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Probabilistic Analysis of Event-Mode Experimental Data

Phillip M. Bentley, Thomas H. Rod

Abstract

Neutron and x-ray scattering experiments traditionally rely upon histogrammed data sets, which are analysed using least-squares curve fitting of multiple probability distribution components to quantify separately the various scientific contributions of interest. The main advantage to these methods is the relative ease of deployment due to their intuitive nature. Despite great popularity, these methods have known drawbacks, which can cause systematic errors and biases in some common scenarios in this field. Improvements over the base methods include dynamic optimisation of histogram bin width and the application of modern numerical optimisation methods that have greater stability, but, whilst reduced, the systematic effects carried by this stack nonetheless remain. In this study, we demonstrate analysis of neutron scattering event data using neither any numerical integration or histogramming steps, nor least squares fitting. The benefits of the new methodology are revealed: more accurate parameter values, orders of magnitude greater efficiency (i.e. fewer data points required for the same parameter accuracy) and a reduced impact of inherent systematic error. The main drawbacks are a less intuitive analysis method and an increase in computation time.

Probabilistic Analysis of Event-Mode Experimental Data

Abstract

Neutron and x-ray scattering experiments traditionally rely upon histogrammed data sets, which are analysed using least-squares curve fitting of multiple probability distribution components to quantify separately the various scientific contributions of interest. The main advantage to these methods is the relative ease of deployment due to their intuitive nature. Despite great popularity, these methods have known drawbacks, which can cause systematic errors and biases in some common scenarios in this field. Improvements over the base methods include dynamic optimisation of histogram bin width and the application of modern numerical optimisation methods that have greater stability, but, whilst reduced, the systematic effects carried by this stack nonetheless remain. In this study, we demonstrate analysis of neutron scattering event data using neither any numerical integration or histogramming steps, nor least squares fitting. The benefits of the new methodology are revealed: more accurate parameter values, orders of magnitude greater efficiency (i.e. fewer data points required for the same parameter accuracy) and a reduced impact of inherent systematic error. The main drawbacks are a less intuitive analysis method and an increase in computation time.

Paper Structure

This paper contains 19 sections, 27 equations, 10 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Event Mode Data Analysis
  3. Maximum Likelihood Estimation
  4. Testing MLE vs LSE with Simple Data
  5. Testing MLE with More Realistic Data
  6. Maximum A Posteriori Estimation
  7. Parameter Uncertainties from MLE and MAP
  8. Sampling the Posterior with Markov-Chain Monte-Carlo
  9. PyMC (Rejected)
  10. Pomegranate (Rejected)
  11. Tensorflow Probability (An option)
  12. Edward (An option)
  13. EMCEE (Tested and deployable)
  14. Implementation of MCMC-Sampled Bayesian Analysis
  15. Testing of MCMC-Sampled Bayesian Analysis
  16. ...and 4 more sections

Figures (10)

  • Figure 1: Fits to random events, generated by a simple gaussian function, using histograms with least squares regression, and maximum likelihood estimation.
  • Figure 2: Boxplot of the extracted parameters from MLE and LSE from fitting test data similar to that in figure \ref{['fig:mle-lse-fits']}.
  • Figure 3: Standard deviations and means of the extracted parameters as a function of number of data events, for both LSE and MLE.
  • Figure 4: Fits to random events, generated by a Cauchy distribution, using histograms with least squares regression, and maximum likelihood estimation.
  • Figure 5: Boxplot of the extracted parameters from MLE and LSE from fitting test data similar to that in figure \ref{['fig:mle-lse-sanslike-fits']}.
  • ...and 5 more figures