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Peak Fitter

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A Matlab peak fitting program for time-series signals, which uses an unconstrained non-linear optimization algorithm to decompose a complex, overlapping-peak signal into its component parts. The objective is to determine whether your signal can be represented as the sum of fundamental underlying peaks shapes. Accepts signals of any length, including those with non-integer and non-uniform x-values. Fits groups of up to six peaks with eight distinct peak shapes (expandable to other shapes). There are two different versions, (1) a command line version (peakfit.m) for Matlab or Octave, (2) a keypress operated interactive version (ipf.m) for Matlab only. The difference between them is that peakfit.m is completely controlled by command-line input arguments and returns its information via command-line output arguments; ipf.m allows interactive control via keypress commands. For automating the fitting of large numbers of signals, peakfit.m is better; ipf.m is best for exploring signals to determine the optimum fitting range, peak shapes, number of peaks, baseline correction mode, etc. Otherwise they have the same curve-fitting capabilities. The peak shape models available are illustrated in this graphic. See Notes and Hints for more information and useful suggestions.

1. Command line function: peakfit.m (Version 5.8, September, 2014)  

Peakfit.m is a user-defined command window peak fitting function for Matlab or Octave, usable from a remote terminal. It is written as a self-contained function in a single m-file. (To view of download, click Peakfit.m). It takes data in the form of a 2 x n matrix that has the independent variables (X-values) in row 1 and the dependent variables (Y-values) in row 2, or as a single dependent variable vector. The syntax is peakfit(signal, center, window, NumPeaks, peakshape, extra, NumTrials, start, AUTOZERO, fixedwidth, plots, bipolar, minwidth, DELTA). Only the first argument, the data matrix, is required.

The screen display is shown on the right; the upper panel shows the data as blue dots, the combined model as a red line (ideally overlapping the blue dots), and the model components as green lines. The dotted magenta lines are the first-guess peak positions for the last fit. The lower panel shows the residuals (difference between the data and the model).

You can download a ZIP file containing peakfit.m, DemoPeakFit.m, ipf.m, Demoipf.m, some sample data for testing, and a test script (testpeakfit.m) that runs all the examples sequentially to test for proper operation. For a discussion of the accuracy and precision of peak parameter measurement using peakfit.m, click here. Version 3.7 and later works in Octave as well as in Matlab. Version 4.0 adds an additional autozero mode that subtracts a flat baseline without requiring that the signal return to the baseline at both ends of the signal segment. Version 4.2 corrects some bugs and adds a Voigt profile peak shape.Version 4.3 adds 12th input argument, for + or +/- peak mode. Version 5 adds the ability to fit multiple peak shapes in one signal. See examples 24 and 25 below. Version 5.1 includes "baseline" as an additional output argument (for the flat baseline correction mode 3). Version 5.3 separates sigmoid peak shape into 'up sigmoid' or logistic function (shape 10) and 'down sigmoid' (shape 23). Version 5.4: June, 2014. Replaces bifurcated Lorentzian with Breit-Wigner-Fano resonance peak (Shape=15).  Version 5.5: July, 2014. Adds shape 21 (triangular) and 25 (Lognormal distribution). Moves peak table to lower panel and plots residuals as red dots, as shown in the figure on the right. Version 5.7: August, 2014. Adds minimum width constraint as 13th input argument (See example 19); Can be a vector for multiple peak shapes. The default if not specified is the independent variable (x) interval. Version 5.8: September, 2014 adds optional 14th input argument DELTA (controls the variance in start values between successive trials when NumTrials>1). The default value is 1.0; larger values give more variance. 

Peakfit can be called with several optional additional arguments. Note: In version 2.5 and later, all input arguments (except the signal itself) can be replaced by zeros to use their default values.

peakfit(signal);     
  Performs an iterative least-squares fit of a single Gaussian  peak to the entire data matrix "signal", which has x values in row 1 and Y values in row 2 (e.g. [x y]) or which may be a single signal vector (in which case the data points are plotted against their index numbers on the x axis).
 
peakfit(signal,center,window);       
  Fits a single Gaussian peak to a portion of the matrix "signal". The portion is centered on the x-value "center" and has width "window" (in x units). 

In this and in all following examples, set "center" and "window" both to 0 to fit the entire signal. 

peakfit(signal,center,window,NumPeaks);
  "NumPeaks" = number of peaks in the model (default is 1 if not specified; may be a large as desired in version 3.1 and above).
 
peakfit(signal,center,window,NumPeaks,peakshape);
   Specifies the peak shape of the model: "peakshape" = 1-23. (1=Gaussian, 2=Lorentzian, 3=logistic distribution, 4=Pearson, 5=exponentially broadened Gaussian; 6=equal-width Gaussians, 7=equal-width Lorentzians, 8=exponentially broadened equal-width Gaussians, 9=exponential pulse, 10= up-sigmoid (logistic function), 11=fixed-width Gaussians, 12=fixed-width Lorentzians, 13=Gaussian/Lorentzian blend; 14=bifurcated Gaussian, 15=Breit-Wigner-Fano resonance; 16=Fixed-position Gaussians; 17=Fixed-position Lorentzians; 18=exponentially broadened Lorentzian; 19=alpha function; 20=Voigt profile; 21=triangular; 22=multiple shapes; 23=down-sigmoid; 25=lognormal distribution. In version 5 and later, the peakshape can be a vector of different shapes for each peak, e.g. [1 2 1] for three peaks in a Gaussian, Lorentzian, Gaussian sequence. (The function ShapeDemo demonstrates most of the basic peak shapes graphically, showing the variable-shape peaks as multiple lines).
  
peakfit(signal,center,window,NumPeaks,peakshape,extra)
  Specifies the value of 'extra', used in the Pearson, exponentially-broadened Gaussian, Gaussian/Lorentzian blend, bifurcated Gaussian, and Breit-Wigner-Fano shapes to fine-tune the peak shape. In version 5, 'extra' can be a vector of different extra values for each peak).
  
peakfit(signal,center,window,NumPeaks,peakshape,extra,NumTrials);
  Performs "NumTrials" trial fits and selects the best one (with lowest fitting error). NumTrials can be any positive integer (default is 1).
 
peakfit(signal,center,window,NumPeaks,peakshape,extra,NumTrials,start)
  Specifies the first guesses vector "firstguess" for the peak positions and widths, e.g. start=[position1 width1 position2 width2 ...]

peakfit(signal,center,window,NumPeaks,peakshape,extra,NumTrials,start,autozero)
As above, but "autozero" sets the baseline correction mode in the last argument:
autozero=0 (default) does not subtract baseline from data segment;. autozero=1 interpolates a linear baseline from the edges of the data segment and subtracts it from the signal (assumes that the peak returns to the baseline at the edges of the signal); autozero=2,  like mode 1 except that it computes a quadratic curved baseline; autozero=3 compensates for a flat baseline without reference to the signal itself (does not require that the signal return to the baseline at the edges of the signal, as does modes 1 and 2).

peakfit(signal,0,0,0,0,0,0,0,2)
  Use zeros as placeholders to use the default values of input arguments.  In this case,
autozero is set to 2, but all others are the default values (Version 2.5 and later).

peakfit(signal,center,window,NumPeaks,peakshape,extra,NumTrials,start,autozero,fixedparameters)
 'fixedparameters' (10th input argument) specifies fixed values for widths (shapes 10, 11) or positions (shapes 16, 17)
 
peakfit(signal,center,window,NumPeaks,peakshape,extra,NumTrials,start,autozero,fixedparameters,plots)
 'plots' (11th input argument) controls graphic plotting: 0=no plot; 1=plots draw as usual (default)
 
peakfit(signal,center,window,NumPeaks,peakshape,extra,NumTrials,start,autozero,fixedparameters,plots,bipolar)
 (12th input argument) 'bipolar' = 0 constrain peaks heights to be positions; 'bipolar' = 1 allows positive ands negative peak heights.
 
peakfit(signal,center,window,NumPeaks,peakshape,extra,NumTrials,start,autozero,fixedparameters,plots,bipolar,minwidth)
  'minwidth' (13th input argument) sets the minimum allowed peak width. The default if not specified is equal to the x-axis interval. Must be a vector of minimum  widths, one value for each peak, if the multiple peak shape is chosen.

peakfit(signal,center,window,NumPeaks,peakshape,extra,NumTrials,start,autozero,fixedparameters,plots,bipolar,minwidth)
 'DELTA' (14th input argument) controls the restart variance when NumTrials>1. Default value is 1.0. Larger values give more variance. Version 5.8 and later only.

[FitResults,FitError]= peakfit(signal,center,window...);
Returns the FitResults vector in the order peak number, peak position, peak height, peak width, and peak area), and the FitError (the percent RMS difference between the data and the model in the selected segment of that data) of the best fit. To display column labels for the FitResults table, execute the following statements after calling peakfit(...);: disp('      Peak number  Position      Height      Width       Peak area');disp(FitResults)

[FitResults,LowestError,baseline,BestStart,xi,yi,BootstrapErrors]= peakfit([x;y],0,0,2,6,0,1,0,0,0);

Displays parameter error estimates; Version 3 only. See DemoPeakfitBootstrap for a self-contained demo of this function.

Optional output parameters:
  1. FitResults: a table of model peak parameters, one row for each peak, listing Peak number, Peak position, Height, Width, and Peak area.
  2. LowestError: The RMS fitting error of the best trial fit.
  3. baseline: the value of the flat baseline determined by baseline correction mode 3.
  4. BestStart: the starting guesses that gave the best fit.
  5. xi: vector containing 600 interpolated x-values for the model peaks.
  6. yi: matrix containing the y values of model peaks at each xi. Type plot(xi,yi(1,:)) to plot peak 1 or plot(xi,yi) to plot all the peaks
  7. BootstrapErrors: a matrix containing bootstrap standard deviations and interquartile ranges for each peak parameter of each peak in the fit.

Examples:
Note: test script testpeakfit.m runs all the following examples automatically; just press Enter to continue to the next one.

Example 1.  Fits computed x vs y data with a single Gaussian peak model.
  > x=[0:.1:10];y=exp(-(x-5).^2);peakfit([x' y'])
  ans =
         Peak number  Peak position   Height     Width      Peak area
              1            5            1        1.665       1.7725

Example 2. Fits small set of manually-entered y data to a single Gaussian peak model.  
  > y=[0 1 2 4 6 7 6 4 2 1 0 ];x=1:length(y);
  > peakfit([x;y],length(y)/2,length(y),0,0,0,0,0,0)

         Peak number  Peak position   Height     Width      Peak area
             1       6.0001       6.9164       4.5213        32.98


Example 3.  Measurement of very noisy peak with signal-to-noise ratio = 1. (Try several times).
> x=[0:.01:10];y=exp(-(x-5).^2)+randn(size(x));peakfit([x;y])
         Peak number  Peak position   Height     Width      Peak area
             1       5.0951       1.0699       1.6668       1.8984

Example 4.  
Fits a noisy two-peak signal with a double Gaussian model (NumPeaks=2).
  > x=[0:.1:10];y=exp(-(x-5).^2)+.5*exp(-(x-3).^2)+.1*randn(1,length(x));
 > peakfit([x' y'],5,19,2,1,0,1)
         Peak number  Peak position   Height     Width      Peak area
              1       3.0001      0.49489        1.642      0.86504
              2       4.9927       1.0016       1.6597       1.7696

Example 5.
Fits a portion of the humps function, 0.7 units wide and centered on x=0.3, with a single (NumPeaks=1) Pearson function (peakshape=4) with extra=3 (controls shape of Pearson function).
    > x=[0:.005:1];y=humps(x);peakfit([x' y'],.3,.7,1,4,3);
    
Example 6. Creates a data matrix 'smatrix', fits a portion to a two-peak Gaussian model, takes the best of 10 trials.  Returns optional output arguments FitResults and FitError.
  > x=[0:.005:1];y=(humps(x)+humps(x-.13)).^3;smatrix=[x' y'];
 > [FitResults,FitError]=peakfit(smatrix,.4,.7,2,1,0,10)
   FitResults =
               1       0.4128  3.1114e+008      0.10448  3.4605e+007
               2       0.3161  2.8671e+008     0.098862  3.0174e+007
   FitError = 0.68048
 
Example 7. As above, but specifies the first-guess position and width of the two peaks, in the order [position1 width1 position2 width2]
  > peakfit([x' y'],.4,.7,2,1,0,10,[.3 .1 .5 .1]);

Supplying a first guess
position and width is also useful if you have one peak on top of another (like example 4, with both peaks at the same position x=5, but with different widths):
>> x=[2:.01:8];y=exp(-((x-5)/.2).^2)+.5.*exp(-(x-5).^2)+.1*randn(1,length(x));
>> peakfit([x' y'],0,0,2,1,0,1,[5 2 5 1])
ans =
            1       4.9977      0.51229        1.639      0.89377
            2       4.9948       1.0017      0.32878      0.35059

Example 8. As above, returns the vector x1 containing 200 interpolated x-values for the model peaks and the matrix y1 containing the y values of each model peak at each xi.  Type plot(xi,yi(1,:)) to plot peak 1 or plot(xi,yi) to plot all peaks.
> [FitResults,LowestError,BestStart,xi,yi]= peakfit(smatrix,.4,.7,2,1,0,10)

Example 9. Fitting a  single Gaussian on a linear background, using the linear autozero mode (9th input argument = 1)
    >>x=[0:.1:10]';y=10-x+exp(-(x-5).^2);peakfit([x y],5,8,0,0,0,0,0,1)

Example 10. Fits a group of three peaks near x=2400 in DataMatrix3 with three equal-width exponentially-broadened Gaussians.

>> load DataMatrix3
>> [FitResults,FitError]= peakfit(DataMatrix3,2400,440,3,8,31,1)

FitResults =
            1       2300.5      0.82546       60.535       53.188
            2       2400.4      0.48312       60.535       31.131
            3       2500.6      0.84799       60.535       54.635
FitError = 0.19975

Example 11.   Example of an unstable fit to a signal consisting of two Gaussian peaks of equal height (1.0). The peaks are too highly overlapped for a stable fit, even though the fit error is small and the residuals are unstructured. Each time you re-generate this signal, it gives a different fit, with the peaks heights varying about 15% from signal to signal.  
>> x=[0:.1:10]'; y=exp(-(x-5.5).^2)+exp(-(x-4.5).^2)+.01*randn(size(x)); [FitResults,FitError]=peakfit([x y],5,19,2,1)
FitResults =
            1       4.4059      0.80119       1.6347       1.3941
            2       5.3931       1.1606       1.7697       2.1864
FitError = 0.598

  Much more stable results can be obtained using the equal-width Gaussian model (
peakfit([x y],5,19,2,6)), but that is justified only if the experiment is legitimately expected to yield peaks of equal width. See http://terpconnect.umd.edu/~toh/spectrum/CurveFittingC.html#Peak_width_constraints.

Example 12.  Demonstration of the four autozero modes, for a single Gaussian on flat baseline, with position=10, height=1, and width=1.66. The autozero mode is specified by the 9th input argument (0,1,2, or 3).

 Autozero=0 means to ignore the baseline (default mode if not specified). In this case, this leads to large errors.
>>
x=8:.05:12;y=1+exp(-(x-10).^2);
>> [FitResults,FitError]=peakfit([x;y],0,0,1,1,0,1,0,0)
  FitResults =
              1           10       1.8561        3.612       5.7641
  FitError =5.387
 Autozero=1 subtracts linear baseline from edge to edge. Does not work well in this case because the signal does not return completely to
the baseline at the edges.
>> [FitResults,FitError]=peakfit([x;y],0,0,1,1,0,1,0,1)
  FitResults =
              1       9.9984      0.96153        1.559       1.5916
  FitError =1.9801

 Autozero=2 subtracts quadratic baseline from edge to edge. Does not work well in this case because the signal does not return
completely to the baseline at the edges.
>> [FitResults,FitError]=peakfit([x;y],0,0,1,1,0,1,0,2)
  FitResults =
              1       9.9996      0.81749       1.4384       1.2503
  FitError =1.8204

 Autozero=3 subtracts a flat baseline automatically, without requiring that the signal returns to baseline at the edges. This mode works best for this signal..
>>
[FitResults,FitError]=peakfit([x;y],0,0,1,1,0,1,0,3)
  FitResults =
              1           10      0.99999       1.6651       1.7641
  FitError =0.0012156

In the following case, the baseline is curved, so the best choice is autozero=2:
>> x=[0:.1:10]';y=1./(1+x.^2)+exp(-(x-5).^2);peakfit([x y],5,5.5,0,0,0,0,0,2)

    ans =
            1       5.0121       1.0007       1.6526       1.7602 

Example 13. Same as example 4, but with fixed-width Gaussian (shape 11), width=1.666.
>> x=[0:.1:10];y=exp(-(x-5).^2)+.5*exp(-(x-3).^2)+.1*randn(size(x));
>> [FitResults,FitError]=peakfit([x' y'],0,0,2,11,0,0,0,0,1.666)

    FitResults =
            1       3.9943      0.49537        1.666      0.87849
            2       5.9924      0.98612        1.666       1.7488

Example 14. Peak area measurements. Same as the example in Figure 15 on Integration and Peak Area Measurement.  All four peaks have the same theoretical peak area (1.772). The four peaks can be fit together in one fitting operation using a 4-peak Gaussian model, with only rough estimates of the first-guess positions and widths.  The peak areas thus measured are much more accurate than the perpendicular drop method:
>> x=[0:.01:18];
>> y=exp(-(x-4).^2)+exp(-(x-9).^2)+exp(-(x-12).^2)+exp(-(x-13.7).^2);
>> peakfit([x;y],0,0,4,1,0,1,[4 2 9 2 12 2 14 2],0,0)

         Peak#    Position       Height       Width         Area
            1            4            1       1.6651       1.7725
            2            9            1       1.6651       1.7725
            3           12            1       1.6651       1.7724
            4         13.7            1       1.6651       1.7725

This works well even in the presence of substantial amounts of random noise:
>> x=[0:.01:18]; y=exp(-(x-4).^2)+exp(-(x-9).^2)+exp(-(x-12).^2)+exp(-(x-13.7).^2)+.1.*randn(size(x));
>> peakfit([x;y],0,0,4,1,0,1,[4 2 9 2 12 2 14 2],0,0)
ans =
            1       4.0086      0.98555       1.6693       1.7513
            2       9.0223       1.0007        1.669       1.7779
            3       11.997       1.0035       1.6556       1.7685
            4       13.701       1.0002       1.6505       1.7573

Sometimes experimental peaks are effected by exponential broadening, which does not by itself change the true peak areas, but does shift peak positions and increases peak width, overlap, and asymmetry, as shown when you try to fit the peaks with Gaussians. Using the same noise signal from above:
>> y1=ExpBroaden(y',-50);
>> peakfit([x;y1'],0,0,4,1,50,1,0,0,0)
ans =
            1       4.4538      0.83851       1.9744       1.7623
            2       9.4291       0.8511       1.9084       1.7289
            3       12.089      0.59632        1.542      0.97883
            4       13.787       1.0181       2.4016       2.6026

Peakfit.m (and ipf.m) have an exponentially-broadened Gaussian peak shape (shape #5) that works better in those cases:, recovering the original peak positions, heights, widths, and areas. (Adding a first-guess vector as the 8th argument may improve the reliability of the fit in some cases).
>> y1=ExpBroaden(y',-50);
>> peakfit([x;y1'],0,0,4,5,50,1,[4 2 9 2 12 2 14 2],0,0)
ans=      Peak#    Position       Height       Width         Area
            1            4            1       1.6651       1.7725
            2            9            1       1.6651       1.7725
            3           12            1       1.6651       1.7725
            4         13.7      0.99999       1.6651       1.7716

Example 15. Displays a table of parameter error estimates; Version 3 only. See DemoPeakfitBootstrap for a self-contained demo of this function.
>> x=0:.05:9; y=exp(-(x-5).^2)+.5*exp(-(x-3).^2)+.01*randn(1,length(x));
>> [FitResults,LowestError,baseline,BestStart,xi,yi,BootstrapErrors]= peakfit([x;y],0,0,2,6,0,1,0,0,0);

Peak #1         Position    Height       Width       Area
Bootstrap Mean: 2.9987      0.49717     1.6657      0.88151
Bootstrap STD:  0.0039508   0.0018756   0.0026267   0.0032657
Bootstrap IQR:  0.0054789   0.0027461   0.0032485   0.0044656
Percent RSD:    0.13175     0.37726     0.15769     0.37047
Percent IQR:    0.18271     0.55234     0.19502     0.50658
 
Peak #2         Position    Height       Width       Area
Bootstrap Mean: 4.9997     0.99466       1.6657      1.7636
Bootstrap STD:  0.001561   0.0014858     0.00262     0.0025372
Bootstrap IQR:  0.002143   0.0023511     0.00324     0.0035296
Percent RSD:    0.031241   0.14938       0.15769     0.14387
Percent IQR:    0.042875   0.23637       0.19502     0.20014

Example 16. (Version 3.2 or later) Fits both peaks of the Humps function with a Gaussian/Lorentzian blend (shape 13) that is 15% Gaussian (Extra=15). The 'Extra' argument sets the percentage of Gaussian shape.
>> x=[0:.005:1];y=humps(x);[FitResults,FitError]= peakfit([x' y'],0.54,0.93,2,13,15,10,0,0,0)
 FitResults =
             1      0.30078       190.41      0.19131       23.064
             2      0.89788       39.552      0.33448       6.1999
  FitError = 0.34502

Example 17.  (Version 3.2 or later)
Fit a slightly asymmetrical peak with a bifurcated Gaussian (shape 14). The 'Extra' argument (=45) controls the peak asymmetry (50 is symmetrical). 
>> x=[0:.1:10];y=exp(-(x-4).^2)+.5*exp(-(x-5).^2)+.01*randn(size(x));

>> [FitResults,FitError]=peakfit([x' y'],0,0,1,14,45,10,0,0,0)
 FitResults =
             1       4.2028       1.2315        4.077       2.6723
  FitError =0.84461

Example 18. 
(Version 3.3 or later) Example 1 without plotting or command window printing (11th input argument = 0, default is 1)
>> x=[0:.1:10]';y=exp(-(x-5).^2);peakfit([x y],0,0,1,1,0,0,0,0,0,
0)

Example 19.  (Version 3.6 or later) Same as example 4, but with fixed-position Gaussian (shape 16), positions=[3 5]
>> x=[0:.1:10];y=exp(-(x-5).^2)+.5*exp(-(x-3).^2)+.1*randn(size(x));
>> [FitResults,FitError]=peakfit([x' y'],0,0,2,16,0,0,0,0,[3 5])
FitResults =
            1            3      0.49153       1.6492      0.86285
            2            5       1.0114       1.6589        1.786
FitError =8.2693

Example 20.  (Version 3.9 or later) Exponentially modified Lorentzian (shape 18) with added noise. As for peak shape 5, peakfit.m recovers the original peak position (9), height (1), and width (1).

>> x=[0:.01:20]; L=lorentzian(x,9,1)+0.02.*randn(size(x));
L1=ExpBroaden(L',-100);
>> peakfit([x;L1'],0,0,1,18,100,1,0,0)

Example 21. (Version 4.2).Humps” function fit with two Voigt profiles (shape 20), alpha=1.7, flat baseline mode (3).
[FitResults,FitError]=peakfit(humps(0:.01:2),71,140,2,20,1.7,1,...[31 4.7 90 8.8],3)
 FitResults =
             1       31.047       96.762       4.6785       2550.1
             2        90.09       22.935       8.8253       1089.5
 FitError =0.80501
Example 22. peakfitdemob.m. Illustrated on the right. Measurement of three weak Gaussian peaks at x=100, 250, 400, superimposed in a very strong Gaussian baseline plus noise. The peakfit function fits four peaks, treating the baseline as a 4th peak whose peak position is negative. This requires specifying a "start" vector, like Example 7. You can test the reliability of this method by changing the peak parameters in lines 11, 12, and 13 and see if the peakfit function will successfully track the changes and give accurate results for the three peaks without having to change the start vector. See Example 9 on iSignal.html for other ways to handle this signal. (The true peaks heights are 1, 2, and 3, respectively).

Example 23. (Version 4.3 or later) 12th input argument (+/- mode) set to 1 (bipolar) to allow negative as well as positive peak heights. (Default is 0)
  >> x=[0:.1:10];y=exp(-(x-5).^2)-.5*exp(-(x-3).^2)+.1*randn(size(x));
  >> peakfit([x' y'],0,0,2,1,0,1,0,0,0,1,1)
  FitResults =
              1       3.1636      -0.5433         1.62      -0.9369
              2       4.9487      0.96859       1.8456       1.9029
  FitError =8.2757

Example 24. Version 5 or later. Fits humps function to a model consisting of one Lorentzian and one Gaussian peak.
  x=[0:.005:1.2];y=humps(x);[FitResults,FitError]=peakfit([x' y'],0,0,2,[2 1],[0 0])
  FitResults =
              1      0.30178       97.402      0.18862       25.116
              2      0.89615       18.787      0.33676       6.6213
  FitError = 1.0744

 
Example 25. Version 5 or later: Five peaks,
five different shapes, all heights = 1, all widths = 3, "extra" vector included for peaks 4 and 5.
  x=0:.1:60;
  y=modelpeaks2(x,[1 2 3 4 5],[1 1 1 1 1],[10 20 30 40 50],[3 3 3 3 3],[0 0 0 2 -20])+.01*randn(size(x));
  peakfit([x' y'],0,0,5,[1 2 3 4 5],[0 0 0 2 -20])  

You can also use this technique to create models with all the same shapes but with different values of  'extra' using a vector of 'extra' values, or (in version 5.7) with different minimum width restrictions by using a vector of  'minwidth' values as input argument 13.

Example 26.
Version 5.7 or later: Minimum width constraint (13th input argument)
  x=1:30;y=gaussian(x,15,8)+.05*randn(size(x));
  No constraint (minwidth=0):
  peakfit([x;y],0,0,5,1,0,10,0,0,0,1,0,0);
  Widths constrained to values 7 or above:
  peakfit([x;y],0,0,5,1,0,10,0,0,0,1,0,7);

Example 27. Noise test with very noisy peak signal: peak height = RMS noise = 1.
  x=[-10:.05:10];y=exp(-(x).^2)+randn(size(x));P=peakfit([x;y],0,0,1,1,0,10);

Note: to display the results table with column labels,  call peakfit.m with output arguments [FitResults,FitError...] and type :
disp('      Peak number  Position      Height      Width       Peak area');disp(FitResults)

How do you find the right input arguments for peakfit?

If you have no idea where to start, you can use the Interactive Peak Fitter (ipf.m) to quickly try different fitting regions, peak shapes, numbers of peaks, baseline correction modes, etc. When you get a good fit, you can press the "W" key to print out the command line statement for peakfit.m that will perform that fit in a single line of code, with or without graphics.

Demonstration script for peakfit.m

DemoPeakFit.m is a demonstration script for peakfit.m. It generates an overlapping Gaussian peak signal, adds normally-distributed noise, fits it with the peakfit.m function (in line 78), repeats this many times ("NumRepeats" in line 20), then compares the peak parameters (position, height, width, and area) of the measurements to their actual values and computes accuracy (percent error) and precision (percent relative standard deviation). You can change any of the initial values in lines 13-30.  Here is a typical result for a two-peak signal with Gaussian peaks:

Percent errors of measured parameters:

Position      Height       Width       Area
 0.048404     0.07906     0.12684     0.19799
 0.023986     0.38235    -0.36158    -0.067655

Average Percent Parameter Error for all peaks:
 0.036195     0.2307      0.24421     0.13282

In these results, you can see that the accuracy and precision (%RSD) of the peak position measurements are always the best, followed by peak height, and then the peak width and peak area, which are usually the worst

DemoPeakFitTime.m is a simple script that demonstrates how to apply multiple curve fits to a signal that is changing with time. The signal contains two noisy Gaussian peaks (similar to the illustration at the right) in which the peak position of the second peak increases with time and the other parameters remain constant (except for the noise).  The script creates a set of 100 noisy signals (on line 5) containing two Gaussian peaks where the position of the second peak changes with time (from x=6 to 8) and the first peak remains the same.  Then it fits a 2-Gaussian model to each of those signals (on line 8), stores the FitResults in a 100 × 2 × 5 matrix, displays the signals and the fits graphically with time (click to play animation), then plots the measured peak position of the two peaks vs time on line 12.

Which to use: iPeak, iSignal, or Peakfit?  Read this comparison of all three.  Or download these Matlab demos that compare iPeak.m with Peakfit.m for signals with a few peaks and signals with many peaks and that shows how to adjust iPeak to detect broad or narrow peaks. DemoPeakfitBootstrap demonstrates the ability of peakfit version 3 to computer estimates of the errors in the measured peak parameters. These are self-contained demos that include all required Matlab functions. Just place them in your path and click Run or type their name at the command prompt.  Or you can download all these demos together in idemos.zippeakfitVSfindpeaks.m  performs a direct comparison of the peak parameter accuracy of findpeaks vs peakfit.


2. Automatically finding and Fitting Peaks.

findpeaksfit.m is
essentially a combination of findpeaks.m and peakfit.m. It uses the number of peaks found and the peak positions and widths determined by findpeaks as input for the peakfit.m function, which then fits the entire signal with the specified peak model.  This combination function is more convenient that using findpeaks and peakfit separately. It yields better values that findpeaks alone, because peakfit fits the entire peak, not just the top part, and because it deals with non-Gaussian and overlapped peaks. However, it fits only those peaks that are found by findpeaks, so you will have to make sure that every peak that contributes to your signal is located by findpeaks. The syntax is

function [P,FitResults,LowestError,BestStart,xi,yi] =
findpeaksfit(x,y,SlopeThreshold,AmpThreshold,smoothwidth,peakgroup,smoothtype,peakshape,extra,NumTrials,autozero,fixedparameters,plot
s)

The first seven input arguments are exactly the same as for the findpeaks.m function; if you have been using findpeaks or iPeak to find and measure peaks in your signals, you can use those same input argument values for findpeaksfit.m. The remaining six input arguments of findpeaksfit.m are for the peakfit function; if you have been using peakfit.m or ipf.m to fit peaks in your signals, you can use those same input argument values for findpeaksfit.m. Type "help findpeaksfit" for more information.  This function is included in the ipf10.zip distribution.


indexprevious next

3. Interactive keypress-operated version: ipf.m (Version 10.7, September 2014)   Animated instructions.

ipf.m is a version of the peak fitter for x,y data that uses keyboard commands and the mouse cursor. It is written as a self-contained Matlab function, in a single m-file. The flexible input syntax allows you to specify the data as separate x,y vectors or as a 2xn matrix, and to optionally define the initial focus by adding “center” and “window” values as additional input arguments, where 'center' is the desired x-value in the center of the upper window and “window” is the desired  width of that window.  For example:
   1 input argument:
    ipf(y) or ipf([x;y]) or ipf([x;y]');
   2 input arguments:
    ipf(x,y) or ipf([x;y],center) or ipf([x;y]',center);
   3 input arguments:
    ipf(x,y,center) or ipf(y,center,window) or
    ipf([x;y],center,window) or ipf([x;y]',center,window);
   4 input arguments:
    ipf(x,y,center,window);
 

Like iPeak and iSignal, ipf starts out by showing the entire signal in the lower panel and the selected region in the upper panel (adjusted by the same cursor controls keys as iPeak and iSignal). After performing a fit (figure on the right), the upper panel shows the data as blue dots, the total model as a red line (ideally overlapping the blue dots), and the model components as green lines. The dotted magenta lines are the first-guess peak positions for the last fit. The lower panel shows the residuals (difference between the data and the model). Version 10.4 moves the peak table to the lower panel and changes the color of the residual plot to cyan.


Recent version history. Version 10.5, August 2014 adds Shift-Ctrl-S and Shift-Ctrl-P keys to transfer the current signal to iSignal and iPeak, respectively, if those functions are installed in your Matlab path; Version 10.4, June, 2014. Moves fitting result text to bottom panel of graph. Log mode: (M key) toggles log mode on/off, fits log(model) to log(y). Replaces bifurcated Lorentzian with the Breit-Wigner-Fano resonance peak (Shift-B key); see http://en.wikipedia.org/wiki/Fano_resonance. Ctrl-A selects all; Shift-N negates signal. Version 10 adds multiple-shape models; Version 9.9 adds '+=' key to flip between + (positive peaks only) and bipolar (+/- peaks) modes; Version 9.8 adds Shift-C to specify custom first guess ('start'). Version 9.7 adds Voigt profile shape. Version 9.6 added an additional autozero mode that subtracts a flat baseline without requiring that the signal return to the baseline at both ends of the signal segment. Version 9.5 added exponentially broadened Lorentzian (peak shape 18) and alpha function (peak shape 19);Version 9.4: added bug fix for height of Gaussian/ Lorentzian blend shape; Version 9.3 added Shift-S to save Figure as a png graphic to the current directory. Version 9.2: bug fixes; Version 9.1 added fixed-position Gaussians (shape 16) and fixed-position Lorentzians (shape 17) and a peak shape selection menu ( activated by the '_-' key).  Version 10.7 corrects bugs in equal-width Lorentzians (shape 7) and in the bipolar (+ and -) mode.

Demoipf.m is a demonstration script for ipf.m, with a
built-in simulated signal generator. To download these m-files, right-click on these links, select Save Link As..., and click Save. You can also download a ZIP file containing ipf.m, Demoipf.m, and peakfit.m. 

Example 1: Test with pure Gaussian function, default settings of all input arguments..
>> x=[0:.1:10];y=exp(-(x-5).^2);ipf(x,y)
In this example, the fit is essentially perfect, no matter what are the pan and zoom settings or the initial first-guess (start) values. However, the peak area (last fit result reported) includes only the area within the upper window, so it does vary.  (But if there were noise in the data or if the model were imperfect, then all the fit results will depend on the exact the pan and zoom settings).

Example 2: Test with "center" and "window" specified.
>> x=[0:.005:1];y=humps(x).^3;
>> ipf(x,y,0.335,0.39) focuses on first peak
>> ipf(x,y,0.91,0.18) focuses on second peak

Example 3:
>> x=1:.1:1000;y=sin(x).^2;ipf(x,y,843.45,5)
Isolates a  narrow segment toward the end of the signal.

KEYBOARD CONTROLS (Version 10.5):
 Pan signal left and right...Coarse: < and >
                             Fine: left and right cursor arrow keys
                             Nudge: [ ]
 Zoom in and out.............Coarse zoom: ?/ and "
                             Fine zoom: up and down cursor arrow keys
 Select entire signal........Crtl-A (Zoom all the way out)
 Resets pan and zoom.........ESC
 Select # of peaks...........Number keys 1-9, or press 0 key to enter number manually
 Select peak shape from menu _- (minus or hyphen), then type number or shape vector and Enter
 Select peak shape by key....g Gaussian
                             h equal-width Gaussians
                             Shift-G fixed-width Gaussians
                             Shift-P fixed-position Gaussians
                             Shift-H bifurcated Gaussians (a,z keys adjust shape)
                             e Exponential-broadened Gaussian (a,z keys adjust broadening)
                             j exponential-broadened equal-width Gaussians
                                 (a,z keys adjust broadening)
                             l Lorentzian
                             :; equal-width Lorentzians
                             Shift [ fixed-position Lorentzians
                             E (Shift-E) Exponential-broadened Lorentzians
                                 (a,z keys adjust broadening)
                             L (Shift-L) Fixed-width Lorentzians (a,z keys adjust broadening)
                             o LOgistic distribution (Use Sigmoid for logistic function)
                             p Pearson (a,z keys adjust shape)
                             u exponential pUlse  y=exp(-tau1.*x).*(1-exp(-tau2.*x))
                             Shift-U Alpha: y=(x-tau2)./tau1.*exp(1-(x-tau2)./tau1)
                             s Up Sigmoid (logistic function): y=.5+.5*erf((x-tau1)/sqrt(2*tau2))
                             Shift-D
Down Sigmoid y=.5-.5*erf((x-tau1)/sqrt(2*tau2))
                             ~` Gauss/Lorentz blend (a/z keys adjust fraction of Gaussian))
                             Shift-V Voigt profile (a/z adjusts shape)
                             Shift
-B Breit-Wigner-Fano (a/z adjusts Fano factor)
 Fit.........................f
 Select autozero mode........t  selects none, linear, quadratic, or flat baseline mode
 + or +/- peak mode..........+=  Flips between + peaks only and +/- peak mode
 Invert (negate) signal......Shift-N
 Toggle log y mode OFF/ON....m  Log mode plots and fits log(model) to log(y).
 2-point Baseline............b, then click left and right baseline
 Set manual baseline.........Backspace, then click baseline at multiple points
 Restore original baseline...|\  to cancel previous background subtraction
 Cursor start positions......c, then click on each peak position
 Type in start vector........Shift-C Type or Paste start vector [p1 w1 p2 w2 ...]
 Print current start vector..Shift-Q
 Enter value of 'Extra'......Shift-x, type value (or vector of values in brackets), press Enter.
 Adjust 'Extra' up/down......a,z: 5% change; upper case A,Z: 0.5% change.
 Print parameters & results..q
 Print fit results only......r
 Compute bootstrap stats.....v  Estimates standard deViations of parameters.
 Fit single bootstrap........n  Extracts and Fits siNgle bootstrap sub-sample.
 Plot signal in figure 2.....y
 Print model data table......d
 Refine fit..................x Takes best of 10 trial fits (change number in line 219)
 Print peakfit function......w  Print peakfit function with all input arguments
 Save Figure as png file.....Shift-S  Saves as Figure1.png, Figure2.png, etc.
 Display current settings....Shift-? displays list of current settings
 Switch to iPeak.............Shift-Ctrl-P transfers current signal to iPeak.m
 Switch to iSignal...........Shift-Ctrl-S transfers current signal to iSignal.m

(The function ShapeDemo demonstrates the basic peak shapes graphically, showing the variable-shape peaks as multiple lines)

Practical examples with experimental data:


 
In this example, pan and zoom controls are used to isolate a segment of a chromatogram that contains three weak peaks near 5.8 minutes The lower plot shows the whole chromatogram and the upper plot shows the segment.  Only the peaks in that segment are subjected to the fit. Pan and zoom are adjusted so that the signal returns to the local baseline at the ends of the segment.


 Pressing T turns on the autozero mode, causing the program to compute and subtract a linear baseline between those points.  Pressing 3, E,
F performs a 3-peak exponentially-broadened Gaussian fit (a common peak shape in chromatography). The A and Z keys are then used to adjust the time constant ("Extra") to obtain the best fit. The randomness and lack of obvious structure of the residuals indicates that the fit is as good as possible at this noise level.

The accuracy of peak position measurement can be good even if the fitting error is rather poor. In this example, an experimental high-resolution atomic emission spectrum is examined in the region of the well-known spectral lines of the element Sodium. Two lines are found there (figure on the right), and when fit to a Lorentzian or Gaussian model, the peak wavelengths are determined to be 588.98 nm and 589.57 nm.

         Percent Fitting Error 6.9922%   
         Peak#   Position       Height       Width         Area
           1       588.98       234.34      0.16079       56.473
           2       589.57       113.18      0.17509       29.63

Compare this to the ASTM recommended wavelengths for this elements (588.995 and 589.59 nm) and you can see that the error is no greater than 0.02 nm, which is less that the interval between the data points (0.05 nm). This despite the fact that the fit is not particularly good, because the peaks shapes are rather oddly shaped. This high degree of absolute accuracy compared to a reliable exterior standard is a testament to the excellent wavelength calibration of the instrument on which these experimental data were obtained, but it also shows that peak position is by far the most precisely measurable parameter in peak fitting, even when the curve fit is not particularly good. The bootstrap standard deviation estimates calculated by ipf.m for both wavelengths is 0.015 nm (see #17 in the next section), so using the 2 x standard deviation rule-of-thumb would have predicted a probable error within 0.03 nm. (An even lower fitting error can be achieved by fitting to 4 peaks, but the position accuracy is virtually unchanged).

Operating instructions for ipf.m (version 10.4). Click here for animated instructions.

  1. At the command line, type ipf(x,y), (x = independent variable, y = dependent variable) or  ipf(datamatrix) where "datamatrix" is a matrix that has x values in row or column 1 and y values in row or column 2. Or if you have only one signal vector y, type ipf(y).  You may optionally add to additional numerical arguments:  ipf(x,y,center,window); where 'center' is the desired x-value in the center of the upper window and “window” is the desired  width of that window.
  2. Use the four cursor arrow keys on the keyboard to pan and zoom the signal to isolate the peak or group of peaks that you want to fit in the upper window. (Use the < and > and ? and " keys for coarse pan and zoom and the square bracket keys [ and ] to nudge one point left and right). The curve fitting operation applies only to the segment of the signal shown in the top plot. The bottom plot shows the entire signal. Try not to get any undesired peaks in the upper window or the program may try to fit them. To select the entire signal, press Ctrl-A.
  3. Press the number keys (19) to choose the number of model peaks, that is, the minimum number of peaks that you think will suffice to fit this segment of the signal. For more than 9 peaks, press 0, type the number, and press Enter.
  4. Select the desired model peak shape by pressing the '-' key and selecting the desired shape by number from the list that is displayed. To select a multi-shape model, type in a vector of shape numbers, including the square brackets (e.g. [1 1 2]). Alternatively, for single shape models, you can select peak shapes directly in one keystroke by pressing the following keys: Gaussian (lower case g), Equal-width Gaussians (lower case h), Fixed-width Gaussians (Shift-G), fixed-position Gaussians (Shift-P), exponentially-broadened Gaussian (e), exponentially-broadened equal-width Gaussians (j); bifurcated Gaussian (Shift-H), Lorentzian (lower case L); Exponential-broadened Lorentzians (Shift-E); Fixed-width Lorentzians (upper case L), fixed-position Lorentzians (Shift [ ), Equal-width Lorentzians (lower case ;); Voigt (Shift-V); Shift-B Breit-Wigner-Fano; Logistic distribution (lower case o), Pearson (lower case p), exponential pulse (u); Alpha function (Shift-U); up sigmoid or logistic function (s); down sigmoid (Shift-D), and Gaussian/Lorentzian blend (`) models. 
      If the peak widths of each group of peaks is expected to be the same or nearly so, select the "equal-width" shapes. If the peak widths or peak positions are known from previous experiments, select the "fixed-width" or "fixed position" shapes. These more constrained fits are faster, easier, and much more stable than regular all-variable fits, especially if the number of model peaks is greater than 3 (because there are fewer variable parameters for the program to adjust - rather than a independent value for each peak).  
  5. A set of vertical dashed lines are shown on the plot, one for each model peak. Try to fine-tune the Pan and Zoom keys so that the signal goes to the baseline at both ends of the upper plot and so that the peaks (or bumps) in the signal roughly line up with the vertical dashed lines, as show in the figures on the left. This does not have to be exact.
  6. If you want to allow negative peaks as well as positive peaks, press the + key to flip to the +/- mode (indicated by the +/- sign in the y-axis label of the upper panel. Press it again to return to the + mode (positive peaks only). You can switch between these modes at any time.  To negate the entire signal, press Shift-N.
  7. Press F to initiate the curve-fitting calculation. Each time you press F, another fit of the selected model to the data is performed with slightly different starting positions, so that you can judge the stability of the fit with respect to changes in starting first guesses. Keep your eye on the residuals plot and on the "Error %" display. Do this several times, trying for the lowest error and the most unstructured random residuals plot. At any time, you can adjust the signal region to be fit (step 2), the baseline position (step 9 and 10), change the number or peaks (step 3), or peak shape (step 4). If the fit seems unstable, try pressing the X key a few times (see #13, below),
  8. The model parameters of the last fit are displayed in the upper window. For example, for a 3-peak fit:
    Peak# Position  Height   Width    Area
     1    5.33329   14.8274  0.262253 4.13361
     2    5.80253   26.825   0.326065 9.31117
     3    6.27707   22.1461  0.249248 5.87425
    The column are, left to right: the peak number, peak position, peak height, peak width, and the peak area. (Note: for exponential pulse (U) and sigmoid (S) shapes, Position and Width are replaced by Tau1 and Tau2). Press R to print this table out in the command window. Peaks are numbered from left to right. (The area of each component peak within the upper window is computed using the trapezoidal method and displayed after the width). Pressing Q prints out a report of settings and results in the command window, like so:

    Peak Shape = Gaussian
    Number of peaks = 3
    Fitted range = 5 - 6.64
    Percent Error = 7.4514    Elapsed time = 0.19741 Sec.
    Peak#  Position      Height       Width        Area
    1      5.33329      14.8274     0.262253      4.13361
    2      5.80253       26.825     0.326065      9.31117
    3      6.27707      22.1461     0.249248      5.87425

  9. To select the baseline correction mode, press the T key repeatedly; it cycles thorough four modes: No baseline correction, linear baseline subtraction, quadratic baseline subtraction, and flat baseline correction.  When baseline subtraction is linear, a straight-line baseline connecting the two ends of the signal segment in the upper panel will be automatically subtracted. When baseline subtraction is quadratic, a parabolic baseline connecting the two ends of the signal segment in the upper panel will be automatically subtracted.  Use the quadratic baseline correction if the baseline is curved, as in these examples:

    Autozero OFF

    Linear autozero

    Quadratic autozero
  10. If you prefer to set the baseline manually, press the B key, then click on the baseline to the LEFT the peak(s), then click on the baseline to the RIGHT the peak(s). The new baseline will be subtracted and the fit re-calculated. (The new baseline remains in effect until you use the pan or zoom controls). Alternatively, you may use the multipoint background correction for the entire signal: press the Backspace key, type in the desired number of background points and press the Enter key, then click on the baseline starting at the left of the lowest x-value and ending to the right of the highest x-value. Press the \ key to restore the previous background to start over.
  11. In some cases it will help to manually specify the first-guess peak positions: press C, then click on your estimates of the peak positions in the upper graph, once for each peak. A fit is automatically performed after the last click. Peaks are numbered in the order clicked. Alternatively, you can type Shift-C and then type in or Paste in the start vector, complete with square brackets, e.g. “[pos1 wid1 pos2 wid2 ...]” where "pos1" is the position of peak 1, "wid1" is the width of peak 1, and so on for each peak. The custom start values remain in effect until you change the number of peaks or use the pan or zoom controls. Hint: if you Copy the start vector and keep it in the Paste buffer, you can use the Shift-C key and Paste it back in after changing the pan or zoom controls.  Note: It's possible to click beyond the x-axis range, to try to fit a peak whose maximum is outside the x-axis range displayed. This is useful when you want to fit a curved baseline by treating it as an additional peak whose peak position if off-scale (example).
  12. The A and Z keys control the "shape" parameter ('extra') that is used only if you are using the Voigt profile, Pearson, exponentially-broadened Gaussian (ExpGaussian), exponentially-broadened Lorentzian (ExpLorentzian), bifurcated Gaussian, Breit-Wigner-Fano, or Gaussian/Lorentzian blend. For the Voigt profile, the "shape" parameter controls alpha, the ratio of the Lorentzian width to the Doppler width. For the Pearson shape, a value of 1.0 gives a Lorentzian shape, a value of 2.0 gives a shape roughly half-way between a Lorentzian and a Gaussian, and a larger values give a nearly Gaussian shape. For the exponentially broadened Gaussian shapes, the "shape" parameter controls the exponential "time constant" (expressed as the number of points). For the Gaussian/Lorentzian blend and the bifurcated Gaussian shape, the "shape" parameter controls the peak asymmetry (a values of 50 gives a symmetrical peak). For the Breit-Wigner-Fano, it controls the Fano factor. You can enter an initial value of the "shape" parameter by pressing Shift-X , typing in a value, and pressing the Enter key. For multi-shape models, enter a vector of  "extra" values, one for every peak, enclosed in square brackets. For single-shape models, you can adjust this value using the A and Z keys (hold down the Shift key to fine tune). Seek to minimize the Error % or set it to a previously-determined value. Note: if fitting multiple overlapping variable-shape peaks, it's easier to fit a single peak first, to get a rough value for the "shape" parameter parameter, then just fine-tune that parameter for the multipeak fit if necessary.
  13. For difficult fits, it may help to press X, which performs ten silent trial fits with slightly different first guesses and takes the one with the lowest fitting error. (You can change the number of trials,"NumTrials", in or near line 214- the default value is 10). The peak positions and widths resulting from this best-of-10 fit then become the starting points for subsequent fits, so the fitting error should gradually get smaller and smaller is you press X again and again, until it settles down to a minimum. If none of the 10 trials gives a lower fitting error than the previous one, nothing is changed. Those starting values remain in effect until you change the number of peaks or use the pan or zoom controls. (Remember: equal-width fits, fixed-width fits, and fixed position shapes are both faster, easier, and much more stable than regular variable fits, so use equal-width fits whenever the peak widths are expected to be equal or nearly so, or fixed-width (or fixed position) fits when the peak widths or positions are known from previous experiments). 
  14. Press Y to display the entire signal full screen without cursors, with the last fit displayed in green. In version 5.6 and later, the residual is displayed in red, on the same y-axis scale as the entire signal.
  15. Press M to switch back and forth between log and linear modes. In log mode, the y-axis of the upper plot switches to semilog-y, and log(model) is fit to log(y), which may be useful if the peaks vary greatly in amplitude.  
  16. Press the D key to print out a table of model data in the command window: x, y1, y2, ..., where x is the column of x values of the fitted region and the y's are the y-values of each component, one for each peak in the model. You can then Copy and Paste this table into a spreadsheet or data plotting program of your choice.
  17. Press W to print out the ipf.m function in the command window with the current values of  'center' and 'window' as input arguments. Useful when you want to return to that specific data segment later. Also prints out peakfit.m with all input arguments, including the last best-fit values of the first guess vector. You can copy and paste the peakfit.m function into your own code. 
  18. Both ipf.m, and peakfit.m are able to estimate the expected variability of the peak position, height, width, and area from the signal, by using the bootstrap sampling method. This involves extracting 100 bootstrap samples from the signal, fitting each of those samples with the model, then computing the percent relative standard deviation (RSD) and the interquartile range (IQR) of the parameters of each peak. Basically this method calculates weighted fits of a single data set, using a different set of different weights for each sample. This process is computationally intensive can take several minutes to complete, especially if the number of peaks in the model and/or the number of points in the signal are high.  

    To activate this process in ipf.m, press the V key. It first asks you to type in the number of "best-of-x" trial fits per bootstrap sample (the default is 1, but you may need higher number here if the fits are occasionally unstable; try 5 or 10 here if the initial results give NaNs or wildly improbable numbers). (To activate this process in peakfit.m, you must use version 3.1 or later and include all six output arguments, e.g. [FitResults, LowestError, BestStart, xi, yi, BootstrapErrors]=peakfit...).  The results are displayed as a table in the command window. For example, for a three-peak fit (to the same three peaks used by the Demoipf demonstration script described in the next section) and using 10 as the number or trials:

    Number of fit trials per bootstrap sample (0 to cancel): 10
    Computing bootstrap sampling statistics....May take several minutes.
         
          Peak #1         Position   Height      Width      Area
          Bootstrap Mean: 800.5387   2.969539    31.0374   98.10405
          Bootstrap STD:  0.20336    0.02848      0.5061    1.2732
          Bootstrap IQR:  0.21933    0.027387     0.5218    1.1555
          Percent RSD:    0.025402   0.95908      1.6309    1.2978
          Percent IQR:    0.027398   0.92226      1.6812    1.1778
         
          Peak #2         Position   Height      Width      Area
          Bootstrap Mean: 850.0491   1.961368    36.4809     75.98684
          Bootstrap STD:  6.4458     0.14622      3.0657      4.7596
          Bootstrap IQR:  0.52358    0.058845     1.4303      3.9161
          Percent RSD:    0.75828    7.4549       8.4035      6.2637
          Percent IQR:    0.061594   3.0002       3.9205      5.1537 
                etc ....
          Elapsed time is 98.394381 seconds.
          Min/Max Fitting Error of the 100 bootstrap samples: 3.0971/2.5747
Observe that the RSD of the peak position is best (lowest), followed by height and width and area. This is a typical pattern, seen before.  Also, be aware that the reliability of the computed variability depends on the assumption that the noise in the signal is representative of the average noise in repeated measurements. If the number of data points in the signal is small, these estimates can be very approximate.

A likely pitfall with the bootstrap method when applied to iterative fits is the possibility that one (or more) of the bootstrap fits will go astray, that is, will result in peak parameters that are wildly different from the norm, causing the estimated variability of the parameters to be too high. For that reason, in ipf 8.12, the interquartile range (IQR) as well as the standard deviation (STD) is reported. The IQR is more robust to outliers. For a normal distribution, the interquartile range is equal to 1.34896 times the standard deviation. But if one or more of the bootstrap sample fits fails, resulting in a distribution of peak parameters with large outliers, the STD will be much greater that the IQR. In that case, a more realistic estimate of variability is IRQ/1.34896. (Note: if you do not have the Matlab Statistics Toolbox installed, you will have to use ipf version 8.62 or later, which includes the necessary IQR function). It's best to try to increase the fit stability by choosing a better model (e.g. using an equal-width or fixed-width model, or a fixed-position shape, if appropriate), adjusting the fitted range (pan and zoom keys), the background subtraction (T or B keys), or the start positions (C key), and/or selecting a higher number of fit trials per bootstrap (which will increase the computation time). As a quick preliminary test of bootstrap fit stability, pressing the N key will perform a single fit to a single random bootstrap sample and plot the result; do that several times to see whether the bootstrap fits are stable enough to be worth computing a 100-sample bootstrap. Note: it's normal for the stability of the bootstrap sample fits (N key) to be poorer than the full-sample fits (F key), because the latter includes only the variability caused by changing the starting positions for one set of data and noise, whereas the N and V keys aim to include the variability caused by the random noise in the sample by fitting bootstrap sub-samples. Moreover, the best estimates of the measured peak parameters are those obtained by the normal fits of the full signal (F and X keys), not the means reported for the bootstrap samples (V and N keys), because there are more independent data points in the full fits and because the bootstrap means are influenced by the outliers that occur more commonly in the bootstrap fits. The bootstrap results are useful only for estimating the variability of the peak parameters, not for estimating their mean values. The N and V keys are also very useful way to determine if you are using too many peaks in your model; superfluous peaks will be very unstable when N is press repeatedly and will have much higher standard deviation of its peak height when the V key is used.

Demoipf.m

Demoipf.m is a demonstration script for ipf.m, with a built-in simulated signal generator. The true values of the simulated peak positions, heights, and widths are displayed in the Matlab command window, for comparison to the FitResults obtained by peak fitting. The default simulated signal contains six independent groups of peaks that you can use for practice: a triplet near x = 150, a singlet at 400, a doublet near 600, a triplet near 850, and two broad single peaks at 1200 and 1700. Run this demo and see how close to the actual true peak parameters you can get. The useful thing about a simulation like this is that you can get a feel for the accuracy of peak parameter measurements, that is, the difference between the true and measured values of peak parameters. To download this m-file, right-click on the links, select Save Link As..., and click Save. To run it it, place both ipf.m and Demoipf in the Matlab path, then type Demoipf at the Matlab command prompt.

An example of the use of this script is shown on the right.  Here we focus in on the 3 fused peaks located near x=850.  The true peak parameters (before the addition of the random noise) are:

                    Position       Height        Width     Area
                       800            3           30       95.808
                       850            2           40       85.163
                       900            1           50       53.227

When these peaks are isolated in the upper window and fit with three Gaussians, the results are

                    Position     Height       Width         Area 
                    800.04       3.0628       29.315       95.583
                    850.15       1.9881       41.014       86.804
                    901.3        0.9699       46.861       48.376

So you can see that the accuracy of the measurements are excellent for peak position, good for peak height, and least good for peak width and area.  It's no surprise that the least accurate measurements are for the smallest peak with the poorest signal-to-noise ratio. Note: in ipf version 8.12, the expected standard deviation of these peak parameters can be determined by the bootstrap sampling method, as described in the previous section. We would expect that the measured values of the peak parameters (comparing the true to the measured values) would be within about 2 standard deviations of the true values listed above).

Demoipf2.m is identical, except that the peaks are superimposed on a strong curved baseline, so you can test the accuracy of the baseline correction methods (# 9 and 10, above).

Execution time

By execution time I mean the time it takes for one fit to be performed, exclusive of plotting or printing the results. The biggest factors that determine the execution time are (a) the speed of the computer, (b) the number of peaks, and (c) the peak shape:
a) The execution time can vary over a factor of 4 or 5 or more between different computers, (e.g. a small laptop with 1.6 GHz, dual core Athlon CPU with 4 Gbytes RAM, vs a desktop with a 3.4 GHz i7 CPU with 16 Gbytes RAM).  Run the Matlab "bench.m" benchmark test to see how your computer stacks up.

b) The execution time increases with the square of the number of peaks in the model. (See PeakfitTimeTest.m).

c) The execution time varies greatly (sometimes by a factor of 100 or more) with the peak shape, with the exponentially-broadened Gaussian being the slowest and the fixed-width Gaussian being the fastest.  See PeakfitTimeTest2.m and PeakfitTimeTest2a.m  The equal-width and fixed-width shape variations are always faster.  

d) The execution time increases directly with NumTrials in peakfit.m. The "Best of 10 trials" function (X key in ipf.m) takes about 10 times longer than a single fit.  
Other factors that are less important are the number of data points in the fitted region (but only if the number of data points is very large; see PeakfitTimeTest3.m) and the starting values (good starting values can reduce execution time slightly; PeakfitTimeTest2.m and PeakfitTimeTest2a.m have examples). Some of these TimeTest scripts need DataMatrix2 and  DataMatrix3.

Matlab and Octave have "tic" and "toc" functions that are useful for determining execution time of your code. Note that the ipf.m version 8.92 displays the elapsed time to the reports printed by the Q and R keys.


Hints and Tips

  1. It's best not to smooth your data prior to curve fitting. Smoothing can distort the signal shape and the noise distribution, making it harder to evaluate the fit by visual inspection of the residuals plot.  Smoothing your data beforehand makes it impossible to achieve the goal of obtaining a random unstructured residuals plot and it increases the chance that you will "fit the noise" rather that the actual signal.
  2. The most important factor in non-linear iterative curve fitting is selecting the underlying model peak function, for example, Gaussian, Equal-width Guardians, Lorentzian, etc. (Some examples of this). It's worth spending some time finding and verifying a suitable function for your data. In particular, if the peak widths of each group of peaks is expected to be the same or nearly so, select the "equal-width" shapes; equal-width fits (available for the Gaussian and Lorentzian shapes in this version) are faster, easier, and much more stable than regular variable-width fits. 
  3. You should always use the minimum number of peaks that adequately fits your data. (Some examples of this). Using too many peaks will result in an unstable fit - the green lines in the upper plot, representing the individual component peaks, will bounce around wildly for each repeated fit, without significantly reducing the Error. A very useful way to determine if you are using too many peaks in your model is to use the N key (see #9, below) to perform a single fit to a bootstrap sub-sample of points; superfluous peaks will be very unstable when N is press repeatedly. (You can get better statistics for this test, at the expense of time, by using the V key to compute the standard deviation of 100 bootstrap sub-samples).
  4. If the peaks are superimposed on a background or baseline, that has to be accounted for before fitting, otherwise the peak parameters (especially height, width and area) will be inaccurate. Either subtract the baseline from the entire signal using the Backspace key (# 10 in Operating Instructions, above) or use the T key to select one of the automatic baseline correction modes (# 9 in Operating Instructions, above).
  5. This program uses an iterative non-linear search function ("modified Simplex") to determine the peak positions and widths that best match the data. This requires first guesses for the peak positions and widths. (The peak heights don't require first guesses, because they are linear parameters; the program determines them by linear regression). The default first guesses for the peak positions are made by the computer on the basis of the pan and zoom settings and are indicated by the magenta vertical dashed lines. The first guesses for the peak widths are computed from the Zoom setting, so the best results will be obtained if you zoom in so that the particular group of peaks is isolated and spread out as suggested by the peak position markers (vertical dashed lines).
  6. If the peak components are very unevenly spaced, you might be better off entering the first-guess peak positions yourself by pressing the C key and then clicking on the top graph where you think the peaks might be. None of this has to be exact - they're just first guesses, but if they are too far off it can throw the search algorithm off. You can also type in the first guesses for position and width manually by pressing Shift-C).
  7. Each time you perform a repeat fit (e.g. pressing the F key), the program adds small random deviations to the first guesses, in order to determine whether an improved fit might be obtained with slightly different first guesses. This is useful for determining the robustness or stability of the fit with respect to starting values. If the error and the values of the peak parameters vary slightly in a tight region, this means that you have a robust fit (for that number of peaks). If the error and the values of the peak parameters bounces around wildly, it means the fit is not robust (try changing the number of peaks, peak shape, and the pan and zoom settings), or it may simply be that the data are not good enough to be fit with that model. Try pressing the X key, which takes the best of 10 fits and also (in version 8.93) uses those best-fit values as the starting first guesses for subsequent fits. So each time you press X, if any of those fits yield a fitting error less that the previous best, that one is taken as the start for the nest fit.  As a result, the fits tend to get better and better gradually as the X key is pressed repeatedly. Often, even if the first fit is terrible, subsequent X-key fits will improve considerably.
  8. The variability in the peak parameters from fit to fit using the X or F keys is only an estimate of the uncertainty caused by the curve fitting procedure (but not of the uncertainty caused by the noise in the data, because this is only for one sample of the data and noise; for that you need the N key fits).
  9. To examine the robustness or stability of the fit with respect to random noise in the data, press the N key. Each time you press N, it will perform a fit on a different subset of data points (called a "bootstrap sample"). If that gives reasonable-looking fits, then you can go on to compute the peak error statistics by pressing the V key.  If on the other hand the N key gives wildly different fits, with highly variable fitting errors and peak parameters, then the fit is not stable, and you might try the X key to take the best of 10 fits and reset the starting guesses, then press N again. In difficult cases, it may be necessary to increase the number of trials when asked (but that will increase the time it takes to complete), or if that does not help, use another model or a better data set. (The N and V keys are a good way to evaluate a multi-peak model for the possibility of superfluous peaks, see # 3 above).
  10. If you don't find the peak shape you need in this program, look at the next section , or write me and I'll see what I can do.


How to add a new peak shape

 It's easier than you think to add your own custom peak shape to peakfit.m or ipf.m The easiest way is to modify an existing peak shape that you do not plan to use, replacing it with your new function. Pick a shape to sacrifice that has the same number of variables are your new shape. For example, if your shape has variable position and width, like the Gaussian or Lorentzian shapes, you could modify the Triangular shape (number 21). If your shape has an 'extra' parameter, like the Voigt, Pearson, BWF, exponentially modified Gaussian or Lorentzian, or blended Gaussian/Lorentzian, use one of those. If you need equal widths or fixed widths, etc, use one of those shapes. This is important; you must have the same number of variables and constraints.

There are just two required steps to the process:
  1.  Open peakfit.m or ipf.m in the Matlab editor and re-write the old shape function (such as triangle.m, located near the end of the code) by changing name of the function and the mathematics of the assignment statement (e.g. g = 1-(1./wid).*abs(x-pos);), using the same variables ('x' for the independent variable, 'pos' for peak position, 'wid' for peak width, etc). Scale your function to have a peak height of 1.0.
  2. Use the search function in Matlab to find all instances of the name of the old function and replace it with the new name, checking "Wrap around" but leaving "Match case" and "Whole word" unchecked in the Search box. If you do it right, for example, all instances of "triangular" and all instances of "fittriangular' will be modified with your new name replacing "triangular"
That's it! Your new shape will now be shape 21 (or whatever was the shape number of the old shape you sacrificed) and, in ipf.m, will be activated by the same keystroke used by the old shape (e.g. Shift-T for the triangular, key number 84). (If you wish, you can change the keystroke assignment in ipf.m; find a key or Shift-Key that is not yet assigned, and therefore gives an "UnassignedKey" error statement when you press it with ipf.m running,  and change the old key number 84 to that unassigned one in the big "switch double(key)," case statement near the beginning of the code. But it's simpler just to use the old key). 

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Tom O'Haver
Professor Emeritus
Department of Chemistry and Biochemistry
The University of Maryland at College Park
toh@umd.edu
http://www.wam.umd.edu/~toh

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