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). 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.

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,fixedwidth,0)

11th input argument set to 0 (default is 1) to suppress plotting
and command window printing.

peakfit(signal,center,window,NumPeaks,peakshape,extra,NumTrials,start,autozero,fixedwidth,plot,1)

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

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

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 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.

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).

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,

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,

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,

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,

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=0means 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=1subtracts 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,FitResults =1)

1 9.9984 0.96153 1.559 1.5916

FitError =1.9801

Autozero=2subtracts 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=3subtracts 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: Prints out 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

>> x=[0:.005:1];y=humps(x);[FitResults,FitError]= peakfit([x' y'],0.54,0.93,2,

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,

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,

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,

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,

Example 21:(Version4.2).“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

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.

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.zip. peakfitVSfindpeaks.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.

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.

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.

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

>> 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.

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........

Resets pan and zoom.........

Select # of peaks...........Number keys

Select peak shape from menu

Select peak shape by key....

(

(

Shift-D

Fit.........................

Select autozero mode........

+ or +/- peak mode..........

Invert (negate) signal......

Toggle log y mode OFF/ON....

2-point Baseline............

Set manual baseline.........

Restore original baseline...

Cursor start positions......

Type in start vector........

Print current start vector..

Enter value of 'Extra'......

Adjust 'Extra' up/down......

Print parameters & results..

Print fit results only......

Compute bootstrap stats.....

Fit single bootstrap........

Plot signal in figure 2.....

Print model data table......

Refine fit..................

Print peakfit function......

Save Figure as png file.....

Display current settings....

Switch to iPeak.............

Switch to iSignal...........

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

**Practical examples with experimental
data:**

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

- 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.
- 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**. - 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
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**s**igmoid 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**.

- 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),

- 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 - 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

- 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. - 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).

- 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, 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). - 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.
- 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.
- 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.
- 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.
- 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

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.

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:

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

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).

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.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.

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.

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.

- 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 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.
- 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 Instructions, above). - 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**key fits).

- 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.

- Load your data set into the vectors x and y (x = independent variable, y = dependent variable) and then execute InteractivePeakFitter.m.
- 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 fit them. - 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:

1 5.33329 14.8274 0.262253 4.13361

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 the 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.

2 5.80253 26.825 0.326065 9.31117

3 6.27707 22.1461 0.249248 5.87425 - 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.

>> ColumnLabels =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.

Peak# Positon Height Width

ActualParameters =

1 100 1 30

2 130 2 40

3 200 3 50

4 400 1 30

5 600 4 20

6 630 1 20

7 800 3 30

8 850 2 40

9 900 1 50

10 1200 2 100

**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
parameter.

**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.

- 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
fitgaussian.m.
function err = fitNewPeakShape(lambda,t,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).

% 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'); - Change FitAndPlot.m to add a 6th case to the switch statement
that starts at at line 20:
case 6

and at to the one that starts at line 52:

FitParameters=FMINSEARCH('fitNewPeakShape',start,options,x,y);

ShapeString='NewPeakShape';case 6

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 InteractivePeakFitter.m.

A(m,:)=NewPeakShape(x,FitParameters(2*m-1),FitParameters(2*m)); - 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 MaxShapes=6.

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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