Crystallography
2. formulation of the proposed framework, 3. formulation of a multicomponent monodisperse spheres model, 4. numerical experiments, 5. discussion, 6. conclusions.
Format | BIBTeX | |
EndNote | ||
RefMan | ||
Refer | ||
Medline | ||
CIF | ||
SGML | ||
Plain Text | ||
Text | ||
JOURNAL OF APPLIED CRYSTALLOGRAPHY |
a Graduate School of Frontier Sciences, University of Tokyo, Kashiwa, Chiba 277-8561, Japan, b Japan Synchrotron Radiation Research Institute, Sayo, Hyogo 679-5198, Japan, c National Institute for Materials Science, Tsukuba, Ibaraki 305-0047, Japan, and d Facalty of Advanced Science and Technology, Kumamoto University, Kumamoto 860-8555, Japan * Correspondence e-mail: [email protected]
Small-angle scattering (SAS) is a key experimental technique for analyzing nanoscale structures in various materials. In SAS data analysis, selecting an appropriate mathematical model for the scattering intensity is critical, as it generates a hypothesis of the structure of the experimental sample. Traditional model selection methods either rely on qualitative approaches or are prone to overfitting. This paper introduces an analytical method that applies Bayesian model selection to SAS measurement data, enabling a quantitative evaluation of the validity of mathematical models. The performance of the method is assessed through numerical experiments using artificial data for multicomponent spherical materials, demonstrating that this proposed analysis approach yields highly accurate and interpretable results. The ability of the method to analyze a range of mixing ratios and particle size ratios for mixed components is also discussed, along with its precision in model evaluation by the degree of fitting. The proposed method effectively facilitates quantitative analysis of nanoscale sample structures in SAS, which has traditionally been challenging, and is expected to contribute significantly to advancements in a wide range of fields.
Keywords: small-angle X-ray scattering ; small-angle neutron scattering ; nanostructure analysis ; model selection ; Bayesian inference .
SAS measurement data are expressed in terms of scattering intensity that corresponds to a scattering vector, a physical quantity representing the scattering angle. Data analysis requires selection and parameter estimation of a mathematical model of the scattering intensity that contains information about the structure of the specimen. This selection process is critical as it involves assumptions about the structure of the specimen.
We conducted numerical experiments to assess the effectiveness of our proposed method. These experiments are based on synthetic data used to estimate the number of distinct components in a specimen, which was modeled as a mixture of monodisperse spheres of varying radii, scattering length densities and volume fractions. The results demonstrate the high accuracy, interpretability and stability of our method, even in the presence of measurement noise. To discuss the utility of the proposed method, we compare our approach with traditional model selection methods based on the reduced χ -squared error.
In this section, we present a detailed formulation of our algorithm for selecting mathematical models for SAS specimens using Bayesian model selection. The pseudocode for this algorithm is provided in Algorithm 1.
The likelihood is thus expressed as
Let φ ( K ) be the prior distribution of the parameter K that characterizes the model, and φ ( Ξ | K ) be the prior distribution of the model parameters Ξ . Then, from Bayes' theorem, the posterior distribution of the parameters given the measurement data can be written as
Sampling from the joint probability distribution at each inverse temperature gives
In this paper, we consider isotropic scattering and focus on the scattering vector's magnitude q , defined as
Monodisperse spheres are spherical particles of uniform radius. The scattering intensity I ( q , ξ ) of a specimen composed of sufficiently dilute monodisperse spheres of a single type for the scattering vector magnitude q is given by
To formulate the scattering intensity of a specimen composed of K types of monodisperse sphere, we assume a dilute system and denote the particle size of the k th component in the sample as R k and the scale as S k . The scattering intensity of a sample composed of K types of monodisperse sphere is then given by
An illustration of a mixture of two types of spherical specimen. This shows scenarios with two components ( = 2), including mixtures of spherical particles of different sizes or volume fractions, and aggregates from a single particle type approximated as a large sphere. |
The numerical experiments reported in this section were conducted with a burn-in period of 10 5 and a sample size of 10 5 for the REMC. We set the number of replicas for REMC, the values of inverse temperature and the step size of the Metropolis method taking into consideration the state exchange rate and the acceptance rate.
(i) Set the number of data points to N = 400 and define the scattering vector magnitudes at N equally spaced points within the interval [0.1, 3] to obtain { q i } i =1 N =400 (nm −1 ).
In this section, we consider cases with pseudo-measurement times of T = 1 and T = 0.1. Generally, smaller values of T indicate greater effects from measurement noise.
In the Bayesian model selection framework, prior knowledge concerning the parameters Ξ and the model-characterizing parameter K is set as their prior distributions.
In this numerical experiment, the prior distributions for the parameters Ξ were set as Gamma distributions based on the pseudo-measurement time T used during data generation, while the prior for K was a discrete uniform distribution over the interval [1, 4].
Plots of the prior distributions for various parameters. ( ) Prior distribution of , φ( ). ( ) Prior distribution of ) Prior distribution of , φ( ). ( ) Prior distribution of , φ( ). |
The ratio of the scale parameters S 1 and S 2 for spheres 1 and 2 during data generation, denoted r S , is defined as
COMMENTS
The results chapter (also referred to as the findings or analysis chapter) is one of the most important chapters of your dissertation or thesis because it shows the reader what you've found in terms of the quantitative data you've collected. It presents the data using a clear text narrative, supported by tables, graphs and charts.
Step 1: Summarize your key findings. Start this section by reiterating your research problem and concisely summarizing your major findings. To speed up the process you can use a summarizer to quickly get an overview of all important findings. Don't just repeat all the data you have already reported—aim for a clear statement of the overall result that directly answers your main research ...
Reporting quantitative research results. If you conducted quantitative research, you'll likely be working with the results of some sort of statistical analysis.. Your results section should report the results of any statistical tests you used to compare groups or assess relationships between variables.It should also state whether or not each hypothesis was supported.
Begin with a clear statement of the principal findings. This will reinforce the main take-away for the reader and set up the rest of the discussion. Explain why the outcomes of your study are important to the reader. Discuss the implications of your findings realistically based on previous literature, highlighting both the strengths and ...
Tips to Write the Results Section. Direct the reader to the research data and explain the meaning of the data. Avoid using a repetitive sentence structure to explain a new set of data. Write and highlight important findings in your results. Use the same order as the subheadings of the methods section.
Reporting Research Results in APA Style | Tips & Examples. Published on December 21, 2020 by Pritha Bhandari.Revised on January 17, 2024. The results section of a quantitative research paper is where you summarize your data and report the findings of any relevant statistical analyses.. The APA manual provides rigorous guidelines for what to report in quantitative research papers in the fields ...
What (exactly) is the discussion chapter? The discussion chapter is where you interpret and explain your results within your thesis or dissertation. This contrasts with the results chapter, where you merely present and describe the analysis findings (whether qualitative or quantitative).In the discussion chapter, you elaborate on and evaluate your research findings, and discuss the ...
Do you want to learn how to write effective results and discussion chapters for quantitative research? This pdf document from Massey University provides clear guidelines and examples for structuring and presenting your findings and implications. You will also find useful tips on how to avoid common pitfalls and errors in your writing.
Research results refer to the findings and conclusions derived from a systematic investigation or study conducted to answer a specific question or hypothesis. These results are typically presented in a written report or paper and can include various forms of data such as numerical data, qualitative data, statistics, charts, graphs, and visual aids.
IMRaD Results Discussion. Results and Discussion Sections in Scientific Research Reports (IMRaD) After introducing the study and describing its methodology, an IMRaD* report presents and discusses the main findings of the study. In the results section, writers systematically report their findings, and in discussion, they interpret these findings.
After that comes a literature review, which ends with a summary of the research question(s) and/or hypotheses. A methods section, which explains the source of data, sample, and variables and quantitative techniques used, follows. Many analysts will include a short discussion of their descriptive statistics in the methods section.
The discussion section is one of the final parts of a research paper, in which an author describes, analyzes, and interprets their findings. They explain the significance of those results and tie everything back to the research question(s). In this handout, you will find a description of what a discussion section does, explanations of how to ...
The results section of a research paper tells the reader what you found, while the discussion section tells the reader what your findings mean. The results section should present the facts in an academic and unbiased manner, avoiding any attempt at analyzing or interpreting the data. Think of the results section as setting the stage for the ...
This table reflects a single point of each ROC curve in Fig. 5.7 which matches the selected threshold. The wavelet filtering algorithm achieved more than 95% recall in detecting close-range calls ('very loud' and 'loud'). Even when the calls were very faded the recall was just below 70%.
Table of contents. What not to include in your discussion section. Step 1: Summarise your key findings. Step 2: Give your interpretations. Step 3: Discuss the implications. Step 4: Acknowledge the limitations. Step 5: Share your recommendations. Discussion section example.
Martin McMorrow. This document provides guidance on writing the results and discussion chapters for quantitative research theses. It discusses the structure and style of these chapters, including how to present tables and figures, summarize results, and compare findings to previous research. Examples are given from published theses.
INTRODUCTION. Scientific research is usually initiated by posing evidenced-based research questions which are then explicitly restated as hypotheses.1,2 The hypotheses provide directions to guide the study, solutions, explanations, and expected results.3,4 Both research questions and hypotheses are essentially formulated based on conventional theories and real-world processes, which allow the ...
Chapter 4-Quantitative Results and Discussion 4.1. Introduction In the previous chapter, the research design used in this study was described in detail. This included both the quantitative data collection involving the two questionnaires: BALLI and PELLEM, and the qualitative data collection which entailed a semistructured interview.
This chapter 5 presents the results of the study. First, an outline of the informants included in the study and an overview of the statistical techniques employed in the data analyses are given ...
Chapter III RESULTS AND DISCUSSION. The presentation, analysis, and interpretation of the data acquired for this study are all included in this chapter. According to the methodology, statistical tools are utilized to determine the student's perceptions towards the preservation and improvement of Central Luzon State University's landmark.
Revised on June 22, 2023. Quantitative research is the process of collecting and analyzing numerical data. It can be used to find patterns and averages, make predictions, test causal relationships, and generalize results to wider populations. Quantitative research is the opposite of qualitative research, which involves collecting and analyzing ...
In the realm of spatial research on traditional villages, a prevailing trend involves supporting qualitative research through quantitative analysis using measurement tools. This trend primarily concentrates on two key aspects: the characterization of spatial patterns and their causal analysis, along with the impacts resulting from these spatial ...
The research questions will be answered in two parts, with a quantitative study focusing on questions 1), 2) and 3) and a qualitative study addressing question 4). The results will be reflected in a joint discussion at the end of this article.
F (K) is referred to as the Bayesian free energy, also known as the stochastic complexity.The posterior probability of the model, , can be rephrased as the validity of model K for the measurement data .In other words, calculating and comparing the value of for all candidate models {K} thus enables quantitative model selection.Note that in Bayesian model selection the parameter K does not need ...
When collecting and analyzing data, quantitative research deals with numbers and statistics, while qualitative research deals with words and meanings. Both are important for gaining different kinds of knowledge. Quantitative research. Quantitative research is expressed in numbers and graphs. It is used to test or confirm theories and assumptions.