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:  ENSO              (ENSO.dat)
   8:  * 
   9:  * Description:   The data are monthly averaged atmospheric pressure 
  10:  *                differences between Easter Island and Darwin, 
  11:  *                Australia.  This difference drives the trade winds in 
  12:  *                the southern hemisphere.  Fourier analysis of the data
  13:  *                reveals 3 significant cycles.  The annual cycle is the
  14:  *                strongest, but cycles with periods of approximately 44
  15:  *                and 26 months are also present.  These cycles
  16:  *                correspond to the El Nino and the Southern Oscillation.
  17:  *                Arguments to the SIN and COS functions are in radians.
  18:  * 
  19:  * Reference:     Kahaner, D., C. Moler, and S. Nash, (1989). 
  20:  *                Numerical Methods and Software.  
  21:  *                Englewood Cliffs, NJ: Prentice Hall, pp. 441-445.
  22:  * 
  23:  * Data:          1 Response  (y = atmospheric pressure)
  24:  *                1 Predictor (x = time)
  25:  *                168 Observations
  26:  *                Average Level of Difficulty
  27:  *                Observed Data
  28:  * 
  29:  * Model:         Miscellaneous Class
  30:  *                9 Parameters (b1 to b9)
  31:  * 
  32:  *                y = b1 + b2*cos( 2*pi*x/12 ) + b3*sin( 2*pi*x/12 ) 
  33:  *                       + b5*cos( 2*pi*x/b4 ) + b6*sin( 2*pi*x/b4 )
  34:  *                       + b8*cos( 2*pi*x/b7 ) + b9*sin( 2*pi*x/b7 )  + e
  35:  *  
  36:  *           Starting values                  Certified Values
  37:  *  
  38:  *         Start 1     Start 2           Parameter     Standard Deviation
  39:  *   b1 =   11.0        10.0          1.0510749193E+01  1.7488832467E-01
  40:  *   b2 =    3.0         3.0          3.0762128085E+00  2.4310052139E-01
  41:  *   b3 =    0.5         0.5          5.3280138227E-01  2.4354686618E-01
  42:  *   b4 =   40.0        44.0          4.4311088700E+01  9.4408025976E-01
  43:  *   b5 =   -0.7        -1.5         -1.6231428586E+00  2.8078369611E-01
  44:  *   b6 =   -1.3         0.5          5.2554493756E-01  4.8073701119E-01
  45:  *   b7 =   25.0        26.0          2.6887614440E+01  4.1612939130E-01
  46:  *   b8 =   -0.3        -0.1          2.1232288488E-01  5.1460022911E-01
  47:  *   b9 =    1.4         1.5          1.4966870418E+00  2.5434468893E-01
  48:  * 
  49:  * Residual Sum of Squares:                    7.8853978668E+02
  50:  * Residual Standard Deviation:                2.2269642403E+00
  51:  * Degrees of Freedom:                               159
  52:  * Number of Observations:                           168
  53:  */
  54: Title "ENSO";
  55: Variables y,x;
  56: Parameter b1 = 11.0;
  57: Parameter b2 =   3.0;
  58: Parameter b3 =   0.5;
  59: Parameter b4 =  40.0;
  60: Parameter b5 =  -0.7;
  61: Parameter b6 =  -1.3;
  62: Parameter b7 =  25.0;
  63: Parameter b8 =  -0.3;
  64: Parameter b9 =   1.4;
  65: Function y = b1 + b2*cos( 2*pi*x/12 ) + b3*sin( 2*pi*x/12 ) 
  66:              + b5*cos( 2*pi*x/b4 ) + b6*sin( 2*pi*x/b4 )
  67:              + b8*cos( 2*pi*x/b7 ) + b9*sin( 2*pi*x/b7 );
  68: Angletype radians;
  69: plot;
  70: 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.

ENSO
Number of observations = 168
Maximum allowed number of iterations = 500
Convergence tolerance factor = 1.000000E-010
Stopped due to: Relative function convergence.
Number of iterations performed = 13
Final sum of squared deviations = 7.8853979E+002
Final sum of deviations = 1.0502255E-006
Standard error of estimate = 2.22696
Average deviation = 1.72826
Maximum deviation for any observation = 5.70664
Proportion of variance explained (R^2) = 0.5983  (59.83%)
Adjusted coefficient of multiple determination (Ra^2) = 0.5781  (57.81%)
Durbin-Watson test for autocorrelation = 1.642


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

 Variable    Minimum value   Maximum value    Mean value     Standard dev.
----------  --------------  --------------  --------------  --------------
         y             0.3            17.6        10.64167        3.428507
         x               1             168            84.5        48.64155


                   ----  Calculated Parameter Values  ----

 Parameter  Initial guess   Final estimate   Standard error      t      Prob(t)
----------  -------------  ----------------  --------------  ---------  -------
        b1             11        10.5107497       0.1748883      60.10  0.00001
        b2              3        3.07621274       0.2431005      12.65  0.00001
        b3            0.5       0.532801467       0.2435469       2.19  0.03015
        b4             40        44.3110746       0.9440801      46.94  0.00001
        b5           -0.7       -1.62314546       0.2807827      -5.78  0.00001
        b6           -1.3       0.525536331       0.4807378       1.09  0.27597
        b7             25        26.8876135       0.4161293      64.61  0.00001
        b8           -0.3       0.212320664       0.5146003       0.41  0.68046
        b9            1.4        1.49668749       0.2543444       5.88  0.00001


                  ----  Analysis of Variance  ----

  Source     DF   Sum of Squares    Mean Square    F value   Prob(F)
----------  ----  --------------  --------------  ---------  -------
Regression     8        1174.489        146.8111      29.60  0.00001
Error        159        788.5398         4.95937
Total        167        1963.028