This function allows you to fit a kinetic trace for the selected wavelength with a sum of convoluted exponentials:
where IRF is the width of instrument response function (full width half maximum), t0 is time zero, Ai and ti are amplitudes and decay times respectively, * is convolution.
Much like in the previous section, before proceeding with the fitting operation one needs to select the desired time profile by using the vertical cursor on the 3D Surface Panel.
NOTE: Surface Xplorer operates with all the data independent of main window kinetics panel range. All the unwanted segments of the kinetics can be trimmed off by using the Crop function from the Surface menu.
To activate the fitting procedure, select Fit Kinetics option from Kinetics drop down menu. The new Fit Kinetics window will open (Figure 34). Central panel displays the kinetic trace to be fitted Sexp (blue dots) and the S(t) fit itself (red solid line), below is a residuals graph (Sexp-S). You can adjust scale range using the graph control pallet (by default it appears automatically when hovering the mouse over graph). Standard deviation value below Residuals graph is the numerical characterization of the fit quality:
Top left portion of the screen is used to control the fit procedure. To run fit you should provide basic information about kinetics: number of components to use (Finite lifetimes control), whether to use component with infinite lifetime or not. This information will be displayed in Current fit table. At first launch all amplitudes (Ainf, Ai) are 0 and fit graph shows straight fit line.
NOTE: In order to obtain good fit results, number of components set should be reasonable. Trying to fit data with excessive number of components can yield inconsistent results.
NOTE: As there is no constant offset in the fit formula, background should be corrected in main window before running the fitting function.
You can press Fit button to run the procedure with default parameters. When the fitting finishes, graphs and Current fit table will be updated with fit results. Table shows the optimal coefficient values ± corresponding confidence intervals found by the algorithm. Confidence interval is a factor that shows both fit quality and model sensitivity to a given parameter. Confidence intervals are always calculated with some probability - in SX it is set to 90%. It roughly means that modifying parameters within confidence intervals will not change fit criteria (standard deviation) more than 10%.
NOTE: Mouse cursor is disabled until fitting procedure is completed.
After fit procedure ends, please review parameters for consistency:
- Fit and experimental data should be close giving low residual values
- t0 can be visually estimated from the graph and found value should be close to it.
- IRF width should be reasonable: it is determined by probe and pump pulse widths.
- Components’ amplitudes should be reasonable. For example, 2 components with very close lifetimes and huge opposite sign amplitudes indicate bad fit setup.
You can save resulting fit parameters into the table (and automatically into FitCoefficients.csv file). Please see file description in Appendix Fit Files. Table shows normalized coefficients: all amplitude modules sum up to 1. To show absolute values, remove checkmark Normalized coefficients in the right corner of the window.
If fit results are not consistent, you may need to use advanced fitting tools to restrict parameters.
Click Current Fit table row to show component control tools (Figure 35).
If you click any of the time parameter (t0, IRF, lifetimes ti) rows in the Current Fit table, a control will pop up allowing you to individually limit these parameters. It contains the following elements:
- Scale with sliders. Contains (left to right) Low limit, Value and High limit sliders. Scale range is determined by graph X scale range. Zoom in to the beginning of the rise on the kinetics and then set IRF, t0, short lifetimes ti more precisely. Click Autoscale once button in the X axis control panel to show all graph and then set long lifetimes. You can also type in scale limits in the control, they do not affect slider values.
- Value numeric field and large slider – current parameter value. Value displayed in the table, fit and residuals graphs will be updated automatically as you change it.
NOTE: You cannot set Value outside limits, change limits first.
- Low limit and High limit numerical fields and left and right sliders’ values. These controls allow you to set the parameter limits during the fit, result will never go out of the defined range. By default High limit is Inf and Low limit is 0, except for t0 - it is determined by the time scale low limit. Type a value into numerical fields to set parameter limit outside scale range. Table row will be colored yellow and header will say limited. Obviously this parameter does not play any role if parameter is fixed. But you can set initial guess and limit parameter at the same time. This is also a very powerful tool – it can be used to order parameter lifetimes, ignore noisy signal in the beginning, etc.
- Fix – fix current value not to be modified during the fit. Table row will be colored red and say fixed in row header.
- Initial guess – use current value as initial guess for the algorithm. Fit results can be improved dramatically if you enter correct initial guess values. If parameter was fixed, it becomes variable automatically and Fix button returns to the original state. Table row is colored blue and row header says Initial guess.
- Reset. Clears all restrictions of the parameter: makes it variable in unlimited ranges (0, inf) with default guess value.
Click anywhere outside the control to hide it. Changes are applied automatically.
The following buttons to the left of the table control all time parameters:
- Fixed. All time parameters are fixed at their current values.
- Initial guess. Initial guesses for all time parameters that are not fixed are set at their current values.
- Variable. Clears all restrictions for all parameters: all become variable in unlimited range with default guess values.
If you click any of the Ainf, or Ai rows in the table a control (Figure 36) will pop up, allowing you to change component amplitude using slider or by entering a value. Fit and residuals graphs will be updated automatically to reflect new amplitude. You can also type slider limits for better slider range.
NOTE: You cannot limit or fix amplitude values.
Figure 36 shows parameters with multiple restrictions
All parameter restrictions and values are saved if you close the Fit Kinetics window and then load it with new data.