Principles of Fluorescence Spectroscopy

PRINCIPLES OF FLUORESCENCE SPECTROSCOPY 131
In this expression the sum extends over the number (n) of
channels or datapoints used for a particular analysis and σ k
in the standard deviation of each datapoint.
In TCSPC it is straightforward to assign the standard
deviations (σ k ). From Poisson statistics the standard devia-
tion is known to be the square root of the number of photon
counts: . Hence, for a channel with 10,000
counts, σk = 100, and for 106 counts, σk = 1000. This relationship
between the standard deviation and the number of
photons is only true if there are no systematic errors and
counting statistics are the only source of uncertainty in the
data. If the data contains only Poisson noise then the relative
uncertainty in the data decreases as the number of photons
increases. The value of χ2 is the sum of the squares
deviations between the measured N(tk ) and expected values
Nc (tk ), each divided by squared deviations expected for the
number of detected photons.
It is informative to compare the numerator and denominator
in eq. 4.21 for a single datapoint. Assume a channel
contains 104 counts. Then the expected deviation for this
measurement is 100 counts. If the assumed model accounts
for the data, then the numerator and denominator of eq. 4.21
are both (102 ) 2 , and this datapoint contributes 1.0 to the
value of χ2 . In TCSPC, and also for frequency-domain
measurements, the number of datapoints is typically much
larger than the number of parameters. For random errors
and the correct model, χ2 is expected to be approximately
equal to the number of datapoints (channels).
Suppose the data are analyzed in terms of the multiexponential
model (eq. 4.8). During NLLS analysis the values
of αi and τi are varied until χ2 is a minimum, which
occurs when N(tk ) and Nc (tk ) are most closely matched. Several
mathematical methods are available for selecting how
αi and τi are changed after each iteration during NLLS fitting.
Some procedures work more efficiently than others,
but all seem to perform adequately. 185–186 These methods
include the Gauss-Newton, modified Gauss-Newton, and
Nelder-Mead algorithms. This procedure of fitting the data
according to eq. 4.21 is frequently referred to as deconvolution,
which is inaccurate. During analysis an assumed decay
law I(t) is convoluted with L(tk ), and the results are compared
with N(tk ). This procedure is more correctly called
iterative reconvolution.
It is not convenient to interpret the values of χ2 because
χ2 depends on the number of datapoints. 1 The value of χ2 will be larger for data sets with more datapoints. For this
reason one uses the value of reduced χ2 σk √N(tk) :
χ 2 R χ2
n p
χ2
ν
(4.22)
where n is the number of datapoints, p is the number of
floating parameters, and ν = n – p is the number of degrees
of freedom. For TCSPC the number of datapoints is typically
much larger than the number of parameters so that (n –
p) is approximately equal to n. If only random errors contribute
to χ R 2, then this value is expected to be near unity.
This is because the average χ 2 per datapoint should be about
one, and typically the number of datapoints (n) is much
larger than the number of parameters. If the model does not
fit, the individual values of χ 2 and χ R 2 are both larger than
expected for random errors.
The value of χ R 2 can be used to judge the goodness-offit.
When the experimental uncertainties σ k are known, then
the value of χ R 2 is expected to be close to unity. This is
because each datapoint is expected to contribute σ k 2 to χ 2 ,
the value of which is in turn normalized by the Σσ k 2, so the
ratio is expected to be near unity. If the model does not fit
the data, then χ R 2 will be significantly larger than unity.
Even though the values of χ R 2 are used to judge the fit, the
first step should be a visual comparison of the data and the
fitted function, and a visual examination of the residuals.
The residuals are the differences between the measured data
and the fitted function. If the data and fitted function are
grossly mismatched there may be a flaw in the program, the
program may be trapped in a local minimum far from the
correct parameter values, or the model may be incorrect.
When the data and fitted functions are closely but not perfectly
matched, it is tempting to accept a more complex
model when a simpler one is adequate. A small amount of
systematic error in the data can give the appearance that the
more complex model is needed. In this laboratory we rely
heavily on visual comparisons. If we cannot visually see a
fit is improved, then we are hesitant to accept the more
complex model.
4.9.3. Meaning of the Goodness-of-Fit
During analysis of the TCSPC data there are frequently two
or more fits to the data, each with a value of χ 2. R The value
of χ 2
R will usually decrease for the model with more
adjustable parameters. What elevation of χ 2
R is adequate to
reject a model? What decrease in χ 2
R is adequate to justify
accepting the model with more parameters? These questions
can be answered in two ways, based on experience
and based on mathematics. In mathematical terms one can