Tools for solving nonlinear least squares problems. UNDER DEVELOPMENT.

Description

The package provides some tools related to using the Nash variant of Marquardt's algorithm for nonlinear least squares.

Details

Package: nlmrt
Type: Package
Version: 1.0
Date: 2012-03-05
License: GPL-2

This package includes methods for solving nonlinear least squares problems specified by a modeling expression and given a starting vector of named paramters. Note: You must provide an expression of the form lhs ~ rhsexpression so that the residual expression rhsexpression - lhs can be computed. The expression can be enclosed in quotes, and this seems to give fewer difficulties with R. Data variables must already be defined, either within the parent environment or else in the dot-arguments. Other symbolic elements in the modeling expression must be standard functions or else parameters that are named in the start vector.

The main functions in nlmrt are:

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   nlfb - Nash variant of the Marquardt procedure for nonlinear least squares,
	with bounds constraints, using a residual and optionally Jacobian
	described as \code{R} functions. 
    20120803: Todo: Make masks more consistent between nlfb and nlxb.

   nlxb - Nash variant of the Marquardt procedure for nonlinear least squares,
	with bounds constraints, using an expression to describe the residual via
        an \code{R} modeling expression. The Jacobian is computed via symbolic
	differentiation.
            
   wrapnls - Uses nlxb to solve nonlinear least squares then calls nls() to
            create an object of type nls.

   model2grfun.R - Generate a gradient vector function from a nonlinear 
        model expression and a vector of named parameters.

   model2jacfun.R - Generate a Jacobian matrix function from a nonlinear 
        model expression and a vector of named parameters.

   model2resfun.R - Generate a residual vector function from a nonlinear 
        model expression and a vector of named parameters.

   model2ssfun.R - Generate a sum of squares objective function from a 
        nonlinear model expression and a vector of named parameters.

   modgr.R - compute gradient of the sum of squares function using the 
        Jacobian and residuals for a nonlinear least squares problem
      
   modss.R - computer the sum of squares function from the residuals of
        a nonlinear least squares problem

   myfn.R, mygr.R, myjac.R, myres.R, myss.R - dummy functions that seem to
       be needed so there is an available handle for output of functions that
       generate various functions from expressions.

For testing purposes, there are also some experimental codes using different internal computations for the linear algebraic sub-problems in the inst/dev-codes/ sub-folder.

Author(s)

John C Nash

Maintainer: <nashjc@uottawa.ca>

References

Nash, J. C. (1979, 1990) _Compact Numerical Methods for Computers. Linear Algebra and Function Minimisation._ Adam Hilger./Institute of Physics Publications

others!!??

See Also

nls

Examples

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rm(list=ls())
# library(nlmrt)

# traceval set TRUE to debug or give full history
traceval  <-  FALSE

## Problem in 1 parameter to ensure methods work in trivial case

cat("Problem in 1 parameter to ensure methods work in trivial case\n")
nobs<-8
tt <- seq(1,nobs)
dd <- 1.23*tt + 4*runif(nobs)

df <- data.frame(tt, dd)

a1par<-nlxb(dd ~ a*tt, start=c(a=1), data=df)
a1par


# Data for Hobbs problem
cat("Hobbs weed problem -- unscaled\n")
ydat  <-  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
y  <-  ydat  # for testing
tdat  <-  seq_along(ydat) # for testing

eunsc  <-   y ~ b1/(1+b2*exp(-b3*tt))

cat("Hobbs unscaled with data in data frames\n")

weeddata1  <-  data.frame(y=ydat, tt=tdat)
# scale the data 
weeddata2  <-  data.frame(y=1.5*ydat, tt=tdat)
start1  <-  c(b1=1, b2=1, b3=1)
anlxb1  <-  try(nlxb(eunsc, start=start1, trace=traceval, data=weeddata1))
print(anlxb1)

anlxb2  <-  try(nlxb(eunsc, start=start1, trace=traceval, data=weeddata2))
print(anlxb2)

# Problem 2 - Gabor Grothendieck 2016-3-2

cat("Gabor G problem with zero residuals\n")

DF <- data.frame(x = c(5, 4, 3, 2, 1), y = c(1, 2, 3, 4, 5))
library(nlmrt)
nlxb1 <- nlxb(y ~ A * x + B, data = DF, start = c(A = 1, B = 6), trace=TRUE)
print(nlxb1)

# tmp <- readline("continue with start at the minimum -- failed on 2014 version. ")

nlxb0 <- nlxb(y ~ A * x + B, data = DF, start = c(A = -1, B = 6), trace=TRUE)
print(nlxb0) 

## Not run: 
# WARNING -- using T can get confusion with TRUE
tt  <-  tdat
# A simple starting vector -- must have named parameters for nlxb, nls, wrapnls.

cat("GLOBAL DATA\n")

anls1g  <-  try(nls(eunsc, start=start1, trace=traceval))
print(anls1g)

cat("GLOBAL DATA AND EXPRESSION -- SHOULD FAIL\n")
anlxb1g  <-  try(nlxb(eunsc, start=start1, trace=traceval))
print(anlxb1g)


## End(Not run) # end dontrun

rm(y)
rm(tt)


startf1  <-  c(b1=1, b2=1, b3=.1)


## Not run: 

## With BOUNDS

anlxb1  <-  try(nlxb(eunsc, start=startf1, lower=c(b1=0, b2=0, b3=0), 
      upper=c(b1=500, b2=100, b3=5), trace=traceval, data=weeddata1))
print(anlxb1)


## End(Not run) # end dontrun


# Check nls too
## Not run: 
cat("check nls result\n")
anlsb1  <-  try(nls(eunsc, start=start1, lower=c(b1=0, b2=0, b3=0), 
     upper=c(b1=500, b2=100, b3=5), trace=traceval, data=weeddata1, 
             algorithm='port'))
print(anlsb1)

# tmp  <-  readline("next")


## End(Not run) # end dontrun

## Not run: 

anlxb2  <-  try(nlxb(eunsc, start=start1, lower=c(b1=0, b2=0, b3=0), 
        upper=c(b1=500, b2=100, b3=.25), trace=traceval, data=weeddata1))
print(anlxb2)


anlsb2  <-  try(nls(eunsc, start=start1, lower=c(b1=0, b2=0, b3=0), 
                upper=c(b1=500, b2=100, b3=.25), trace=traceval, 
                data=weeddata1, algorithm='port'))
print(anlsb2)

# tmp  <-  readline("next")

## End(Not run) # end dontrun


## Not run: 
cat("UNCONSTRAINED\n")
an1q  <-  try(nlxb(eunsc, start=start1, trace=traceval, data=weeddata1))
print(an1q)
# tmp  <-  readline("next")

## End(Not run) # end dontrun


## Not run: 
cat("TEST MASKS\n")

anlsmnqm  <-  try(nlxb(eunsc, start=start1, lower=c(b1=0, b2=0, b3=0), 
   upper=c(b1=500, b2=100, b3=5), masked=c("b2"), trace=traceval, data=weeddata1))
print(anlsmnqm)

## End(Not run) # end dontrun


## Not run: 

cat("MASKED\n")

an1qm3  <-  try(nlxb(eunsc, start=start1, trace=traceval, data=weeddata1, 
                masked=c("b3")))
print(an1qm3)
# tmp  <-  readline("next")

# Note that the parameters are put in out of order to test code.
an1qm123  <-  try(nlxb(eunsc, start=start1, trace=traceval, data=weeddata1, 
                  masked=c("b2","b1","b3")))
print(an1qm123)
# tmp  <-  readline("next")


## End(Not run) # end dontrun


cat("BOUNDS test problem for Hobbs")
start2  <-  c(b1=100, b2=10, b3=0.1)
an1qb1  <-  try(nlxb(eunsc, start=start2, trace=traceval, data=weeddata1, 
                     lower=c(0,0,0), upper=c(200, 60, .3)))
print(an1qb1)

## tmp  <-  readline("next")


cat("BOUNDS and MASK")

## Not run: 

an1qbm2  <-  try(nlxb(eunsc, start=start2, trace=traceval, data=weeddata1, 
                      lower=c(0,0,0), upper=c(200, 60, .3), masked=c("b2")))
print(an1qbm2)

# tmp  <-  readline("next")


## End(Not run) # end dontrun


escal  <-   y ~ 100*b1/(1+10*b2*exp(-0.1*b3*tt))
suneasy  <-  c(b1=200, b2=50, b3=0.3)
ssceasy  <-  c(b1=2, b2=5, b3=3)
st1scal  <-  c(b1=100, b2=10, b3=0.1)


cat("EASY start -- unscaled")
anls01  <-  try(nls(eunsc, start=suneasy, trace=traceval, data=weeddata1))
print(anls01)
anlmrt01  <-  try(nlxb(eunsc, start=ssceasy, trace=traceval, data=weeddata1))
print(anlmrt01)

## Not run: 

cat("All 1s start -- unscaled")
anls02  <-  try(nls(eunsc, start=start1, trace=traceval, data=weeddata1))
if (class(anls02) == "try-error") {
   cat("FAILED:")
   print(anls02)
} else {
   print(anls02)
}
anlmrt02  <-  nlxb(eunsc, start=start1, trace=traceval, data=weeddata1)
print(anlmrt02)

cat("ones start -- scaled")
anls03  <-  try(nls(escal, start=start1, trace=traceval, data=weeddata1))
print(anls03)
anlmrt03  <-  nlxb(escal, start=start1, trace=traceval, data=weeddata1)
print(anlmrt03)

cat("HARD start -- scaled")
anls04  <-  try(nls(escal, start=st1scal, trace=traceval, data=weeddata1))
print(anls04)
anlmrt04  <-  nlxb(escal, start=st1scal, trace=traceval, data=weeddata1)
print(anlmrt04)

cat("EASY start -- scaled")
anls05  <-  try(nls(escal, start=ssceasy, trace=traceval, data=weeddata1))
print(anls05)
anlmrt05  <-  nlxb(escal, start=ssceasy, trace=traceval, data=weeddata1)
print(anlmrt03)


## End(Not run) # end dontrun


## Not run: 

shobbs.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  <-  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
    return(jj)
}

cat("try nlfb\n")
st  <-  c(b1=1, b2=1, b3=1)
low  <-  -Inf
up <- Inf

ans1 <- nlfb(st, shobbs.res, shobbs.jac, trace=traceval)
ans1
cat("No jacobian function -- use internal approximation\n")
ans1n <- nlfb(st, shobbs.res, trace=TRUE, control=list(watch=TRUE)) # NO jacfn
ans1n

# tmp <- readline("Try with bounds at 2")
time2 <- system.time(ans2 <- nlfb(st, shobbs.res, shobbs.jac, upper=c(2,2,2), 
                                  trace=traceval))
ans2
time2



## End(Not run) # end dontrun

## Not run: 

cat("BOUNDS")
st2s <- c(b1=1, b2=1, b3=1)

an1qb1 <- try(nlxb(escal, start=st2s, trace=traceval, data=weeddata1, 
  lower=c(0,0,0), upper=c(2, 6, 3), control=list(watch=FALSE)))
print(an1qb1)

# tmp <- readline("next")

ans2 <- nlfb(st2s,shobbs.res, shobbs.jac, lower=c(0,0,0), upper=c(2, 6, 3), 
   trace=traceval, control=list(watch=FALSE))
print(ans2)

cat("BUT ... nls() seems to do better from the TRACE information\n")
anlsb <- nls(escal, start=st2s, trace=traceval, data=weeddata1, lower=c(0,0,0),
     upper=c(2,6,3), algorithm='port')
cat("However, let us check the answer\n")
print(anlsb)
cat("BUT...crossprod(resid(anlsb))=",crossprod(resid(anlsb)),"\n")


## End(Not run) # end dontrun


# tmp <- readline("next")

cat("Try wrapnls\n")
traceval <- FALSE
# Data for Hobbs problem
ydat <- 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
tdat <- seq_along(ydat) # for testing
start1 <- c(b1=1, b2=1, b3=1)
escal <-  y ~ 100*b1/(1+10*b2*exp(-0.1*b3*tt))
up1 <- c(2,6,3)
up2 <- c(1, 5, 9)

weeddata1 <- data.frame(y=ydat, tt=tdat)

an1w <- try(wrapnls(escal, start=start1, trace=traceval, data=weeddata1))
print(an1w)


## Not run: 

cat("BOUNDED wrapnls\n")

an1wb <- try(wrapnls(escal, start=start1, trace=traceval, data=weeddata1, upper=up1))
print(an1wb)


cat("BOUNDED wrapnls\n")

an2wb <- try(wrapnls(escal, start=start1, trace=traceval, data=weeddata1, upper=up2))
print(an2wb)

cat("TRY MASKS ONLY\n")

an1xm3 <- try(nlxb(escal, start1, trace=traceval, data=weeddata1, 
                   masked=c("b3")))
printsum(an1xm3)
an1fm3 <- try(nlfb(start1, shobbs.res, shobbs.jac, trace=traceval, 
                   data=weeddata1, maskidx=c(3)))
printsum(an1fm3)

an1xm1 <- try(nlxb(escal, start1, trace=traceval, data=weeddata1, 
                   masked=c("b1")))
printsum(an1xm1)
an1fm1 <- try(nlfb(start1, shobbs.res, shobbs.jac, trace=traceval, 
                   data=weeddata1, maskidx=c(1)))
printsum(an1fm1)


## End(Not run) # end dontrun

# Need to check when all parameters masked.??

## Not run: 


cat("\n\n Now check conversion of expression to function\n\n")
cat("K Vandepoel function\n")

x <- c(1,3,5,7) # data
y <- c(37.98,11.68,3.65,3.93)
penetrationks28 <- data.frame(x=x,y=y)

cat("Try nls() -- note the try() function!\n")

fit0  <-  try(nls(y ~ (a+b*exp(1)^(-c * x)), data = penetrationks28, 
    start = c(a=0,b = 1,c=1), trace = TRUE))
print(fit0)

cat("\n\n")

fit1  <-  nlxb(y ~ (a+b*exp(-c*x)), data = penetrationks28, 
   start = c(a=0,b=1,c=1), trace = TRUE) 
printsum(fit1)

mexprn <- "y ~ (a+b*exp(-c*x))"
pvec <- c(a=0,b=1,c=1)
bnew <- c(a=10,b=3,c=4)

k.r <- model2resfun(mexprn , pvec)
k.j <- model2jacfun(mexprn , pvec)
k.f <- model2ssfun(mexprn , pvec)
k.g <- model2grfun(mexprn , pvec)


cat("At pvec:")
print(pvec)
rp <- k.r(pvec, x=x, y=y)
cat(" rp=")
print(rp)
rf <- k.f(pvec, x=x, y=y)
cat(" rf=")
print(rf)
rj <- k.j(pvec, x=x, y=y)
cat(" rj=")
print(rj)
rg <- k.g(pvec, x=x, y=y)
cat(" rg=")
print(rg)
cat("modss at pvec gives ")
print(modss(pvec, k.r, x=x, y=y))
cat("modgr at pvec gives ")
print(modgr(pvec, k.r, k.j, x=x, y=y))
cat("\n\n")

cat("At bnew:")
print(bnew)
rb <- k.r(bnew, x=x, y=y)
cat(" rb=")
print(rb)
rf <- k.f(bnew, x=x, y=y)
cat(" rf=")
print(rf)
rj <- k.j(bnew, x=x, y=y)
cat(" rj=")
print(rj)
rg <- k.g(bnew, x=x, y=y)
cat(" rg=")
print(rg)
cat("modss at bnew gives ")
print(modss(bnew, k.r, x=x, y=y))
cat("modgr at bnew gives ")
print(modgr(bnew, k.r, k.j, x=x, y=y))
cat("\n\n")


## End(Not run)  ## end of dontrun