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>  # 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
[[1]]
 [1]  47.3  87.0 120.1 130.4  58.8 111.9 136.5 132.0  68.8 138.1 145.7 143.0

[[2]]
 [1] 0.0 0.4 0.8 1.6 0.0 1.0 2.0 4.0 0.0 1.5 3.0 5.9

>  #
>  #               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)
[1] 143.000000 -88.203926   1.475360
>  #
>  #  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 )
630.1842 :  143.000000 -88.203926   1.475360 
555.4333 :  141.701493 -83.987048   1.333616 
555.1984 :  141.585853 -83.960237   1.349271 
555.1962 :  141.608235 -83.973907   1.347612 
555.1962 :  141.605989 -83.972601   1.347799 
555.1962 :  141.606243 -83.972750   1.347778 
>   that1
Nonlinear regression model
  model:  y ~ th1 + th2 * exp(-th3 * x) 
   data:  fulnls 
    th1     th2     th3 
141.606 -83.973   1.348 
 residual sum-of-squares: 555.2

Number of iterations to convergence: 5 
Achieved convergence tolerance: 2.944e-06 
>   thfin <- coef(that1)
>   print("coefficients")
[1] "coefficients"
>   print(thfin)
       th1        th2        th3 
141.606243 -83.972750   1.347778 
>  #
>  #               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 )
  0:     315.09208:  143.000 -88.2039  1.47536
  1:     288.81805:  142.296 -86.7454  1.40558
  2:     277.61497:  141.668 -83.9996  1.34196
  3:     277.59829:  141.598 -83.9679  1.34841
  4:     277.59810:  141.607 -83.9732  1.34771
  5:     277.59809:  141.606 -83.9727  1.34779
  6:     277.59809:  141.606 -83.9727  1.34778
  7:     277.59809:  141.606 -83.9727  1.34778
>   that1
Nonlinear regression model
  model:  y ~ th1 + th2 * exp(-th3 * x) 
   data:  fulnls 
    th1     th2     th3 
141.606 -83.973   1.348 
 residual sum-of-squares: 555.2

Algorithm "port", convergence message: relative convergence (4) 
>  #
>  #               cov matrix of coefficients
>   print("Cov(coefficients)")
[1] "Cov(coefficients)"
>   vcov(that1)
            th1         th2         th3
th1  20.2976505 -18.6255768 -0.85709424
th2 -18.6255768  36.8083599  0.50091546
th3  -0.8570942   0.5009155  0.07038162
>   print("Residuals, etc.")
[1] "Residuals, etc."
>   resid(that1)
 [1] -10.333482  -5.627781   7.061237  -1.487593   1.166518  -7.888724
 [7]   0.562326  -9.223563  11.166518   7.614637   5.566564   1.423342
attr(,"label")
[1] "Residuals"
>   sum(resid(that1)**2)
[1] 555.1962
>  #               done
>   rm(list=ls())
>   q()