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  Non-Linear Least Squares Regression example

  Erwin Kalvelagen,  may 2008

  Reference:
   http://www.statsci.org/data/general/chloride.html


Transport of Sulfite Ions from Blood Cells
--------------------------------------------------------------------------------

Description
The data related to the transport of sulfite ions from blood cells suspended in a salt solution.
The chloride concentration (%) was measured over a period of about 8 minutes as a continuous curve
generated from electrical potentials. The data given here were digitized from the curve at 10 second intervals.

The theory of ion transport suggested that the concentration asymptote exponentially, i.e.,

   Chloride = q1 {1 - q2 exp(- q3 Time)}



--------------------------------------------------------------------------------
 Variable        Description
--------------------------------------------------------------------------------
 Time            Elapsed time in minutes
 Chloride        Chloride concentration (%)
--------------------------------------------------------------------------------


Source
Sredni, J. (1970). Problems of Design, Estimation and Lack of Fit in Model Building. PhD Thesis, University of Wisconsin-Madison.

Bates, D. M., and Watts, D. G. (1988). Nonlinear Regression Analysis and Its Applications. Wiley, New York.

The data were originally collected by W. H. Dennis and P. Wood at the University of Wisconsin. The data were obtained electronically from
the web site Data sets from "Nonlinear Regression Analysis" maintained by Douglas Bates


Analysis
> chloride <- read.table("chloride.txt",header=T)
> attach(chloride)
> plot(Time,Chloride)
> out <- nls(Chloride~cbind(1,exp(-b*Time)),start=list(b=0.1),algorithm="plinear")
> summary(out)

Formula: Chloride ~ cbind(1, exp( - b * Time))

Parameters:
       Value Std. Error  t value
b   0.158523   0.010444  15.1784
   39.094900   0.974427  40.1209
  -32.387600   0.561816 -57.6481

Residual standard error: 0.191962 on 51 degrees of freedom

Correlation of Parameter Estimates:
      b
 -0.994
  0.911 -0.950
> lines(Time,fitted(out),col=2)

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set i /i1*i54/;

table data(i,*)

        Time        Chloride
 i1     2.45        17.3
 i2     2.55        17.6
 i3     2.65        17.9
 i4     2.75        18.3
 i5     2.85        18.5
 i6     2.95        18.9
 i7     3.05        19
 i8     3.15        19.3
 i9     3.25        19.8
 i10    3.35        19.9
 i11    3.45        20.2
 i12    3.55        20.5
 i13    3.65        20.6
 i14    3.75        21.1
 i15    3.85        21.5
 i16    3.95        21.9
 i17    4.05        22
 i18    4.15        22.3
 i19    4.25        22.6
 i20    4.35        22.8
 i21    4.45        23
 i22    4.55        23.2
 i23    4.65        23.4
 i24    4.75        23.7
 i25    4.85        24
 i26    4.95        24.2
 i27    5.05        24.5
 i28    5.15        25
 i29    5.25        25.4
 i30    5.35        25.5
 i31    5.45        25.9
 i32    5.55        25.9
 i33    5.65        26.3
 i34    5.75        26.2
 i35    5.85        26.5
 i36    5.95        26.5
 i37    6.05        26.6
 i38    6.15        27
 i39    6.25        27
 i40    6.35        27
 i41    6.45        27
 i42    6.55        27.3
 i43    6.65        27.8
 i44    6.75        28.1
 i45    6.85        28.1
 i46    6.95        28.1
 i47    7.05        28.4
 i48    7.15        28.6
 i49    7.25        29
 i50    7.35        29.2
 i51    7.45        29.3
 i52    7.55        29.4
 i53    7.65        29.4
 i54    7.75        29.4

;

parameters
   Time(i)            'Elapsed time in minutes'
   Chloride(i)        'Chloride concentration (%)'
;

Time(i) = data(i,'Time');
Chloride(i) = data(i,'Chloride');


variables
   a1  'parameter to be estimated'
   a2  'parameter to be estimated'
   a3  'parameter to be estimated'
   sse 'sum of squared errors'
;

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

sumsq..    sse =n= 0;
fit(i)..   Chloride(i) =e= a1+a2*exp(-a3*Time(i));

a3.l=1;

option nlp=nls;



models m /sumsq,fit/;
solve m minimizing sse using nlp;