An Optimization-Based Supervised Learning Algorithm for PXRD Phase Fraction Estimation

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ANOPTIMIZATION-BASED SUPERVISED LEARNING

ALGORITHM FOR PXRD PHASE FRACTION ESTIMATION

A PREPRINT

Patrick Hosein

Department of Computer Science

The University of the West Indies

St. Augustine, Trinidad

patrick.hosein@sta.uwi.edu

Jaimie Greasley

Department of Physics

The University of the West Indies

St. Augustine, Trinidad

jaimie.greasley@gmail.com

October 21, 2022

ABSTRACT

In powder diffraction data analysis, phase identiﬁcation is the process of determining the crystalline

phases in a sample using its characteristic Bragg peaks. For multiphasic spectra, we must also de-

termine the relative weight fraction of each phase in the sample. Machine Learning algorithms (e.g.,

Artiﬁcial Neural Networks) have been applied to perform such difﬁcult tasks in powder diffraction

analysis, but typically require a signiﬁcant number of training samples for acceptable performance.

We have developed an approach that performs well even with a small number of training samples.

We apply a ﬁxed-point iteration algorithm on the labelled training samples to estimate monophasic

spectra. Then, given an unknown sample spectrum, we again use a ﬁxed-point iteration algorithm

to determine the weighted combination of monophase spectra that best approximates the unknown

sample spectrum. These weights are the desired phase fractions for the sample. We compare our

approach with several traditional Machine Learning algorithms.

Keywords machine learning, x-ray diffraction, phase identiﬁcation, quantitative phase analysis

Main

The assessment of powder X-ray diffraction (PXRD) spectra is central to many materials investigations. X-ray scatter-

ing data reveals important structural and micro-structural parameters for characterizing a material[7]. The collection

of scattered intensities attributed to labelled crystallographic planes, in fact serves as a ﬁngerprint reference for the

material structure[15]. Phase identiﬁcation is performed by matching observed Bragg peaks to a powder pattern ref-

erence in a database, usually with the aid of a search-match program. Finding the corresponding reference may not

be easy as different instrument settings or any slight deviation in structure cause variant diffraction proﬁles for any

given phase. For multi-phase analysis, phase fraction estimation is possible by presuming some relationship between

the observed intensities for each phase in the spectrum. Several method varying in complexity, computational rigor

and sample preparation requirements, are available [20]. These include the use of internal standard calibrations as in

the Reference Intensity Ratio method [9], or whole powder pattern ﬁtting as with full-pattern summation [17,5,4,12]

and Rietveld reﬁnement [16].

Recently, Machine Learning has been used to characterize PXRD spectra [1,6]. Some success has been reported

for phase identiﬁcation by both conventional Machine Learning models [2,3] as well as Deep Learning architectures

[11,19,10,13,18]. However, few studies have investigated phase quantiﬁcation [11,10,14].

Previously, we performed PXRD Rietveld characterization of mineral phases for a small batch of urinary stones [8].

Rietveld reﬁnement is a powerful, but time-consuming, pattern ﬁtting procedure which employs least-squares mini-

mization to obtain reﬁned crystallographic parameters for a material, including phase fractions. While full crystallo-

graphic characterization may be useful for research in stone formation, it is not required for a clinical stone analysis

program. Yet, estimated weight fractions serve to aid the analyst in differentiating between primary and secondary

arXiv:2210.10867v1 [cs.LG] 19 Oct 2022

Supervised Learning Algorithm for PXRD Phase Fraction Estimation A PREPRINT

mineralization events. We therefore developed an approach, described in the next section, for estimating phase frac-

tions given the spectrum of an unknown sample.

Proposed Algorithm

We present the proposed model by ﬁrst describing the training process and then showing how data derived from

training is used for estimating phase-fractions for unknown samples. Note that this is a supervised Machine Learning

approach, unlike full-pattern summation [5,4,12] which requires building a reference library prior to ﬁtting.

We used Leave-One-Out cross-validation by reserving one sample at a time for testing while all others are used for

training. Let yi(s)represent the intensity value for sample sat angle i. Let αj(s)represent the percentage of phase j

in sample s. Let xi(j)denote the reading that would be obtained at angle iif the sample consisted solely of phase j.

We denote the number of samples N, the number of phases M, and the number of angles K.

Consider the model whereby the intensity achieved for sample sat angle iconsists of a weighted combination of the

intensities obtained for the individual components j. As such the reading at angle ifor sample sis approximated by

ˆyi(s) =

j=1

αj(s)xi(j)(1)

Note that in the training set, we know αj(s)and yi(s)but we do not know xi(j)so we use the training set to obtain

approximations to xi(j)that provides the lowest error. The sum squared error for the approximation for sample sat

angle iis given by

E2=X

i,s

(yi(s)−ˆyi(s))2=X

i,s



yi(s)−

j=1

αj(s)xi(j)



(2)

We then need to ﬁnd values for xi(j)for all iand jthat minimizes E2. If we take the partial derivative of E2with

respect to xi(j)we get

∂E2

∂xi(j)=X

s−2αj(s)

yi(s)−

j0=1

αj0(s)xi(j0)

(3)

and setting to zero we get

αj(s)

yi(s)−

j0=1

αj0(s)xi(j0)

= 0 (4)

We use a coordinate gradient descent approach. For each variable, we take the derivative and update to the value that

achieves a zero gradient. Note that this is a constrained optimization problem so if the zero gradient is achieved at a

negative value then we instead set to zero.

xi(j)←max 





αj(s)2X

αj(s)

yi(s)−

j0=1,j06=j

αj0(s)xi(j0)

,0





(5)

Next, we describe how testing was performed. Once xi(j)is solved for all iand j, these can be used to compute our

estimate of αjfor a reserved test sample s. These estimated values are denoted as ˆαj. Knowing now xi(j)estimated

from the training set, we have yi(s). We now need to ﬁnd αj(s). We let:

ˆyi(s) =

j=1

ˆαj(s)xi(j)

The sum squared error is again given by

E2=X

i,s



yi(s)−

j=1

αj(s)xi(j)



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摘要：

ANOPTIMIZATION-BASEDSUPERVISEDLEARNINGALGORITHMFORPXRDPHASEFRACTIONESTIMATIONAPREPRINTPatrickHoseinDepartmentofComputerScienceTheUniversityoftheWestIndiesSt.Augustine,Trinidadpatrick.hosein@sta.uwi.eduJaimieGreasleyDepartmentofPhysicsTheUniversityoftheWestIndiesSt.Augustine,Trinidadjaimie.greasley@g...

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An Optimization-Based Supervised Learning Algorithm for PXRD Phase Fraction Estimation

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