J. N. Shaumeyer, Matthew E. Briggs, and Robert W. Gammon, "Statistical fitting accuracy in photon correlation spectroscopy," Appl. Opt. 32, 3871-3879 (1993)
Continuing our experimental investigation of the fitting accuracy associated with photon correlation spectroscopy, we collect 150 correlograms of light scattered at 90° from a thermostated sample of 91-nm-diameter, polystyrene latex spheres in water. The correlograms are taken with two correlators: one with linearly spaced channels and one with geometrically spaced channels. Decay rates are extracted from the single-exponential correlograms with both nonlinear, least-squares fits and second-order cumulant fits. We make several statistical comparisons between the two fitting techniques and verify an earlier result that there is no sample-time dependence in the decay rate errors. We find, however, that the two fitting techniques give decay rates that differ by 1%.
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Polydispersity μ/Γ2 for All the Cumulant Fits of the Correlograms in Set 1, Grouped by Run Number (and Value of α)
1–10 (α = l)
11–20 (α = 2)
21–30 (α = 4)
31–40 (α = 8)
41–50 (α = 16)
−0.030
−0.044
−0.050
0.003
0.009
−0.033
0.005
−0.017
−0.038
0.073
0.014
−0.013
0.053
−0.013
0.039
−0.027
−0.069
0.004
0.119
−0.034
0.068
−0.046
0.064
−0.026
−0.024
0.062
0.017
−0.044
−0.007
−0.093
0.010
−0.037
0.025
0.055
−0.044
−0.039
−0.057
−0.015
0.126
0.096
0.059
0.045
−0.014
0.142
−0.011
0.082
−0.049
−0.002
0.133
0.076
Table 2
Average Values of Polydispersity μ/Γ2 with Associated Standard Deviations σ- for All Three Data Setsa
Run
Set 1
Set 2
Set 3
μ/Γ2
σ
μ/Γ2
σ
μ/Γ2
σ
1–10
0.017
0.045
−0.002
0.068
0.048
0.078
11–20
−0.025
0.035
−0.025
0.062
0.029
0.059
21–30
0.000
0.036
0.001
0.079
0.045
0.082
31–40
0.049
0.070
0.034
0.036
0.009
0.068
41–50
0.009
0.058
0.122
0.068
0.006
0.093
1–50
0.010
0.048
0.026
0.063
0.074
0.076
Results are given for 10-run averages (at each value of α for sets 1 and 2), and for the 50-run average for each set.
Table 3
Comparison of Estimators and Measured Estimators of the Error of the Mean Decay Rates δΓ/Γ, given in Percenta
Ensemble (s)
Predicted (%)
Nonlinear Fit (%)
Cumulant Fit (%)
Set 1
20
1.6
1.5
2.7
200
0.52
0.52
1.7
1000
0.23
0.22
0.38
Set 2
10
2.3
2.3
4.4
100
0.73
0.72
3.1
500
0.33
0.32
0.62
Set 3
10
2.3
1.5
2.4
100
0.73
0.22
0.74
500
0.33
0.22
0.35
The values in the Predicted column are calculated for the appropriate run time from Eq. (2). There are three ensembles to consider for each data set: the individual correlograms are drawn from the ensemble of the 20-s (set 1) or 10-s (sets 2 and 3) runs; the decay rates obtained from the 10-run averages at each α are drawn from the ensemble of 200-s (set 1) or 100-s (sets 2 and 3) runs; and the 50-run averages are drawn from either the ensemble of 1000-s (set 1) or 500-s (sets 2 and 3) runs. In this instance, the values given in the third row for each set are not independent, being derived from the first row by dividing by
; the values are included to indicate the precision of the measurements.
Table 4
Average Values of
, with Error Estimators for the Mean, for All Three Data Setsa
Runs
Set l
Set 2
Set 3
1–10
1632 ± 5
1638 ± 6
1641 ± 11
1641 ± 11
1653 ± 9
1679 ± 13
11–20
1637 ± 5
1624 ± 8
1651 ± 10
1638 ± 16
1645 ± 7
1659 ± 11
21–30
1633 ± 9
1633 ± 9
1637 ± 9
1640 ± 16
1647 ± 6
1678 ± 15
31–40
1647 ± 12
1696 ± 17
1655 ± 15
1682 ± 18
1652 ± 8
1657 ± 14
41–50
1651 ± 6
1656 ± 17
1625 ± 12
1758 ± 27
1649 ± 11
1653 ± 10
1–50
1640 ± 4
1650 ± 6
1642 ± 5
1672 ± 10
1649 ± 4
1665 ± 6
Results are given for 10-run averages (at each value of α for sets 1 and 2), and for the 50-run average for each set.
Tables (4)
Table 1
Polydispersity μ/Γ2 for All the Cumulant Fits of the Correlograms in Set 1, Grouped by Run Number (and Value of α)
1–10 (α = l)
11–20 (α = 2)
21–30 (α = 4)
31–40 (α = 8)
41–50 (α = 16)
−0.030
−0.044
−0.050
0.003
0.009
−0.033
0.005
−0.017
−0.038
0.073
0.014
−0.013
0.053
−0.013
0.039
−0.027
−0.069
0.004
0.119
−0.034
0.068
−0.046
0.064
−0.026
−0.024
0.062
0.017
−0.044
−0.007
−0.093
0.010
−0.037
0.025
0.055
−0.044
−0.039
−0.057
−0.015
0.126
0.096
0.059
0.045
−0.014
0.142
−0.011
0.082
−0.049
−0.002
0.133
0.076
Table 2
Average Values of Polydispersity μ/Γ2 with Associated Standard Deviations σ- for All Three Data Setsa
Run
Set 1
Set 2
Set 3
μ/Γ2
σ
μ/Γ2
σ
μ/Γ2
σ
1–10
0.017
0.045
−0.002
0.068
0.048
0.078
11–20
−0.025
0.035
−0.025
0.062
0.029
0.059
21–30
0.000
0.036
0.001
0.079
0.045
0.082
31–40
0.049
0.070
0.034
0.036
0.009
0.068
41–50
0.009
0.058
0.122
0.068
0.006
0.093
1–50
0.010
0.048
0.026
0.063
0.074
0.076
Results are given for 10-run averages (at each value of α for sets 1 and 2), and for the 50-run average for each set.
Table 3
Comparison of Estimators and Measured Estimators of the Error of the Mean Decay Rates δΓ/Γ, given in Percenta
Ensemble (s)
Predicted (%)
Nonlinear Fit (%)
Cumulant Fit (%)
Set 1
20
1.6
1.5
2.7
200
0.52
0.52
1.7
1000
0.23
0.22
0.38
Set 2
10
2.3
2.3
4.4
100
0.73
0.72
3.1
500
0.33
0.32
0.62
Set 3
10
2.3
1.5
2.4
100
0.73
0.22
0.74
500
0.33
0.22
0.35
The values in the Predicted column are calculated for the appropriate run time from Eq. (2). There are three ensembles to consider for each data set: the individual correlograms are drawn from the ensemble of the 20-s (set 1) or 10-s (sets 2 and 3) runs; the decay rates obtained from the 10-run averages at each α are drawn from the ensemble of 200-s (set 1) or 100-s (sets 2 and 3) runs; and the 50-run averages are drawn from either the ensemble of 1000-s (set 1) or 500-s (sets 2 and 3) runs. In this instance, the values given in the third row for each set are not independent, being derived from the first row by dividing by
; the values are included to indicate the precision of the measurements.
Table 4
Average Values of
, with Error Estimators for the Mean, for All Three Data Setsa
Runs
Set l
Set 2
Set 3
1–10
1632 ± 5
1638 ± 6
1641 ± 11
1641 ± 11
1653 ± 9
1679 ± 13
11–20
1637 ± 5
1624 ± 8
1651 ± 10
1638 ± 16
1645 ± 7
1659 ± 11
21–30
1633 ± 9
1633 ± 9
1637 ± 9
1640 ± 16
1647 ± 6
1678 ± 15
31–40
1647 ± 12
1696 ± 17
1655 ± 15
1682 ± 18
1652 ± 8
1657 ± 14
41–50
1651 ± 6
1656 ± 17
1625 ± 12
1758 ± 27
1649 ± 11
1653 ± 10
1–50
1640 ± 4
1650 ± 6
1642 ± 5
1672 ± 10
1649 ± 4
1665 ± 6
Results are given for 10-run averages (at each value of α for sets 1 and 2), and for the 50-run average for each set.