Advantages of the TFit method compared to conventional methods are: (a) 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 optical ideality.
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 (on a 3 Ghz Pentium Windows machine, calculations take about 70 microseconds per spectral data point for a three-component analysis); (b) it requires knowledge of the instrument function, i.e., the slit function or the resolution function of an optical spectrometer. (However, this is easily measured 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 the ZIP file "TFit.zip" that also includes supporting functions, self-contained demos to show how it works. You can also download it from the Matlab File Exchange. The following is a discription of the main functions and programs in this zip file:
Typical use:
lam=FMINSEARCH('fitM',start,options,yobsd,TrueSpectrum,InstFunction,straylight);
where start is the first guess (or guesses) and options = optimset('TolX',0.1);. The other arguments are passed on to FitM. In this example FMINSEARCH returns the value of the absorbance determined by the TFit method (or a vector of absorbances, if it is a multi-component analysis).
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Interactive demonstration of the Tfit method for a single Lorentzian absorbing peak, with sliders that allow you to adjust the true absorbance (Peak A), spectral width of the absorption peak (AbsWidth), width of the instrument function (InstWidth), stray light and the noise level (Noise) continuously while observing the effect dynamically. Simulates the effect of photon noise, unabsorbed stray light and random background intensity shifts. Compares observed absorbances computed by the single-wavelength, weighted multilinear regression (sometimes called Classical Least Squares in the chemometrics literature), and TFit methods. Requires Matlab 6.5. |
True A SingleW SimpleR WeightR TFit
MeanResult =
0.0010 0.0004 0.0005 0.0006 0.0010
PercentRelativeStandardDeviation =
1.0e+003 *
0.0000 1.0318 1.4230 0.0152 0.0140
PercentAccuracy =
0.0000 -60.1090 -45.1035 -38.6300 0.4898
MeanResult =
100.0000 2.0038 3.7013 57.1530 99.9967
PercentRelativeStandardDeviation =
0 0.2252 0.2318 0.0784 0.0682
PercentAccuracy =
0 -97.9962 -96.2987 -42.8470 -0.0033
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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 displayed in the MATLAB command window. |
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Comparison of analytical curves for single-wavelength, simple regression, weighted regression, and TFit methods 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. The parameters can be changed in lines 20 - 27. |
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Note: In the most common type of absorption spectrometer, the instrument width (InstWidth) is changed by changing the slit width, which also effects the light intensity and the signal-to-noise ratio. Therefore, in DemoTFit, Tfit3, TfitStats and TfitCalCurve, when you change InstWidth, the photon noise is automatically changed accordingly.
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 absorbance, defined as
A = log10(Izero./I)
Ideally, absorbance is proportional to concentration, which simplifies analytical calibration. However, any real spectrometer has a finite spectral resolution, meaning that an intensity reading at one wavelength setting is actually an average over a small spectral interval. Specifically, 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 precision for most instruments is in the range near 1.0 absorbance. Therefore it would seem that the optimum operating range would be centered on an absorbance of 1.0, for example .1 – 10, or 0.01 – 100. 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 close to that. In practice, it is extremely difficult to achieve an instrument 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).
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 width, InstWidth). However, this has a serious undesired side effect: in dispersive instruments, reducing the slit width to increase the resolution degrades the signal-to-noise substantially. The optimum signal-to-noise is often achieved when the resolution of the instrument approximately matches the width of the analyte absorption, but this would result in very substantial non-linearity over most of the absorbance range.
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 instrument 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 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 FMINSEARCH function.
October 03, 2006.
Tom O'Haver
Professor Emeritus
Department of Chemistry and Biochemistry
The University of Maryland at College Park
toh@umd.edu
http://www.wam.umd.edu/~toh