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:  Gauss1            (Gauss1.dat)
   8:  * 
   9:  * Procedure:     Nonlinear Least Squares Regression
  10:  * 
  11:  * Description:   The data are two well-separated Gaussians on a 
  12:  *                decaying exponential baseline plus normally 
  13:  *                distributed zero-mean noise with variance = 6.25.
  14:  * 
  15:  * Reference:     Rust, B., NIST (1996).
  16:  * 
  17:  * Data:          1 Response  (y)
  18:  *                1 Predictor (x)
  19:  *                250 Observations
  20:  *                Lower Level of Difficulty
  21:  *                Generated Data
  22:  *  
  23:  * Model:         Exponential Class
  24:  *                8 Parameters (b1 to b8) 
  25:  *  
  26:  *                y = b1*exp( -b2*x ) + b3*exp( -(x-b4)**2 / b5**2 )
  27:  *                                    + b6*exp( -(x-b7)**2 / b8**2 ) + e
  28:  *  
  29:  *  
  30:  *           Starting values                  Certified Values
  31:  *  
  32:  *         Start 1     Start 2           Parameter     Standard Deviation
  33:  *   b1 =    97.0        94.0         9.8778210871E+01  5.7527312730E-01
  34:  *   b2 =     0.009       0.0105      1.0497276517E-02  1.1406289017E-04
  35:  *   b3 =   100.0        99.0         1.0048990633E+02  5.8831775752E-01
  36:  *   b4 =    65.0        63.0         6.7481111276E+01  1.0460593412E-01
  37:  *   b5 =    20.0        25.0         2.3129773360E+01  1.7439951146E-01
  38:  *   b6 =    70.0        71.0         7.1994503004E+01  6.2622793913E-01
  39:  *   b7 =   178.0       180.0         1.7899805021E+02  1.2436988217E-01
  40:  *   b8 =    16.5        20.0         1.8389389025E+01  2.0134312832E-01
  41:  * 
  42:  * Residual Sum of Squares:                    1.3158222432E+03
  43:  * Residual Standard Deviation:                2.3317980180E+00
  44:  * Degrees of Freedom:                               242
  45:  * Number of Observations:                           250
  46:  */
  47: Title "Gauss1";
  48: Variables y,x;
  49: Parameter b1 = 97.0;
  50: Parameter b2 = 0.009;
  51: Parameter b3 = 100.0;
  52: Parameter b4 = 65.0;
  53: Parameter b5 = 20.0;
  54: Parameter b6 = 70.0;
  55: Parameter b7 = 178.0;
  56: Parameter b8 = 16.5;
  57: Function   y = b1*exp( -b2*x ) + b3*exp( -(x-b4)**2 / b5**2 ) + b6*exp( -(x-b7)**2 / b8**2 );
  58: plot;
  59: data;

Beginning computation...
Stopped due to: Singular convergence.  Mutually dependent parameters?


   ----  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.

Gauss1
Number of observations = 250
Maximum allowed number of iterations = 500
Convergence tolerance factor = 1.000000E-010
Stopped due to: Singular convergence.  Mutually dependent parameters?
Warning: All data points are on one side of the curve.
This indicates the model does not fit the data well.
Number of iterations performed = 1
Final sum of squared deviations = 4.7460231E+103
Final sum of deviations = -9.1298813E+051
Standard error of estimate = 4.42851E+050
Average deviation = 3.65195E+049
Maximum deviation for any observation = 6.62623E+051


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

 Variable    Minimum value   Maximum value    Mean value     Standard dev.
----------  --------------  --------------  --------------  --------------
         y        1.182746        152.0519         60.5314        41.70884
         x               1             250           125.5        72.31298


  ----  Calculated Parameter Values  ----

 Parameter  Initial guess   Final estimate 
----------  -------------  ----------------
        b1             97                97
        b2          0.009             0.009
        b3            100               100
        b4             65                65
        b5             20                20
        b6             70                70
        b7            178               178
        b8           16.5              16.5


                  ----  Analysis of Variance  ----

  Source     DF   Sum of Squares    Mean Square    F value   Prob(F)
----------  ----  --------------  --------------  ---------  -------
Regression     7               0               0       0.00  1.00000
Error        242   4.746023E+103   1.961167E+101
Total        249        433167.1