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:  Lanczos1          (Lanczos1.dat)
   8:  * 
   9:  * Description:   These data are taken from an example discussed in
  10:  *                Lanczos (1956).  The data were generated to 14-digits
  11:  *                of accuracy using
  12:  *                f(x) = 0.0951*exp(-x) + 0.8607*exp(-3*x) 
  13:  * 
  14:  * Reference:     Lanczos, C. (1956).
  15:  *                Applied Analysis.
  16:  *                Englewood Cliffs, NJ:  Prentice Hall, pp. 272-280.
  17:  * 
  18:  * Data:          1 Response  (y)
  19:  *                1 Predictor (x)
  20:  *                24 Observations
  21:  *                Average Level of Difficulty
  22:  *                Generated Data
  23:  * 
  24:  * Model:         Exponential Class
  25:  *                6 Parameters (b1 to b6)
  26:  * 
  27:  *                y = b1*exp(-b2*x) + b3*exp(-b4*x) + b5*exp(-b6*x)  +  e
  28:  *
  29:  *  
  30:  *           Starting values                  Certified Values
  31:  *  
  32:  *         Start 1     Start 2           Parameter     Standard Deviation
  33:  *   b1 =   1.2         0.5           9.5100000027E-02  5.3347304234E-11
  34:  *   b2 =   0.3         0.7           1.0000000001E+00  2.7473038179E-10
  35:  *   b3 =   5.6         3.6           8.6070000013E-01  1.3576062225E-10
  36:  *   b4 =   5.5         4.2           3.0000000002E+00  3.3308253069E-10
  37:  *   b5 =   6.5         4             1.5575999998E+00  1.8815731448E-10
  38:  *   b6 =   7.6         6.3           5.0000000001E+00  1.1057500538E-10
  39:  * 
  40:  * Residual Sum of Squares:                    1.4307867721E-25
  41:  * Residual Standard Deviation:                8.9156129349E-14
  42:  * Degrees of Freedom:                                18
  43:  * Number of Observations:                            24
  44:  */
  45: Title "Lanczos1";
  46: Variables y,x;
  47: Parameter b1 = 1.2;
  48: Parameter b2 = 0.3;
  49: Parameter b3 = 5.6;
  50: Parameter b4 = 5.5;
  51: Parameter b5 = 6.5;
  52: Parameter b6 = 7.6;
  53: Function y = b1*exp(-b2*x) + b3*exp(-b4*x) + b5*exp(-b6*x);
  54: Plot;
  55: Data;

Beginning computation...
Stopped due to: Absolute 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.

Lanczos1
Number of observations = 24
Maximum allowed number of iterations = 500
Convergence tolerance factor = 1.000000E-010
Stopped due to: Absolute function convergence.
Number of iterations performed = 98
Final sum of squared deviations = 3.2347651E-025
Final sum of deviations = -1.6729673E-012
Standard error of estimate = 1.34056E-013
Average deviation = 1.00349E-013
Maximum deviation for any observation = 2.80664E-013
Proportion of variance explained (R^2) = 1.0000  (100.00%)
Adjusted coefficient of multiple determination (Ra^2) = 1.0000  (100.00%)
Durbin-Watson test for autocorrelation = 1.097


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

 Variable    Minimum value   Maximum value    Mean value     Standard dev.
----------  --------------  --------------  --------------  --------------
         y      0.06239313          2.5134       0.5998777       0.6802231
         x               0            1.15           0.575       0.3535534


                   ----  Calculated Parameter Values  ----

 Parameter  Initial guess   Final estimate   Standard error      t      Prob(t)
----------  -------------  ----------------  --------------  ---------  -------
        b1            1.2            0.0951   8.021168E-011   1.2E+009  0.00001
        b2            0.3                 1   4.130781E-010   2.4E+009  0.00001
        b3            5.6            0.8607   2.041255E-010   4.2E+009  0.00001
        b4            5.5                 3   5.008136E-010   6.0E+009  0.00001
        b5            6.5            1.5576   2.829078E-010   5.5E+009  0.00001
        b6            7.6                 5   1.662572E-010   3.0E+010  0.00001


                  ----  Analysis of Variance  ----

  Source     DF   Sum of Squares    Mean Square    F value   Prob(F)
----------  ----  --------------  --------------  ---------  -------
Regression     5        10.64218        2.128436   1.2E+026  0.00001
Error         18   3.234765E-025   1.797092E-026
Total         23        10.64218