64
S. Ghosh et al.
This method is implemented for the microstructure with 6000 γ precipitates to
generate a database contributing to the 12-dimensional parameter space of Y par in
Eq. (4). It has been further observed in [30, 47] that the parameter set can be further
reduced using the constraint n = n 1 = n 2 = n 3 with minimal loss in statistical
error. In addition, the principal axis half-lengths are replaced by the normalized
aspect ratios
a
b ,
b
c and the half-length of the intermediate principal axis b. These
modifications yield the re-parametrization:
ˆ
Y par =
x 0 , y 0 , z 0 , n,
a
b
,
b
c
, b, φ 1 , ,, φ 2
(9)
Optimally selected analytical forms for the probability density functions of each
parameter of ˆ
Y par are chosen and fit the experimental data. For example, the aspect
ratios are approximately fit to a shifted beta prime distribution function B (s) =
s −α−β (s−1) α−1
¯
B(α,β)
, where s corresponds to an aspect ratio, (α, β) are fitting parameters,
and ¯
B is the beta function, as shown in Fig. 4a. The shape exponent, on the other
hand, is fit to a log-normal distribution function f (n) =
1
(n−2)
√
2πσ
e
−
(ln(n−2)−μ) 2
2σ
with the origin shifted to n = 2, and (μ, σ ) are fitting parameters as shown in
Fig. 4b.
(a)
(b)
Fig. 4 (a) Comparison of the cumulative distribution of the evaluated aspect ratio
b
c for 6000
precipitates with the fitted shifted beta prime distribution and (b) probability distribution function
of the reduced shape exponent n for all 6000 precipitates and a maximum likelihood estimation of
a shifted log-normal distribution. (Reprinted from: Pinz et al. [30], with permission from Elsevier)
S. Ghosh et al.
This method is implemented for the microstructure with 6000 γ precipitates to
generate a database contributing to the 12-dimensional parameter space of Y par in
Eq. (4). It has been further observed in [30, 47] that the parameter set can be further
reduced using the constraint n = n 1 = n 2 = n 3 with minimal loss in statistical
error. In addition, the principal axis half-lengths are replaced by the normalized
aspect ratios
a
b ,
b
c and the half-length of the intermediate principal axis b. These
modifications yield the re-parametrization:
ˆ
Y par =
x 0 , y 0 , z 0 , n,
a
b
,
b
c
, b, φ 1 , ,, φ 2
(9)
Optimally selected analytical forms for the probability density functions of each
parameter of ˆ
Y par are chosen and fit the experimental data. For example, the aspect
ratios are approximately fit to a shifted beta prime distribution function B (s) =
s −α−β (s−1) α−1
¯
B(α,β)
, where s corresponds to an aspect ratio, (α, β) are fitting parameters,
and ¯
B is the beta function, as shown in Fig. 4a. The shape exponent, on the other
hand, is fit to a log-normal distribution function f (n) =
1
(n−2)
√
2πσ
e
−
(ln(n−2)−μ) 2
2σ
with the origin shifted to n = 2, and (μ, σ ) are fitting parameters as shown in
Fig. 4b.
(a)
(b)
Fig. 4 (a) Comparison of the cumulative distribution of the evaluated aspect ratio
b
c for 6000
precipitates with the fitted shifted beta prime distribution and (b) probability distribution function
of the reduced shape exponent n for all 6000 precipitates and a maximum likelihood estimation of
a shifted log-normal distribution. (Reprinted from: Pinz et al. [30], with permission from Elsevier)