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 built-in demo fitdemo.m and its fitting function fitfun.m, which model the sum of two exponential decays. (To see this, just type "fitdemo" in the Matlab command window).
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 least-squares 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 non-linear 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"). This is easily extended to fitting two overlapping Gaussians in Demofitgauss2.m (shown on the left) using the same fitting function (which easily adapts to any number of peaks, depending on the length of the first-guess "start" vector). These functions call the user-defined peak shape function gaussian.m. Similar procedures can be defined for other peak shapes simply by calling the corresponding peak shape function, such as lorentzian.m. (Note: in order for scripts like Demofitgauss.m or Demofitgauss2.m to work on your version of Matlab, all the functions that they call must be loaded into Matlab beforehand, in this case fitgauss.m and gaussian.m).
You can create your own fitting functions for any purpose; they are not limited to single algebraic expressions, but can be arbitrarily complex multi-step algorithms. For example, in the TFit method for quantitative absorption spectroscopy, a model of the instrumentally-broadened transmission spectrum is fit to the observed transmission data, using Matlab's fminsearch function, in order to extend the dynamic range and calibration linearity of absorption spectroscopy beyond the normal limits, using a fitting function that performs Fourier convolution of the transmission spectrum with the known slit function of the spectrometer.
The uncertainty of the peak parameters determined by iterative least-squares fitting is readily estimated by the bootstrap sampling method. A simple demonstration of bootstrap estimation of the variability of an iterative least-squares 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 x-axis value at the peak maximum, TrueWidth is the true half-width (FWHM) of the peak, NumPoints is the number of points taken for the least-squares fit, Noise is the standard deviation of (normally-distributed) 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.
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 non-normal, 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 multi-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 deviaiton 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.
It's instructive to compare the iterative least-squares method with classical least-squares 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 non-linear 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
that the non-linear 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 least-squares (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 right-click on any of the m-file 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.
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 user-defined 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).
>> [FitResults,MeanFitError]=peakfit([x' y'],5,10,1,1)
Peak# Position Height Width Area
1 5 1 1.6649 1.7724
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 best-fit 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(-x2), which is the square root of p, or 1.7725...).
But this same peak, when fit with a Logistic model (peak shape number 3), gives a fitting error of 1.4% and height and width errors of 3% and 6%, respectively:
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)
Peak# Position Height Width Area
1 5 1.0876 1.3139 2.0579
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, clearly 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.
Number of peaks. Another
source of model error occurs if you have the wrong number of peaks in your
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 signal-to-noise ratio of 20):
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 single-peak Gaussian model, you get:
>> [FitResults,MeanFitError]=peakfit([x' y'],5,10,1,1)
Peak# Position Height Width Area
1 5.5291 0.86396 2.9789 2.7392
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
But a fit with two peaks yields much better results (The 4th input argument for the peakfit function specifies the number of peaks to be used in the fit).
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). We see 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
averaging effect of fitting to multiple data points in the
But if going from one peak to two peaks gave
us a better fit, why not go to three peaks? Changing the number of
peaks to three gives these results:
>> [FitResults,MeanFitError]=peakfit([x' y'],5,10,3,1)
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
The fitting algorithm has now tried to fit an additional low-amplitude peak (numbered peak 2 in this case) located at x=6.78. The fitting error is actually lower that for the 2-peak fit, but only slightly lower, and the residuals are no less visually random that with a 2-peak fit. So, knowing nothing else, a 3-peak fit might be rejected on that basis alone. In fact, there is a serious downside to fitting more peaks than are 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 2-peak fit.
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 3-peak 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).
>> [FitResults,MeanFitError]=peakfit([x' y'],5,10,3,1)
Peak# Position Height Width Area
1 4.115 0.44767 1.8768 0.89442
2 5.3118 0.093402 2.6986 0.26832
3 6.0681 0.91085 1.5116 1.4657
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 3-peak model: the 3-peak 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 2-peak fit on the same new data, we
get much better measurements of the peak heights.
>> [FitResults,MeanFitError]=peakfit([x' y'],5,10,2,1)
Peak# Position Height Width Area
1 4.1601 0.49981 1.9108 1.0167
2 6.0585 0.97557 1.548 1.6076
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 2-peak 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
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 equal-width constraint into the fit by
using peak shape 6 (Equal width Gaussians) or peak shape 7 (Equal-width 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)
Peak# Position Height Width Area
1 4.0293 0.52818 1.5666 0.8808
2 5.9965 1.0192 1.5666 1.6997
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 equal-width fit than for an unconstrained-width 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 fixed-width Gaussian or Lorentzian shapes (shape numbers 11 and 12), in which the width of the peaks are not only equal to each other but are known beforehand and are specified in input argument 10, rather than being determined from the data as in the equal-width fit above. Introducing this constraint onto the previous example, and supplying an almost-accurate width as the 10th input argument:
>> [FitResults,MeanFitError]=peakfit([x' y'],0,0,2,11,0,0,0,0,1.666)
Peak# Position Height Width Area
1 3.9943 0.49537 1.666 0.8785
2 5.9924 0.98612 1.666 1.7488
Comparing to the previous equal-width 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 equal-width 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”).
b. Background correction.
The peaks that are measured in many scientific instruments are sometimes superimposed on a non-specific background. 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
sophisticated methods described in the literature for estimating and
subtracting the background in such cases. The simplest assumption is
that the background is locally linear,
that is, can be approximated as a straight line in the local region of
group of peaks being fit together. This is easily accomplished if
each group of peaks is sufficiently isolated from other groups that the
signal returns to the local background between the groups of peaks, and
this is the basis of the "autozero" mode in the ipf.m and iPeak.m functions. When the autozero mode of these functions is turned on (T key), a straight-line baseline connecting
the two ends of the signal segment in the upper panel will be
automatically subtracted as the pan and zoom controls are used to isolate the group of overlapping peaks to be fit.
Example of an
experimental chromatographic signal. From
left to right, (1) Raw data with peaks superimposed on 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 is turned on in ipf.m;
Fit with a three-peak 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
futher 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 multi-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 whole-signal 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.
left to right, (1) Raw data with peaks superimposed on baseline. (2)
Background subtracted from the entire signal using the multi-point
background subtraction function in iPeak.m (ipf.m and iSignal have the same function).
In some cases the background may be
caused by a broad peak whose maximum falls outside of the range of data
acquired. This is not an uncommon situation with peaks shapes
that have gradually sloping sides, such as the Lorentzian. There is no
explicit provision for this in the current version of my programs;
however, if the background peak has the same shape as the measured
peaks, it is possible to fit the off-screen peak in ipf.m
simply by increasing the number of peaks by one. It may help to the
cursor-defined start position (C key) to define the start position for
the background peak beyond the x-axis range of the plot, but still
inside the Figure window. (Don't use the equal-width shapes for
this, because it's likely that the background peak is broader than the
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 long-term 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 long-term 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 best-fit 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 left-most peaks in this example are not only weaker but also more overlapped that the right-most 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.
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 signal-to-noise 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 if 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 peak further 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 x-axis interval between
points, or to take multiple readings at each x-axis 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.
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
half-width of 10, and normally-distributed random white noise with a
standard deviation of 0.1 has been added to the entire signal. The
x-axis sampling interval, however, is different for the two peaks; it's
0.1 for the first peaks and 1.0 for the second peak. This means
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 9.9415
d. Iterative fitting errors.
Unlike multiple linear regression curve fitting, iterative methods may not converge on the exact same model parameters each time the fit is repeated with slightly different starting values (first guesses). The Interactive Peak Fitter 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 RMS difference between the model and the data) is minimized, the parameter errors (the difference between the actual parameters and the parameters of the best-fit 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 left-most 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 sample-to-sample variations are likely to be much greater than the within-sample 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.
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 Area
But 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:
1 4.4117 0.83282 1.61 1.4273
2 5.4022 1.1486 1.734 2.12
The fit is stable with any one sample of noise (if peakfit was run again with slightly different starting values, or if the F key were pressed several times in ipf.m), so the probem is not iterative fitting errors casued 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 sub-sample 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 (equal-width 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 numbver ot trials (NumTrials) to a number above 1): peakfit([x,y],5,10,2,6,0,10,[4 2 5 2],0,0).
to sum up, we can make the following observations about the accuracy of
model parameters: (1) the parameter errors depend on the peak shape and
number of peaks; (2) the parameter errors
are directly proportional to the noise in the data and to the fitting
error; (3) the errors are
typically least for peak position and worse for peak width and area;
(4) 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); (5) 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.
|Actual peak parameters
|Gaussian fit to
|ExpGaussian fit tosignal||2800.1302||0.51829906