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.
automating the fitting of large numbers of signals, peakfit.m
is better; ipf.m is best for exploring signals to
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 5.4, June, 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). 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).
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.
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).
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 0to fit the entire
"NumPeaks" = number of peaks in the model (default is 1 if
not specified; may be a large as desired in version 3.1 and
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
Lorentzians, 8=exponentially broadenedequal-width
Gaussians, 9=exponential pulse, 10= up-sigmoid (logistic function),11=fixed-width
blend; 14=bifurcated Gaussian, 15=Breit-Wigner-Fano
Gaussians; 17=Fixed-position Lorentzians; 18=exponentially
broadened Lorentzian; 19=alpha function; 20=Voigt profile;
21=triangular; 22=multiple shapes; 23=down-sigmoid.
In version 5, 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).
Specifies the first guesses vector "firstguess" for the
peak positions and widths, e.g. start=[position1 width1 position2
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).
Use zeros as placeholders to use the default values of input
arguments. In this case, autozerois set to 2, but all
others are the default values (Version 2.5 and later).
11th input argument set to 0 (default is 1) to suppress plotting
and command window printing.
12th input argument set to 1 (default is 0) to allow negative as
well as positive peak heights in the fit.
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
Prints out 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
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
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
7. BootstrapErrors: a matrix
containing bootstrap standard deviations and interquartile ranges
for each peak parameter of each peak in the fit.
Note: test script testpeakfit.mruns 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
Width Peak area
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
Width Peak area
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
Width Peak area
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'
number Peak position
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']; >
0.3161 2.8671e+008 0.098862
= 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): >>
>> peakfit([x' y'],0,0,2,1,0,1,[5 2 5 1])
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 orplot(xi,yi)to plot all peaks. >
Example 9: Fitting
a single Gaussian on a linear background, using the linear
(9th input argument = 1)
a group of three peaks near x=2400 inDataMatrix3with
three equal-width exponentially-broadened Gaussians.
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]';
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. Seehttp://terpconnect.umd.edu/~toh/spectrum/CurveFittingC.html#Peak_width_constraints.
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,
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)
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 =
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
[FitResults,FitError]=peakfit([x;y],0,0,1,1,0,1,0,2) FitResults =
subtracts a flat baseline automatically, without requiring
that the signal returns to
baseline at the edges. This mode works best for this
In the following
case, the baseline is curved, so the best choice is
as example 4, but withfixed-widthGaussian (shape
11), width=1.666. >>
>> [FitResults,FitError]=peakfit([x' y'],0,0,2,11,0,0,0,0,1.666) FitResults
area measurements. Same as the example in Figure 15 onIntegration 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: >>
>> peakfit([x;y],0,0,4,1,0,1,[4 2 9 2 12 2 14
works well even in the presence of substantial amounts of random
noise: >> x=[0:.01:18];
>> peakfit([x;y],0,0,4,1,0,1,[4 2 9 2 12 2 14 2],0,0)
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)
(and ipf.m) have an exponentially-broadened Gaussian peak shape
(shape #5) that works better in those cases:, recovering theoriginalpeak 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
Example 15: Prints out a table of
parameter error estimates; Version 3 only. See DemoPeakfitBootstrap for a
self-contained demo of this function.
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. >>
y'],0.54,0.93,2,13,15,10,0,0,0) FitResults =
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).
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)
Example 19: (Version 3.6 or later)
Same as example 4, but withfixed-positionGaussian (shape 16),
positions=[3 5]. >>
>> [FitResults,FitError]=peakfit([x' y'],0,0,2,16,0,0,0,0,[3
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];
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)
1 31.047 96.762 4.6785 2550.1
2 90.09 22.935 8.8253 1089.5
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 =
1.8456 1.9029 FitError =8.2757
Version 5 or later. Fits humps function to a model
consisting of one Lorentzian and one Gaussian peak.
y'],0,0,2,[2 1],[0 0]) FitResults =
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
Note: to display the results table with column labels,
call peakfit.m with output arguments [FitResults,FitError...] and type : disp(' Peak number
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.
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
is a typical result for a two-peak signal with Gaussian peaks:
Percent errors of measured
Width Area 0.048404
Average Percent Parameter Error for all peaks:
In these results, you can see that the accuracy and precision (%RSD)
of the peak positionmeasurements are
always the best, followed by peak height, and then the peak width and peak
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
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.
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.zip.
finding and Fitting Peaks.
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] =
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.
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
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:
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 10.4, 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; Reorganized peak shape table ('-'
key). 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).
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
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.
CONTROLS (Version 10.4):
Pan signal left and right...Coarse: <
Fine: left and right cursor arrow keys
Nudge: [ ]
Zoom in and out.............Coarse zoom: /
Fine zoom: up and down cursor arrow keys
Select entire signal........Crtl-A
Resets pan and zoom.........ESC
Select # of peaks...........Number keys 1-9,
or press 0 key to enter 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 G (Shift-G) fixed-width Gaussians P Shift-P fixed-position Gaussians H (Shift-H) bifurcated Gaussians (a,z keys adjust
shape) e Exponential-broadened Gaussian 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 L (Shift-L) Fixed-width Lorentzians 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)) U (Shift-U) Alpha:
y=(x-tau2)./tau1.*exp(1-(x-tau2)./tau1) s Up Sigmoid (logistic function):
Shift-D Down Sigmoid
y=.5-.5*erf((x-tau1)/sqrt(2*tau2)) ~ Gauss/Lorentz blend (a/z keys adjust fraction
of Gaussian)) V (Shift-V) Voigt profile (a/z adjusts
Breit-Wigner-Fano (a/z adjusts Fano factor)
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........C (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
function ShapeDemo demonstrates
the basic peak shapes graphically,
showing the variable-shape peaks as multiple lines)
Practical examples with experimental
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
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.
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).
At the command line, typeipf(x,y),
(x = independent variable, y = dependent variable) or ipf(datamatrix)
where "datamatrix" is a matrix that hasx 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
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 the number keys (1–
9) 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.
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
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).
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.
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.
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),
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
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:
If you prefer to set the baseline manually, press theB 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.
In some cases it will help to manually specify the
first-guess peak positions: pressC, 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
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.
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, 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
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
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.
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.
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.
and peakfit.m are able to estimate the
expected variability of thepeak
position, height, width, and area from the signal, by using
thebootstrap sampling method. This
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.
Bootstrap Mean: 800.5387
2.969539 31.0374 98.10405
Mean: 850.0491 1.961368
3.9205 5.1537 etc ....
Elapsed time is 98.394381 seconds.
Min/Max Fitting Error of the 100 bootstrap samples:
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
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:
When these peaks are isolated in the upper window and fit with three
Gaussians, the results are
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).
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
The equal-width and fixed-width shape variations are always
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.
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
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.
The most important factor in non-linear iterative curve
fitting is selecting the underlying
peak function, for example, Gaussian, Equal-width
Guardians, Lorentzian, etc. (Some examples of this).
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
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 %.
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
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 slider 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).
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
(clicking on the Custom slider) 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. In version 9.8,
you can enter the first guesses for position and width
manually by pressing Shift-C).
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.
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
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.
If you don't find the peak shape you need in this program, write
me and I'll see what I can do.
3. Obsolete version with mouse-controlled
sliders (2006 version, for Matlab 6.5): InteractivePeakFitter.m
This is an older version
that used mouse-controlled sliders for control. However,
as the program grew and more functions were added, it soon
turned out that the slider version became impractical because
the screen became cluttered more and more screen space would
have to have been devoted to controls display, reducing the
screen space available for plotting the all-important
data. Moreover, the slider function used does not work
reliably in more recent versions of Matlab. Development was
halted in 2008 and it therefore lacks all the important newer
features of ipf.m. It does work in
Matlab 6.5. Click here to download the
ZIP file "PeakFitter.zip" that also includes supporting
functions and a self-contained demo to show how it works. You
can also download it from the Matlab
The opening screen view shows the entire signal
below and the zoomed-in segment above. Pan and zoom the
signal to select the desired group of peaks to be fit,
adjusting so that the signal goes to the baseline at
both ends of the upper plot and so that the peaks in the
signal roughly line up with the vertical dashed lines.
After selecting a speak shape and number of of
peaks, press the F key to perform a fit.
A table of fit results are shown in the upper
window, with the component peaks shown in
green. The baseline can be subtracted
automatically or manually.
The bottom graph shows the residuals and standard
deviation of the error. Pressing F again repeats
the fit with slightly different start positions.
Adjusting the zoom pan and zoom settings,
refining the background subtraction, and using custom
start positions may help reduce the fitting
This figure shows a peak fit with an exponentially-broadened
Gaussian shape. The time constant of the
exponential is controlled by the "Extra" slider. Other
shapes include Gaussian ('g'),
Lorentzian ('l'), Logistic ('o'), and Pearson ('p'). Other shapes can be
added if you know the mathematical form.
Operating Instructions for the old InteractivePeakFitter.m (2006
version, for Matlab 6.5)
Load your data set into the vectors x and y (x = independent
variable, y = dependent variable) and then execute
Use the Pan and Zoom sliders on the left and
right to isolate the peak or group of peaks that you want to
fit. You can fine-tune the Pan and Zoom by using the cursor
arrow keys on the keyboard. 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 will try to
Use the # Peaks slider (or press the 1,2,3,4 or 5
keys) to choose the number of model peaks, that is, the minimum
number of peaks that you think might suffice to fit this segment
of the signal.
Use the Shape slider or press the G,L,O,P, or E keys to select the desired
model peak shape. In this version there are five choices:
Gaussian ('g'), Lorentzian ('l'), Logistic ('o'),
Pearson ('p'), and Exponentially-broadened Gaussian ('e').
A set of vertical dashed lines are shown on the plot, one for
each model peak. Try to fine-tune the Pan and Zoom
using the sliders or the cursor arrow 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.
Click on the Re-fit slider or press 'f' to
initiate the curve-fitting calculation. Each time you click on
this slider, another fit of the selected model to the data is
performed. 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 refine the baseline position (step 8), change the number
or peaks (step 3), or peak shape (step 4).
The model parameters of the last fit are displayed in the
upper window. For example, for a 3-peak fit:
The column are, left to right: the peak number, peak position,
peak height, peak width, and peak area. (The area of each
component peak within the upper window is computed using the
trapezoidal method and displayed after the width). Peaks are
numbered from left to right. To print this out in the Matlab
command window, type "FitResults"
in the Matlab command window and press Enter. FitResults(:,2) lists
peak positions, FitResults(:,3)
lists the peak heights, FitResults(:,4)
lists the peak widths, and FitResults(:,5) lists the peak areas
falling within the upper window.
If you wish 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 to
change the fitting region; then you have to re-do the baseline
for that new region).
If you want to manually specify the first-guess peak
positions, click the Custom button or 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.
(The custom start positions remain in effect until you
change the number of peaks or use the pan or zoom controls).
The Extra slider (or the A and Z
keys, in the keyboard-operated version ipf.m)
is used only if you are using the Pearson or the
exponentially-broadened Gaussian (ExpGaussian) shapes. It
controls the extra shape factor of the Pearson (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) and it also controls the
time constant of the exponentially broadened Gaussian. You can
either adjust this to minimize the Error % or set it to a
previously-determined value. You can change this slider's range
to suit your proposes in lines 47-48 of InteractivePeakFitter.m.
Demonstration script for the Interactive Peak Fitter with a built-in
simulated signal generator. Requires Matlab 6.5. 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 at x = 150, a singlet
at 400, a doublet near 600, a triplet at 850, and two broad single
peaks at 1200 and 1700. You can change the character of the
simulated signal in lines 23-28.
Note 1: When a fit is performed, the lower half of the
figure window displays the residuals (differences between the
model and the data). The "Error" or "Percent Error" reported
by this program is the RMS difference between the best-fit model
and the raw data over the fitted segment.
Note 2: When the number of peaks is 2 or greater, the
green lines displayed in the upper plot after a fit is performed
are the individual model peaks; the red line is the total model,
i.e. the sum of the components, which is a least-squares best-fit
to the blue data points.
Note 3: In the slider version, if the range of the sliders
is inappropriate for your signal, you can adjust the slider ranges
in lines 45-52 of InteractivePeakFitter.m.
Note 4: The Extra slider (or the A and Z keys, in the
keypress-operated version ipf.m) is used to
manually control the global variable "extra", which could be used
for a variety of purposes. If you add your own peak shape
(see "How to add a new peak shape", below) , you can use the extra
variable for any purpose, is you need another adjustable
Note 5: To select the Autozero mode, press the T key repeatedly; it cycles
thorough Autozero OFF, Linear Autozero, and Quadratic Autozero.
When autozero is linear, a straight-line baseline connecting
the two ends of the signal segment in the upper panel will be
automatically subtracted. When autozero is quadratic, a
parabolic baseline connecting the two ends of the signal segment
in the upper panel will be automatically
subtracted. Try to adjust the pan and zoom to include
some of the baseline at the beginning and end of the segment in
the upper window, allowing the automatic baseline subtract gets a
good reading of the baseline If that's not possible, you can
still refine the baseline subtraction manually if you wish, by
using the the BG button or press 'B' as described in step
8, above. To select the autozero mode, press T.
Note 6: The peak areas
are calculated only for the portion of the component peaks falling
within the upper window. If some of the component peaks are
close enough to the edges of the window that they tail off outside
the window, their peak areas will be inaccurate.
Note 7: The "Error"
or "Percent Error" reported by this program is simply the RMS
difference between the best-fit model and the raw data over the
fitted segment. It is not
the error in the parameters derived from the fit (peak position,
height, and width). The accuracy of those parameters (the
difference between the true and measured values of peak
parameters) is much harder to estimate, because of course you
don't know the true values of the peak parameters. However, the precision of measurement can
be estimated if you can obtain several repeat samples of the same
signal (with independent random noise), fit each sample using the
same procedure, and calculate the standard deviation of the
parameters from all those fits. If the random errors arising
from random noise in the signal or random variations is the
accuracy of background subtraction are the dominant sources of
error, then an estimate of the precision is a good indication of
the accuracy of parameter measurement. More on accuracy and precision of
peak parameter measurement.
How to add a new peak shape
To add a new peak shape to the five initial shapes, you must
have a function for that peak shape, similar to gaussian.m, that
takes as arguments a vector of x-values, a peak-center value,
and a peak width, and returns a vector of that function
evaluated at each value of x. Let's call it NewPeakShape. Add
NewPeakShape to the PeakFitter folder (or if you are using the
keypress version ipf.m, there is already
a "NewPeakShape" function at the end of the code; just
replace "g=sin(a.*x+b)" with your own function of x, a, and b,
and assign your calculated function to the variable g).
Create a fitting function for this peak shape, similar to
function err = fitNewPeakShape(lambda,t,y) % Fitting function for NewPeakShape. global c A = zeros(length(t),round(length(lambda)/2)); for j = 1:length(lambda)/2, A(:,j) = NewPeakShape(t,lambda(2*j-1),lambda(2*j))'; end c = A\y'; z = A*c; err = norm(z-y');
Save this function as "fitNewPeakShape" into the PeakFitter
folder (or if you are using the single-function keypress version
ipf.m, there is
already a "fitNewPeakShape" rear the end of the code; just
modify its name to suit).
Change FitAndPlot.m to add a 6th case to the switch statement
that starts at at line 20:
case 6 FitParameters=FMINSEARCH('fitNewPeakShape',start,options,x,y); ShapeString='NewPeakShape';
and at to the one that starts at line 52:
case 6 A(m,:)=NewPeakShape(x,FitParameters(2*m-1),FitParameters(2*m));
Note: If your function requires three adjustable arguments
rather than two, you can assign one to the Extra slider, as is
done here for the exponentially-broadened Gaussian and
Pearson-type shape. In that case, add "extra" as the last
argument of the function calls above, as is done in case 4 and 5
of the two switch statements in FitAndPlot.m. Change this
slider's range to suit your proposes in lines 47-48 of
Change ipfshape.m to add a 6th case to the switch statement:
case 6 ShapeString='NewPeakShape';
Change ipfpeaks.m to add a 6th case to the switch statement:
case 6 ShapeString='NewPeakShape';
Assign a key to the new peak shape and change KeyPressTest.m
to add a case to the switch statement in KeyPressTest to
interpret this keypress, similar to the other shapes.
Change line 39 in InteractivePeakFitter.m to read