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All Books(/doc)Origin Help(/doc/Origin-Help)Regression and Curve Fitting(/doc/Origin-Help/Regression-Curve-Fitting)Nonlinear Curv e Fitting(/doc/Origin-Help/Nonlinear-Curv e-Fit)

Origin Help Search 17.7.4Algorithms(Nonlinear Curve Fitting)

Contents

1How Origin Fits the Curve

1.1The Fitting Model for Explicit Functions

1.2The Fitting Model for Implicit Functions

1.3Weighted Fitting

2Parameters

2.1The Fitted Value

2.2Parameter Standard Errors

2.3The Standard Error for Derived Parameter

2.4Confidence Intervals

2.5t Value

2.6Prob>|t|

2.7Dependency

2.8CI Half Width

3Statistics

3.1Degree of Freedom

3.2Residual Sum of Squares

3.3Reduced Chi-Sqr

3.4R-Square(COD)

3.5Adj.R-Square

3.6R Value

3.7Root-MSE(SD)

4ANOVA Table

5Confidence and Prediction Bands

5.1Confidence Band

5.2Prediction Band

6Reference

How Origin Fits the Curve

The Fitting Model for Explicit Functions

A general nonlinear model can be expressed as follows:

(1)

where is the independent variables and is the parameters.

The aim of nonlinear fitting is to estimate the parameter values which best describe the data.The standard way of finding the best fit is to choose the parameters that would minimize the deviations of the theoretical curve(s)from the experimental points.This method is also called chi-square minimization,defined as follows:

(2)

where is the row vector for the i th(i=1,2,...,n)observation.

To estimate the value with the least square method,we need to solve the normal equations which are set to be zero for the partial derivatives o with respect to each.

(3)

Since there are no explicit solutions to the normal equations,we employ an iterative strategy to estimate the parameter values.This process starts with some initial values,.With each iteration,a value is computed and then the parameter values are adjusted to reduce the.When the values computed in two successive iterations are small enough(compared with the tolerance),we can say that the fitting procedure has converged.In the NLFit output messages,you can see the reduced chi-square,which is the mean deviation for all data points,as shown below:

(4)

Origin uses the Levenberg-Marquardt(L-M)algorithm to adjust the parameter values in the iterative procedure.This algorithm,which combines the Gauss-Newton method and the steepest descent method,works for most cases.You may wish to consult other sources for details of the L-M algorithm.Origin's fitter additionally offers the Simplex method and orthogonal distance regression algorithm.

The Fitting Model for Implicit Functions

A general implicit function could be expressed as:

originerror

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