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S. Ghosh et al.
morphological parameters of the GSE is determined by a sampling error associated
with finite number of sampling from a distribution, whereas the convergence of
spatial microstructural statistics is calculated by generating 30 instantiations of
SEVMs with N p = 10, 50, 100, and 200 precipitates and subsequently comparing
their spatial distribution with those from experimental data.
2.4.1 Convergence of Morphological Distributions
Within the SEVM generation procedure framework, a small sample size may incur
large discrepancies between the sampled dataset and representative distributions.
This sampling error is reliably determined with respect to N p by the KS test
statistic D n , defined as the maximum absolute difference between two cumulative
distribution functions. By specifying the number of samples N p and the frequency
f b of observing a D n or greater, the number of required samples N p is solved for as:
f b =
√
2π
N p D n
∞
k=1
exp
− (2k − 1)
2 π 2
8N p D 2
n
(13)
The cases of f b = 0.5, 0.95 are displayed in Fig. 7a. At Np ≈ 50, 95% of all
sampled datasets have a KS statistic of less than 0.2, corresponding to the minimum
acceptable M-SERVE population for morphological distributions.
2.4.2 Convergence of Spatial Distributions
While morphological distributions can all be characterized similarly with the KS
test, the spatial distributions are unique, and each test may require a different error
metric. Therefore, each test is analyzed independently, and the largest required
minimum size is taken to be the M-SERVE for all spatial metrics. Where applicable,
a 2% error threshold is applied, and the resultant number of precipitates required for
that threshold are given.
For non-ellipsoidal precipitates the local volume fraction is calculated by
assigning γ matrix voxels to the nearest γ precipitate and then by dividing the
number of γ precipitate voxels by the total number of voxels assigned to that
precipitate. This method generates N p data-points per microstructure, which gives
rise to large variability across microstructural instantiations of the same statistics.
The ensemble statistics of its distribution is employed to estimate convergence.
The ensemble distribution of the local volume fraction is computed for the same
30 instantiations of SEVMs for N p = 10, 20, 50, 100, 200. Their cumulative
distribution functions are shown in Fig. 7d. The KS test values, computed between
distribution for N p = 200 and those for N p = 10, 20, 50, 100, are, respectively,
0.0743, 0.0493, 0.0224, 0.0213. For 2% error, the local volume fraction measure
requires N p > 100 to converge.
S. Ghosh et al.
morphological parameters of the GSE is determined by a sampling error associated
with finite number of sampling from a distribution, whereas the convergence of
spatial microstructural statistics is calculated by generating 30 instantiations of
SEVMs with N p = 10, 50, 100, and 200 precipitates and subsequently comparing
their spatial distribution with those from experimental data.
2.4.1 Convergence of Morphological Distributions
Within the SEVM generation procedure framework, a small sample size may incur
large discrepancies between the sampled dataset and representative distributions.
This sampling error is reliably determined with respect to N p by the KS test
statistic D n , defined as the maximum absolute difference between two cumulative
distribution functions. By specifying the number of samples N p and the frequency
f b of observing a D n or greater, the number of required samples N p is solved for as:
f b =
√
2π
N p D n
∞
k=1
exp
− (2k − 1)
2 π 2
8N p D 2
n
(13)
The cases of f b = 0.5, 0.95 are displayed in Fig. 7a. At Np ≈ 50, 95% of all
sampled datasets have a KS statistic of less than 0.2, corresponding to the minimum
acceptable M-SERVE population for morphological distributions.
2.4.2 Convergence of Spatial Distributions
While morphological distributions can all be characterized similarly with the KS
test, the spatial distributions are unique, and each test may require a different error
metric. Therefore, each test is analyzed independently, and the largest required
minimum size is taken to be the M-SERVE for all spatial metrics. Where applicable,
a 2% error threshold is applied, and the resultant number of precipitates required for
that threshold are given.
For non-ellipsoidal precipitates the local volume fraction is calculated by
assigning γ matrix voxels to the nearest γ precipitate and then by dividing the
number of γ precipitate voxels by the total number of voxels assigned to that
precipitate. This method generates N p data-points per microstructure, which gives
rise to large variability across microstructural instantiations of the same statistics.
The ensemble statistics of its distribution is employed to estimate convergence.
The ensemble distribution of the local volume fraction is computed for the same
30 instantiations of SEVMs for N p = 10, 20, 50, 100, 200. Their cumulative
distribution functions are shown in Fig. 7d. The KS test values, computed between
distribution for N p = 200 and those for N p = 10, 20, 50, 100, are, respectively,
0.0743, 0.0493, 0.0224, 0.0213. For 2% error, the local volume fraction measure
requires N p > 100 to converge.