobbs.res, jacfn=shobbs.jac)), unit="us")
tnlfb
## minpack.lm::nls.lm() on Hobbs scaled
tnls.lm<-microbenchmark((anls.lm<-minpack.lm::nls.lm(par=st1, fn=shobbs.res, jac=shobbs.jac)))
tnls.lm

Marquardt stabilization

All three of the R functions under consideration try to minimize a sum of squares. If the model is provided in the form

y ~ (some expression)

then the residuals are computed by evaluating the difference between (some expression) and y. My own preference, and that of K F Gauss, is to use (some expression) - y. This is to avoid having to be concerned with the negative sign -- the derivative of the residual defined in this way is the same as the derivative of the modelling function, and we avoid the chance of a sign error. The Jacobian matrix is made up of elements where element i, j is the partial derivative of residual i w.r.t. parameter j.

nls() attempts to minimize a sum of squared residuals by a Gauss-Newton method. If we compute a Jacobian matrix J and a vector of residuals r from a vector of parameters x, then we can define a linearized problem

$$ J^T J \delta = - J^T r $$

This leads to an iteration where, from a set of starting parameters x0, we compute

$$ x_{i+1} = x_i + \delta$$

This is commonly modified to use a step factor step

$$ x_{i+1} = x_i + step * \delta$$

It is in the mechanisms to choose the size of step and to decide when to terminate the iteration that Gauss-Newton methods differ. Indeed, though I have tried several times, I find the very convoluted code behind nls() very difficult to decipher. Unfortunately, its authors have (at 2018 as far as I am aware) all ceased to maintain the code.

Both nlsr::nlxb() and minpack.lm::nlsLM use a Levenberg-Marquardt stabilization of the iteration. (@Marquardt1963, @Levenberg1944), solving

$$ (J^T J + \lambda D) \delta = - J^T r $$

where $D$ is some diagonal matrix and lambda is a number of modest size initially. Clearly for $\lambda = 0$ we have a Gauss-Newton method. Typically, the sum of squares of the residuals calculated at the "new" set of parameters is used as a criterion for keeping those parameter values. If so, the size of $\lambda$ is reduced. If not, we increase the size of $\lambda$ and compute a new $\delta$. Note that a new $J$, the expensive step in each iteration, is NOT required.

As for Gauss-Newton methods, the details of how to start, adjust and terminate the iteration lead to many variants, increased by different possibilities for specifying $D$. See @jncnm79. There are also a number of ways to solve the stabilized Gauss-Newton equations, some of which do not require the explicit $J^T J$ matrix.

Criterion used to terminate the iteration

nls() and nlsr use a form of the relative offset convergence criterion, @BatesWatts81. minpack.lm uses a somewhat different and more complicated set of tests. Unfortunately, the relative offset criterion as implemented in nls() is unsuited to problems where the residuals can be zero. As of R 4.1.0, there is a work-around in providing a non-zero value to the control element scaleOffset as documented in the manual page of nls(). See An illustrative nonlinear regression problem below.

Output of the modelling functions

nls() and nlsLM() return the same solution structure. Let us examine this for one of our example results (we will choose one that does NOT have small residuals, so that all the functions "work").

# Here we set up an example problem with data
# Define independent variable
t0 <- 0:19
t0a<-t0
t0a[1]<-1e-20 # very small value
# Drop first value in vectors
t0t<-t0[-1]
y1 <- 4 * (t0^0.25)
y1t<-y1[-1]
n <- length(t0)
fuzz <- rnorm(n)
range <- max(y1)-min(y1)
## add some "error" to the dependent variable
y1q <- y1 + 0.2*range*fuzz
edta <- data.frame(t0=t0, t0a=t0a, y1=y1, y1q=y1q)
edtat <- data.frame(t0t=t0t, y1t=y1t)

start1 <- c(a=1, b=1)
try(nlsy0t0ax <- nls(formula=y1~a*(t0a^b), start=start1, data=edta, control=nls.control(maxiter=10000)))
nlsry1t0a <- nlxb(formula=y1~a*(t0a^b), start=start1, data=edta)
library(minpack.lm)
nlsLMy1t0a <- nlsLM(formula=y1~a*(t0a^b), start=start1, data=edta)

str(nlsy0t0ax)

The minpack.lm::nlsLM output has the same structure, which could be revealed by the R command str(nlsLMy1t0a). Note that this structure has a lot of special functions in the sub-list m. By contrast, the nlsr() output is much less flamboyant. There are, in fact, no functions as part of the structure. 

str(nlsry1t0a)

Which of these approaches is "better" can be debated. My preference is for the results of optimization computations to be essentially data, including messages, though some tools within some of my packages will return functions for specific reasons, e.g., to return a function from an expression. However, I prefer to use specified functions such as predict.nlsr() below to obtain predictions. I welcome comment and discussion, as this is not, in my view, a closed topic.

Prediction

Let us predict our models at the mean of the data. Because nlxb() returns a different structure from that found by nls() and nlsLM() the code for predict() for an object from nlsr is different. minpack.lm uses predict.nls since the output structure of the modelling step is equivalent to that from nls().

nudta <- colMeans(edta)
predict(nlsy0t0ax, newdata=nudta)
predict(nlsLMy1t0a, newdata=nudta)
predict(nlsry1t0a, newdata=nudta)

An illustrative nonlinear regression problem

So we can illustrate some of the issues, let us create some example data for a seemingly straightforward computational problem.

# Here we set up an example problem with data
# Define independent variable
t0 <- 0:19
t0a<-t0
t0a[1]<-1e-20 # very small value
# Drop first value in vectors
t0t<-t0[-1]
y1 <- 4 * (t0^0.25)
y1t<-y1[-1]
n <- length(t0)
fuzz <- rnorm(n)
range <- max(y1)-min(y1)
## add some "error" to the dependent variable
y1q <- y1 + 0.2*range*fuzz
edta <- data.frame(t0=t0, t0a=t0a, y1=y1, y1q=y1q)
edtat <- data.frame(t0t=t0t, y1t=y1t)

Let us try this example modelling y0 against t0. Note that this is a zero-residual problem, so nls() should complain or fail, which it appears to do by exceeding the iteration limit, which is not very communicative of the underlying issue. The nls() documentation warns

"Warning

Do not use nls on artificial "zero-residual" data."

It goes on to recommend that users add "error" to the data to avoid such problems. I feel this is a very unsatisfactory kludge. It is NOT due to a genuine mathematical issue, but due to the relative offset convergence criterion used to terminate the method. Using the contr

Here is the output.

nls

cprint <- function(obj){
   # print object if it exists
  sobj<-deparse(substitute(obj))
  if (exists(sobj)) {
      print(obj)
  } else {
      cat(sobj," does not exist\n")
  }
#  return(NULL)
}
start1 <- c(a=1, b=1)
try(nlsy0t0 <- nls(formula=y1~a*(t0^b), start=start1, data=edta))
cprint(nlsy0t0)
# Since this fails to converge, let us increase the maximum iterations
try(nlsy0t0x <- nls(formula=y1~a*(t0^b), start=start1, data=edta,
                    control=nls.control(maxiter=10000)))
cprint(nlsy0t0x)
try(nlsy0t0ax <- nls(formula=y1~a*(t0a^b), start=start1, data=edta, 
                     control=nls.control(maxiter=10000)))
cprint(nlsy0t0ax)
try(nlsy0t0t <- nls(formula=y1t~a*(t0t^b), start=start1, data=edtat))
cprint(nlsy0t0t)

nlsr

nlsry1t0 <- try(nlxb(formula=y1~a*(t0^b), start=start1, data=edta))
cprint(nlsry1t0)
nlsry1t0a <- nlxb(formula=y1~a*(t0a^b), start=start1, data=edta)
cprint(nlsry1t0a)
nlsry1t0t <- nlxb(formula=y1t~a*(t0t^b), start=start1, data=edtat)
cprint(nlsry1t0t)

minpack.lm

library(minpack.lm)
nlsLMy1t0 <- nlsLM(formula=y1~a*(t0^b), start=start1, data=edta)
nlsLMy1t0
nlsLMy1t0a <- nlsLM(formula=y1~a*(t0a^b), start=start1, data=edta)
nlsLMy1t0a
nlsLMy1t0t <- nlsLM(formula=y1t~a*(t0t^b), start=start1, data=edtat)
nlsLMy1t0t

We have seemingly found a workaround for our difficulty, but I caution that initially I found very unsatisfactory results when I set the "very small value" to 1.0e-7. The correct approach is clearly to understand what is going on. Getting computers to provide that understanding is a serious challenge.

Problems that are NOT regressions

Some nonlinear least squares problems are NOT nonlinear regressions. That is, we do not have a formula y ~ (some function) to define the problem. This is another reason to use the residual in the form (some function) - y In many cases of interest we have no y.

The Brown and Dennis test problem (@More1981TUO, problem 16) is of this form. Suppose we have m observations, then we create a scaled index t which is the "data" for the function. To run the nonlinear least squares functions that use a formula, we do, however, need a "y" variable. Clearly adding zero to the residual will not change the problem, so we set the data for "y" as all zeros. Note that nls() and nlsLM() need some extra iterations to find the solution to this somewhat nasty problem.

m <- 20
t <- seq(1, m) / 5
y <- rep(0,m)
library(nlsr)
library(minpack.lm)

bddata <- data.frame(t=t, y=y)
bdform <- y ~ ((x1 + t * x2 - exp(t))^2 + (x3 + x4 * sin(t) - cos(t))^2)
prm0 <- c(x1=25, x2=5, x3=-5, x4=-1)
fbd <-model2ssgrfun(bdform, prm0, bddata)
cat("initial sumsquares=",as.numeric(crossprod(fbd(prm0))),"\n")
nlsrbd <- nlxb(bdform, start=prm0, data=bddata, trace=FALSE)
nlsrbd

nlsbd10k <- nls(bdform, start=prm0, data=bddata, trace=FALSE, 
                control=nls.control(maxiter=10000))
nlsbd10k

nlsLMbd10k <- nlsLM(bdform, start=prm0, data=bddata, trace=FALSE, 
                    control=nls.lm.control(maxiter=10000, maxfev=10000))
nlsLMbd10k

Let us try predicting the "residual" for some new data.

ndata <- data.frame(t=c(5,6), y=c(0,0))
predict(nlsLMbd10k, newdata=ndata)

# now nls
predict(nlsbd10k, newdata=ndata)

# now nlsr
predict(nlsrbd, newdata=ndata)

We could, of course, try setting up a different formula, since the "residuals" can be computed in any way such that their absolute value is the same. Therefore we could try moving the exponential part of the function for each equation to the left hand side as in bdf2 below.

bdf2 <-  (x1 + t * x2 - exp(t))^2 ~ - (x3 + x4 * sin(t) - cos(t))^2

However, we discover that the parsing of the model formula fails for this formulation.

A check on the Brown and Dennis calculation via function minimization

We can attack the Brown and Dennis problem by applying nonlinear function minimization programs to the sum of squared "residuals" as a function of the parameters. The code below does this. We omit the output for space reasons.

#' Brown and Dennis Function
#'
#' Test function 16 from the More', Garbow and Hillstrom paper.
#'
#' The objective function is the sum of \code{m} functions, each of \code{n}
#' parameters.
#'
#' \itemize{
#'   \item Dimensions: Number of parameters \code{n = 4}, number of summand
#'   functions \code{m >= n}.
#'   \item Minima: \code{f = 85822.2} if \code{m = 20}.
#' }
#'
#' @param m Number of summand functions in the objective function. Should be
#'   equal to or greater than 4.
#' @return A list containing:
#' \itemize{
#'   \item \code{fn} Objective function which calculates the value given input
#'   parameter vector.
#'   \item \code{gr} Gradient function which calculates the gradient vector
#'   given input parameter vector.
#'   \item \code{fg} A function which, given the parameter vector, calculates
#'   both the objective value and gradient, returning a list with members
#'   \code{fn} and \code{gr}, respectively.
#'   \item \code{x0} Standard starting point.
#' }
#' @references
#' More', J. J., Garbow, B. S., & Hillstrom, K. E. (1981).
#' Testing unconstrained optimization software.
#' \emph{ACM Transactions on Mathematical Software (TOMS)}, \emph{7}(1), 17-41.
#' \url{https://doi.org/10.1145/355934.355936}
#'
#' Brown, K. M., & Dennis, J. E. (1971).
#' \emph{New computational algorithms for minimizing a sum of squares of
#' nonlinear functions} (Report No. 71-6).
#' New Haven, CT: Department of Computer Science, Yale University.
#'
#' @examples
#' # Use 10 summand functions
#' fun <- brown_den(m = 10)
#' # Optimize using the standard starting point
#' x0 <- fun$x0
#' res_x0 <- stats::optim(par = x0, fn = fun$fn, gr = fun$gr, method =
#' "L-BFGS-B")
#' # Use your own starting point
#' res <- stats::optim(c(0.1, 0.2, 0.3, 0.4), fun$fn, fun$gr, method =
#' "L-BFGS-B")
#'
#' # Use 20 summand functions
#' fun20 <- brown_den(m = 20)
#' res <- stats::optim(fun20$x0, fun20$fn, fun20$gr, method = "L-BFGS-B")
#' @export
#`
brown_den <- function(m = 20) {
  list(
    fn = function(par) {
      x1 <- par[1]
      x2 <- par[2]
      x3 <- par[3]
      x4 <- par[4]

      ti <- (1:m) * 0.2
      l <- x1 + ti * x2 - exp(ti)
      r <- x3 + x4 * sin(ti) - cos(ti)
      f <- l * l + r * r
      sum(f * f)
    },
    gr = function(par) {
      x1 <- par[1]
      x2 <- par[2]
      x3 <- par[3]
      x4 <- par[4]

      ti <- (1:m) * 0.2
      sinti <- sin(ti)
      l <- x1 + ti * x2 - exp(ti)
      r <- x3 + x4 * sinti - cos(ti)
      f <- l * l + r * r
      lf4 <- 4 * l * f
      rf4 <- 4 * r * f
      c(
        sum(lf4),
        sum(lf4 * ti),
        sum(rf4),
        sum(rf4 * sinti)
      )
    },
    fg = function(par) {
      x1 <- par[1]
      x2 <- par[2]
      x3 <- par[3]
      x4 <- par[4]

      ti <- (1:m) * 0.2
      sinti <- sin(ti)
      l <- x1 + ti * x2 - exp(ti)
      r <- x3 + x4 * sinti - cos(ti)
      f <- l * l + r * r
      lf4 <- 4 * l * f
      rf4 <- 4 * r * f

      fsum <- sum(f * f)
      grad <- c(
        sum(lf4),
        sum(lf4 * ti),
        sum(rf4),
        sum(rf4 * sinti)
      )

      list(
        fn = fsum,
        gr = grad
      )
    },
    x0 = c(25, 5, -5, 1)
  )
}
mbd <- brown_den(m=20)
mbd
mbd$fg(mbd$x0)
bdsolnm <- optim(mbd$x0, mbd$fn, control=list(trace=0))
bdsolnm
bdsolbfgs <- optim(mbd$x0, mbd$fn, method="BFGS", control=list(trace=0))
bdsolbfgs

library(optimx)
methlist <- c("Nelder-Mead","BFGS","Rvmmin","L-BFGS-B","Rcgmin","ucminf")

solo <- opm(mbd$x0, mbd$fn, mbd$gr, method=methlist, control=list(trace=0))
summary(solo, order=value)

## A failure above is generally because a package in the 'methlist' is not installed.

\pagebreak

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