1: /*
   2:  * Statistical Reference Datasets  (Nonlinear Regression)
   3:  * Statistical Engineering Division
   4:  * National Institute of Standards and Technology
   5:  * http://www.nist.gov/itl/div898/strd/
   6:  *
   7:  * Dataset Name:  MGH09             (MGH09.dat)
   8:  * 
   9:  * Description:   This problem was found to be difficult for some very 
  10:  *                good algorithms.  There is a local minimum at (+inf,
  11:  *                -14.07..., -inf, -inf) with final sum of squares 
  12:  *                0.00102734....
  13:  * 
  14:  *                See More, J. J., Garbow, B. S., and Hillstrom, K. E. 
  15:  *                (1981).  Testing unconstrained optimization software.
  16:  *                ACM Transactions on Mathematical Software. 7(1): 
  17:  *                pp. 17-41.
  18:  * 
  19:  * Reference:     Kowalik, J.S., and M. R. Osborne, (1978).  
  20:  *                Methods for Unconstrained Optimization Problems.  
  21:  *                New York, NY:  Elsevier North-Holland.
  22:  * 
  23:  * Data:          1 Response  (y)
  24:  *                1 Predictor (x)
  25:  *                11 Observations
  26:  *                Higher Level of Difficulty
  27:  *                Generated Data
  28:  *  
  29:  * Model:         Rational Class (linear/quadratic)
  30:  *                4 Parameters (b1 to b4)
  31:  *  
  32:  *                y = b1*(x**2+x*b2) / (x**2+x*b3+b4)  +  e
  33:  *  
  34:  *           Starting values                  Certified Values
  35:  * 
  36:  *         Start 1     Start 2           Parameter     Standard Deviation
  37:  *   b1 =   25          0.25          1.9280693458E-01  1.1435312227E-02
  38:  *   b2 =   39          0.39          1.9128232873E-01  1.9633220911E-01
  39:  *   b3 =   41.5        0.415         1.2305650693E-01  8.0842031232E-02
  40:  *   b4 =   39          0.39          1.3606233068E-01  9.0025542308E-02
  41:  * 
  42:  * Residual Sum of Squares:                    3.0750560385E-04
  43:  * Residual Standard Deviation:                6.6279236551E-03
  44:  * Degrees of Freedom:                                7
  45:  * Number of Observations:                           11
  46:  */
  47: Title "MGH09";
  48: Variables y,x;
  49: Parameter b1 = 25;
  50: Parameter b2 = 39;
  51: Parameter b3 = 41.5;
  52: Parameter b4 = 39;
  53: Function y = b1*(x**2+x*b2) / (x**2+x*b3+b4);
  54: Plot;
  55: Data;

Beginning computation...
Stopped due to: Relative function convergence.


   ----  Final Results  ----

NLREG version 4.0
Copyright (c) 1992-1997 Phillip H. Sherrod.  All rights reserved.
This is a registered copy of NLREG that may not be redistributed.

MGH09
Number of observations = 11
Maximum allowed number of iterations = 500
Convergence tolerance factor = 1.000000E-010
Stopped due to: Relative function convergence.
Number of iterations performed = 61
Final sum of squared deviations = 3.0750560E-004
Final sum of deviations = 2.7014482E-003
Standard error of estimate = 0.00662792
Average deviation = 0.00373222
Maximum deviation for any observation = 0.0111094
Proportion of variance explained (R^2) = 0.9941  (99.41%)
Adjusted coefficient of multiple determination (Ra^2) = 0.9915  (99.15%)
Durbin-Watson test for autocorrelation = 3.354


             ----  Descriptive Statistics for Variables  ----

 Variable    Minimum value   Maximum value    Mean value     Standard dev.
----------  --------------  --------------  --------------  --------------
         y          0.0235          0.1957      0.09374545      0.07193252
         x          0.0625               4       0.7599273        1.225952


                   ----  Calculated Parameter Values  ----

 Parameter  Initial guess   Final estimate   Standard error      t      Prob(t)
----------  -------------  ----------------  --------------  ---------  -------
        b1             25       0.192806936      0.01143531      16.86  0.00001
        b2             39       0.191282315       0.1963322       0.97  0.36238
        b3           41.5       0.123056508      0.08084202       1.52  0.17178
        b4             39       0.136062324      0.09002555       1.51  0.17445


                  ----  Analysis of Variance  ----

  Source     DF   Sum of Squares    Mean Square    F value   Prob(F)
----------  ----  --------------  --------------  ---------  -------
Regression     3      0.05143536      0.01714512     390.29  0.00001
Error          7    0.0003075056   4.392937E-005
Total         10      0.05174287