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fit_formulas [Thu Nov 24 10:21:25 2011] Alexander |
fit_formulas [Tue May 30 16:28:13 2017] (current) Alexander |
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====== Fitting Algorithm and Computational Formulas ====== | ====== Fitting Algorithm and Computational Formulas ====== | ||
- | MagicPlot uses iterative [[wp> | + | MagicPlot uses iterative [[wp> |
- | MagicPlot | + | MagicPlot |
===== Nonlinear Least Squares Algorithm | ===== Nonlinear Least Squares Algorithm | ||
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* //f//(//x, β< | * //f//(//x, β< | ||
* //p// is the number of fit parameters // | * //p// is the number of fit parameters // | ||
- | * // | + | * // |
- | An initial guess for the parameters has to be provided to start minimization. Calculation of the new guess of parameters on each fit iteration is based on the fit function partial derivatives for current values of fit parameters | + | An initial guess for the parameters has to be provided to start minimization. Calculation of the new guess of parameters on each fit iteration is based on the fit function partial derivatives for current values of fit parameters for each //x// value: |
< | < | ||
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===== Weighting of Data Points Using Y Errors ===== | ===== Weighting of Data Points Using Y Errors ===== | ||
MagicPlot can use weighting of //y// values based on y errors // | MagicPlot can use weighting of //y// values based on y errors // | ||
- | * If standard //y// errors are **not** specified: all // | ||
- | * If standard //y// errors // | ||
- | <m>w_i=C 1/{{s_i}^2}</m> | + | * If standard //y// errors //s<sub>i</ |
- | + | * Otherwise: all //w<sub>i</sub>// = 1. | |
- | here //C// is normalizing coefficient | + | |
- | + | ||
- | <m>C=N sum{i=1}{N}{{s_i}^2}</m> | + | |
In '' | In '' | ||
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* If the number of iterations exceeds maximum number of iterations, which is 100 by default | * If the number of iterations exceeds maximum number of iterations, which is 100 by default | ||
- | You can change the minimum allowable deviation decrement and maximum number of iterations in Fitting tab of MagicPlot Preferences. | + | You can change the minimum allowable deviation decrement and maximum number of iterations in '' |
- | ===== Formulas ===== | + | ===== Formulas |
In the table below you can find the formulas which MagicPlot uses to calculate fit parameters and values in '' | In the table below you can find the formulas which MagicPlot uses to calculate fit parameters and values in '' | ||
Because of some confusion in the names of the parameters in different sources (books and software), we also give many different names of same parameter in //note// column. | Because of some confusion in the names of the parameters in different sources (books and software), we also give many different names of same parameter in //note// column. | ||
+ | |< 100% 15% 10% 45% 30% >| | ||
^ Parameter Name ^ Symbol | ^ Parameter Name ^ Symbol | ||
^ Original Data and Fit Model Properties | ^ Original Data and Fit Model Properties | ||
- | | Number of used data points | + | | Number of used data points |
- | | Fit parameters | + | | Fit parameters |
- | | Number of fit function parameters //β// | < | + | | Number of fit function parameters //β// | < |
| [[wp> | | [[wp> | ||
| Estimated mean of data | < | | Estimated mean of data | < | ||
| Estimated variance of data | < | | Estimated variance of data | < | ||
- | | Data total sum of squares, TSS | TSS | < | + | | Data total sum of squares |
^ Fit Result | ^ Fit Result | ||
- | | Residual sum of squares, < | + | | Residual sum of squares |
- | | Reduced // | + | | Reduced // |
- | | Residual standard deviation | + | | Residual standard deviation |
| [[wp> | | [[wp> | ||
- | | Adjusted // | + | | Adjusted // |
- | | Covariance matrix of parameters // | + | | Covariance matrix of parameters // |
- | | Standard deviation of parameters // | + | | Standard deviation of parameters // |
- | | Correlation matrix of parameters // | + | | Correlation matrix of parameters // |
===== See Also ===== | ===== See Also ===== | ||
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* [[guess_peaks]] | * [[guess_peaks]] | ||
* [[fit_equations]] | * [[fit_equations]] | ||
- | * [[transform_xy]] | ||
* [[interval_statistics]] | * [[interval_statistics]] |