=Paper=
{{Paper
|id=Vol-2964/article_157
|storemode=property
|title=Bayesian-Inference-based Inverse Estimation of Small Angle Scattering
|pdfUrl=https://ceur-ws.org/Vol-2964/article_157.pdf
|volume=Vol-2964
|authors=Akinori Asahara,Hidekazu Morita,Kanta Ono,Masao Yano,Chiharu Mitsumata,Tetsuya Shoji,Kotaro Saito
|dblpUrl=https://dblp.org/rec/conf/aaaiss/AsaharaMOYMSS21
}}
==Bayesian-Inference-based Inverse Estimation of Small Angle Scattering==
https://ceur-ws.org/Vol-2964/article_157.pdf
Bayesian-Inference-based Inverse Estimation
of Small Angle Scattering
Akinori Asahara,1 Hidekazu Morita,1 Masao Yano,2 Tetsuya Shoji,2
Kanta Ono3 , Chiharu Mitsumata4 , Kotaro Saito5
1
Hitachi Ltd., 2 Toyota Motor Corporation, 3 High Energy Accelerator Research Organization
4
National Institute for Materials Science, 5 Paul Scherrer Institute
Abstract SAS pattern
y
As an application of machine-learning algorithms, we im- Incidental Beam
proved SAS (Small Angle Scattering), which is common 𝜃
experiment in material science, by developing a Bayesian
inference and deriving the confidence-level contour. In the
SAS experiment, the grain-size of the sample material has Material x
to be estimated from the distribution of the scattered beam.
A stochastic model and maximum-likelihood inference with Detector
EM-algorithm are often used, but the result is noisy due to plane
data noise. With the proposed method, the grain-size distri-
bution can be estimated similarly to the maximum-likelihood Figure 1: SAS Experiment
inference method and the confidence levels can be visualized.
Thus, researchers can determine estimation reliability and de-
cide whether there are sufficient data. Simulation-generated
tection events on a plane during SAS form a pattern on the
datasets were processed with the proposed method to evalu-
ate its effectiveness, and it was confirmed that it is useful for plane (called an SAS pattern), reflecting the features of the
automatic SAS data analysis. microstructures. Various particles, such as those of x-ray,
ion-beam, etc, can be applied to SAS.
One of the objectives of SAS is estimation of microscale-
Introduction grain-size distribution in material samples. Material science
Information technology for making material development researchers carefully observe SAS patterns to obtain grain-
faster, sometimes called Materials Informatics (MI), being size information about the microstructure of the sample ma-
investigated(National Institute of Standards and Technology terial.
2019). This technology will help researchers to extract new Therefore, several automatic estimation methods of grain-
knowledge of materials. size distribution with SAS pattern data have been proposed.
One of the usecases of the technology is to automatically One of the such methods, called Indirect Fourier Transfor-
find features of new materials characteristics from exper- mation (IFT), is a based on function optimization to fit the
imental data. Traditionally, researchers carefully inspected grain-size distribution to SAS pattern. However that requires
experimental data to find such features. However, it is time- lots of effort for parameter adjustment by material science
consuming and the researchers might miss such features. researchers. To reduce such effort, a maximum-likelihood
To solve this problem, methods known as ”data mining” (ML) inference (Asahara et al. 2020), is a stochastic method
are applicable to finding such features. With these methods, for machine learnings, was proposed. The ML inference
knowledge extraction from experimental data can be carried frees researchers from such effort due to the probabilistic
out automatically. Therefore, experiments are made faster. modeling of SAS experimental processes but its reliability
This paper focuses on small-angle scattering (SAS) (Hig- is insufficient. ML inference tends to fit to the noise caused
gins and Benoı̂t 1994)(Asahara et al. 2019). SAS is a scat- by observation because it is point-wise, that is, the result is
tering experiment for observing the microstructures of ma- only one certain parameter setting.
terials. In this experiment, the particle beam incident upon A method of Bayesian inference and a derivation of the
the sample interacts with the microstructures inside. An in- confidence level contour are proposed in this work. For
strument setting of SAS is illustrated in Fig. 1. The direc- Bayesian inference, parameters of the model for SAS are
tions of the particles change due to interactions with the mi- also stochastic, i.e., the probabilities of certain grain-size
crostructures of the materials. The number of particle de- distributions can be evaluated when an SAS pattern is ob-
Copyright © 2021, 2021 for this paper by its authors. Use permit- tained. Because all grain-size distributions derived from the
ted under Creative Commons License Attribution 4.0 International proposed method can be reliably considered as possible so-
(CC BY 4.0) lutions, the contour with a confidence level higher than the
Wave number (nm-1)
(a) low-q 1
I(r, q); however, this is difficult, relating to the phase prob-
0.1 1 10 100
lem(Feigin, Svergun et al. 1987). Thus, material science re-
0.1 (c) intermediate
searchers have tried to guess f (r) with clues from small fea-
Scattering amplitude
0.01
tures latent in the plot of I(r, q), as shown in Fig. 2. This fig-
0.001
ure presents a log-log plot of an SAS pattern, and its domain
0.0001
is separated into three parts (a), (b), and (c). In (a) (q → 0)
(b)high-q
0.00001
and (b) (q → ∞) , S(q) behaves linearly, being indepen-
0.000001
dent from r as shown the graph. Only in (c), I(r, q) oscil-
0.0000001
lates and it’s frequency depends on r. Material science re-
Figure 2: SAS pattern analysis with graphs searchers accordingly have to discover the fluctuation at (c)
because it gives implicit hints to determine f (r). Therefore,
f (r) is only roughly estimated. If f (r) is estimated directly,
given threshold can be visualized. the SAS could provide much more information of the sam-
ple.
Threfore, a method for automatic estimation of f (r) is
Problem Settings needed.
Small Angle Scattering
In SAS, particle beam incident upon the sample interacts Related Works
with the microstructures inside. The directions of the parti- Indirect Fourier Transformation
cles thus change due to the interactions. The angle θ between
a straight beam and the changed direction of the scattered Parametric function fitting is a known automatic grain-size
beam depends on the interaction. Finally, detectors arranged estimation method for SAS. With this method, parameters
on a plane detect the scattered beam. The number of detec- of the function f (r) are adjusted to fit to the obtained SAS
tion events form SAS pattern on the plane. pattern(Joachim and Ingo 2018). However, the form of f (r)
The particle behavior during SAS is stochastic, modeled is required and the true f (r) is generally unknown in actual
with a differential equation known as the Schödinger equa- situations.
tion. Because the distance between the sample and plane is To avoid such difficulty, a function having a more general
large enough, the coordinate values on the plane x = (x, y) formula should be used. One method using such a function
are approximately in proportion to θ. The probability den- is Indirect Fourier Transform (IFT) (Otto 1977). With IFT,
sity function (PDF) P (x) of detection corresponds to the weighted summation of multiple stepwise functions θn (x),
probability that a particle goes in the direction of θ, which is where θn (r) returns 1 when rn < r < rn+1 , and 0 other-
related to microscopic structures called grains. wise, is assumed as the formula. The integral of S(q) un-
Assume a simple case in which the grains are balls. Inten- der this assumption is decomposed into definite integrations
sity I(r, q) of an SAS pattern scattered with balls of radius which can be carried out analytically and reformed as a lin-
r (grain size) is proportional to I(r, q) as follows, ear combination of the weights, denoted an . After minimiz-
2 ing the difference between the linear combination of an and
the SAS pattern, f (r) is obtained as the sum of an θn (r).
1 sin qr r cos qr
I(r, q) ∝ I(r, q) = 3 − , (1) The resolution of f (r) is determined by θn with IFT, as
r q3 q2
shown above. Therefore, the range of θn should be small to
where the q indicates a quantity called wave number, which improve the resolution of f (r). However the higher resolu-
is the frequency of the wave function multiplied by 2π. The tion setting makes estimation error larger because more an s
frequency of the wave function is three dimensional because have to be determined when the range of θn is set smaller.
it is derived with the Fourier transformation of the wave The SAS pattern must be more accurate because the number
function in three dimensional space. The θ depends on the of detection events in the small range are few and sensitive
frequency, so the size of q = q along the vertical vector to to small errors.
the incident beam (”q = (qx , qy )” in Fig 1) appears in the A method for avoiding this problem is to add regulariza-
formula. Therefore, q indicates the location x on the detec- tion terms to suppress over fitting. However, the regulariza-
tion plane, derived from distance between the incident beam tion terms must be adjusted manually. To automate regular-
center and that location. That is, we can obtain actual SAS ization, complicated methods for determining the regular-
intensity corresponding to I(r, q) by converting x to q. ization terms have been proposed, but they are not in wide
An SAS pattern formed by multiple grain sizes is the use yet.
weighted sum of I(r, q) over r, and the weight is the grain-
size distribution f (r) of the material. Accordingly the scat- Maximum Likelihood Inference
tering pattern S(q) is derived as Another method is ML inference, often used for machine
Z learning. The SAS process is modeled as a stochastic pro-
S(q) ∝ f (r)I(r, q)dr. (2) cess with latent variables which indicates the r of particle
interaction. The likelihood derived from the stochastic pro-
To estimate f (r), S(q), which is the integration of cess is maximized to fit the SAS pattern. As a result, (r) is
f (r)I(r, q), should be decomposed to the summation of obtained as the optimal model parameter of the stochastic
detection events at qk in the SAS pattern. Grain-size param-
𝑟0 𝑞0
eters to be estimated are as denoted {π0 , · · · , πL }, where πi
𝑟1 𝑞1 indicates the ratio of ri grains to all grains and assumed pro-
𝑟2 𝐼(𝑟, 𝑞) 𝑞2 portional to f (r), which indicates the grain-size ratio. Ac-
source cordingly, πi posterior is maximized after obtaining the SAS
pattern, which is denoted as P ({πi }|{nk }).
𝑟𝑛 𝑞𝑘
The P ({πi }|{nk }) can be easily rewritable with Bayes
material detectors theorem:
P ({nk }|{πi })P ({πi })
Figure 3: Probabilistic solution of scattering problems P ({πi }|{nk }) = , (3)
P ({nk })
where P ({nk }) is a prior regarding the events related to the
process. No assumption is required for this method if a non- wavenumber qk . Since it is independent from f (r), P ({nk })
parametric model (that is, a very general stochastic model will be canceled with a normalization constant. Thus the
such as a Gaussian mixture) is applied for the SAS process. P ({nk }|{πi }) and the P ({πi }) should be handled carefully.
The expectaion-maximizedion (EM) algorithm(Bishop The P ({nk }|{πi }) is the posterior of the number of de-
2006) is a well known algorithm for non-parametric ML tection events after determining {πi }. Note the probabil-
inference (Zhang 1993)(Demoment 1989) (Nagata, Sugita, ity of one-particle detection at q after interaction with r
and Okada 2012). Similar methods are used in astrophysics is proportional to I(r, q) (as shown in (1)). Therefore, the
(William 1972) (Leon 1974), bioinformatics (Lustig et al. SAS process is modeled as an N -times P iteration of the ran-
2008) (Lustig, Donoho, and Pauly 2007) and compressed dom sampling (the probability is i πi I(ri , qk )). That is,
sensing (Donoho 2006). An application for grain-size esti- P({nk }|{πi }) is a multinomial function with the parameter
P
mation was also proposed(Asahara et al. 2020). The SAS i πi I(ri , qk ):
process is modeled as a combination of two random choice
X
P ({nk }|{πi }) = Mul({nk }; { I(ri , qk )πk }). (4)
processes for the EM algorithm. In the first process, the in- i
cident beam interacts with grains, and in second process, the
where ηjk = I(qj , rk ) is defined for ease of reference here-
incident beam changes its direction and arrives at a point on
after.
the detector plane. E-step to obtain the expectation value of
The P ({πi }) is a prior regarding r. Generally, a prior
interacting grains and M-step to obtain f (r) with ML are
is determined as the conjugate prior of the posterior (i.e.
iteratively carried out to derive the solution of the ML infer-
P ({nk }|{πi })). Because the conjugate prior of a multi-
ence.
nomial distribution is the Dirichlet distribution, the prior
The EM algorithm automatically derives f (r), though the
of P ({nk }|{πi }) should be similar to Dirichlet distribu-
result is noisy when a noisy SAS pattern is input. As shown
tion. Accordingly the parameter-transformed the Dirichlet
in Formula (1), the rate of event detection at high-q de- ˆ is defined for P ({nk }|{πi }) as the prior
creases in proportion to 1/q 4 . Thus, a long time to collect distribution Dir
detection events is required to obtain the detection events at P ({πi }):
!αk −1
higher q. An SAS pattern does not reflect probability at high- Y X
q when the experimental time is limited. This causes noisy ˆ
P ({πi }) = Dir({π k }; {αk }) ∝ π̂l ηlk , (5)
SAS patterns. K k
Such noise can be reduced by obtaining more detection where αk is a hyperparameter of Dirichlet distribution,
events; however, the cost is extremely high. Since the gen- which indicates knowlege obtained in advance. This leads
eration of incidental particle beam, such as a neutron beam, to
is costly, researchers have to save on beam time. Therefore P ({nk }|{πi })P ({πi })
estimation reliability should be evaluated, since the exper- X
iment should be finished as soon as a sufficient amount of = Mul({nk }; { ηik πk })D̂ir({πk }; {αk })
data is obtained. i
= D̂ir({πk }; {nk + αk }). (6)
Proposed Method For estimating {πi }, P ({πi }|{nk }) should be max-
MAP Inference imized with the formulation as P ({πi }|{nk }) ∝
P ({nk }|{πi })P ({πi }). The procedure to estimate this
Maximum A Posteriori (MAP) inference is Bayesian pa- is similar to that of the EM algorithm. For simplicity, the
rameter estimation method. With MAP inference, a PDF
of parameters (called a prior) is defined and revised after
logarithm of P the posterior is maximized by {πk } under
constraint πk = 1. Therefore, the maximization is
obtaining observations (the revised PDF is called a poste- carried out with the Lagrange multiplier method. Finally, by
rior). The parameter setting that gives the maximum poste- iterating the following formula until the convergence, {π̂lt }
rior is adopted as the inference result. The number of de- at t → ∞ is obtained as f (r).
tection events in an SAS pattern is denoted as K integers:
{n0 , · · · nK }, where each integer corresponds to the particle
X (nk + α0 ) π̂ t η
π̂lt+1 = P P l tl,k . (7)
detector for each wavenumber. That is, nk is the number of (nk + α0 ) j π̂j ηj,k
k
P
Algorithm 1: MAP inference of grain size be merged with ηl,k as j Alj ηl,k where Alj is the smooth-
P
Input: SAS pattern intensity nk ≥ 0, wavenumber qk ≥ 0 ing matrix. This change indicates that l Alj πl not {πl }
(k = 0, 1, · · · , K) corresponds to r. In this setting, {πl } corresponds to the
resolution of grain size rl ≥ 0 where (l = 0, 1, · · · , L) weight of a component such as a Gaussian packet. That is,
Output: {πl } f (r) is represented as the combination of these components.
N ⇐ k (nk + αk ), {ηlk } ⇐ { PI(rI(r l ,qk )
P
},
m l ,qk ) Experiments
{πl } ⇐ 1/L
repeat Experimental Settings
+αk Pπl ηlk
{πl } ⇐ k nkN
P
We conducted an experiment to evaluate whether the pro-
j πj ηjk
until convergence posed method can be used to estimate f (r) consistent with
an SAS pattern. In the experiment, simulation-generated
SAS pattern datasets were processed to compare the results
with the ground truth.
The algorithm of MAP inference is shown in Algorithm 1. The datasets were processed with the proposed method,
and ML inference method by the EM algorithm for com-
Uncertainty of Parameters parison. Ten thousands iterations of the MAP inference and
To evaluate the confidence level of the inference, the accu- the EM algorithm were carried out instead of checking con-
mulation of the probability around the MAP-inference result vergence, to simulate the situation in which the processing
should be evaluated. The following pl is the accumulation time is limited during the SAS experiment. For the proposed
from the MAP inference result π̂l to π̂l + δ. method, a Gaussian-smoothing matrix is multiplied by ηl,k ;
Z π̂l +δ Z Z therefore, the result was expected to be smooth and the con-
pl = dπl dπ0 · · · dπN P ({πi }|{nqk }) (8) fidence level contour to be consistent with it.
π̂l In the experiment, three types of diffrent f (r) were de-
From this definition, the contour can be visualized with δ de- fined. Each pattern is one gamma distribution or the sum of
termined by pl , for instance, to visualize the 95% confidence three Gamma distributions having the most frequent point
level contour, δ is determined P to satisfy pl = 0.95. around 10nm. The f (r) was discretized by 0.05 nm, and its
To derive δ, the constraint πl = 1 should be satisfied domain is set from 0 to 20 nm (i.e., 400 values), correspond-
during integration. Therefore, integration is difficult to be ing to f (r) in (2).
carried out. For the problem of the difficulty, an approxi- To obtain the SAS patterns, random sampling was car-
mation is introduced. In P ({πi }|{nqk }), i.e., the Dirichlet ried out. The detection-event number was set to 10,000, and
distribution, the effect of {πi } changes exponentially. There- the SAS patterns of the f (r)s were generated. First, q’s do-
fore, the contribution from {πi } far from the MAP inference main, which is from 0.1nm−1 to 10nm−1 , was discretized
result can be ignored because it decays exponentially. into 200 lots denoted as qk . The S(qk ) was calculated by
Consequently πi s are fixed to the following π̃, which is evaluating the integration of (2). Random sampling along
near from {π̂i }, instead of integration. S(qk ), i.e. the probability of detection, was carried out to
simulate particle-detection events and the event number was
π̂i counted to generate SAS patterns.
π̃i = P × (1 − (π̂l + δ)). (9)
i6=l π̂i
A computer with Intel(R) Core(TM) i3-4150 3.50GHz
CPU and 11 GB RAM and Cent OS was used for the exper-
With the π̃i , iment. The implementation was based on Python 3.6.5, and
Z π̂l +δ numpy library (Oliphant 2006) was used to improve the effi-
pl ' P (π̃0 , · · · , πl · · · π̃L |{nqk })dπl . (10) ciency of the calculation. Each calculation time lasted about
π̂l 1 minute, which was short enough for carrying out before
The procedure to calculate pl is simple: iter- SAS is finished.
ating πl is shifted by a very small value, and Settings for the proposed method is as follows: αk , which
P (π̃0 , · · · , πl · · · π̃L |{nqk }) is added to pl with re- is a hyperparameter used in the proposed method, was 1.0;
2
calculated π̃i until pl becomes higher than the threshold.
r −r
Gaussian-smoothing matrix was Aij = exp − 12 i0.5 j ;
Smoothing, which is used to reduce noise, should be taken
into account by the uncertainty calculation. Smoothing to re- the threshold of the confidence contour was 95%.
move noise from the estimation result involves multiply a fil-
ter matrix by πi . A Gaussian filter is often used for this pur- Experimental Results
pose. However, as discussed above, since uncertainty is es- Figure 4 shows the results. Figure 4 (a) plots the SAS pattern
timated based on probability, non-stochastic smoothing may by log-log plot, Fig. 4 (b) plots f (r) estimated with MAP
make conflicts with the MAP inference result, e.g., negative inference, and Fig.4 (c) plots the f (r) estimated with ML
πl . inference for comparison. The blue lines in Fig. 4 (a) plot
To avoid such conflicts, smoothing should be done with a 10000 × S(qk ) and orange points shows SAS pattern gen-
stochastic model. Remember ηl,k is multiplied by {πi } in- erated with the S(qk ); The black lines in Figs. 4 (b) and (c)
side P ({πi }|{nqk }). Therefore the matrix for smoothing can plot the truth, i.e. the original f (r); the red lines in Figs.
(a) Input (b) MAP results (c) ML results
Pattern 1
Relative frequency
Relative frequency
number of events
Wave number (nm-1) Grain size (nm) Grain size (nm)
Pattern 2
Relative frequency
number of events
Relative frequency
Wave number (nm-1) Grain size (nm) Grain size (nm)
Pattern 3
Relative frequency
number of events
Relative frequency
Wave number (nm-1) Grain size (nm) Grain size (nm)
Figure 4: Results of Pattern 1, 2, 3
(b) and (c) plot the estimation results; and red areas indicate q. As shown in Fig. 4 (a), the number of events at high q
95% confidence. SAS pattern is almost zero or one because S(q) in the high
The f (r) of Pattern 1 has only one peak at the center of q is very small. The high q corresponds to the low r com-
the q-range. In Fig. 4 (b), 95% confidence level area indi- ponent of the distribution because the low r corresponds to
cates that r around the center is quite uncertain. As shown a large wave number due to I(q, r). Therefore, the results
in Fig. 4 (c), the SAS pattern of Pattern 1 is so noisy that in Fig. 4 (c) oscillate due to the loss of the high-frequency
the ML inference results become noisy. The MAP inference component.
results are better and the confidence-level area can be drawn One problem with MAP inference is the behavior at the
without conflicts. low-r region. The estimated fluctuation is small in spite
The f (r) of Pattern 2 has a peak at lower q. In Fig. 4 (b), of the estimated f (r) being inaccurate. Because I(r, q) is
the 95% confidence level area around the peak is extremely small when both r and q are low, a high q is considered to
high. The MAP inference curve (red curve) differs from the contribute to this behavior. As mentioned above, the obser-
truth curve (black line). However, at low q, the truth curve is vations at a high q are extremely low. Because sparseness
outside the 95% confidence level area, showing that the 95% might cause inconsistent visualization, more experiments
confidence level area is not perfect. are required to specify this cause.
The f (r) of Pattern 3 has three peaks. As shown in Fig. Figure 5 shows the results with ML inference when SAS
4 (c), it is difficult to observe the three peaks from the ML Pattern3 is ideal, i.e. SAS pattern which is completely in
inference results. The three peaks from the MAP inference proportion to S(q). This shows that the f (r) can be recon-
results in Fig. 4 (b) are more readable than those from the structed if SAS pattern is perfect. Because the low q of Fig.
ML inference results Fig..4 (c). Though the MAP inference 4 (a) is quite similar to S(q), the difference is considered to
curve differs from the truth curve, most of the truth curve is ogirin from the high q loss.
inside the 95% confidence level area, except for low q. Figure 6 shows the results without a Gaussian filter. The
confidence level contour is also consistent with the MAP in-
Discussion ference results. Similarly to that with the Gaussian filter, the
All results in Fig. 4 (b) are similar to tge truth. In contrast, truth is within the 95% confidence level area, except for low
the results in Fig. 4 (c), results include noise. The 95% con- r. The results agree but difficult to read due to oscillation.
fidence level area works well, but there are conflicts in low From these results, MAP inference improves the f (r) es-
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Relative frequency
tion: overview of common estimation structures and prob-
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