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>  # nllsqt2.r                         October 2007, 2008
>  #
>  # test problem for nonlinear least squares
>  # from Gallant (1975)
>  #
>   all <- matrix(scan("gallant.dat"),30,5, byrow=TRUE)
>  #
>  # make a data frame
>   gallant <- data.frame( y=all[,2], x1=all[,3], x2=all[,4], x3=all[,5] )
>   print(gallant)                    # see data frame
         y x1 x2   x3
1  0.98610  1  1 6.28
2  0.95482  1  1 9.11
3  1.02324  1  1 8.11
4  0.96263  1  1 6.58
5  0.98861  1  1 6.52
6  0.98982  1  1 9.86
7  0.66768  1  1 0.47
8  0.96822  1  1 4.07
9  0.59759  1  1 0.17
10 1.01962  1  1 4.39
11 1.04255  1  1 4.73
12 0.97526  1  1 8.90
13 0.80219  1  1 0.77
14 0.95196  1  1 4.51
15 0.50811  1  1 0.08
16 1.03848  0  1 9.86
17 1.04184  0  1 8.43
18 0.90475  0  1 1.82
19 1.05026  0  1 5.02
20 1.03437  0  1 3.75
21 1.01214  0  1 7.31
22 0.55107  0  1 0.07
23 0.98823  0  1 4.61
24 0.99418  0  1 6.99
25 0.69163  0  1 0.39
26 1.04343  0  1 9.42
27 1.04969  0  1 3.02
28 1.01046  0  1 3.31
29 0.97658  0  1 2.65
30 0.91840  0  1 6.11
>  #
>  #               another way is to define residual function
>    res <- function(th1,th2,th3,th4,y,x1,x2,x3,x4) {
+       res <- y - th1*x1 - th2*x2 - th4*exp(th3*x3) } 
>  #
>  #               nonlinear least squares
>   that1 <- nls(  ~res(th1,th2,th3,th4,y,x1,x2,x3) , data=gallant,
+          start=list( th1=-.05, th2=1.04, th3=-1.2, th4=-.53), trace=TRUE )
0.04025717 :  -0.05  1.04 -1.20 -0.53 
0.03049629 :  -0.02582531  1.01558996 -1.11355727 -0.50431161 
0.03049554 :  -0.02588913  1.01567783 -1.11577342 -0.50490932 
0.03049554 :  -0.02588972  1.01567973 -1.11569442 -0.50490262 
>  #               results
>   that1
Nonlinear regression model
  model:  0 ~ res(th1, th2, th3, th4, y, x1, x2, x3) 
   data:  gallant 
     th1      th2      th3      th4 
-0.02589  1.01568 -1.11569 -0.50490 
 residual sum-of-squares: 0.03050

Number of iterations to convergence: 3 
Achieved convergence tolerance: 3.376e-06 
>  #
>   thfin <- coef(that1)
>   print("coefficients")
[1] "coefficients"
>   print(thfin)
        th1         th2         th3         th4 
-0.02588972  1.01567973 -1.11569442 -0.50490262 
>  #              summary
>   summary(that1)

Formula: 0 ~ res(th1, th2, th3, th4, y, x1, x2, x3)

Parameters:
     Estimate Std. Error t value Pr(>|t|)    
th1 -0.025890   0.012624  -2.051   0.0505 .  
th2  1.015680   0.009938 102.202  < 2e-16 ***
th3 -1.115694   0.163542  -6.822 3.06e-07 ***
th4 -0.504903   0.025657 -19.679  < 2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 

Residual standard error: 0.03425 on 26 degrees of freedom

Number of iterations to convergence: 3 
Achieved convergence tolerance: 3.376e-06 

>  #
>  #               done
>   rm(list=ls())
>   q()