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Introduction
where np is the number of data points in the measured decay curve and Wj is
a weighting factor. In the case of TCSPC methods, Wj follows Poisson statis-
tics [44] .
Because χ 2 depends on the number of data points (the larger the number of data
points the larger the value of χ 2 ), the reduced form of χ 2 is used. In this case:
χ 2 r = χ2
n − p
= χ2
ν
(1.18)
For a good fit, the value of χ 2 r is expected to be close to unity. This is because each
data point is expected to contribute to the deviation of χ 2 , which is normalised by
the summation term in Equation 1.17. Another diagnostic for the goodness of fit is
the autocorrelation function [45]. This is given by Equation 1.19 where Dk is the
deviation in the k th data point and Dk+j is the deviation in the (k + j) th data point.
AC(tj) =
1 m
m k=1 DkDk+j
1 n
n k=1 D2 k
(1.19)
Figure 1.15: Time domain intensity decay data for a dilute solution of HPTS dissolved
in 0.2M phosphate buffer. Top: Time-resolved decay trace (blue) and IRF
in red. The best fit to a mono-exponential decay function (black) is shown.
The recovered residuals are given below. Bottom: The recovered autocorrelation
function. λex = 405 nm, λem = 510 nm, τ = 5.4 ns, χ2 R = 1.01.
A Poisson probability distribution is appropriate for counting experiments where the data represent
the number of items or events observed per unit interval. For a Poisson distribution, the
standard deviation is given by : σj = N(tj) where σj is the standard deviation of each data
point, N(tj) is the measured data [47].
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