Code covered by the BSD License

### Highlights from Demo functions for peak detection and fitting.

• DemoPeakfitBootstrap Demonstration script for peakfit.m, with built-in signal generator.
• PeakFitDemo11Lor Demonstration script for ipf.m, with built-in signal generator.
• idemo Self-contained demonstration function for comparing the ipeak and
• idemo1 Self-contained demonstration function for comparing the iPeak and
• idemo2 Self-contained demonstration function for comparing the iPeak and.
• View all files

# Demo functions for peak detection and fitting.

### Tom O'Haver (view profile)

04 Sep 2012 (Updated )

A collection of self-contained demonstration functions for iPeak.m and peakfit.m.

PeakFitDemo11Lor
```function PeakFitDemo11Lor
% Demonstration script for ipf.m, with built-in signal generator.
% You can change the character of the signal in lines 11-117.
% T. C. O'Haver (toh@umd.edu).  Version 3: August 23, 2008.

format short g
format compact
warning off all

% Generate simulated signal
n=2000;  % maximum x-value
increment=2; % Difference between adjacent x values
x=[1:increment:n];
amp=[1 2 3 1 4 1 3 2 1 2 1];  % Amplitudes of the peaks (one entry for each peak)
pos=[400 430 500 600 700 730 800 850 900 1000 1200];   % Positions of the peaks (one entry for each peak)
wid=[20 40 50 40 25 30 30 40 60 100 200];   % Widths of the peaks (one entry for each peak)
noise=.02;
% A = matrix containing one of the unit-amplidude functions in each of its rows
clear A
A=zeros(length(pos),length(x));
for k=1:length(pos),
A(k,:)=lorentzian(x,pos(k),wid(k));  % Gaussian function
area(k)=amp(k).*trapz(x,A(k,:)); % Computes peak areas via trapaziod method
end
TrueY=amp*A;  % Multiplies each row by the corresponding amplitude and adds them up
% TrueY=TrueY+lorentzian(x,0,1000); % Adds sloping curved baseline
y = TrueY + noise .* randn(size(x));

% Arrange the peak parameters of the simulated signal into a matrix
ColumnLabels='           Peak#   Position     Height     Width       Area';
NumPeaks=length(pos);
for m=1:NumPeaks,
if m==1,
ActualParameters=[m pos(m) amp(m) wid(m) area(m) ];
else
ActualParameters=[ActualParameters; [m pos(m) amp(m) wid(m) area(m) ]];
end
end
plot(x,y)
drawnow
disp('Test signal with 11 overlapping Lorentzian peaks ')
% Display the peak parameters of the simulated signal
ColumnLabels
ActualParameters

disp(' ')
disp('Peakfit 3.1 can fit any number of peaks, in theory. However, a simple ')
disp('default fit for 11 peaks does not work at all with this many peaks: ')
disp(' ')
disp('>> [FitResults,FitError]=peakfit([x;y],0,0,11)')
[FitResults,FitError]=peakfit([x;y],0,0,11)
disp(' ')
disp('The only way to get a good fit in this case is to give it good "start" estimate:')
disp(' ')
disp('>> [FitResults,FitError]=peakfit([x;y],0,0,11,2,0,1,[400 20 430 40 500 50 600 40 700 20 730 20 800 30 850 40 900 60 1000 100 1200 200],0)')
[FitResults,FitError]=peakfit([x;y],0,0,11,2,0,1,[400 20 430 40 500 50 600 40 700 20 730 20 800 30 850 40 900 60 1000 100 1200 200],0)
disp(' ')
disp('Now you can compare these FitResults to the ActualParameters.')
disp('As an experiment, make a small change to the peak amplitudes (amp),')
disp('position (pos), or widths (wid), in lines 14-16, to see if the ')
disp('fitting results track your changes. ')
% Internal functions
function [FitResults,LowestError,BestStart,xi,yi,BootResults]=peakfit(signal,center,window,NumPeaks,peakshape,extra,NumTrials,start,AUTOZERO,fixedwidth)
% Version 3.1: September, 2012. Unlimited Number of peaks. Bug fixes.
% A command-line peak fittingds program for time-series signals,
% written as a self-contained Matlab function in a single m-file.
% Uses an 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
% Gaussian, equal-width Gaussians, exponentially-broadened Gaussian,
% Lorentzian, equal-width Lorentzians, Pearson, Logistic, exponential
% pulse, and sigmoid shapes (expandable to other shapes). This is a command
% line version, usable from a remote terminal. It is capable of making
% multiple trial fits with sightly different starting values and taking
% the one with the lowest mean fit error.  Version 3.1: September, 2012,
%
% PEAKFIT(signal);
% Performs an iterative least-squares fit of a single Gaussian
% peak to the data matrix "signal", which has x values
% in column 1 and Y values in column 2 (e.g. [x y])
%
% 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).
%
% PEAKFIT(signal,center,window,NumPeaks);
% "NumPeaks" = number of peaks in the model (default is 1 if not
% specified). No limit to maximum number of peaks in version 3.1
%
% PEAKFIT(signal,center,window,NumPeaks,peakshape);
% Specifies the peak shape of the model: "peakshape" = 1-12.
% (1=Gaussian (default), 2=Lorentzian, 3=logistic, 4=Pearson,
% 5=exponentionally broadened Gaussian; 6=equal-width Gaussians;
% 7=Equal-width Lorentzians; 8=exponentionally broadened equal-width
% Gaussian, 9=exponential pulse, 10=sigmoid, 11=Fixed-width Gaussian,
% 12=Fixed-width Lorentzian;).
%
% PEAKFIT(signal,center,window,NumPeaks,peakshape,extra)
% Specifies the value of 'extra', used in the Pearson and the
% exponentionally broadened Gaussian shapes to fine-tune the peak shape.
%
% 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 ...]
%
% [FitResults,MeanFitError]=PEAKFIT(signal,center,window...)
% Returns the FitResults vector in the order peak number, peak
% position, peak height, peak width, and peak area), and the MeanFitError
% (the percent RMS difference between the data and the model in the
% selected segment of that data) of the best fit.
%
% 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. BestStart: the starting guesses that gave the best fit.
% 4. xi: vector containing 100 interploated x-values for the model peaks.
% 5. yi: matrix 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
% 6. BootResults: a table of bootstrap precision results for a each peak
%    and [peak parameter.
% T. C. O'Haver (toh@umd.edu). Version 3
%
% Example 1:
% >> x=[0:.1:10]';y=exp(-(x-5).^2);peakfit([x y])
% Fits exp(-x)^2 with a single Gaussian peak model.
%
%        Peak number  Peak position   Height     Width      Peak area
%             1            5            1        1.665       1.7725
%
% Example 2:
% x=[0:.1:10]';y=exp(-(x-5).^2)+.1*randn(size(x));peakfit([x y])
% Like Example 1, except that random noise is added to the y data.
% ans =
%             1         5.0279       0.9272      1.7948      1.7716
%
% Example 3:
% x=[0:.1:10];y=exp(-(x-5).^2)+.5*exp(-(x-3).^2)+.1*randn(size(x));
% peakfit([x' y'],5,19,2,1,0,1)
% Fits a noisy two-peak signal with a double Gaussian model (NumPeaks=2).
% ans =
%             1       3.0001      0.49489        1.642      0.86504
%             2       4.9927       1.0016       1.6597       1.7696
%
% Example 4:
% >> x=1:100;y=ones(size(x))./(1+(x-50).^2);peakfit(y,50,100,1,2)
% Create and fit Lorentzian located at x=50, height=1, width=2.
% ans =
%            1           50      0.99974       1.9971       3.1079
% Example 5:
%   >> x=[0:.005:1];y=humps(x);peakfit([x' y'],.3,.7,1,4,3);
%   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).
%
% Example 6:
%  >> x=[0:.005:1];y=(humps(x)+humps(x-.13)).^3;smatrix=[x' y'];
%  >> [FitResults,MeanFitError]=peakfit(smatrix,.4,.7,2,1,0,10)
%  Creates a data matrix 'smatrix', fits a portion to a two-peak Gaussian
%  model, takes the best of 10 trials.  Returns FitResults and MeanFitError.
%  FitResults =
%              1      0.31056  2.0125e+006      0.11057  2.3689e+005
%              2      0.41529  2.2403e+006      0.12033  2.8696e+005
%  MeanFitError =
%         1.1899
%
% Example 7:
% >> peakfit([x' y'],.4,.7,2,1,0,10,[.3 .1 .5 .1]);
% As above, but specifies the first-guess position and width of the two
% peaks, in the order [position1 width1 position2 width2]
%
% Example 8:
% >> peakfit([x' y'],.4,.7,2,1,0,10,[.3 .1 .5 .1],0);
% As above, but sets AUTOZERO mode in the last argument.
% AUROZERO=0 does not subtract baseline from data segment.
% AUROZERO=1 (default) subtracts linear baseline from data segment.
% AUROZERO=2, subtracts quadratic baseline from data segment.
%
% Example 8:
% >> 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       5.0078      0.97001       1.6079       1.6598
% Fitting single Gaussian on a curved background, using quadratic
% autozero (2) and specifying center and window, but using placeholders (zeros)
% to use default values for NumPeaks, peakshape, extra, NumTrials, and start.
%
% Example 9:
% >> x=[0:.1:10];y=exp(-(x-5).^2)+.5*exp(-(x-3).^2)+.1*randn(size(x));
% [FitResults,MeanFitError]=peakfit([x' y'],0,0,2,11,0,0,0,0,1.666)
% Same as example 3, fit with fixed-widrh Gaussian (shape 11), width=1.666
%
% Example 10: (Prints out parameter error estimates; Version 3 only)
% >> x=0:.05:9;y=exp(-(x-5).^2)+.5*exp(-(x-3).^2)+.01*randn(1,length(x));
% >> [FitResults,LowestError,BestStart,xi,yi,BootstrapErrors]=peakfit([x;y],0,0,2,6,0,1,0,0,0);
%
% For more details, see
% http://terpconnect.umd.edu/~toh/spectrum/CurveFittingC.html and
% http://terpconnect.umd.edu/~toh/spectrum/InteractivePeakFitter.htm
%
global AA xxx PEAKHEIGHTS FIXEDWIDTH

format short g
format compact
warning off all

NumArgOut=nargout;
datasize=size(signal);
if datasize(1)<datasize(2),signal=signal';end
datasize=size(signal);
if datasize(2)==1, %  Must be isignal(Y-vector)
X=1:length(signal); % Create an independent variable vector
Y=signal;
else
% Must be isignal(DataMatrix)
X=signal(:,1); % Split matrix argument
Y=signal(:,2);
end
X=reshape(X,1,length(X)); % Adjust X and Y vector shape to 1 x n (rather than n x 1)
Y=reshape(Y,1,length(Y));
% If necessary, flip the data vectors so that X increases
if X(1)>X(length(X)),
disp('X-axis flipped.')
X=fliplr(X);
Y=fliplr(Y);
end

% Isolate desired segment from data set for curve fitting
if nargin==1 || nargin==2,center=(max(X)-min(X))/2;window=max(X)-min(X);end
xoffset=center-window/2;
n1=val2ind(X,center-window/2);
n2=val2ind(X,center+window/2);
if window==0,n1=1;n2=length(X);end
xx=X(n1:n2)-xoffset;
yy=Y(n1:n2);
ShapeString='Gaussian';

% Define values of any missing arguments
switch nargin
case 1
NumPeaks=1;
peakshape=1;
extra=0;
NumTrials=1;
xx=X;yy=Y;
start=calcstart(xx,NumPeaks,xoffset);
AUTOZERO=1;
case 2
NumPeaks=1;
peakshape=1;
extra=0;
NumTrials=1;
xx=signal;yy=center;
start=calcstart(xx,NumPeaks,xoffset);
AUTOZERO=1;
case 3
NumPeaks=1;
peakshape=1;
extra=0;
NumTrials=1;
start=calcstart(xx,NumPeaks,xoffset);
AUTOZERO=1;
FIXEDWIDTH=0;
case 4
peakshape=1;
extra=0;
NumTrials=1;
start=calcstart(xx,NumPeaks,xoffset);
AUTOZERO=1;
FIXEDWIDTH=0;
case 5
extra=0;
NumTrials=1;
start=calcstart(xx,NumPeaks,xoffset);
AUTOZERO=1;
FIXEDWIDTH=0;
case 6
NumTrials=1;
start=calcstart(xx,NumPeaks,xoffset);
AUTOZERO=1;
FIXEDWIDTH=0;
case 7
start=calcstart(xx,NumPeaks,xoffset);
AUTOZERO=1;
FIXEDWIDTH=0;
case 8
AUTOZERO=1;
FIXEDWIDTH=0;
case 10
FIXEDWIDTH=fixedwidth;
otherwise
end % switch nargin

% Default values for placeholder zeros
if NumTrials==0;NumTrials=1;end
if peakshape==0;peakshape=1;end
if NumPeaks==0;NumPeaks=1;end
if start==0;start=calcstart(xx,NumPeaks,xoffset);end
if FIXEDWIDTH==0, FIXEDWIDTH=length(xx)/10;end

% Remove linear baseline from data segment if AUTOZERO==1
X1=min(xx);
X2=max(xx);
bkgsize=round(length(xx)/10);
if bkgsize<2,bkgsize=2;end
if AUTOZERO==1, % linear autozero operation
Y1=mean(yy(1:bkgsize));
Y2=mean(yy((length(xx)-bkgsize):length(xx)));
yy=yy-((Y2-Y1)/(X2-X1)*(xx-X1)+Y1);
end % if
lxx=length(xx);
if AUTOZERO==2, % Quadratic autozero operation
XX1=xx(1:round(lxx/bkgsize));
XX2=xx((lxx-round(lxx/bkgsize)):lxx);
Y1=yy(1:round(length(xx)/bkgsize));
Y2=yy((lxx-round(lxx/bkgsize)):lxx);
bkgcoef=polyfit([XX1,XX2],[Y1,Y2],2);  % Fit parabola to sub-group of points
bkg=polyval(bkgcoef,xx);
yy=yy-bkg;
end % if autozero

PEAKHEIGHTS=zeros(1,NumPeaks);
n=length(xx);
newstart=start;
for peaks=1:NumPeaks,
peakindex=2*peaks-1;
newstart(peakindex)=start(peakindex)-xoffset;
end

% Assign ShapStrings
switch peakshape
case 1
ShapeString='Gaussian';
case 2
ShapeString='Lorentzian';
case 3
ShapeString='Logistic';
case 4
ShapeString='Pearson';
case 5
ShapeString='ExpGaussian';
case 6
ShapeString='Equal width Gaussians';
case 7
ShapeString='Equal width Lorentzians';
case 8
ShapeString='Exp. equal width Gaussians';
case 9
ShapeString='Exponential Pulse';
case 10
ShapeString='Sigmoid';
case 11
ShapeString='Fixed-width Gaussian';
case 12
ShapeString='Fixed-width Lorentzian';
otherwise
end % switch peakshape

% Perform peak fitting for selected peak shape using fminsearch function
options = optimset('TolX',.00001,'Display','off' );
LowestError=1000; % or any big number greater than largest error expected
FitParameters=zeros(1,NumPeaks.*2);
BestStart=zeros(1,NumPeaks.*2);
height=zeros(1,NumPeaks);
bestmodel=zeros(size(yy));
for k=1:NumTrials,
% disp(['Trial number ' num2str(k) ] ) % optionally prints the current trial number as progress indicator
switch peakshape
case 1
TrialParameters=fminsearch(@fitgaussian,newstart,options,xx,yy);
case 2
TrialParameters=fminsearch(@fitlorentzian,newstart,options,xx,yy);
case 3
TrialParameters=fminsearch(@fitlogistic,newstart,options,xx,yy);
case 4
TrialParameters=fminsearch(@fitpearson,newstart,options,xx,yy,extra);
case 5
zxx=[zeros(size(xx)) xx zeros(size(xx)) ];
zyy=[zeros(size(yy)) yy zeros(size(yy)) ];
TrialParameters=fminsearch(@fitexpgaussian,newstart,options,zxx,zyy,-extra);n';
case 6
cwnewstart(1)=newstart(1);
for pc=2:NumPeaks,
cwnewstart(pc)=newstart(2.*pc-1);
end
cwnewstart(NumPeaks+1)=(max(xx)-min(xx))/5;
TrialParameters=fminsearch(@fitewgaussian,cwnewstart,options,xx,yy);
case 7
cwnewstart(1)=newstart(1);
for pc=2:NumPeaks,
cwnewstart(pc)=newstart(2.*pc-1);
end
cwnewstart(NumPeaks+1)=(max(xx)-min(xx))/5;
TrialParameters=fminsearch(@fitlorentziancw,cwnewstart,options,xx,yy);
case 8
cwnewstart(1)=newstart(1);
for pc=2:NumPeaks,
cwnewstart(pc)=newstart(2.*pc-1);
end
cwnewstart(NumPeaks+1)=(max(xx)-min(xx))/5;
TrialParameters=fminsearch(@fitexpewgaussian,cwnewstart,options,xx,yy,-extra);
case 9
TrialParameters=fminsearch(@fitexppulse,newstart,options,xx,yy);
case 10
TrialParameters=fminsearch(@fitsigmoid,newstart,options,xx,yy);
case 11
fixedstart=[];
for pc=1:NumPeaks,
fixedstart(pc)=min(xx)+pc.*(max(xx)-min(xx))./(NumPeaks+1);
end
TrialParameters=fminsearch(@FitFWGaussian,fixedstart,options,xx,yy);
case 12
fixedstart=[];
for pc=1:NumPeaks,
fixedstart(pc)=min(xx)+pc.*(max(xx)-min(xx))./(NumPeaks+1);
end
TrialParameters=fminsearch(@FitFWLorentzian,fixedstart,options,xx,yy);
otherwise
end % switch peakshape

% Construct model from Trial parameters
A=zeros(NumPeaks,n);
for m=1:NumPeaks,
switch peakshape
case 1
A(m,:)=gaussian(xx,TrialParameters(2*m-1),TrialParameters(2*m));
case 2
A(m,:)=lorentzian(xx,TrialParameters(2*m-1),TrialParameters(2*m));
case 3
A(m,:)=logistic(xx,TrialParameters(2*m-1),TrialParameters(2*m));
case 4
A(m,:)=pearson(xx,TrialParameters(2*m-1),TrialParameters(2*m),extra);
case 5
A(m,:)=expgaussian(xx,TrialParameters(2*m-1),TrialParameters(2*m),-extra)';
case 6
A(m,:)=gaussian(xx,TrialParameters(m),TrialParameters(NumPeaks+1));
case 7
A(m,:)=lorentzian(xx,TrialParameters(m),TrialParameters(NumPeaks+1));
case 8
A(m,:)=expgaussian(xx,TrialParameters(m),TrialParameters(NumPeaks+1),-extra)';
case 9
A(m,:)=exppulse(xx,TrialParameters(2*m-1),TrialParameters(2*m));
case 10
A(m,:)=sigmoid(xx,TrialParameters(2*m-1),TrialParameters(2*m));
case 11
A(m,:)=gaussian(xx,TrialParameters(m),FIXEDWIDTH);
case 12
A(m,:)=lorentzian(xx,TrialParameters(m),FIXEDWIDTH);
otherwise
end % switch
for parameter=1:2:2*NumPeaks,
newstart(parameter)=newstart(parameter)*(1+randn/50);
newstart(parameter+1)=newstart(parameter+1)*(1+randn/10);
end
end % for

% Multiplies each row by the corresponding amplitude and adds them up
model=PEAKHEIGHTS'*A;

% Compare trial model to data segment and compute the fit error
MeanFitError=100*norm(yy-model)./(sqrt(n)*max(yy));
% Take only the single fit that has the lowest MeanFitError
if MeanFitError<LowestError,
if min(PEAKHEIGHTS)>0,  % Consider only fits with positive peak heights
LowestError=MeanFitError;  % Assign LowestError to the lowest MeanFitError
FitParameters=TrialParameters;  % Assign FitParameters to the fit with the lowest MeanFitError
height=PEAKHEIGHTS; % Assign height to the PEAKHEIGHTS with the lowest MeanFitError
bestmodel=model; % Assign bestmodel to the model with the lowest MeanFitError
end % if min(PEAKHEIGHTS)>0
end % if MeanFitError<LowestError
end % for k (NumTrials)
%
% Construct model from best-fit parameters
AA=zeros(NumPeaks,200);
xxx=linspace(min(xx),max(xx),200);
for m=1:NumPeaks,
switch peakshape
case 1
AA(m,:)=gaussian(xxx,FitParameters(2*m-1),FitParameters(2*m));
case 2
AA(m,:)=lorentzian(xxx,FitParameters(2*m-1),FitParameters(2*m));
case 3
AA(m,:)=logistic(xxx,FitParameters(2*m-1),FitParameters(2*m));
case 4
AA(m,:)=pearson(xxx,FitParameters(2*m-1),FitParameters(2*m),extra);
case 5
AA(m,:)=expgaussian(xxx,FitParameters(2*m-1),FitParameters(2*m),-extra*length(xxx)./length(xx))';
case 6
AA(m,:)=gaussian(xxx,FitParameters(m),FitParameters(NumPeaks+1));
case 7
AA(m,:)=lorentzian(xxx,FitParameters(m),FitParameters(NumPeaks+1));
case 8
AA(m,:)=expgaussian(xxx,FitParameters(m),FitParameters(NumPeaks+1),-extra*length(xxx)./length(xx))';
case 9
AA(m,:)=exppulse(xxx,FitParameters(2*m-1),FitParameters(2*m));
case 10
AA(m,:)=sigmoid(xxx,FitParameters(2*m-1),FitParameters(2*m));
case 11
AA(m,:)=gaussian(xxx,FitParameters(m),FIXEDWIDTH);
case 12
AA(m,:)=lorentzian(xxx,FitParameters(m),FIXEDWIDTH);
otherwise
end % switch
end % for

% Multiplies each row by the corresponding amplitude and adds them up
heightsize=size(height');
AAsize=size(AA);
if heightsize(2)==AAsize(1),
mmodel=height'*AA;
else
mmodel=height*AA;
end
% Top half of the figure shows original signal and the fitted model.
subplot(2,1,1);plot(xx+xoffset,yy,'b.'); % Plot the original signal in blue dots
hold on
for m=1:NumPeaks,
plot(xxx+xoffset,height(m)*AA(m,:),'g')  % Plot the individual component peaks in green lines
area(m)=trapz(xxx+xoffset,height(m)*AA(m,:)); % Compute the area of each component peak using trapezoidal method
yi(m,:)=height(m)*AA(m,:); % (NEW) Place y values of individual model peaks into matrix yi
end
xi=xxx+xoffset; % (NEW) Place the x-values of the individual model peaks into xi

% Mark starting peak positions with vertical dashed lines
for marker=1:NumPeaks,
markx=BestStart((2*marker)-1);
subplot(2,1,1);plot([markx+xoffset markx+xoffset],[0 max(yy)],'m--')
end % for
plot(xxx+xoffset,mmodel,'r');  % Plot the total model (sum of component peaks) in red lines
hold off;
axis([min(xx)+xoffset max(xx)+xoffset min(yy) max(yy)]);
switch AUTOZERO,
case 0
title('Peakfit 3.1 Autozero OFF.')
case 1
title('Peakfit 3.1 Linear autozero.')
case 2
end
if peakshape==4||peakshape==5||peakshape==8, % Shapes with Extra factor
xlabel(['Peaks = ' num2str(NumPeaks) '     Shape = ' ShapeString '     Error = ' num2str(round(100*LowestError)/100) '%    Extra = ' num2str(extra) ] )
else
xlabel(['Peaks = ' num2str(NumPeaks) '     Shape = ' ShapeString '     Error = ' num2str(round(100*LowestError)/100) '%' ] )
end

% Bottom half of the figure shows the residuals and displays RMS error
% between original signal and model
residual=yy-bestmodel;
subplot(2,1,2);plot(xx+xoffset,residual,'b.')
axis([min(xx)+xoffset max(xx)+xoffset min(residual) max(residual)]);
xlabel('Residual Plot')

% Put results into a matrix, one row for each peak, showing peak index number,
% position, amplitude, and width.
for m=1:NumPeaks,
if m==1,
if peakshape==6||peakshape==7||peakshape==8, % equal-width peak models
FitResults=[[round(m) FitParameters(m)+xoffset height(m) abs(FitParameters(NumPeaks+1)) area(m)]];
else
if peakshape==11||peakshape==12,
FitResults=[[round(m) FitParameters(m)+xoffset height(m) FIXEDWIDTH area(m)]];
else
FitResults=[[round(m) FitParameters(2*m-1)+xoffset height(m) abs(FitParameters(2*m)) area(m)]];
end
end % if peakshape
else
if peakshape==6||peakshape==7||peakshape==8, % equal-width peak models
FitResults=[FitResults ; [round(m) FitParameters(m)+xoffset height(m) abs(FitParameters(NumPeaks+1)) area(m)]];
else
if peakshape==11||peakshape==12,
FitResults=[FitResults ; [round(m) FitParameters(m)+xoffset height(m) FIXEDWIDTH area(m)]];
else
FitResults=[FitResults ; [round(m) FitParameters(2*m-1)+xoffset height(m) abs(FitParameters(2*m)) area(m)]];
end
end % if peakshape
end % m==1
end % for m=1:NumPeaks

% Display Fit Results on upper graph
subplot(2,1,1);
startx=min(xx)+xoffset+(max(xx)-min(xx))./20;
dxx=(max(xx)-min(xx))./10;
dyy=(max(yy)-min(yy))./10;
starty=max(yy)-dyy;
FigureSize=get(gcf,'Position');
if peakshape==9||peakshape==10,
text(startx,starty+dyy/2,['Peak #          tau1           Height           tau2             Area'] );
else
text(startx,starty+dyy/2,['Peak #          Position        Height         Width             Area'] );
end
% Display FitResults using sprintf
for peaknumber=1:NumPeaks,
for column=1:5,
itemstring=sprintf('%0.4g',FitResults(peaknumber,column));
xposition=startx+(1.7.*dxx.*(column-1).*(600./FigureSize(3)));
yposition=starty-peaknumber.*dyy.*(400./FigureSize(4));
text(xposition,yposition,itemstring);
end
end
if NumArgOut==6,
disp('Computing bootstrap sampling statistics.....')
BootstrapResultsMatrix=zeros(5,100,NumPeaks);
BootstrapErrorMatrix=zeros(1,100,NumPeaks);
clear bx by
cutoff=0.5;
% offset=xoffset
% sizexx=size(xx)
% sizeyy=size(yy)
tic;
for trial=1:100,
n=1;
bx=xx;
by=yy;
while n<length(xx)-1,
if rand>cutoff,
bx(n)=xx(n+1);
by(n)=yy(n+1);
end
n=n+1;
end
bx=bx+xoffset;
[FitResults,BootFitError]=fitpeaks(bx,by,NumPeaks,peakshape,extra,NumTrials,start,AUTOZERO,FIXEDWIDTH);
for peak=1:NumPeaks,
BootstrapResultsMatrix(:,trial,peak)=FitResults(peak,:);
BootstrapErrorMatrix(:,trial,peak)=BootFitError;
end
end
toc;
for peak=1:NumPeaks,
disp(' ')
disp(['Peak #',num2str(peak) '         Position    Height       Width       Area']);
BootstrapMean=mean(real(BootstrapResultsMatrix(:,:,peak)'));
BootstrapSTD=std(BootstrapResultsMatrix(:,:,peak)');
BootstrapIQR=iqr(BootstrapResultsMatrix(:,:,peak)');
PercentRSD=100.*BootstrapSTD./BootstrapMean;
PercentIQR=100.*BootstrapIQR./BootstrapMean;
BootstrapMean=BootstrapMean(2:5);
BootstrapSTD=BootstrapSTD(2:5);
BootstrapIQR=BootstrapIQR(2:5);
PercentRSD=PercentRSD(2:5);
PercentIQR=PercentIQR(2:5);
disp(['Bootstrap Mean: ', num2str(BootstrapMean)])
disp(['Bootstrap STD:  ', num2str(BootstrapSTD)])
disp(['Bootstrap IQR:  ', num2str(BootstrapIQR)])
disp(['Percent RSD:    ', num2str(PercentRSD)])
disp(['Percent IQR:    ', num2str(PercentIQR)])
BootResults(peak,:)=[BootstrapMean BootstrapSTD PercentRSD BootstrapIQR PercentIQR];
end
end
% ----------------------------------------------------------------------
function [FitResults,LowestError]=fitpeaks(xx,yy,NumPeaks,peakshape,extra,NumTrials,start,AUTOZERO,fixedwidth)
% Based on peakfit Version 3: June, 2012.
global PEAKHEIGHTS FIXEDWIDTH
format short g
format compact
warning off all
FIXEDWIDTH=fixedwidth;
xoffset=0;
if start==0;start=calcstart(xx,NumPeaks,xoffset);end
PEAKHEIGHTS=zeros(1,NumPeaks);
n=length(xx);
newstart=start;

for peaks=1:NumPeaks,
peakindex=2*peaks-1;
newstart(peakindex)=start(peakindex)-xoffset;
end

% Perform peak fitting for selected peak shape using fminsearch function
options = optimset('TolX',.00001,'Display','off' );
LowestError=1000; % or any big number greater than largest error expected
FitParameters=zeros(1,NumPeaks.*2);
BestStart=zeros(1,NumPeaks.*2);
height=zeros(1,NumPeaks);
bestmodel=zeros(size(yy));
for k=1:NumTrials,
switch peakshape
case 1
TrialParameters=fminsearch(@fitgaussian,newstart,options,xx,yy);
case 2
TrialParameters=fminsearch(@fitlorentzian,newstart,options,xx,yy);
case 3
TrialParameters=fminsearch(@fitlogistic,newstart,options,xx,yy);
case 4
TrialParameters=fminsearch(@fitpearson,newstart,options,xx,yy,extra);
case 5
zxx=[zeros(size(xx)) xx zeros(size(xx)) ];
zyy=[zeros(size(yy)) yy zeros(size(yy)) ];
TrialParameters=fminsearch(@fitexpgaussian,newstart,options,zxx,zyy,-extra);n';
case 6
cwnewstart(1)=newstart(1);
for pc=2:NumPeaks,
cwnewstart(pc)=newstart(2.*pc-1);
end
cwnewstart(NumPeaks+1)=(max(xx)-min(xx))/5;
TrialParameters=fminsearch(@fitewgaussian,cwnewstart,options,xx,yy);
case 7
cwnewstart(1)=newstart(1);
for pc=2:NumPeaks,
cwnewstart(pc)=newstart(2.*pc-1);
end
cwnewstart(NumPeaks+1)=(max(xx)-min(xx))/5;
TrialParameters=fminsearch(@fitlorentziancw,cwnewstart,options,xx,yy);
case 8
cwnewstart(1)=newstart(1);
for pc=2:NumPeaks,
cwnewstart(pc)=newstart(2.*pc-1);
end
cwnewstart(NumPeaks+1)=(max(xx)-min(xx))/5;
TrialParameters=fminsearch(@fitexpewgaussian,cwnewstart,options,xx,yy,-extra);
case 9
TrialParameters=fminsearch(@fitexppulse,newstart,options,xx,yy);
case 10
TrialParameters=fminsearch(@fitsigmoid,newstart,options,xx,yy);
case 11
fixedstart=[];
for pc=1:NumPeaks,
fixedstart(pc)=min(xx)+pc.*(max(xx)-min(xx))./(NumPeaks+1);
end
TrialParameters=fminsearch(@FitFWGaussian,fixedstart,options,xx,yy);
case 12
fixedstart=[];
for pc=1:NumPeaks,
fixedstart(pc)=min(xx)+pc.*(max(xx)-min(xx))./(NumPeaks+1);
end
TrialParameters=fminsearch(@FitFWLorentzian,fixedstart,options,xx,yy);
otherwise
end % switch peakshape

% Construct model from Trial parameters
A=zeros(NumPeaks,n);
for m=1:NumPeaks,
switch peakshape
case 1
A(m,:)=gaussian(xx,TrialParameters(2*m-1),TrialParameters(2*m));
case 2
A(m,:)=lorentzian(xx,TrialParameters(2*m-1),TrialParameters(2*m));
case 3
A(m,:)=logistic(xx,TrialParameters(2*m-1),TrialParameters(2*m));
case 4
A(m,:)=pearson(xx,TrialParameters(2*m-1),TrialParameters(2*m),extra);
case 5
A(m,:)=expgaussian(xx,TrialParameters(2*m-1),TrialParameters(2*m),-extra)';
case 6
A(m,:)=gaussian(xx,TrialParameters(m),TrialParameters(NumPeaks+1));
case 7
A(m,:)=lorentzian(xx,TrialParameters(m),TrialParameters(NumPeaks+1));
case 8
A(m,:)=expgaussian(xx,TrialParameters(m),TrialParameters(NumPeaks+1),-extra)';
case 9
A(m,:)=exppulse(xx,TrialParameters(2*m-1),TrialParameters(2*m));
case 10
A(m,:)=sigmoid(xx,TrialParameters(2*m-1),TrialParameters(2*m));
case 11
A(m,:)=gaussian(xx,TrialParameters(m),FIXEDWIDTH);
case 12
A(m,:)=lorentzian(xx,TrialParameters(m),FIXEDWIDTH);
otherwise
end % switch
for parameter=1:2:2*NumPeaks,
newstart(parameter)=newstart(parameter)*(1+randn/50);
newstart(parameter+1)=newstart(parameter+1)*(1+randn/10);
end
end % for
% Multiplies each row by the corresponding amplitude and adds them up
model=PEAKHEIGHTS'*A;

% Compare trial model to data segment and compute the fit error
MeanFitError=100*norm(yy-model)./(sqrt(n)*max(yy));
% Take only the single fit that has the lowest MeanFitError
if MeanFitError<LowestError,
if min(PEAKHEIGHTS)>0,  % Consider only fits with positive peak heights
LowestError=MeanFitError;  % Assign LowestError to the lowest MeanFitError
FitParameters=TrialParameters;  % Assign FitParameters to the fit with the lowest MeanFitError
%         BestStart=newstart; % Assign BestStart to the start with the lowest MeanFitError
height=PEAKHEIGHTS; % Assign height to the PEAKHEIGHTS with the lowest MeanFitError
%         bestmodel=model; % Assign bestmodel to the model with the lowest MeanFitError
end % if min(PEAKHEIGHTS)>0
end % if MeanFitError<LowestError
end % for k (NumTrials)

for m=1:NumPeaks,
area(m)=trapz(xx+xoffset,height(m)*A(m,:)); % Compute the area of each component peak using trapezoidal method
end

for m=1:NumPeaks,
if m==1,
if peakshape==6||peakshape==7||peakshape==8, % equal-width peak models
FitResults=[[round(m) FitParameters(m)+xoffset height(m) abs(FitParameters(NumPeaks+1)) area(m)]];
else
if peakshape==11||peakshape==12,
FitResults=[[round(m) FitParameters(m)+xoffset height(m) FIXEDWIDTH area(m)]];
else
FitResults=[[round(m) FitParameters(2*m-1)+xoffset height(m) abs(FitParameters(2*m)) area(m)]];
end
end % if peakshape
else
if peakshape==6||peakshape==7||peakshape==8, % equal-width peak models
FitResults=[FitResults ; [round(m) FitParameters(m)+xoffset height(m) abs(FitParameters(NumPeaks+1)) area(m)]];
else
if peakshape==11||peakshape==12,
FitResults=[FitResults ; [round(m) FitParameters(m)+xoffset height(m) FIXEDWIDTH area(m)]];
else
FitResults=[FitResults ; [round(m) FitParameters(2*m-1)+xoffset height(m) abs(FitParameters(2*m)) area(m)]];
end
end % if peakshape
end % m==1
end % for m=1:NumPeaks
% ----------------------------------------------------------------------
function start=calcstart(xx,NumPeaks,xoffset)
n=max(xx)-min(xx);
start=[];
startpos=[n/(NumPeaks+1):n/(NumPeaks+1):n-(n/(NumPeaks+1))]+min(xx);
for marker=1:NumPeaks,
markx=startpos(marker)+ xoffset;
start=[start markx n/5];
end % for marker
% ----------------------------------------------------------------------
function [index,closestval]=val2ind(x,val)
% Returns the index and the value of the element of vector x that is closest to val
% If more than one element is equally close, returns vectors of indicies and values
% Tom O'Haver (toh@umd.edu) October 2006
% Examples: If x=[1 2 4 3 5 9 6 4 5 3 1], then val2ind(x,6)=7 and val2ind(x,5.1)=[5 9]
% [indices values]=val2ind(x,3.3) returns indices = [4 10] and values = [3 3]
dif=abs(x-val);
index=find((dif-min(dif))==0);
closestval=x(index);
% ----------------------------------------------------------------------
function err = fitgaussian(lambda,t,y)
% Fitting function for a Gaussian band signal.
global PEAKHEIGHTS
numpeaks=round(length(lambda)/2);
A = zeros(length(t),numpeaks);
for j = 1:numpeaks,
A(:,j) = gaussian(t,lambda(2*j-1),lambda(2*j))';
end
PEAKHEIGHTS = abs(A\y');
z = A*PEAKHEIGHTS;
err = norm(z-y');
% ----------------------------------------------------------------------
function err = fitewgaussian(lambda,t,y)
% Fitting function for a Gaussian band signal with equal peak widths.
global PEAKHEIGHTS
numpeaks=round(length(lambda)-1);
A = zeros(length(t),numpeaks);
for j = 1:numpeaks,
A(:,j) = gaussian(t,lambda(j),lambda(numpeaks+1))';
end
PEAKHEIGHTS = abs(A\y');
z = A*PEAKHEIGHTS;
err = norm(z-y');
% ----------------------------------------------------------------------
function err = FitFWGaussian(lambda,t,y)
%	Fitting function for a fixed width Gaussian
global PEAKHEIGHTS FIXEDWIDTH
numpeaks=round(length(lambda));
A = zeros(length(t),numpeaks);
for j = 1:numpeaks,
A(:,j) = gaussian(t,lambda(j),FIXEDWIDTH)';
end
PEAKHEIGHTS = abs(A\y');
z = A*PEAKHEIGHTS;
err = norm(z-y');
% ----------------------------------------------------------------------
function err = FitFWLorentzian(lambda,t,y)
%	Fitting function for a fixed width Gaussian
global PEAKHEIGHTS FIXEDWIDTH
numpeaks=round(length(lambda));
A = zeros(length(t),numpeaks);
for j = 1:numpeaks,
A(:,j) = lorentzian(t,lambda(j),FIXEDWIDTH)';
end
PEAKHEIGHTS = abs(A\y');
z = A*PEAKHEIGHTS;
err = norm(z-y');
% ----------------------------------------------------------------------
function err = fitlorentziancw(lambda,t,y)
% Fitting function for a Lorentzian band signal with equal peak widths.
global PEAKHEIGHTS
numpeaks=round(length(lambda)-1);
A = zeros(length(t),numpeaks);
for j = 1:numpeaks,
A(:,j) = lorentzian(t,lambda(j),lambda(numpeaks+1))';
end
PEAKHEIGHTS = abs(A\y');
z = A*PEAKHEIGHTS;
err = norm(z-y');
% ----------------------------------------------------------------------
function g = gaussian(x,pos,wid)
%  gaussian(X,pos,wid) = gaussian peak centered on pos, half-width=wid
%  X may be scalar, vector, or matrix, pos and wid both scalar
% Examples: gaussian([0 1 2],1,2) gives result [0.5000    1.0000    0.5000]
% plot(gaussian([1:100],50,20)) displays gaussian band centered at 50 with width 20.
g = exp(-((x-pos)./(0.6005615.*wid)).^2);
% ----------------------------------------------------------------------
function err = fitlorentzian(lambda,t,y)
%	Fitting function for single lorentzian, lambda(1)=position, lambda(2)=width
%	Fitgauss assumes a lorentzian function
global PEAKHEIGHTS
A = zeros(length(t),round(length(lambda)/2));
for j = 1:length(lambda)/2,
A(:,j) = lorentzian(t,lambda(2*j-1),lambda(2*j))';
end
PEAKHEIGHTS = A\y';
z = A*PEAKHEIGHTS;
err = norm(z-y');
% ----------------------------------------------------------------------
function g = lorentzian(x,position,width)
% lorentzian(x,position,width) Lorentzian function.
% where x may be scalar, vector, or matrix
% position and width scalar
% T. C. O'Haver, 1988
% Example: lorentzian([1 2 3],2,2) gives result [0.5 1 0.5]
g=ones(size(x))./(1+((x-position)./(0.5.*width)).^2);
% ----------------------------------------------------------------------
function err = fitlogistic(lambda,t,y)
%	Fitting function for logistic, lambda(1)=position, lambda(2)=width
%	between the data and the values computed by the current
%	function of lambda.  Fitlogistic assumes a logistic function
%  T. C. O'Haver, May 2006
global PEAKHEIGHTS
A = zeros(length(t),round(length(lambda)/2));
for j = 1:length(lambda)/2,
A(:,j) = logistic(t,lambda(2*j-1),lambda(2*j))';
end
PEAKHEIGHTS = A\y';
z = A*PEAKHEIGHTS;
err = norm(z-y');
% ----------------------------------------------------------------------
function g = logistic(x,pos,wid)
% logistic function.  pos=position; wid=half-width (both scalar)
% logistic(x,pos,wid), where x may be scalar, vector, or matrix
% pos=position; wid=half-width (both scalar)
% T. C. O'Haver, 1991
n = exp(-((x-pos)/(.477.*wid)) .^2);
g = (2.*n)./(1+n);
% ----------------------------------------------------------------------
function err = fitlognormal(lambda,t,y)
%	Fitting function for lognormal, lambda(1)=position, lambda(2)=width
%	between the data and the values computed by the current
%	function of lambda.  Fitlognormal assumes a lognormal function
%  T. C. O'Haver, May 2006
global PEAKHEIGHTS
A = zeros(length(t),round(length(lambda)/2));
for j = 1:length(lambda)/2,
A(:,j) = lognormal(t,lambda(2*j-1),lambda(2*j))';
end
PEAKHEIGHTS = A\y';
z = A*PEAKHEIGHTS;
err = norm(z-y');
% ----------------------------------------------------------------------
function g = lognormal(x,pos,wid)
% lognormal function.  pos=position; wid=half-width (both scalar)
% lognormal(x,pos,wid), where x may be scalar, vector, or matrix
% pos=position; wid=half-width (both scalar)
% T. C. O'Haver, 1991
g = exp(-(log(x/pos)/(0.01.*wid)) .^2);
% ----------------------------------------------------------------------
function err = fitpearson(lambda,t,y,shapeconstant)
%   Fitting functions for a Pearson 7 band signal.
% T. C. O'Haver (toh@umd.edu),   Version 1.3, October 23, 2006.
global PEAKHEIGHTS
A = zeros(length(t),round(length(lambda)/2));
for j = 1:length(lambda)/2,
A(:,j) = pearson(t,lambda(2*j-1),lambda(2*j),shapeconstant)';
end
PEAKHEIGHTS = A\y';
z = A*PEAKHEIGHTS;
err = norm(z-y');
% ----------------------------------------------------------------------
function g = pearson(x,pos,wid,m)
% Pearson VII function.
% g = pearson7(x,pos,wid,m) where x may be scalar, vector, or matrix
% pos=position; wid=half-width (both scalar)
% m=some number
%  T. C. O'Haver, 1990
g=ones(size(x))./(1+((x-pos)./((0.5.^(2/m)).*wid)).^2).^m;
% ----------------------------------------------------------------------
function err = fitexpgaussian(lambda,t,y,timeconstant)
%   Fitting functions for a exponentially-broadened Gaussian band signal.
%  T. C. O'Haver, October 23, 2006.
global PEAKHEIGHTS
A = zeros(length(t),round(length(lambda)/2));
for j = 1:length(lambda)/2,
A(:,j) = expgaussian(t,lambda(2*j-1),lambda(2*j),timeconstant);
end
PEAKHEIGHTS = A\y';
z = A*PEAKHEIGHTS;
err = norm(z-y');
% ----------------------------------------------------------------------
function err = fitexpewgaussian(lambda,t,y,timeconstant)
% Fitting function for exponentially-broadened Gaussian bands with equal peak widths.
global PEAKHEIGHTS
numpeaks=round(length(lambda)-1);
A = zeros(length(t),numpeaks);
for j = 1:numpeaks,
A(:,j) = expgaussian(t,lambda(j),lambda(numpeaks+1),timeconstant);
end
PEAKHEIGHTS = abs(A\y');
z = A*PEAKHEIGHTS;
err = norm(z-y');
% ----------------------------------------------------------------------
function g = expgaussian(x,pos,wid,timeconstant)
%  Exponentially-broadened gaussian(x,pos,wid) = gaussian peak centered on pos, half-width=wid
%  x may be scalar, vector, or matrix, pos and wid both scalar
%  T. C. O'Haver, 2006
g = exp(-((x-pos)./(0.6005615.*wid)) .^2);
% ----------------------------------------------------------------------
% ExpBroaden(y,t) convolutes y by an exponential decay of time constant t
% by multiplying Fourier transforms and inverse transforming the result.
ly=length(y);
ey=[zeros(size(y));y;zeros(size(y))];
fy=fft(ey);
a=exp(-(1:length(fy))./t);
fa=fft(a);
fy1=fy.*fa';
ybz=real(ifft(fy1))./sum(a);
yb=ybz(ly+2:length(ybz)-ly+1);
% ----------------------------------------------------------------------
function err = fitexppulse(tau,x,y)
% Iterative fit of the sum of exponental pulses
% of the form Height.*exp(-tau1.*x).*(1-exp(-tau2.*x)))
global PEAKHEIGHTS
A = zeros(length(x),round(length(tau)/2));
for j = 1:length(tau)/2,
A(:,j) = exppulse(x,tau(2*j-1),tau(2*j));
end
PEAKHEIGHTS =abs(A\y');
z = A*PEAKHEIGHTS;
err = norm(z-y');
% ----------------------------------------------------------------------
function g = exppulse(x,t1,t2)
% Exponential pulse of the form
% Height.*exp(-tau1.*x).*(1-exp(-tau2.*x)))
e=(x-t1)./t2;
p = 4*exp(-e).*(1-exp(-e));
p=p .* [p>0];
g = p';
% ----------------------------------------------------------------------
function err = fitsigmoid(tau,x,y)
% Fitting function for iterative fit to the sum of
% sigmiods of the form Height./(1 + exp((t1 - t)/t2))
global PEAKHEIGHTS
A = zeros(length(x),round(length(tau)/2));
for j = 1:length(tau)/2,
A(:,j) = sigmoid(x,tau(2*j-1),tau(2*j));
end
PEAKHEIGHTS = A\y';
z = A*PEAKHEIGHTS;
err = norm(z-y');
% ----------------------------------------------------------------------
function g=sigmoid(x,t1,t2)
g=1./(1 + exp((t1 - x)./t2))';
% ----------------------------------------------------------------------
function b=iqr(a)
% b = IQR(a)  returns the interquartile range of the values in a.  For
%  vector input, b is the difference between the 75th and 25th percentiles
%  of a.  For matrix input, b is a row vector containing the interquartile
%  range of each column of a.
mina=min(a);
sizea=size(a);
NumCols=sizea(2);
for n=1:NumCols,b(:,n)=a(:,n)-mina(n);end
Sorteda=sort(b);
lx=length(Sorteda);
SecondQuartile=round(lx/4);
FourthQuartile=3*round(lx/4);
b=abs(Sorteda(FourthQuartile,:)-Sorteda(SecondQuartile,:));

```