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:  Lanczos3          (Lanczos3.dat)
   8:  * 
   9:  * Description:   These data are taken from an example discussed in
  10:  *                Lanczos (1956).  The data were generated to 5-digits
  11:  *                of accuracy using
  12:  *                f(x) = 0.0951*exp(-x) + 0.8607*exp(-3*x) 
  13:  *                                      + 1.5576*exp(-5*x).
  14:  * 
  15:  * Reference:     Lanczos, C. (1956).
  16:  *                Applied Analysis.
  17:  *                Englewood Cliffs, NJ:  Prentice Hall, pp. 272-280.
  18:  * 
  19:  * Data:          1 Response  (y)
  20:  *                1 Predictor (x)
  21:  *                24 Observations
  22:  *                Lower Level of Difficulty
  23:  *                Generated Data
  24:  *  
  25:  * Model:         Exponential Class
  26:  *                6 Parameters (b1 to b6)
  27:  *  
  28:  *                y = b1*exp(-b2*x) + b3*exp(-b4*x) + b5*exp(-b6*x)  +  e
  29:  * 
  30:  *           Starting values                  Certified Values
  31:  * 
  32:  *         Start 1     Start 2           Parameter     Standard Deviation
  33:  *   b1 =   1.2         0.5           8.6816414977E-02  1.7197908859E-02
  34:  *   b2 =   0.3         0.7           9.5498101505E-01  9.7041624475E-02
  35:  *   b3 =   5.6         3.6           8.4400777463E-01  4.1488663282E-02
  36:  *   b4 =   5.5         4.2           2.9515951832E+00  1.0766312506E-01
  37:  *   b5 =   6.5         4             1.5825685901E+00  5.8371576281E-02
  38:  *   b6 =   7.6         6.3           4.9863565084E+00  3.4436403035E-02
  39:  * 
  40:  * Residual Sum of Squares:                    1.6117193594E-08
  41:  * Residual Standard Deviation:                2.9923229172E-05
  42:  * Degrees of Freedom:                                18
  43:  * Number of Observations:                            24
  44:  */
  45: Title "Lanczos3";
  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: 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.

Lanczos3
Number of observations = 24
Maximum allowed number of iterations = 500
Convergence tolerance factor = 1.000000E-010
Stopped due to: Relative function convergence.
Number of iterations performed = 103
Final sum of squared deviations = 1.6117194E-008
Final sum of deviations = 4.4413376E-008
Standard error of estimate = 2.99232E-005
Average deviation = 2.25095E-005
Maximum deviation for any observation = 4.38193E-005
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 = 2.354


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

 Variable    Minimum value   Maximum value    Mean value     Standard dev.
----------  --------------  --------------  --------------  --------------
         y          0.0624          2.5134       0.5998708       0.6802196
         x               0            1.15           0.575       0.3535534


                   ----  Calculated Parameter Values  ----

 Parameter  Initial guess   Final estimate   Standard error      t      Prob(t)
----------  -------------  ----------------  --------------  ---------  -------
        b1            1.2      0.0868163166      0.01719805       5.05  0.00008
        b2            0.3        0.95498047      0.09704252       9.84  0.00001
        b3            5.6       0.844007514      0.04148894      20.34  0.00001
        b4            5.5        2.95159454        0.107664      27.41  0.00001
        b5            6.5        1.58256895      0.05837199      27.11  0.00001
        b6            7.6        4.98635629      0.03443663     144.80  0.00001


                  ----  Analysis of Variance  ----

  Source     DF   Sum of Squares    Mean Square    F value   Prob(F)
----------  ----  --------------  --------------  ---------  -------
Regression     5        10.64207        2.128414   2.4E+009  0.00001
Error         18   1.611719E-008   8.953996E-010
Total         23        10.64207