This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.
PLEASE CITE THIS ARTICLE AS DOI:10.1063/5.0071166

Bayesian sparse modeling of extended X-ray absorption fine structure to
determine interstitial oxygen positions in yttrium oxyhydride epitaxial thin
film
Hiroyuki Kumazoe,1, a) Yasuhiko Igarashi,2, 3 Fabio Iesari,4, 5 Ryota Shimizu,6, 3 Yuya Komatsu,6 Taro Hitosugi,6Daiju Matsumura,7 Hiroyuki Saitoh,8 Kazunori Iwamitsu,9 Toshihiro Okajima,4 Yoshiki Seno,5 Masato Okada,10, 11
and Ichiro Akai1, b)
1)Institute of Industrial Nanomaterials, Kumamoto University, Kumamoto 860-8555,
Japan
2)
Faculty of Engineering, Information and Systems, University of Tsukuba, Tsukuba 305-8573,
Japan
3)
Japan Science and Technology Agency, PRESTO, Saitama 332-0012, Japan
4)
Aichi Synchrotron Radiation Center, Aichi 489-0965, Japan
5)
Kyushu Synchrotron Light Research Center, Saga 841-0005, Japan
6)
School of Materials and Chemical Technology, Tokyo Institute of Technology, Tokyo 152-8552,
Japan
7)
Materials Sciences Research Center, Japan Atomic Energy Agency, Hyogo 679-5148,
Japan
8)
Quantum Beam Science Research Directorate, National Institute for Quantum and Radiological Science andTechnology, Hyogo 679-5148, Japan
9)
Technical Division, Kumamoto University, Kumamoto 860-8555, Japan10)
Department of Complexity Science and Engineering, The University of Tokyo, Chiba 277-8561,
Japan
11)
Research and Services Division of Materials Data and Integrated System, National Institute for Materials Science,
Ibaraki, 305-0047, Japan(Dated: 4 November 2021)

This letter presents a Bayesian sparse modeling method to analyze extended X-ray absorption fine structure(EXAFS) data with basis functions built on two-body signals. This method does not require any structuralmodel and allows us to evaluate regression coefficients proportional to the radial distribution functions of
the respective elements and their errors and is very effective for analysis of EXAFS with weak absorption
intensity and severe signal-to-noise ratio. As an application example, we used it to analyze EXAFS ofan yttrium oxyhydride (YOx Hy ) epitaxial thin film. This EXAFS data shows weak absorption intensity
and its signal-to-noise ratio becomes severe due to the small amount of X-ray absorption in the thin film
sample. However, this approach revealed that the radial distance ratio of the second neighbor yttrium to thefirst neighbor oxygen coincides with that of a tetrahedral configuration. This result demonstrates that theinterstitial oxygen position is tetrahedral in the YOx Hy thin film.
The extended X-ray absorption fine structure (EXAFS)1,2 is a primary method to study atomic-scale geometry around an element specified by the energy of the
X-ray absorption edge. While the peak-fitting approachand the regularization method3,4 have often been used toanalyze EXAFS, deep-learning applications5,6 to EXAFS
would afford the key to improve the study of atomic-scale
geometry. However, there is another alternative machinelearning approach for measured EXAFS data analyses
to avoid overfitting that attempts to reproduce even thenoise. Although, of course, efforts for reducing measurement noise are important, the overfitting becomes a serious problem especially when the absorption intensity is
weak, such as in thin films.
In this letter, a unified method of sparse modeling(SpM)7–9 and Bayesian inference (BI)10–12 is proposed
to analyze EXAFS data.

a) kumazoe@kumamoto-u.ac.jpb) iakai@kumamoto-u.ac.jp

Our method has the following advantages: (1) only theelemental species information is required for the analysis, but no structural information is needed; (2) the phaseshift of the electron wave due to back-scattering is incorporated, so that the radial distance can be accurately
estimated; (3) a model for k-dependent noise (due to k w
weight) is introduced, so that the noise intensity, which
varies with k, can be properly evaluated; and (4) as aresult, even EXAFS signals with a severe signal-to-noise
ratio can be analyzed with high noise immunity.
We apply this method onto the EXAFS data measured
on an yttrium oxyhydride YOx Hy epitaxial thin film inorder to determine the atomic-scale structure of the oxygen (O) atom around yttrium (Y) atom. This material ispromising for switchable optical properties such as smart
windows and optical sensors13,14 because it exhibits a reversible light-induced insulator-to-metal transition. In
addition, by fabricating thin films, the photoresponsive
character is expected to be enhanced than that of polycrystalline thin films. Recent studies have shown that Y
atom in YOx Hy thin films belongs to a face-centered cu-

2
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0.1
0.0
-0.1
-0.2

-1

k (Å )

tion:

1.0
0.0

k

2

Analysis region Noise region

c

k

3

-3

(Å )

-1.0
20
10

c

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-2

(Å )

c

χα (k) =

0

-10
0

2

4

6
-1
k (Å )

8

10

FIG. 1. Measured EXAFS data of YOx Hy . Blue and redareas indicate the regions for analysis and noise estimation,
respectively.

bic (fcc) lattice15 and its lattice constant takes 5.34 Å16
or 5.35 Å15 . It is also reported that the oxygen coordinates around Gd atom for gadolinium oxyhydride thinfilms17 . However, how the oxygen coordinates around Y
atom for YOx Hy thin films remain elusive.
Epitaxial thin films of YOx Hy (111)/CaF2 (111) substrates were grown using reactive magnetron sputtering.
Details for sample preparation are provided in Ref. 18.
Yttrium K-edge EXAFS measurements were performedat room temperature in air at BL14B1 in SPring-8. A
Si (311) double-crystal monochromator and a 36-element
solid-state detector were used for EXAFS observation
with fluorescence mode. Rh-coated double mirrors wereapplied for cutting higher-order reflection in incident Xray.
Figure 1 shows EXAFS data k w χ(k) (w = 1, 2, 3) of an
YOx Hy epitaxial thin film. Abscissa is the wavenumber
k of photoelectrons. Although the EXAFS data is analyzed with the absorption intensity χ(k) multiplied byk w to emphasize the oscillation at larger k, the weightw has been empirically selected. As seen in Fig. 1, thesignal fluctuations are large in the noise region at larger
k because of weak absorption intensity due to a thin film
sample and measurement in fluorescence mode: in suchcases, the EXAFS analysis is difficult with conventional
Fourier transform methods. As discussed later, the proposed method makes possible to determine which orderw should be used to interpret the data as well as to study
atomic-scale geometry even in such a tough analysis withhigh noise intensity.
Although the EXAFS oscillation of χ(k) comes frominterferences of photoelectron waves, we consider twobody terms19 which provide dominant EXAFS oscilla-

XZ ∞
β

0

(2)

γαβ (r, k)nαβ (r) dr,(1)

where α and β mean elements of photo-absorber and scattering atoms, respectively. nαβ (r) is the coordination
number distribution of β as a function of the distance(2)
r from α, and γαβ (r, k) contains the two-body multiple scattering term corresponding to the same geometrical configuration.
Here, the wavenumber k is givenpby k =
2m(E − E0 + ∆E)/~ as a function of photon energy E, where E0 , ∆E and m are the absorption edge energy, the energy offset of the theoretical zerofor the k-space and electron mass, respectively. In our
method, this ∆E can be optimized as described below.
(2)Each signal γαβ (r, k) is written as an oscillating function,Aαβ (r, k) sin Φαβ (r, k)19 . The amplitude Aαβ (r, k) andthe phase shift Φαβ (r, k) are calculated by GNXAS20–22 ,in which a complex Hedin-Lundqvist potential23 is used
for inelastic loss effects in the framework of the multiple
scattering theory.
We convert the integral of r in Eq. (1) into a Riemann
sum with N subintervals whose each length is ∆r:χα (k) =NXXβAαβ (rj , k) sin Φαβ (rj , k)∆r nαβ (rj ).

j=1

(2)
Let y be {yi = kiw χα (ki ) | i = 1, . . . , p}, the EXAFS
analysis becomes a linear regression problem with a given
design matrix Mij = kiw Aαβ (rj , ki ) sin Φαβ (rj , ki )∆r. We
introduce regression coefficients xβ = {xβ (rj )} for anelement β and x = {xβ } for all elements in proportional
to the coordination numbers.
This linear regression is an ill-posed problem. However, the L1 regularization is particularly effective in
EXAFS analysis of crystalline solid materials7 because
neighbor atoms will coordinate at discrete (sparse) distances from the photo-absorbers on the basis of symmetry and chemical structure.
Therefore, we employ the least absolute shrinkage and selection operator(LASSO)24 method, and a sparse solution xSpM is obtained by the formula at some regularization parameterλ (> 0):
1
2
xSpM = arg min
ky − M xk2 + λ kxk1 .
(3)
2pxFabio et al. used this LASSO method for EXAFS signalsof Cu foil, Fe foil and cuprous oxide and determined λ bycross-validation9 . As a result, they succeeded in obtaining peak positions of the radial distribution for their materials and structural parameters, like average distancesand Debye–Waller factors without using any prior information about the structure.
However, the LASSO would select jointly correlatedxβ (rj ) when their features in M are highly correlated24,25
and λ is often chosen heuristically. To overcome theseWe use the Bayes free energy (BFE)10,27,28 , F (c) =− ln P (y | c) for model selection of c.
To calculate Eq. (4), we introduce appropriate models for the prior probability P (x | c) and the likelihood
P (y | x, c)10 . For P (x | c), we use a normal distribution
N (0, σxβ ) for the nonzero components in xβ , where we
incorporate the sparsity of xβ by being the mean to zero.
On the other hand, the coordination numbers of nonzero
components are expected to increase with rj2 because thevolume of a spherical shell increases with rj2 . So we set
to σxβ = rj2 (zxβ 0 )−1/2 , where zxβ 0 of each element β willbe estimated by BFE minimization.
For P (y | x, c), we assume a multivariate normal distribution N (M (c ◦ x), Σ) for the superimposed noise in
y and also introduce a model for its variance-covariancematrix Σ. Since we analyze {yi = kiw χα (ki )}, yi is considered to include a uniform k-independent noise and a
k-dependent noise in proportion to k w , where the variance of the uniform noise and the standard deviationof the k-dependent noise are 1/z and σ(k), respectively.
Diagonal elements of Σ are the sum of 1/z and σ(k)2 ,and off-diagonal elements are zero. The 1/z and σ(k)2
are formulated as follows. First, the fluctuation in thenoise region (k ≥ knoise ) of Fig. 1 is considered to comefrom noises because the EXAFS oscillation almost decays, and the root mean square deviation of the fluctuation is obtained as σdata . Subsequently, σ(k) is given2by σ(k) = (k/knoise )w (σdata− 1/z)1/2 for k < knoise . We
also estimate 1/z by minimizing BFE.
By these modeling of P (x | c) and P (y | x, c), we canobtain the posterior probability and the BFE analytically. The regression coefficients x are optimized by
the maximum a posteriori (MAP) estimation10 under
non-negative constraint. As the posterior probability
P (x | y, c) is the Gaussian-type function, the reliabilityof x can be also estimated from the diagonal elements ofthe variance-covariance matrix of P (x | y, c).
The EXAFS oscillation shown in the analysis region ofFig. 1 is chosen for analysis because some X-ray absorption near edge structures would remain in the small k−1
region. The analysis region is k = 3.0–8.0 Å , and the−1
region to obtain σdata is k = 8.0–10.5 Å . For radial
distances rj , we use a linear gird of rj = 2.000–9.975 Å

60

w=1

w=2

w=3

55
F^( E)

problems by BI10 , we introduced a binary vector c indicating nonzero components of x and applied linear regression y = M (c ◦ x) + , where  is a noise superimposed on y and the operator ◦ means the Hadamard
product between vectors. In BI, we maximize the posterior probability P (c | y), where it can be expanded as
P (c | y) = P (y | c)P (c)/P (y) on the basis of Bayes’ theorem26 . To perform unbiased analysis, the prior probability P (c) should be uniform. Consequently, P (c | y) is
proportional to the likelihood P (y | c), and it is given by
Eq. (4).
Z
P (y | c) = P (y | x, c)P (x | c) dx.
(4)

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3

50
45
0

1

2

3

DE (eV)

4

5

6

FIG. 2. BFE F̂ (∆E) for w = 1, 2, 3.

with the step, ∆r = 0.025 Å.
In BI, the number of combinations for the binary vectorc that represents the nonzero components of x is enormous, and their exhaustive search is impractical. So wefirst use Eq. (3) to get a sparse solution xSpM , i.e. cSpM .
The sparsity control parameter λ is set to be in the rangeof λmax to λmax /100 in a geometric sequence, where all
xSpM are zero for λ ≥ λmax 29 .
Since the design matrix M contains the parameter ∆E
to be optimized, we obtain the BFE as a function of∆E for each w as follows: (i) Obtain xSpM (cSpM ) bysolving Eq. (3) at given ∆E and λ and calculate the
MAP solution xBI for the obtained cSpM ; (ii) Exhaustive search in the power set, C := Power(cSpM ) of cSpM
through the minimization of the BFE: ĉBI (∆E, λ) =arg minc⊆C F (c; ∆E, λ). The exhaustive search is performed under kck0 < 20 for c ⊆ C because of the computational cost; (iii) Finally, we obtain the minimized BFE[F̂ (∆E) = minλ F (ĉBI (∆E, λ))] for each ∆E. In theseprocedures, the exhaustive search is necessary because
the basis functions are not strictly orthogonal each otherand there is concern about excess extraction of correlated
components.
Figure 2 shows F̂ (∆E) for w = 1, 2, 3 in the range of∆E = 0.0 − 10.0 eV with 0.1 eV step. The information
criterion minimizing the BFE selects ∆E = 1.0 eV andw = 1. Although the same ∆E is selected for w = 2,
∆E for w = 3 differs from others. This is because thenoise intensity is overestimated for large exponent forhigh k-weighting. In Fig. 1, the amplitude of k 3 χ(k)in the analysis region is smaller than them in the noise
region.
The red curve in Fig. 3(a) is the regression curve forthe selected model x̂BI with ∆E = 1.0 eV and w = 1, and
it is found that the red curve explains appropriately themeasured data indicated by open circles without overfitting. In EXAFS analysis, we have to pay attention to thecontributions from three- or more-many body multiple
scattering effects30 . However, as seen from the scatteredopen circles in Fig. 3(a), it is considered that such contributions are not important in the case of severe signal-tonoise ratio because of a thin film sample. On the other

4

(a)

-1

k (k) (Å )0.2(b) Y–Y

60

0.1
0.0

c

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80

Measured
Regression
The 1st (O) and 2nd (Y)nearest neighbor atoms

Magnitude (arb. unit)

0.3

-0.1

d5

40
d2

20
0
80
60

d4

(c) Y–O
d1

d3

2

4

40
20

-0.2

3

4

5

-1 6

k (Å )

7

8

0

0

r (Å)

6

8

10

FIG. 3. (a) Open circles indicate the measured EXAFS data kχ(k). Red and blue curves are the regression results with x̂BI
at ∆E = 1.0 eV and only with the first and second nearest neighbor atoms in the x̂BI , respectively. (b) and (c) Magnitude of
the regression coefficients for Y and O atoms, respectively, as functions of r, where the coefficients are labeled by di in orderof distance of the photo-absorber Y atom.

hand, the blue-dashed curve is drawn only by components of the two nearest neighbor atoms (O: 2.275 Å, Y:
3.750 Å) and presents a good overview of the EXAFS
oscillation. In our method, the elements can be treatedseparately as β in Eq. (1), and the radial distributionof each element can be obtained separately as shown in
Figs. 3(b) and (c), which are for Y and O atoms, respectively. Error bars show the reliability estimated from thevariance-covariance matrix of P (x | y, c). Since we do notconsider the effect of the reduction factor S0 , the regression coefficients will be underestimated. Nevertheless,the coefficients take large values. This is because that
thermal effect is excluded by the exhaustive search. As
the EXAFS measurements were performed at room temperature, each peak in Figs. 3(b) and (c) should be distributed with a finite width near the equilibrium atomic
position. However, it is very difficult to estimate thewidth of the radial distribution function from the datawith a severe signal-to-noise ratio. Because the exhaustive search excludes the correlated coefficients and thecontributions of the thermal effect due to a tradeoff between excluding correlated coefficients and considering
the thermal effect. As a result, the coefficients standonly at the equilibrium position despite the finite temperature, and become large.

mean value with the weights of the inverse squares of theerror bars in Fig. 3(b) is 5.306 ± 0.025 Å. It is slightly
smaller than the previous studies15,16 (5.34−5.35 Å), andthis difference might come from the epitaxial growth.

In the used basis functions, the phase shift due to photoelectron wave scatterings are incorporated. Therefore,
the distance di for the ith neighbor atom can be evaluated correctly, and we can estimate the atomic-scale
three dimensional structure based on the distance ratio
between the neighbor atoms. When we focus on Y atoms
in Fig.
√ 3(b), d4 /d2 = 5.325/3.750 ≈ 1.420 is very close
to 2 ≈ 1.414 of that characterizes the fcc structure.
This result confirms a previous work15 that Y atom takes
the fcc structure. Additionally,a lattice constant of fcc√
is also obtained from 2d2 and d4 , and their weighted-

In summary, we proposed a Bayesian SpM for analysis of EXAFS data with GNXAS basis functions incorporating two-body multiple scatterings. Model selection for the sparse solution of the regression coefficients proportional to the radial distribution function was
performed by BFE minimization through the exhaustive
search within nonzero components obtained by L1 regularization. Using this method, we succeeded in determining that the sublattice of Y atom is fcc and the interstitial O position is tetrahedral in the YOx Hy epitaxial
thin film.

The coordination structure of O atoms around Y atom
can be determined from the ratio d2 /d1 , where there are
two candidates for the interstitial O-sites: tetrahedral
and octahedral sites. If the interstitial
p sites form a regular tetrahedron, the ratio d2 /d1 is 8/3 ≈ 1.633.
On the√other hand, d2 /d1 for octahedral becomes 2 ≈ 1.414.
In our analysis, d2 /d1 ≈ 1.648 (= 3.750/2.275) is quite
close to the tetrahedral one, and we can conclude that theinterstitial O sites in YOx Hy are tetrahedral. A similar
result has been obtained in a previous study17 for EXAFS of Gadolinium oxyhydride thin films, in which theconventional Fourier transform was used with a structure
model predicted by the X-ray diffraction. In contrast,
the advantage of our method is obvious, since it is basedon the accurate atomic distances compensated with thephase shift and any structure models are not required.

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ACKNOWLEDGMENTS

The synchrotron radiation experiments were performed using a JAEA experimental station at JAEA
(QST) beamline BL14B1, SPring-8, with the approvalof the Japan Synchrotron Radiation Research Institute(JASRI) (Proposal No. 2020A3648); The JAEA Advanced Characterization Nanotechnology Platform under the remit of “Nanotechnology Platform” of the Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan (Grant No. JPMXP09A20AE0006).
This work was supported by JST CREST: JPMJCR1861 and JPMJCR1761, Japan, JST PRESTO:JPMJPR17N2 and JPMJPR17N6, Japan, JSPS KAKENHI: JP19H02596, JP19H04689, JP18H05513, andJP18H05514, Japan, and AGC Inc.: KC31AGC08p.

DATA AVAILABILITY

The data that support the findings of this study areavailable from the corresponding authors upon reasonable request.
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55
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(a)

Measured
Regression
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0.2-1(b) Y–Y

60

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