Advantages of the TFit method compared to conventional methods are: (a) much wider dynamic range; (b) greatly improved calibration linearity; (c) ability to operate under conditions that are optimized for signal-to-noise ratio rather than for Beer's Law ideality (e.g. small spectrometers with lower dispersion and larger slit widths).
Just like the multilinear regression (classical least squares) methods conventionally used in absorption spectroscopy, the Tfit method (a) requires an accurate reference spectrum of each analyte, (b) utilizes multi-wavelength data such as would be acquired on diode-array, Fourier transform, or automated scanning spectrometers, and (c) applies both to single-component and multi-component mixture analysis.
The disadvantages of the Tfit method are: (a) it is computationally more intensive than the multilinear regression methods (but, on a typical personal computer, calculations take only a fraction of a second, even for the analysis of a mixture of three components); (b) it requires knowledge of the instrument function, i.e., the slit function or the resolution function of an optical spectrometer (however, this is a fixed characteristic of the instrument and is easily measured beforehand by scanning the spectrum of a narrow atomic line source such as a hollow cathode lamp); and (c) it is an iterative method that can under unfavorable circumstances converge on a local optimum (but this is handled by proper selection of the starting values, based on preliminary approximations calculated by conventional methods).
Click here to download a self-contained demo m-file that works in recent versions of Matlab. (You can also download a ZIP file "TFit.zip" that contains both the newver version and the older one for Matlab 6.5) and you can also download it from the Matlab File Exchange. The following is a discription of the main functions and programs:
Typical use:
options = optimset('TolX',0.1);
absorbance=fminsearch(@fitM,start,options,yobsd,ReferenceSpectra,InstFunction,straylight);
where start is the first guess (or guesses) of the absorbance(s)
of the analyte(s); it's convenient to use the conventional log10(Izero/I) estimate of absorbance for start. The other arguments (described above) are passed on to FitM. In
this example, fminsearch
returns the value of absorbance that would have been measured in the
absence of stray light and polychromatic light errors (which is either
a single value or a vector of absorbances, if it is a
multi-component analysis). The absorbance can then be converted into
concentration by any of the usual calibration procedures (Beer's Law, external standards, standard addition, etc.)
Iterative least-squares methods are ordinarily considered to be more difficult and less reliable than multilinear regression methods,
and this can be true if there are more than one nonlinear variable that
must be iterated, especially if those variables are correlated.
However, in the TFit method, there is only one
iterated variable (absorbance) per measured component, and reasonable
first guesses are easily available from the conventional
single-wavelength absorbance calculation or multiwavelength regression
methods. As a
result, an iterative least-squares method works very well in this
case. Comparing the expression for absorbance given above for the
TFit method to that for the weighted regression method:
absorbance=([weight weight].*[Background ReferenceSpectra])\(-log10(yobsd).*weight)
you can see that, in addition to the ReferenceSpectra and observed transmission spectrum (yobsd), the TFit method also requires a measurement of the Instrument
function (spectral bandpass) and the stray light (which the linear regression methods assume to be zero), but these are
characteristics of the spectrometer and need be done only once for a
given spectrometer. Finally, although the TFit method does
make the computer work harder, the computation time on a typical
laboratory personal computer is only a fraction of a second (roughly 25
µsec per spectral data point per component analyzed), using Matlab as
the computational environment.
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Interactive explorer for the Tfit method, applied to the
measurememt of a
single component with a Lorentzian (or Gaussian) absorption peak, with controls that allow you
to adjust the
true absorbance (Peak A), spectral width of the absorption peak
(AbsWidth), spectral width of the instrument function (InstWidth),
stray light,
and the noise level (Noise) continuously while observing the effects
graphically and numerically. Simulates the effect of photon noise,
unabsorbed stray
light, and random background intensity shifts (light source flicker). Compares observed
absorbances by the single-wavelength, weighted multilinear
regression (sometimes called Classical Least Squares in the
chemometrics literature), and the TFit methods. To run this file, right-click TFitDemo.m click "Save link as...", save it in a folder in the Matlab path, then
type "TFitDemo" at the Matlab command prompt. With the figure
window topmost on the screen, press K to get a list of the keypress
functions. Version 2.1, November 2011, adds SNR calculation; W key to Switch between Transmission and Absorbance display. In the example shown on the left, the true peak absorbance is exactly 1.000, the absorption widths and instrument function widths are equal, the unabsorbed stray light is 0.5%., and the photon noise is 5%. The results below the graphs show that the TFit method gives a much more accurate measurement (1.0046) than the single-wavelength method (0.617) or weighted multilinear regression method (0.954). Note: The original version of this demo, which uses sliders, works only on Matlab 6.5, but I recommend that you use TFitDemo.m instead, as it is more up-to-date and includes the statistics and calibration curve. KEYBOARD COMMANDS Peak shape....Q Toggles between Gaussian and Lorentzian absorption peak shape True peak A...A/Z True absorbance of analyte at peak center, without instrumental broadening, stray light, or noise. AbsWidth......S/X Width of absorption peak SlitWidth.....D/C Width of instrument function (spectral bandpass) Straylight....F/V Fractional unabsorbed stray light. Noise.........G/B Random noise level Re-measure....Spacebar Re-measure signal Switch mode...W Switch between Transmission and Absorbance display Statistics....Tab Prints table of statistics of 50 repeats Cal. Curve....M Displays analytical calibration curve in Figure 2 Keys..........K Print this list of keyboard commands Why does the noise on the graph change if I change the instrument function (slit width or InstWidth)? In the most common type of absorption spectrometer, the spectrometer's spectral bandwidth (InstWidth) is changed by changing the slit width, which also effects the light intensity and thus the signal-to-noise ratio. Therefore, in all these programs, when you change InstWidth, the photon noise is automatically changed accordingly just as it would in a real spectrophotometer. |
True A SingleW SimpleR WeightR TFit |
Simple script that computes the statistics of the TFit method compared to
single- wavelength (SingleW), simple regression (SimpleR), and weighted
regression (WeightR) methods. Simulates photon noise, unabsorbed stray
light and random background intensity shifts. Estimates the precision
and accuracy of the four methods by repeating the calculations 50 times
with different random noise samples. Computes the mean, relative
percent standard deviation, and relative percent deviation from true
absorbance. Parameters are easily changed in lines 19 - 26. Results
are displayed in the MATLAB command window. In the sample output shown on the left, results for true absorbances of 0.001 and 100 are computed, demonstrating that the accuracy and the precision of the TFit method is superior to the other methods over a 10,000-fold range. This statistics function is included as a keypress command (Tab key) in TFitDemo.m. |
 |
Function that compares the analytical curves for single-wavelength, simple regression, weighted
regression, and the TFit method over any specified absorbance range
(specified by the vector “absorbancelist” in line 20). Simulates photon
noise, unabsorbed stray light and random background intensity shifts.
Plots a log-log scatter plot with each repeat measurement plotted as a
separate point (so you can see the scatter of points at low absorbances). The parameters can be changed in lines 20 - 27. In the sample result shown on the left, analytical curves for the four methods are computed over a 10,000-fold range, up to a peak absorbance of 100, demonstrating that the TFit method (shown by the green circles) is much more nearly linear over the whole range than the single-wavelength, simple regression, or weighted regression methods. This calibration curve function is included as a keypress command (M key) in TFitDemo.m. |
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The original version of this demo, which uses sliders, works only on Matlab 6.5, but you can also download the newer self-contained keyboard-operated version that works in recent versions of Matlab:
KEYBOARD COMMANDS
A1 A/Z Increase/decrease true absorbance of component 1
A2 S/X Increase/decrease true absorbance of component 2
A3 D/C Increase/decrease true absorbance of component 3
Sepn F/V Increase/decrease spectral separation of the components
InstWidth G/B Increase/decrease width of instrument function (spectral bandpass)
Noise H/N Increase/decrease random noise level when InstWidth = 1
Peak shape Q Toggles between Gaussian and Lorentzian absorption peak shape
Table Tab Print table of results
K Print this list of keyboard commands
Sample table of results (by pressing the Tab key):
---------------------------------------------------------------
True
Weighted TFit
Absorbance Regression
method
Component 1 3
2.06
3.001
Component 2 0.1
0.4316
0.09829
Component 3 5
2.464
4.998
I = Izero.*10^-(alpha*L*c)
where “Izero” is the intensity of the light incident on the sample, “alpha” is the absorption coefficient of the absorber, “L” is the distance that the light travels through the material (the path length), and “c” is the concentration of absorber in the sample. The variables I, Izero, and alpha are all functions of wavelength; L and c are scalar.
In conventional applications, measured values of I and Izero are used to compute the quantity called "absorbance", defined as
A = log10(Izero./I)
Absorbance is defined in this way so that, when you combine this definition with the Beer-Lambert law, you get:
A = alpha*L*c
So, absorbance is proportional to concentration, ideally, which simplifies analytical calibration. However, any real spectrometer has a finite spectral resolution, meaning that the light beam passing through the sample is not truly monochromatic, with the result that an intensity reading at one wavelength setting is actually an average over a small spectral interval. More exactly, what is actually measured is a convolution of the true spectrum of the absorber and the instrument function. If the absorption coefficient "alpha" varies over that interval, then the calculated absorbance will no longer be linearly proportional to concentration (this is called the “polychromicity” error). The effect is most noticeable at high absorbances. In practice, many instruments will become non-linear starting at an absorbance of 2 (~1% Transmission). As the absorbance increases, the effect of unabsorbed stray light and instrument noise also becomes more significant.
The theoretical best signal-to-noise ratio and absorbance precision for a photon-noise limited optical absorption instrument can be shown to be close to an absorbance of 1.0. However, if one attempts to arrange sample dilutions and absorption cell path lengths to obtain a working range centered on an absorbance of 1.0 , for example over the range .1 – 10, or 0.01 – 100, the measurments will obviously fail at the high end. (Clearly, the direct measurement of an absorbance of 100 is unthinkable, as it implies the measurement of light attenuation of 100 powers of ten - no real measurement system has a dynamic range even close to that). In practice, it is difficult to achieve an dynamic range even as high as 5 or 6 absorbance, so that much of the theoretically optimum absorbance range is actually unusable. (c.f. http://en.wikipedia.org/wiki/Absorbance). So, in conventional practice, greater sample diltuions and shorter path lengths are required to force the absorbance range to lower values, even if this means poorer signal-to-noise ratio and measurement precision at the low end.
It is true that the non-linearity caused by polychromicity can be reduced by operating the instrument at the highest resolution setting (reducing the instrumental slit width). However, this has a serious undesired side effect: in dispersive instruments, reducing the slit width to increase the spectral resolution degrades the signal-to-noise substantially. The optimum signal-to-noise ratio is often achieved when the resolution of the instrument approximately matches the width of the analyte absorption, but operating the instrument in that way would result in very substantial non-linearity over most of the absorbance range becasue of the “polychromicity” error.
The TFit method sidesteps the above problems by calculating the
absorbance in a completely different way: it starts with the reference
spectra (an accurate absorption spectrum for each analyte, also
required by the multilinear regression methods), normalizes them to
unit height, multiplies each by an adjustable coefficient, adds them
up, computes the antilog, and convolutes it with the
previously-measured slit function. The result, representing the
instrumentally broadened transmission spectrum, is compared to the
observed transmission spectrum. The coefficients (one for each unknown
component in the mixture) are adjusted by the program until the computed transmission
model is a least-squares best fit to the observed transmission
spectrum. The best-fit coefficients are then equal to the absorbances
determined under ideal optical conditions. Provision is also made to
compensate for unabsorbed stray light and changes in background
intensity (background absorption). These calculations are performed by
the function fitM, which is used as a fitting function for Matlab's iterative non-linear fitting function fminsearch. The TFit
method gives measurements of absorbance that are much closer to the
"true" peak absorbance (in the absence of stray light and polychromatic
light errors) and it allows linear and wide dynamic range measurements
to be made, even if the slit width of the instrument is increased to
optimize the signal-to-noise ratio.
Note:
It's important to understand that the use of the TFit method does not
guarantee a perfectly linear analytical curve at all absorbances,
despite the impression given by these simulations. The TFit
method simply removes the non-linearity caused by unabsorbed stray
light and the polychromatic light effect. Other sources of
non-linearity remain - in particular, chemical effects
such as photolysis, equilibrium shifts, temperature effects, pH effects, binding,
dimerization, polymerization, molecular phototropism, fluorescence,
etc.
Bottom line: The TFit method is based on the Beer-Lambert Law,
but it calculates the absorbance in a different way that does not
require the assumption that stray light and polychromatic radiation
effects are zero. Because it allows larger slit widths to be used, it
is possible to obtain greater signal-to-noise ratios, while still
achieving a much wider linear dynamic range than previous
methods.
Created October 03, 2006. Revised November, 2011.
Tom O'Haver
Professor Emeritus
Department of Chemistry and Biochemistry
The University of Maryland at College Park
toh@umd.edu
http://terpconnect.umd.edu/~toh/
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