NIST - Lanczos2 Dataset

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NIST - Lanczos2 Dataset

   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:  Lanczos2          (Lanczos2.dat)
   8:  * 
   9:  * Description:   These data are taken from an example discussed in
  10:  *                Lanczos (1956).  The data were generated to 6-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:  *                Average 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           9.6251029939E-02  6.6770575477E-04
  34:  *   b2 =   0.3         0.7           1.0057332849E+00  3.3989646176E-03
  35:  *   b3 =   5.6         3.6           8.6424689056E-01  1.7185846685E-03
  36:  *   b4 =   5.5         4.2           3.0078283915E+00  4.1707005856E-03
  37:  *   b5 =   6.5         4             1.5529016879E+00  2.3744381417E-03
  38:  *   b6 =   7.6         6.3           5.0028798100E+00  1.3958787284E-03
  39:  * 
  40:  * Residual Sum of Squares:                    2.2299428125E-11
  41:  * Residual Standard Deviation:                1.1130395851E-06
  42:  * Degrees of Freedom:                                18
  43:  * Number of Observations:                            24
  44:  */
  45: Title "Lanczos2";
  46: Variables y,x;
  47: // Note: Using Start 2 parameters
  48: Parameter b1 = 0.5;
  49: Parameter b2 = 0.7;
  50: Parameter b3 = 3.6;
  51: Parameter b4 = 4.2;
  52: Parameter b5 = 4;
  53: Parameter b6 = 6.3;
  54: Function y = b1*exp(-b2*x) + b3*exp(-b4*x) + b5*exp(-b6*x);
  55: Plot;
  56: Data;

Beginning computation...
Stopped due to: Both parameter and 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.

Lanczos2
Number of observations = 24
Maximum allowed number of iterations = 500
Convergence tolerance factor = 1.000000E-010
Stopped due to: Both parameter and relative function convergence.
Number of iterations performed = 9
Final sum of squared deviations = 2.2299428E-011
Final sum of deviations = 1.3581594E-009
Standard error of estimate = 1.11304E-006
Average deviation = 5.45365E-007
Maximum deviation for any observation = 3.73724E-006
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 = 3.139


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

 Variable    Minimum value   Maximum value    Mean value     Standard dev.
----------  --------------  --------------  --------------  --------------
         y       0.0623931          2.5134       0.5998774       0.6802225
         x               0            1.15           0.575       0.3535534


                   ----  Calculated Parameter Values  ----

 Parameter  Initial guess   Final estimate   Standard error      t      Prob(t)
----------  -------------  ----------------  --------------  ---------  -------
        b1            0.5      0.0962509964    0.0006677151     144.15  0.00001
        b2            0.7        1.00573312     0.003399014     295.89  0.00001
        b3            3.6       0.864246802     0.001718608     502.88  0.00001
        b4            4.2        3.00782818      0.00417076     721.17  0.00001
        b5              4        1.55290181     0.002374471     654.00  0.00001
        b6            6.3        5.00287974     0.001395898    3583.99  0.00001


                  ----  Analysis of Variance  ----

  Source     DF   Sum of Squares    Mean Square    F value   Prob(F)
----------  ----  --------------  --------------  ---------  -------
Regression     5        10.64216        2.128432   1.7E+012  0.00001
Error         18   2.229943E-011   1.238857E-012
Total         23        10.64216