Regress.NLinRegress Function

General non-linear regression.

Pascal

function NLinRegress(const X: TVec; const Y: TVec; RegFun: TRegressFun; DeriveProc: TDeriveProc; const B: TVec; Method: TOptMethod; out StopReason: TOptStopReason; const Weights: TVec = nil; const YCalc: TVec = nil; SoftSearch: boolean = false; MaxIter: Integer = 500; Tol: double = 0.00000001; GradTol: double = 0.00000001; const Verbose: TStrings = nil): Integer; overload;

File

Regress

Parameters

Parameters	Description
X	Vector of the independent variable.
Y	Vector of dependent variable.
RegFun	Regression function.
DeriveProc	Procedure to calculate the derivatives of RegFun. You can define the exact derivative or use NumericDerive routine as numerical approximation.
B	Holds initial estimate for regression parameters. After the call to NLinRegress b returns calculated regression parameters.
Method	Defines which optimization method will be used to find regression parameters (see MtxVec.hlp TOptMethod type to learn more about this).
StopReason	Returns why regression parameters search stopped (see MtxVec.hlp TOptStopReason type to learn more about different stop reasons).
Weights	Weights (optional).
YCalc	Returns calculated values (optional).
SoftSearch	If true, internal line search algoritm will use soft line search method. Set this parameter to true if you're using numerical approximation for derivative. If this parameter is set to false, internal line search algorithm will use exact line search method. Set this parameter to false if you're using exact derivative.
MaxIter	Maximum allowed numer of allowed iterations.
Tol	Desired regression parameters tolerance.
GradTol	Minimum allowed gradient C-Norm.
Verbose	If assigned, stores Fun, evaluated at each iteration step. Optionally, you can also pass TOptControl object to the Verbose parameter. This allows the optimization procedure to be interrupted from another thread and optionally also allows logging and iteration count monitoring.

Returns

Number of iterations needed to calculate regression parameters with specified tolerance.

Description

The routine fits equations to data by minimizing the sum of squared residuals :

SS = Sum [y(k) - ycalc(k)]^2 ,

where y(k) and ycalc(k) are respectively the observed and calculated value of the dependent variable for observation k. ycalc(k) is a function of the regression parameters b(0), b(1) ... Here the observed values obey the following (non-linear) equation:

y(k) = RegFun[x(k), b(0), b(1), ... ]

Y = RegFun[X,b(0),b(1), ...]

where RegFun is the regression function and b(0),..b(i) are the regression parameters.

See Also

NumericDerive

Example

The following example uses data from NIST study involving circular interference transmittance. The response variable is transmittance, and the predictor variable is wavelength. First we setup the regression function Eckerle4 with three regression parameters (b0,b1,b2). Then we setup data and specify initial estimate for regression parameters (see below):

Uses MtxExp, Math387, Regress, Optimization, MtxVecTee;
// function definition
function Eckerle4(const B: TVec; X: double): double;
begin
  Eckerle4 := B[0]/B[1] * Exp(-0.5*Sqr((X-B[2])/B[1]));
end;
procedure Example;
var x,y,b,yhat: Vector;
  StopReason: TOptStopReason;
begin
  x.SetIt(false,[400.0, 405.0, 410.0, 415.0,
                 420.0, 425.0, 430.0, 435.0,
                 436.5, 438.0, 439.5, 441.0,
                 442.5, 444.0, 445.5, 447.0,
                 448.5, 450.0, 451.5, 453.0,
                 454.5, 456.0, 457.5, 459.0,
                 460.5, 462.0, 463.5, 465.0,
                 470.0, 475.0, 480.0, 485.0,
                 490.0, 495.0, 500.0]);
  y.SetIt(false,[0.0001575, 0.0001699, 0.0002350, 0.0003102,
                 0.0004917, 0.0008710, 0.0017418, 0.0046400,
                 0.0065895, 0.0097302, 0.0149002, 0.0237310,
                 0.0401683, 0.0712559, 0.1264458, 0.2073413,
                 0.2902366, 0.3445623, 0.3698049, 0.3668534,
                 0.3106727, 0.2078154, 0.1164354, 0.0616764,
                 0.0337200, 0.0194023, 0.0117831, 0.0074357,
                 0.0022732, 0.0008800, 0.0004579, 0.0002345,
                 0.0001586, 0.0001143, 0.0000710]);
  b.SetIt(false,[1.0, 10.0, 500.0]); // initial estimates
  NLinRegress(x,y,Eckerle4,nil,b,optMarquardt, StopReason,
              nil,yhat,false,300,1e-8,1e-10);
  DrawValues(x,y,Series1,false); // draw data
  DrawValues(x,yhat,Series2,false); // draw fitted values
end;

#include "MtxExpr.hpp"
#include "Regress.hpp"
#include "MtxVecTee.hpp"  //add TChart to the form

double __fastcall Eckerle4(TVec * const B, double x)
{
  double a = (x - B->Values[2])/B->Values[1];
  return B->Values[0]/B->Values[1] * Exp(-0.5*a*a);
}

void __fastcall Test(TLineSeries* series1, TLineSeries* series2)
{
  sVector x,y,b,yhat;
  TOptStopReason StopReason;

  const int xLen = 35;
  double xData[xLen] = {400.0, 405.0, 410.0, 415.0,
                       420.0, 425.0, 430.0, 435.0,
                       436.5, 438.0, 439.5, 441.0,
                       442.5, 444.0, 445.5, 447.0,
                       448.5, 450.0, 451.5, 453.0,
                       454.5, 456.0, 457.5, 459.0,
                       460.5, 462.0, 463.5, 465.0,
                       470.0, 475.0, 480.0, 485.0,
                       490.0, 495.0, 500.0};

  x.SetIt(false, xData, xLen-1);

  const int yLen = 35;
  double yData[yLen] = {0.0001575, 0.0001699, 0.0002350, 0.0003102,
                       0.0004917, 0.0008710, 0.0017418, 0.0046400,
                       0.0065895, 0.0097302, 0.0149002, 0.0237310,
                       0.0401683, 0.0712559, 0.1264458, 0.2073413,
                       0.2902366, 0.3445623, 0.3698049, 0.3668534,
                       0.3106727, 0.2078154, 0.1164354, 0.0616764,
                       0.0337200, 0.0194023, 0.0117831, 0.0074357,
                       0.0022732, 0.0008800, 0.0004579, 0.0002345,
                               0.0001586, 0.0001143, 0.0000710};

  y.SetIt(false, yData, yLen-1);

  b.SetIt(false,OPENARRAY(double,(1.0, 10.0, 500.0))); // initial estimates
  NLinRegress(x,y,Eckerle4,NULL,b,optMarquardt, StopReason,
                           NULL,yhat,false,300,1e-8,1e-10);
  DrawValues(x,y,series1,false); // draw data
  DrawValues(x,yhat,series2,false); // draw fitted values
}

Regress, Functions, Example, See Also

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