(1) Select a model for the data;This continues until the fitting error is less than the specified error. One popular technique for doing this is called the NelderMead Modified Simplex. This is essentially a way of organizing and optimizing the changes in parameters (step 4, above) to shorten the time required to fit the function to the required degree of accuracy. With contemporary personal computers, the entire process typically takes only a fraction of a second to a few seconds, depending on the complexity of the model and the number of independently adjustable parameters in the model.
(2) Make first guesses of all the nonlinear parameters;
(3) A computer program computes the model and compares it to the data set, calculating a fitting error;
(4) If the fitting error is greater that the required fitting accuracy, the program systematically changes one or more of the parameters and loops back around to step 3.
A simple example is fitting the blackbody equation to the spectrum of an incandescent body for the purpose of estimating its color temperature. In this case there is only one nonlinear parameter, temperature. The script BlackbodyDataFit.m demonstrates the technique, placing the experimentally measured spectrum in the vectors "wavelength" and "radiance" and then calling fminsearch with the fitting function fitblackbody.m.
Another application is demonstrated by Matlab's builtin demo
fitdemo.m and its corresponding
fitting function fitfun.m, which model
the sum of two exponential decays. To see this, just type
"fitdemo" in the Matlab command window. (Octave does not have
this demo function).
Fitting peaks. Many instrumental methods of measurement produce signals in the form of peaks of various shapes; a common requirement is to measure the positions, heights, widths, and/or areas of those peaks, even when they are noisy or overlapped with one another. This cannot be done by linear leastsquares methods, because such signals can not be modeled as polynomials with linear coefficients (the positions and widths of the peaks are not linear functions), so iterative curve fitting techniques are used instead, often using Gaussian, Lorentzian, or some other fundamental simple peak shapes as a model.
The Matlab/Octave demonstration script Demofitgauss.m demonstrates fitting a Gaussian function to a set of data, using the fitting function fitgauss.m. In this case there are two nonlinear parameters: peak position and peak width (the peak height is a linear parameter and is determined by regression in a single step in line 9 of the fitting function fitgauss.m and is returned in the global variable "c"). Compared to the simpler polynomial leastsquares methods for measuring peaks, the iterative method has the advantage of using all the data points across the entire peak, including zero and negative points, and it can be applied to multiple overlapping peaks as demonstrated in in Demofitgauss2.m (shown on the left.
To accommodate the possibility that the baseline may shift, we can add a column of 1s to the A matrix, just as was done in the CLS method. This has the effect of introducing an additional "peak" into the model that has an amplitude but no position or width. The baseline amplitude is returned along with the peak heights in the global vector “c”; Demofitgaussb.m and fitgauss2b.m illustrates this addition. (Demofitlorentzianb.m and fitlorentzianb.m for Lorentzian peaks).
This peak fitting technique is easily extended to any number of
overlapping peaks of the same type using the same
fitting function fitgauss.m, which easily adapts to any number
of peaks, depending on the length of the firstguess "start"
vector lambda that
is passed to the function as input arguments, along with the
data vectors t and y:
1 function err = fitgauss(lambda,t,y) 2 % Fitting functions for a Gaussian band spectrum. 3 % T. C. O'Haver, 2006 Updated to Matlab 6, March 2006 4 global c 5 A = zeros(length(t),round(length(lambda)/2)); 6 for j = 1:length(lambda)/2, 7 A(:,j) = gaussian(t,lambda(2*j1),lambda(2*j))'; 8 end 9 c = A\y'; 10 z = A*c; 11 err = norm(zy');If there are n peaks in the model, then the length of lambda is 2n, one entry for each iterated variable ([position1 width1 position2 width2....etc]). The "for" loop (lines 57) constructs a n × length(t) matrix containing the model for each peak separately, using a userdefined peak shape function (in this case gaussian.m), then it computes the nlength peak height vector c by leastsquares regression in line 9, using the Matlab shortcut "\" notation. The resulting peak heights are used to compute z, the sum of all n model peaks, by matrix multiplication in line 10, and then "err", the rootmeansquare difference between the model z and the actual data y, is computed in line 11 by the Matlab 'norm' function and returned to the calling function ('fminsearch'), which repeats the process many times, trying different values of the peak positions and the peak widths until the value of "err" is low enough.
Variable shape types, such as the Voigt
profile, Pearson, BreitWignerFano,
GaussLorentz blend, and the exponentiallybroadened Gaussian
and Lorentzian, are defined not only by a peak position,
height, and width, but also by an additional parameter that
finetunes the shape of the peak. If that parameter is equal
and known for all peaks in a group, it can be passed as
an additional input argument to the shape function, as shown
in VoigtFixedAlpha.m. If the
shape parameter is allowed to be different for each
peak in the group and is to be determined by iteration (just
as is position and width), then the routine must be modified
to accommodate three, rather than two, iterated
variables, as shown in VoigtVariableAlpha.m.
Although the fitting error is lower with variable alphas, the
execution time is longer and the alphas values so determined
are not very stable, with respect to noise in the data and the
starting guess values, especially for multiple peaks. (These
are selfcontained functions). Version 7 of the downloadable
Matlab/Octave function peakfit.m
includes variable shape types for the Pearson, ExpGaussian,
Voigt, and Gaussian/Lorentzian blend.
Signals with peaks of different shape types
in one signal can be fit by the fitting function fitmultiple.m,
which takes as input arguments a vector of peak types and a
vector of shape variables. The sequence of peak types and
shape parameters must be determined beforehand. To see how
this is used, see Demofitmultiple.m.
You can create your own fitting functions for any purpose; they are not limited to single algebraic expressions, but can be arbitrarily complex multiple step algorithms. For example, in the TFit method for quantitative absorption spectroscopy, a model of the instrumentallybroadened transmission spectrum is fit to the observed transmission data, using Matlab's fminsearch function, using a fitting function that performs Fourier convolution of the transmission spectrum model with the known slit function of the spectrometer. The result is an alternative method of calculating absorbance that extends the dynamic range and calibration linearity of absorption spectroscopy far beyond the normal limits.
The uncertainty of the peak parameters determined by iterative leastsquares fitting is readily estimated by the bootstrap sampling method. A simple demonstration of bootstrap estimation of the variability of an iterative leastsquares fit to a single noisy Gaussian peak is given by the custom downloadable Matlab/Octave function "BootstrapIterativeFit.m", which creates a single x,y data set consisting of a single noisy Gaussian peak, extracts bootstrap samples from that data set, performs an iterative fit to the peak on each of the bootstrap samples, and plots the histograms of peak height, position, and width of the bootstrap samples. The syntax is BootstrapIterativeFit(TrueHeight, TruePosition, TrueWidth, NumPoints, Noise, NumTrials) where TrueHeight is the true peak height of the Gaussian peak, TruePosition is the true xaxis value at the peak maximum, TrueWidth is the true halfwidth (FWHM) of the peak, NumPoints is the number of points taken for the leastsquares fit, Noise is the standard deviation of (normallydistributed) random noise, and NumTrials is the number of bootstrap samples. An typical example for BootstrapIterativeFit(100,100,100,20,10,100); is displayed in the figure on the right.
>>
BootstrapIterativeFit(100,100,100,20,10,100);
Peak Height Peak
Position Peak Width
mean:
99.27028 100.4002
94.5059
STD: 2.8292
1.3264 2.9939
IQR: 4.0897
1.6822 4.0164
IQR/STD Ratio: 1.3518
A similar demonstration function
for two overlapping
Gaussian peaks is available in "BootstrapIterativeFit2.m".
Type "help BootstrapIterativeFit2" for
more information. In both these simulations, the standard
deviation (STD) as well as the interquartile
range (IQR) of each of the peak parameters are
computed. This is done because the interquartile range is
much less influenced by outliers. The distribution
of peak parameters measured by iterative fitting is often
nonnormal, exhibiting a greater fraction of large deviations
from the mean than is expected for a normal distribution. This
is because the iterative procedure sometimes converges on an
abnormal result, especially for multiple peak fits with a large
number of variable parameters. (You may be able to see this in
the histograms plotted by these simulations, especially for the
weaker peak in BootstrapIterativeFit2).
In those cases the standard deviation will be too high
because of the outliers, and the IQR/STD
ratio will be much less than the value of 1.34896 that is expected for a normal distribution. In that
case a better estimate of the standard deviation of the
central portion of the distribution (without the outliers)
is IQR/1.34896.
For the quantitative measurement of peaks, it's
instructive
to compare the iterative leastsquares method with
simpler, less computationallyintensive, methods. For example,
the measurement of the peak height of a single peak of uncertain
width and position could be done simply by taking the maximum of
the signal in that region. If the signal is noisy, a more
accurate peak height will be obtained if the signal is smoothed beforehand. But smoothing
can distort the signal and reduce peak heights. Using an
iterative peak fitting method, assuming only that the peak shape
is known, can give the best possible accuracy and precision,
without requiring smoothing even under high noise conditions,
e.g. when the signaltonoise ratio is 1, as in the demo
script SmoothVsFit.m:
True peak height = 1 NumTrials =
100 SmoothWidth = 50
Method Maximum
y Max Smoothed y Peakfit
Average peak height
3.65
0.96625 1.0165
Standard deviation 0.36395
0.10364 0.11571
If peak area is measured rather than peak height,
smoothing is unnecessary (unless to locate the peak beginning
and end) but peak fitting still yields the best precision. See SmoothVsFitArea.m.
It's
also instructive to compare the iterative leastsquares method
with classical
leastsquares curve fitting, discussed in the previous section, which can
also fit peaks in a signal. The difference is that in
the classical least squares method, the positions, widths, and
shapes of all the individual components are all known
beforehand; the only
unknowns are the amplitudes (e.g. peak heights) of the
components in the mixture. In nonlinear iterative curve
fitting, on the other hand, the positions, widths, and heights
of the peaks are all unknown
beforehand; the only
thing that is known is the fundamental underlying shape of the
peaks. It turns out that the nonlinear iterative curve
fitting is much more difficult to do and more prone to error,
but it's necessary if you need to track shifts in peak
position or width or to decompose a complex overlapping peak
signal into fundamental components knowing only their shape.
The Matlab/Octave script “CLSvsINLS.m”
compares the classical leastsquares (CLS) method with three
different variations of the iterative method (INLS) method for
measuring the peak heights of three Gaussian peaks in a noisy
test signal, demonstrating that the fewer the number of
unknown parameters, the faster and more accurate is the peak
height calculation.
Another comparison of multiple measurement techniques is presented in Case Study C.
Note: you
can rightclick on any of the mfile links on this page and
select Save Link As... to download them
to your computer, then place them in the Matlab path for use
within Matlab.
[Model errors] [Number of peaks] [Peak width] [Background correction] [Random noise] [Iterative fitting errors] [Exponential broadening] [Effect of smoothing]
Iterative curve fitting is often used to measure the
position, height, and width of peaks in a signal, especially
when they overlap significantly. There are four major sources
of error in measuring these peak parameters by iterative curve
fitting:
a. Model errors.
Peak shape. If you have the wrong model for your
peaks, the results can not be expected to be accurate; for
instance, if your actual peaks are Lorentzian in shape, but
you fit them with a Gaussian model, or vice versa. For example,
a single isolated Gaussian peak at x=5, with a height of 1.000
fits a Gaussian model virtually perfectly, using the Matlab
userdefined peakfit
function, as shown on the right. (The 5th input argument for
the peakfit function specifies the shape of peaks to be used
in the fit; "1" means Gaussian).
>>
x=[0:.1:10];y=exp((x5).^2);
>> [FitResults,MeanFitError]=peakfit([x' y'],5,10,1,1)
Peak#
Position
Height
Width
Area
FitResults =
1
5
1
1.6649
1.7724
MeanFitError =0.001679
The "FitResults" are, from left to right, peak number,
peak position, peak height, peak width, and peak area. The
MeanFitError, or just "fitting error", is the square root
of the sum of the squares of the differences between the data and the
bestfit model, as a percentage of the maximum signal in the
fitted region. (Note the good agreement of the area
measurement (1.7724) with the theoretical area
under the curve of exp(x^{2}), which is the square
root of p,
or about 1.7725...).
But this same peak, when fit with a Logistic model (peak shape number 3, above right), gives a fitting error of 1.4% and height and width errors of 3% and 6%, respectively:
>> [FitResults,MeanFitError]=peakfit([x'
y'],5,10,1,3)
Peak#
Position
Height
Width
Area
FitResults =
1
5.0002
0.96652
1.762
1.7419
MeanFitError =1.4095
When fit with a Lorentzian model
(peak shape 2, shown on the right), this peak gives a 6%
fitting error and height and width errors of 8% and 20%,
respectively.
>> [FitResults,MeanFitError]=peakfit([x'
y'],5,10,1,2)
FitResults =
Peak#
Position
Height
Width Area
1
5
1.0876
1.3139
2.0579
MeanFitError =5.7893
So clearly the larger the fitting
errors, the larger are the parameter errors, but the parameter
errors are of course not equal
to the fitting error (that would just be too easy). Also,
it's clear that the peak width
and area are
the parameters most susceptible to errors. The peak positions, as you can see
here, are measured accurately even if the model is way wrong,
as long as the peak is symmetrical and not highly overlapping
with other peaks.
If you do not know the
shape of your peaks, you can use use peakfit.m or ipf.m to try
different shapes to see if one of the standard shapes included
in those programs fits the data; try to find a peak in your
data that is typical, isolated, and that has a good
signaltonoise ratio. For example, the Matlab function ShapeTest.m creates a test signal
consisting of a single (asymmetrical) peak, adds random white
noise, fits it with six different candidate model peak shapes
using peakfit.m, plots each fit in a separate figure window,
and prints out a table of fitting errors in the command
window. In this particular case, the last two model shapes
almost equally well. That function has only six potential
candidate shape; the Matlab function peakfit.m has many more
builtin shapes to choose from, but still it is a finite list
and there is always the possibility that the actual underlying
peak shape is not available in the software you are using.
A good fit is not by itself
proof that the shape function you have chose is the correct
one; in some cases the wrong function can give a fit
that looks perfect. For example, this fit of a real data set
to a 5peak Gaussian model has a low percent fitting error and
the residuals look random  usually an indicator of a good
fit. But in fact in this case the model is wrong; that
data came from an experimental domain where the underlying
shape is fundamentally nonGaussian but in some cases can look
very like a Gaussian. As another example, a data set
consisting of peaks with a Voigt profile
peak shape can be fit with a weighted
sum of a Gaussian and a Lorentzian almost as well as an
with an actual Voigt model, even
though those models are not the same mathematically; the
difference in fitting error is so small that it would likely
be obscured by the random noise
if it were a real experimental signal. In such cases you can
not depend solely on the goodness of fit.
Number of peaks. Another
source
of model error occurs if you have the wrong number of peaks in your
model, for example if the signal actually has two peaks but
you try to fit it with only one peak. In the example below, a
line of Matlab code generates a simulated signal with of two
Gaussian peaks at x=4 and x=6 with peaks heights of 1.000 and
0.5000 respectively and widths of 1.665, plus random noise
with a standard deviation 5% of the height of the largest peak
(a signaltonoise ratio of 20):
>>x=[0:.1:10];y=exp((x6).^2)+.5*exp((x4).^2)+.05*randn(size(x));
In a real experiment you would not usually know the peak positions, heights, and widths; you would be using curve fitting to measure those parameters. Let's assume that, on the basis of previous experience or some preliminary trial fits, you have established that the optimum peak shape model is Gaussian, but you don't know for sure how many peaks are in this group. If you start out by fitting this signal with a singlepeak Gaussian model, you get:
>>
[FitResults,MeanFitError]=peakfit([x' y'],5,10,1,1)
FitResults
Peak#
Position Height
Width Area
1
5.5291 0.86396
2.9789 2.7392
MeanFitError = 10.467
The residual plot shows a "wavy"
structure that's visible in the random scatter of points due
to the random noise in the signal. This means that the fitting
error is not limited by the random noise; it is a clue that
the model is not quite right.
But a fit with two peaks yields much better results (The 4^{th} input argument for the peakfit function specifies the number of peaks to be used in the fit).
>> [FitResults,MeanFitError]=peakfit([x'
y'],5,10,2,1)
FitResults =
Peak# Position
Height Width Area
1
4.0165
0.50484
1.6982
0.91267
2
5.9932 1.0018
1.6652 1.7759
MeanFitError = 4.4635
Now the residuals have a random
scatter of points, as would be expected if the signal is
accurately fit except for the random noise. Moreover, the
fitting error is much lower (less that half) of the error with
only one peak. In fact, the fitting error is just about what
we would expect in this case based on the 5% random noise in
the signal (estimating the relative standard deviation of the
points in the baseline visible at the edges of the signal).
Because this is a simulation in which we know beforehand the
true values of the peak parameters (peaks at x=4 and x=6 with
peaks heights of 1.0 and 0.50 respectively and widths of
1.665), we can actually calculate the parameter errors (the
difference between the real peak positions, heights, and
widths and the measured values). Note that they are quite
accurate (in this case within about 1% relative on the peak
height and 2% on the widths), which is actually better than
the 5% random noise in this signal because of the averaging
effect of fitting to multiple data points in the signal.
But if going from one peak to two
peaks gave us a better fit, why not go to three peaks? If
there were no noise in the data, and if the underlying peak
shape were perfectly matched by the model, then the fitting
error would have already been essentially zero with two model
peaks, and adding a third peak to the model would yield a
vanishingly small height for that third peak. But in our
examples here, as in real data, there is always some random
noise, and the result is that the third peak height will not
be zero. Changing the number of peaks to three gives these
results:
>> [FitResults,MeanFitError]=peakfit([x'
y'],5,10,3,1)
FitResults =
Peak# Position
Height Width Area
1 4.0748
0.51617 1.7874 0.98212
2 6.7799
0.089595 2.0455 0.19507
3 5.9711
0.94455 1.53 1.5384
MeanFitError = 4.3878
The fitting algorithm has now tried to fit an additional lowamplitude peak (numbered peak 2 in this case) located at x=6.78. The fitting error is actually lower that for the 2peak fit, but only slightly lower, and the residuals are no less visually random that with a 2peak fit. So, knowing nothing else, a 3peak fit might be rejected on that basis alone. In fact, there is a serious downside to fitting more peaks than are actually present in the signal: it increases the parameter measurement errors of the peaks that are actually present. Again, we can prove this because we know beforehand the true values of the peak parameters: clearly the peak positions, heights, and widths of the two real peaks than are actually in the signal (peaks 1 and 3) are significantly less accurate that of the 2peak fit.
Moreover, if we repeat that fit
with the same signal
but with a different
sample of random noise (simulating a repeat measurement of a
stable experimental signal in the presence or random noise),
the additional third peak in the 3peak fit will bounce around
all over the place (because the third peak is actually fitting
the random noise,
not an actual peak in the signal).
>> x=[0:.1:10];
>>
y=exp((x6).^2)+.5*exp((x4).^2)+.05*randn(size(x));
>> [FitResults,MeanFitError]=peakfit([x' y'],5,10,3,1)
FitResults =
Peak#
Position Height
Width Area
1
4.115
0.44767
1.8768 0.89442
2
5.3118 0.09340
2.6986 0.26832
3
6.0681
0.91085
1.5116 1.4657
MeanFitError = 4.4089
With this new set of data, two of
the peaks (numbers 1 and 3) have roughly the same position,
height, and width, but peak number 2 has changed substantially
compared to the previous run. Now we have an even more
compelling reason to reject the 3peak model: the 3peak
solution is not stable.
And because this is a simulation in which we know the right
answers, we can verify that the accuracy of the peak heights
is substantially poorer (about 10% error) than expected with
this level of random noise in the signal (5%). If we were to
run a 2peak fit on the same new data, we get much better
measurements of the peak heights.
>> [FitResults,MeanFitError]=peakfit([x'
y'],5,10,2,1)
FitResults =
Peak# Position Height
Width
Area
1
4.1601
0.49981 1.9108 1.0167
2
6.0585
0.97557 1.548
1.6076
MeanFitError = 4.4113
If this is repeated several times,
it turns out that the peak parameters of the peaks at x=4
and x=6 are, on average, more accurately measured by the
2peak fit. In practice, the best way to evaluate a proposed
fitting model is to fit several repeat measurements
of the same signal (if that is practical experimentally) and
to compute the standard deviation of the peak parameter
values.
In real experimental work, of course, you usually don't know the right answers beforehand, so that's why it's important to use methods that work well when you do know. The real data example mentioned above was fit with a succession of 2, 3, 4 and 5 Gaussian models, until the residuals became random. Beyond that point, there is little to be gained by adding more peaks to the model. Another way to determine the minimum number of models peaks needed is to plot the fitting error vs the number of model peaks; the point at which the fitting error reaches a minimum, and increases afterward, would be the fit with the "ideal combination of having the best fit without excess/unnecessary terms" (http://en.wikipedia.org/wiki/Coefficient_of_determination#Adjusted_R2). The Matlab/Octave function testnumpeaks.m (R = testnumpeaks(x, y, peakshape, extra, NumTrials, MaxPeaks)) applies this idea by fitting the x,y data to a series of models of shape peakshape containing 1 to MaxPeaks model peaks. The correct number of peaks underlying peaks is either the fit with the lowest fitting error, or, if two or more fits have about the same fitting error, the fit with the least number of peaks. The Matlab/Octave demo script NumPeaksTest.m uses this function with noisy computergenerated signals containing a userselected 3, 4, 5 or 6 underlying peaks. With very noisy data, however, the technique is not always reliable.
Peak width constraints. Finally, there is one more thing that we can do that might improve the peak parameter measurement accuracy, and it concerns the peak widths. In all the above simulations, the basic assumption that all the peak parameters were unknown and independent of one another. In some types of measurements, however, the peak widths of each group of adjacent peaks are all expected to be equal to each other, on the basis of first principles or previous experiments. This is a common situation in analytical chemistry, especially in atomic spectroscopy and in chromatography, where the peak widths are determined largely by instrumental factors. In the current simulation, the true peaks widths are in fact both equal to 1.665, but all the results above show that the measured peak widths are close but not quite equal, due to random noise in the signal.
But we can introduce an equalwidth constraint
into the fit by using peak shape 6 (Equal width Gaussians) or
peak shape 7 (Equalwidth Lorentzians) in peakfit
version 1.4 and later. Using peak shape 6 on the same
set of data as the previous example:
>> [FitResults,MeanFitError]=peakfit([x'
y'],5,10,2,6)
FitResults =
Peak#
Position Height
Width Area
1 4.0293 0.52818
1.5666 0.8808
2 5.9965
1.0192 1.5666 1.6997
MeanFitError = 4.5588
This "equal width" fit forces all the peaks within one group
to have exactly the same width, but that width is determined
by the program from the data. The result is a slightly higher fitting
error (in this case 4.5 rather than 4.4), but  perhaps
surprisingly  the peak parameter measurements are usually more accurate and more reproducible (Specifically,
the relative standard deviations are on average lower for the
equalwidth fit than for an unconstrainedwidth fit to the
same data, assuming of course that the true underlying peak
widths are really equal). This is an exception to the general
expectation that lower fitting errors result in lower peak
parameter errors. It is an illustration of the general rule
that the more you know about the nature of your signals, and
the closer your chosen model adheres to that knowledge, the
better the results. In this case we knew that the peak
shape was Gaussian (although we could have verified that
choice by trying other candidate peaks shapes). We
determined that the number of peaks was two by inspecting the
residuals and fitting errors for 1, 2, and 3 peak
models. And then we introduced the constraint of equal
peak widths within each group of peaks (based on prior
knowledge of the experiment rather than on inspection of
residuals and fitting errors). Not every experiment yields
peaks of equal width, but when it does, it's better to make
use of that constraint.
Going one step beyond equal widths, in peakfit version 2.6 (and ipf 8.6) and later, you can also specify a fixedwidth Gaussian or Lorentzian shapes (shape numbers 11 and 12), in which the width of the peaks are known beforehand, but not necessarily equal, and are specified as a vector in input argument 10, one element for each peak, rather than being determined from the data as in the equalwidth fit above. Introducing this constraint onto the previous example, and supplying an almostaccurate width as the 10th input argument:
>> [FitResults,MeanFitError]=peakfit([x'
y'],0,0,2,11,0,0,0,0,[1.666 1.666])
FitResults =
Peak#
Position
Height
Width Area
1
3.9943
0.49537
1.666
0.8785
2
5.9924
0.98612
1.666
1.7488
MeanFitError = 4.8128
Comparing to the previous equalwidth fit, the fitting error is larger here (because there are fewer degrees of freedom to minimize the error), but the parameter errors, particularly the peaks heights, are more accurate because the width information provided in the input argument was more accurate (1.666) than the width determined by the equalwidth fit (1.5666). Again, not every experiment yields peaks of known width, but when it does, it's better to make use of that constraint.
Note that if the peak positions are also known,
and only the peak heights
are unknown, you don't even need to use the iterative fitting
method at all; you can use the much easier and faster
multilinear regression technique (“classical least squares”).
Here's a a
screen video (MorePeaksLowerFittingError.mp4)
of an experiment with a complex experimental signal in which
several different fits were performed using models from 4 to 9
variablewidth, equalwidth, and fixedwidth Gaussian peaks.
The fitting error gradually decreases from 11% to 1.4% as more
peaks are used, but is that really justified? If the objective
is simply to get a good fit, then do whatever it takes. But if
the objective is to extract some useful information from the
model peak parameters, then more specific knowledge about that
particular experiment is needed: how many peaks are really
expected; are the peak widths really expected to be
constrained? Note that in this particular case the
residuals (bottom panel) are never really random and
always have a distinct "wavy" character, suggesting that the
data may have been smoothed before curve fitting
(usually not a good idea). Thus there is a real possibility
that some of those 9 peaks are simply "fitting the noise", as
will be discussed in Appendix A.
b. Background
correction.
The peaks that are measured in many scientific instruments are
sometimes superimposed on a nonspecific background or
baseline. Ordinarily the experiment protocol is designed to
minimize the background or to compensate for the background,
for example by subtracting a "blank" signal from the signal of
an actual specimen. But even so there is often a residual
background that can not be eliminated completely
experimentally. The origin and shape of that background
depends on the specific measurement method, but often this
background is a broad, tilted, or curved shape, and the peaks
of interest are comparatively narrow features superimposed on
that background. The presence of the background has little
effect on the peak positions, but it is impossible to measure
the peak heights, width, and areas accurately unless something
is done to account for the background.
There are various methods
described in the literature for estimating and subtracting the
background in such cases. The simplest assumption is that the
background can be approximated as a simple function in the
local region of group of peaks being fit together, for example
as a constant (flat), straight line (linear) or curved line
(quadratic). This is the basis of the "autozero" modes in the
ipf.m,
iSignal.m, and iPeak.m functions,
which are selected by the T key to cycle thorough OFF,
linear, quadratic, and flat modes. In
the flat mode, a constant baseline is included in the
curve fitting calculation, as described above.
In linear mode, a straightline baseline connecting
the two ends of the signal segment in the upper panel will be
automatically subtracted. In quadratic mode, a
parabolic baseline is subtracted. In the last two modes, you
must adjust the pan and zoom controls to isolate the group of
overlapping peaks to be fit, so that the signal returns to the
local background at the left and right ends of the window.
Example of an
experimental chromatographic signal. From left to right, (1)
Raw data with peaks superimposed on a tilted baseline. One
group of peaks is selected using the the pan and zoom
controls, adjusted so that the signal returns to the local
background at the edges of the segment displayed in the upper
window; (2) The linear baseline is subtracted when the
autozero mode set to 1 in ipf.m by pressing the T key;
(3) Fit with a threepeak Gaussian model, activated by
pressing 3, G, F (3 peaks, Gaussian, Fit).
Alternatively, it may
be better to subtract the background from the entire
signal first, before further operations are performed. As
before, the simplest assumption is that the background is
piecewise linear, that is, can be approximated as a series of
small straight line segments. This is the basis of the
multiple point background subtraction mode in ipf.m,
iPeak.m, and in iSignal. The user enters the
number of points that is thought to be sufficient to define
the baseline, then clicks where the baseline is thought to be
along the entire length of the signal in the lower
wholesignal display (e.g. on the valleys between the
peaks). After the last point is clicked, the program
interpolates between the clicked points and subtracts the
piecewise linear background from the original signal.
From left to right, (1) Raw
data with peaks superimposed on baseline. (2) Background
subtracted from the entire signal using the multipoint
background subtraction function in iPeak.m (ipf.m and iSignal have the
same function).
Sometimes, even without an actual baseline
present, the peaks may overlap enough so that the signal never
return to the baseline, making it seem that there is a
baseline to be corrected. This can occur especially with peaks
shapes that have gradually sloping sides, such as the
Lorentzian, as shown in
this example. Curve fitting without baseline
correction will work in that case.
In some cases the background may
be modeled as a broad peak whose maximum falls outside of
the range of data acquired, as in the real data example on the
left. It may be possible to fit the offscreen peak simply by
including extra peak in the model to account for the baseline.
(Don't use the equalwidth shapes for this, because it's
likely that the background peak is broader than the measured
peaks). In the example on the left, there are three clear
peaks visible, superimposed on a tilted baseline. In
this case the signal was fit nicely with four, rather than
three, variablewidth Gaussians, with an error of only 1.3%.
The additional broad Gaussian, with a peak at x = 38.7,
serves as the baseline.
c.
Random noise in the signal.
Any experimental signal has a certain amount of random noise,
which means that the individual data points scatter randomly
above and below their mean values. The assumption is
ordinarily made that the scatter is equally above and below
the true signal, so that the longterm average approaches the
true mean value; the noise "averages to zero" as it is often
said. The practical problem is that any given recording of the
signal contains only one finite sample of the noise. If
another recording of the signal is made, it will contain
another independent sample of the noise. These noise samples
are not infinitely long and therefore do not represent the
true longterm nature of the noise. This presents two
problems: (1) an individual sample of the noise will not
"average to zero" and thus the parameters of the
bestfit model will not necessarily equal the true values, and
(2) the magnitude of the noise during one sample might not be
typical; the noise might have been randomly greater or smaller
than average during that time. This means that the
mathematical "propagation of error" methods, which seek to
estimate the likely error in the model parameters based on the
noise in the signal, will be subject to error (underestimating the error
if the noise happens to be lower
than average and overestimating
the errors if the noise happens to be larger than average).
A better way to estimate the parameter errors is to record multiple samples of the signal, fit each of those separately, compute the models parameters from each fit, and calculate the standard error of each parameter. But if that is not practical, it is possible to simulate such measurements by adding random noise to a model with known parameters, then fitting that simulated noisy signal to determine the parameters, then repeating the procedure over and over again with different sets of random noise. This is exactly what the script DemoPeakfit.m (which requires the peakfit.m function) does for simulated noisy peak signals such as those illustrated below. It's easy to demonstrate that, as expected, the average fitting error precision and the relative standard deviation of the parameters increases directly with the random noise level in the signal. But the precision and the accuracy of the measured parameters also depend on which parameter it is (peak positions are always measured more accurately than their heights, widths, and areas) and on the peak height and extent of peak overlap (the two leftmost peaks in this example are not only weaker but also more overlapped that the rightmost peak, and therefore exhibit poorer parameter measurements). In this example, the fitting error is 1.6% and the percent relative standard deviation of the parameters ranges from 0.05% for the peak position of the largest peak to 12% for the peak area of the smallest peak.
Overlap matters: The errors in the values of peak parameters measured by curve fitting depend not only on the characteristics of the peaks in question and the signaltonoise ratio, but also upon other peaks that are overlapping it. From left to right: (1) a single peak at x=100 with a peak height of 1.0 and width of 30 is fit with a Gaussian model, yielding a relative fit error of 4.9% and relative standard deviation of peak position, height, and width of 0.2%, 0.95%, and 1.5% , respectively. (2) The same peak, with the same noise level but with another peak overlapping it, reduces the relative fit error to 2.4% (because the addition of the second peak increases overall signal amplitude), but increases the relative standard deviation of peak position, height, and width to 0.84%, 5%, and 4%  a seemingly better fit, but with poorer precision for the first peak. (3) The addition of a third (nonoverlapping) peak reduces the fit error to 1.6% , but the relative standard deviation of peak position, height, and width of the first peak are still 0.8%, 5.8%, and 3.64%, about the same as with two peaks, because the third peak does not overlap the first one significantly.
If the average noise noise in the signal is not known or its probability distribution is uncertain, it is possible to use the bootstrap sampling method to estimate the uncertainty of the peak heights, positions, and widths, as described above. Version 8 of the keypress operated interactive version of ipf.m has added a function (activated by the 'v' key) that estimates the expected standard deviation of the peak parameters using this method.
One way to reduce the effect of
noise is to take more data. If the experiment makes it
possible to reduce the xaxis interval between points, or to
take multiple readings at each xaxis values, then the
resulting increase in the number of data points in each peak
should help reduce the effect of noise. As a
demonstration, using the script DemoPeakfit.m
to create a simulated overlapping peak signal like that shown
above right, it's possible to change the interval between x
values and thus the total number of data points in the signal.
With a noise level of 1% and 75 points in the signal, the
fitting error is 0.35 and the average parameter error is 0.8%.
With 300 points in the signal and the same noise level, the
fitting error is essentially the same, but the average
parameter error drops to 0.4%, suggesting that the accuracy of
the measured parameters varies inversely with the square root
of the number of data points in the peaks.
The figure
on the right illustrates the importance of sampling interval
and data density. You can download the data file "udx" in TXT format or in Matlab MAT format. The signal consists of two
Gaussian peaks, one located at x=50 and the second at x=150.
Both peaks have a peak height of 1.0 and a peak halfwidth of
10, and normallydistributed random white noise with a
standard deviation of 0.1 has been added to the entire signal.
The xaxis sampling interval, however, is different for the
two peaks; it's 0.1 for the first peak and 1.0 for the second
peak. This means that the first peak is characterized by
ten times more points that the second peak. When you fit
these peaks separately to a Gaussian model (e.g., using
peakfit.m or ipf.m), you will find that all the parameters of
the first peak are measured more accurately than the second,
even though the fitting error is not much different:
First
peak:
Second peak:
Percent Fitting Error =
7.6434% Percent Fitting Error = 8.8827%
Peak# Position
Height Width Peak#
Position Height Width
1
49.95 1.0049
10.111 1
149.64 1.0313 9.941
So far this discussion has applied
to white noise. But other noise colors have
different effects. Lowfrequency weighted (“pink”) noise has a
greater effect on the accuracy of peak parameters
measured by curve fitting, and, in a nice symmetry,
highfrequency “blue” noise has a smaller effect on
the accuracy of peak parameters that would be expected on the
basis of its standard deviation, because the information in a
smooth peak signal
is concentrated at low frequencies. An example of this
occurs when curve fitting is applied to a signal that has been
deconvoluted to remove a
broadening effect.
Sometime you may notice that the residuals in a curve fitting operation are structured into bands or lines rather than being completely random. This can occur if either the independent variable or the dependent variable is quantized into discrete steps rather than continuous. It may look strange, but it has little effect on the results as long as the random noise is larger than the steps.
d. Iterative fitting
errors.
Unlike multiple linear regression curve fitting, iterative
methods may not always converge on the exact same model
parameters each time the fit is repeated with slightly
different starting values (first guesses). The Interactive Peak Fitter
ipf.m makes it easy to test this, because it uses slightly
different starting values each time the signal is fit (by
pressing the F key in
ipf.m,
for example). Even better, by pressing the X key, the ipf.m function
silently computes 10 fits with different starting values and
takes the one with the lowest fitting error. A basic
assumption of any curve fitting operation is that the fitting
error (the rootmeansquare difference between the model and
the data) is minimized, the parameter errors (the difference
between the actual parameters and the parameters of the
bestfit model) will also be minimized. This is generally a
good assumption, as demonstrated by the graph to the
right, which shows typical percent parameters errors as a
function of fitting error for the leftmost peak in one sample
of the simulated signal generated by DemoPeakfit.m (shown in the
previous section). The variability of the fitting error here
is caused by random small variations in the first guesses,
rather than by random noise in the signal. In many
practical cases there is enough random noise in the signals
that the iterative fitting errors within one sample of the
signal are small compared to the random noise errors between
samples.
Remember that the variability in measured peak parameters from fit to fit of a single sample of the signal is not a good estimate of the precision or accuracy of those parameters, for the simple reason that those results represent only one sample of the signal, noise, and background. The sampletosample variations are likely to be much greater than the withinsample variations due to the iterative curve fitting. (In this case, a "sample" is a single recording of signal). To estimate the contribution of random noise to the variability in measured peak parameters when only a single sample if the signal is available, the bootstrap method can be used.
e. A difficult case. As
a dramatic example of the ideas in parts c and d, consider this
simulated example signal, consisting of two Gaussian peaks of
equal height = 1.00, overlapping closely enough so that their
sum is a single symmetrical peak that looks like a
single Gaussian.
If there were no noise in the signal, the peakfit.m or ipf.m routines could easily extract the two equal Gaussian components to an accuracy of 1 part in 1000.
>> peakfit([x y],5,19,2,1)
Peak# Position Height Width AreaBut in the presence of even a little noise (for example, 1% RSD), the results are uneven; one peak is almost always significantly higher than the other:
Peak#
Position Height
Width Area
1
4.4117
0.83282
1.61 1.43
2
5.4022
1.1486 1.734 2.12
The fit is stable with any one sample of noise (if peakfit.m was run again with slightly different starting values, for example by pressing the F key several times in ipf.m), so the problem is not iterative fitting errors caused by different starting values. The problem is the noise: although the signal is completely symmetrical, any particular sample of the noise is not perfectly symmetrical (e.g. the first half of the noise usually averages a slightly higher or lower than the second half, resulting in an asymmetrical fit result). The surprising thing is that the error in the peak heights are much larger (about 15% relative, on average) than the random noise in the data (1% in this example). So even though the fit looks good  the fitting error is low (less than 1%) and the residuals are random and unstructured  still the model parameters can still be very far off. If you were to make another measurement (i.e. generate another independent set of noise), the results would be different but still inaccurate (the first peak has an equal chance of being larger or smaller than the second). Unfortunately, the expected error is not accurately predicted by the bootstrap method, which seriously underestimates the standard deviation of the peak parameters with repeated measurements of independent signals (because a bootstrap subsample of asymmetrical noise is likely to remain asymmetrical). A Monte Carlo simulation would give a more reliable estimation of uncertainty in such cases.
Better results can be obtained in cases where the peak widths are expected to be equal, in which case you can use peak shape 6 (equalwidth Gaussian) instead of peak shape 1: peakfit([x y],5,19,2,6). It also helps to provide decent first guesses (start) and to set the number of trials (NumTrials) to a number above 1): peakfit([x,y],5,10,2,6,0,10,[4 2 5 2],0,0).
So, to sum up, we can make the following observations about
the accuracy of model parameters: (1) the parameter errors
depend on the accuracy of the model chosen and on number of
peaks; (2) the parameter errors are directly proportional to
the noise in the data (and worse for lowfrequency or pink
noise); (3) all else being equal, parameter errors are
proportional to the fitting error, but a model that fits the
underlying reality better, e.g. equal or fixed widths or
shapes) often gives lower parameter errors even if the fitting
error is larger; (4) the errors are typically least for peak
position and worse for peak width and area; (5) the errors
depend on the data density (number of independent data
points in the width of each peak) and on the extent of
peak overlap (the parameters of isolated peaks are
easier to measure than highly overlapped peaks); (6) if only a
single signal is available, the effect of noise on the
standard deviation of the peak parameters in many cases can be
predicted approximately by the bootstrap method, but
if the overlap of the peaks is too great, the error of the
parameter measurements can be much greater than predicted.
Appendix a:
Fitting signals that are subject to exponential
broadening.
DataMatrix2 (figure on the right) is a computergenerated test signal consisting of 16 symmetrical Gaussian peaks with random white noise added. The peaks occur in groups of 1, 2, or 3 overlapping peaks, but the peak maxima are located at exactly integer values of x from 300 to 3900 (on the 100's) and the peak widths are always exactly 60 units. The peak heights vary from 0.06 to 1.85. The standard deviation of the noise is 0.01. You can use this signal to test curvefitting programs and to determine the accuracy of their measurements of peak parameters. Rightclick and select "Save" to download this signal, put it in the Matlab path, then type "load DataMatrix2" at the command prompt to load it into the Matlab workspace.
DataMatrix3 (figure on the left) is a exponentially broadened version of DataMatrix2, with a "time constant" of 33 points on the xaxis. The result of the exponential broadening is that all the peaks in this signal are asymmetrical, their peak maxima are shifted to longer x values, and their peak heights are smaller and their peak widths are larger than the corresponding peaks in DataMatrix2. Also, the random noise is damped in this signal compared to the original and is no longer "white", as a consequence of the broadening. This type of effect is common in physical measurements and often arises from some physical or electrical effect in the measurement system that is apart from the fundamental peak characteristics. In such cases it is usually desirable to compensate for the effect of the broadening, either by deconvolution or by curve fitting, in an attempt to measure what the peak parameters would have been before the broadening (and also to measure the broadening itself). This can be done for Gaussian peaks that are exponentially broadened by using the "ExpGaussian" peak shape in peakfit.m and ipf.m (or the "ExpLorentzian", if the underlying peaks are Lorentzian). Rightclick and select "Save" to download this signal, put it in the Matlab path, then type "load DataMatrix3" to load it into the Matlab workspace.
The example illustrated on the
right focuses on the single isolated peak whose "true" peak position, height,
width, and area in the original unbroadened signal, are 2800, 0.52, 60, and 33.2 respectively. (The relative standard
deviation of the noise is 0.01/0.52=2%.) In the broadened
signal, the peak is visibly asymmetrical, the peak maximum
is shifted to larger x values, and it has a shorter height
and larger width, as demonstrated by the attempt to fit a
normal (symmetrical) Gaussian to the broadened peak. (The
peak area, on
the other hand, is not much effected by the
broadening).
>> load DataMatrix3
The
large "wavy" residual in the plot above is a tipoff that the
model is not quite right. Moreover, the fitting error (1.2%)
is larger than expected for a peak with a halfwidth of 60
points and a 2% noise RSD (approximately 2%/sqrt(60)=0.25%).
Fitting to an
exponentiallybroadened Gaussian (pictured on the right)
gives a much lower fitting error ("Percent error") and a
more nearly random residual plot. But the interesting
thing is that it also recovers the original peak
position, height, and width to an accuracy of a fraction
of 1%. In performing this fit, the time
constant ("extra") was experimentally determined from the
broadened signal by adjusting it with the A and Z keys to
give the lowest fitting error; that also results in a reasonably good measurement of the
broadening factor (32.6, vs the actual value of 33). Of
course, had the original signal been nosier, these
measurements would not be so accurate. Note: When using
peakshape 5 (exponentially broadened Gaussian) you have to
give it a reasonably good value for the time
constant ('extra'), the input argument right after the
peakshape number. If the value is too far off, the
fit may fail completely, returning all zeros. A little
trial and error suffice. (Also, peakfit.m
version 7 has a variable time constant exponentiallybroadened
Gaussian, shape number 31, that will measure the time
constant as an iterated variable).
Peak Shape = Exponentiallybroadened
Gaussian
Autozero ON
Number of peaks = 1
Extra = 32.6327
Fitted range = 2640
 2979.5 (339.5) (2809.75)
Percent Error =
0.21696
Peak# Position
Height
Width Area
1
2800.1302
0.51829906 60.086295 33.152429
Comparing the two methods, the exponentiallybroadened Gaussian
fit recovers all the underlying peak parameters quite
accurately:
Position 
Height 
Width 
Area 

Actual peak parameters 
2800 
0.52 
60 
33.2155 
Gaussian fit to broadened
signal 
2814.832 
0.45100549 
68.441262 
32.859436 
ExpGaussian fit to broadened signal  2800.1302  0.51829906 
60.086295 
33.152429 
Other peaks in the same signal,
under the broadening influence of the same time constant,
can be fit with similar settings, for example the set of
three overlapping peaks near x=2400. As before, the
peak positions are recovered almost exactly and even the
width measurements are reasonably accurate (1% or better).
If the exponential
broadening time constant is not the same for all
the peaks in the signal, for example if it gradually
increases for larger x values, then the time constant
setting can be optimized for each group of peaks.
The smaller fitting error
evident here is just a reflection of the larger peak
heights in this particular group of peaks  the noise is
the same everywhere in this signal.
Peak Shape = Exponentiallybroadened
Gaussian
Autozero OFF
Number of peaks = 3
Extra = 31.9071
Fitted range = 2206
 2646.5 (440.5) (2426.25)
Percent Error =
0.11659
Peak#
Position Height
Width Area
1 2300.2349 0.83255884
60.283214
53.422354
2 2400.1618 0.4882451 60.122977 31.24918
3
2500.3123
0.85404245
60.633532 55.124839
The residual plots in both of these examples still have some "wavy" character, rather than being completely random and "white". The exponential broadening smooths out any white noise in the original signal that is introduced before the exponential effect, acting as a lowpass filter in the time domain and resulting in a lowfrequency dominated "pink" noise, which is what remains in the residuals after the broadened peaks have been fit as well as possible. On the other hand, white noise that is introduced after the exponential effect would continue to appear white and random on the residuals. In real experimental data, both types of noise may be present in varying amounts.
One final caveat: peak asymmetry
similar to exponential broadening could possibly be the
result a pair of closelyspaced peaks of
different peak heights. In fact, a single
exponential broadened Gaussian peak can sometimes be fit
with two symmetrical
Gaussians to a fitting error at least as low as a single exponential broadened Gaussian
fit. This makes it hard to distinguish between these
two models on the basis of fitting error alone. However,
this can usually be decided by inspecting the other peaks
in the signal: in most experiments, exponential broadening
applies to every peak in the signal, and the broadening is
either constant or changes gradually over the length of
the signal. On the other hand, it is relatively unlikely
that every peak in the signal will be accompanied by a
smaller side peak that varies in this way. So, if a only
one or a few of the peaks exhibit asymmetry, and the
others are symmetrical, it's most likely that the asymmetry is due to closelyspaced peaks of
different peak heights. If all peaks have the same or similar asymmetry, it's more likely to
be a broadening factor that applies to the entire signal.
The two figures here provide an example from real
experimental data. On the left, three asymmetrical peaks
are each fit with two symmetrical Gaussians (six peaks
total). On the right, those three peaks are fit with one
exponentially broadened Gaussian each (three peaks total).
In this case, the three asymmetrical peaks all
have the same asymmetry and can be fit with the same time
constant ("extra"). Moreover, the fitting error is
slightly lower for the threepeak exponentially broadened fit.
Both of these observations argue for the threepeak exponentially broadened fit
rather than the sixpeak fit.
Note: if your peaks are trailing
off to the left, rather that to the right as in the above
examples, simply use a negative value for the
time constant (in ipf.n, press ShiftX and type a negative
values)
Appendix b: The Effect of Smoothing before leastsquares analysis.
In general, it is not advisable to smooth a signal before applying leastsquares fitting, because doing so might distort the signal, make it hard to evaluate the residuals properly, and bias the results of bootstrap sampling estimations of precision, causing it to underestimate the betweensignal variations in peak parameters. SmoothOptimization.m is a Matlab/Octave script that compares the effect of smoothing on the measurements of peak height of a Gaussian peak with a halfwidth of 166 points, plus white noise with a signaltonoise ratio (SNR) of 10, using three different methods: (a) simply taking the single point at the center of the peak as the peak height; (b) using the gaussfit method to fit the top half of the peak (see CurveFitting.html#Transforming), and (c) fitting the entire signal with a Gaussian using the iterative method. The results of 150 trials with independent white noise samples are shown on the left: a typical raw signal is shown in the upper left. The other three plots show the effect of the SNR of the measured peak height vs the smooth ratio (the ratio of the smooth width to the halfwidth of the peak) for those three measurement methods. The results show that the simple singlepoint measurement is indeed much improved by smoothing, as is expected; however, the optimum SNR (which improves by roughly the square root of the peak width of 166 points) is achieved only when the smooth ratio approaches 1.0, and that much smoothing distorts the peak shape significantly, reducing the peak height by about 40%. The curvefitting methods are much less effected by smoothing and the iterative method hardly at all. So the bottom line is that you should not smooth prior to curvefitting, because it will distort the peak and will not gain any significant SNR advantage. The only situations where it might be advantageous so smooth before fitting are when the noise in the signal is highfrequency weighted (i.e. "blue" noise), where lowpass filtering will make the peaks easier to see for the purpose of setting the staring points for an iterative fit, or if the signal is contaminated with highamplitude narrow spike artifacts, in which case a medianbased prefilter can remove the spikes without much change to the rest of the signal.