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  Linear Least Squares Regression

  NIST test data

  Erwin kalvelagen, dec 2004

  Reference:
      http://www.itl.nist.gov/div898/strd/lls/lls.shtml

      Norris, J., NIST.
      Calibration of Ozone Monitors.


  Model:       Linear Class
               2 Parameters (B0,B1)

               y = B0 + B1*x + e



               Certified Regression Statistics

                                          Standard Deviation
     Parameter          Estimate             of Estimate

        B0        -0.262323073774029     0.232818234301152
        B1         1.00211681802045      0.429796848199937E-03

     Residual
     Standard Deviation   0.884796396144373

     R-Squared            0.999993745883712


               Certified Analysis of Variance Table

Source of Degrees of    Sums of             Mean
Variation  Freedom      Squares            Squares           F Statistic

Regression    1     4255954.13232369   4255954.13232369   5436385.54079785
Residual     34     26.6173985294224   0.782864662630069

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set i 'cases' /i1*i36/;

table data(i,*)
            y          x
  i1       0.1        0.2
  i2     338.8      337.4
  i3     118.1      118.2
  i4     888.0      884.6
  i5       9.2       10.1
  i6     228.1      226.5
  i7     668.5      666.3
  i8     998.5      996.3
  i9     449.1      448.6
  i10    778.9      777.0
  i11    559.2      558.2
  i12      0.3        0.4
  i13      0.1        0.6
  i14    778.1      775.5
  i15    668.8      666.9
  i16    339.3      338.0
  i17    448.9      447.5
  i18     10.8       11.6
  i19    557.7      556.0
  i20    228.3      228.1
  i21    998.0      995.8
  i22    888.8      887.6
  i23    119.6      120.2
  i24      0.3        0.3
  i25      0.6        0.3
  i26    557.6      556.8
  i27    339.3      339.1
  i28    888.0      887.2
  i29    998.5      999.0
  i30    778.9      779.0
  i31     10.2       11.1
  i32    117.6      118.3
  i33    228.9      229.2
  i34    668.4      669.1
  i35    449.2      448.9
  i36      0.2        0.5
;

variables
   b0  'constant term'
   b1
   sse 'sum of squared errors'
;

equation
   fit(i)    'equation to fit'
   sumsq
;

sumsq..   sse =n= 0;
fit(i)..  data(i,'y') =e= b0 + b1*data(i,'x');

option lp = ls;
model leastsq /fit,sumsq/;
solve leastsq using lp minimizing sse;
option decimals=8;
display b0.l,b1.l;

scalar B0cert /  -0.262323073774029  /;
scalar B1cert /   1.00211681802045   /;

scalar err "Sum of squared errors in estimates";
err = sqr(b0.l-B0cert) + sqr(b1.l-B1cert);
display err;
abort$(err>0.0001) "Solution not accurate";