# nllsqt1.r                         October 2007
 #
 # test problem for nonlinear least squares
 # from Fuller (1976)
 #
  both <- matrix(scan("fulnls.dat"),12,2, byrow=TRUE)
  y <- both[,1]
  x <- both[,2]
  fulnls <- list(y,x)              # need data frame or list
  print(fulnls)                    # see list
 #
 #               nonlinear least squares
 #
 #  find better starting values
  t1 <- y[ x==max(x)]             # max value as asymptote
  xs <- x[ x < 2 ]                # use smallest x's
  ys <- log( t1 - y[ x < 2 ] )    # transform reversed y's
 #  simple linear regression
  t3 <- sum( (xs - mean(xs) )*ys )/ sum( (xs-mean(xs))**2 ) # slope
  t2 <- mean(ys)-t3*mean(xs)                                # intercept
 #  transform back from log and reverse
  t2 <- -exp(t2)
  t3 <- - t3
 #  done -- now have better starting values
  c(t1,t2,t3)
 #
 #  call   nls using defaults + trace
  that1 <- nls( y ~ th1 + th2*exp(-th3*x), data=fulnls,
             start=list( th1=t1, th2=t2, th3=t3), trace=TRUE )
  that1
  thfin <- coef(that1)
  print("coefficients")
  print(thfin)
 #
 #               nonlinear least squares using nl2sol
  that1 <- nls( y ~ th1 + th2*exp(-th3*x), data=fulnls, algorithm="port",
             start=list( th1=t1, th2=t2, th3=t3), trace=TRUE )
  that1
 #
 #               cov matrix of coefficients
  print("Cov(coefficients)")
  vcov(that1)
  print("Residuals, etc.")
  resid(that1)
  sum(resid(that1)**2)
 #               done
  rm(list=ls())
  q()