Principal Component Analysis (PCA) for Powder Diffraction Data: Towards Unblinded Applications

Dmitry Chernyshov,Vadim Dyadkin,Wouter Van Beek,Iurii Dovgaliuk

doi:10.3390/cryst10070581

Abstract

We analyze the application of Principal Component Analysis (PCA) for untangling the main contributions to changing diffracted intensities upon variation of site occupancy and lattice dimensions induced by external stimuli. The information content of the PCA output consists of certain functions of Bragg angles (loadings) and their evolution characteristics that depend on external variables like pressure or temperature (scores). The physical meaning of the PCA output is to date not well understood. Therefore, in this paper, the intensity contributions are first derived analytically, then compared with the PCA components for model data; finally PCA is applied for the real data on isothermal gas uptake by nanoporous framework γ –Mg(BH 4 ) 2 . We show that, in close agreement with previous analysis of modulation diffraction, the variation of intensity of Bragg lines and the displacements of their positions results in a series of PCA components. Every PCA extracted component may be a mixture of terms carrying information on the average structure, active sub-structure, and their cross-term. The rotational ambiguities, that are an inherently part of PCA extraction, are at the origin of the mixing. For the experimental case considered in the paper, the extraction of the physically meaningful loadings and scores can only be achieved with a rotational correction. Finally, practical recommendations for non-blind applications, i.e., what boundary conditions to apply for the the rotational correction, of PCA for diffraction data are given.

Highlights

Thanks to the increased brightness of X-rays at modern synchrotron sources and the appearance of fast and nearly noise-free detectors, it becomes possible to collect diffraction data with fine time sampling or with fine steps in pressure, temperature or anisotropic external fields
A first comparison of Modulation-Enhanced Diffraction (MED) and Principal Component Analysis (PCA) was done in [15] where it was shown that the first principal component may correspond to a cross product of the time average and time changing parts of the structure factor, while the second component is proportional to the square of the time changing part of the structure factor, in close similarity with MED
We have performed a theoretical analysis on how specific changes in time resolved diffraction data are represented into PCA output

Summary

Introduction

Thanks to the increased brightness of X-rays at modern synchrotron sources and the appearance of fast and nearly noise-free detectors, it becomes possible to collect diffraction data with fine time sampling or with fine steps in pressure, temperature or anisotropic external fields. A first comparison of MED and PCA was done in [15] where it was shown that the first principal component may correspond to a cross product of the time average and time changing parts of the structure factor, while the second component is proportional to the square of the time changing part of the structure factor, in close similarity with MED This observation and conclusion (for both MED and PCA) is correct only for data with a negligibly small variation of the unit cell dimensions. The effective approach to do that, Optimal Constrained Component Rotation (OCCR) is considered in details in [17] As it stated in [17], “if the hypothesis that the peak shape and position do not change with the stimulus does not hold, the PCA decomposition cannot be accomplished”. The goal of the present paper is to fill this gap and propose a rigorous scheme for a unambiguous interpretation of the PCA output on diffraction data

Theory

Simulated Data and PCA Analysis

Variation of Occupancy

Variation of Lattice Dimension

Findings

Conclusions