title: Estimating the size of the cosmic-ray halo using particle distribution moments

authors: Litvinenko, Yuri E.

subjects: cosmic rays – galaxies

abstract: Context: Particle transport in many astrophysical problems can be described either by the Fokker–Planck equation or by an equivalent system of stochastic differential equations. Aims: It is shown that the latter method can be applied to the problem of defining the size of the cosmic-ray galactic halo. Methods: Analytical expressions for the leading moments of the pitch-angle distribution of relativistic particles are determined. Particle scattering and escape are analyzed in terms of the moments. Results: In the case of an anisotropic distribution, the first moment leads to an expression for the halo size, identified with the particle escape from the region of strong scattering. Previous studies are generalized by analyzing the case of a strictly isotropic initial distribution. A new expression for the variance of the distribution is derived, which illustrates the anisotropization of the distribution. Conclusions: Stochastic calculus tools allow one to analyze physically motivated forms for the scattering rate, so that a detailed realistic model can be developed.

text: Estimating the size of the cosmic-ray halo using particle distribution moments Context: Particle transport in many astrophysical problems can be described either by the Fokker–Planck equation or by an equivalent system of stochastic differential equations. Aims: It is shown that the latter method can be applied to the problem of defining the size of the cosmic-ray galactic halo. Methods: Analytical expressions for the leading moments of the pitch-angle distribution of relativistic particles are determined. Particle scattering and escape are analyzed in terms of the moments. Results: In the case of an anisotropic distribution, the first moment leads to an expression for the halo size, identified with the particle escape from the region of strong scattering. Previous studies are generalized by analyzing the case of a strictly isotropic initial distribution. A new expression for the variance of the distribution is derived, which illustrates the anisotropization of the distribution. Conclusions: Stochastic calculus tools allow one to analyze physically motivated forms for the scattering rate, so that a detailed realistic model can be developed.

year: 2009

title: A four-stage framework for assessing statistical literacy

authors: Sharma, Sashi

abstract: Despite statistical literacy being relatively new in statistics education research, it needs special attention as attempts are being made to enhance the teaching, learning and assessing of this strand. It is important that teachers are aware of the challenges of teaching and assessing of literacy. The growing importance of statistics in today’s information world and conceptions of statistical literacy are outlined and models for developing statistical literacy from research literature are considered. A four-stage framework for assessing statistical literacy from our design research is proposed. Responses to tasks used in our research are provided to explain the levels of thinking and the article concludes with some implications for practice and research.

text: A four-stage framework for assessing statistical literacy Despite statistical literacy being relatively new in statistics education research, it needs special attention as attempts are being made to enhance the teaching, learning and assessing of this strand. It is important that teachers are aware of the challenges of teaching and assessing of literacy. The growing importance of statistics in today’s information world and conceptions of statistical literacy are outlined and models for developing statistical literacy from research literature are considered. A four-stage framework for assessing statistical literacy from our design research is proposed. Responses to tasks used in our research are provided to explain the levels of thinking and the article concludes with some implications for practice and research.

year: 2012

title: Visualising categorical data: Linguistic case studies from te Reo Māori and New Zealand English

authors: Trye, David

subjects: visualisation

abstract: Categorical variables are prevalent in real-world datasets across numerous domains, yet few visualisation techniques accommodate them effectively. This is especially true of datasets comprising three or more categorical variables, termed multivariate categorical data. Visualising such data is challenging due to the lack of inherent ordering of nominal categories, the so-called ‘curse of dimensionality’, and the potential variability in the number of categories per variable. Corpus linguistics, which involves the study of large digital collections of naturally occurring language, serves as the primary application domain in this thesis. This domain was chosen because it is rich in multivariate categorical data and, at the same time, is often visualised using only basic techniques.

This thesis contributes to the area of categorical data visualisation in several ways. First, we propose a taxonomy of techniques for visualising categorical data, highlight limitations of existing solutions, and identify relevant analysis tasks. Building on this foundation, the thesis introduces novel techniques and enhancements for visualising datasets involving multiple categorical variables. We focus on adapting the layout and interactive capabilities of an existing technique that uses a matrix of heatmaps to represent pairwise category intersections. These modifications show that directly visualising statistical test results for categorical data can be beneficial for exploring bivariate patterns and associations. Furthermore, we contribute the design, implementation and evaluation of a novel technique called MultiCat, which is not restricted to pairwise intersections but rather facilitates analysis of relationships among multiple variables simultaneously. Both these techniques are interactive and offer greater scalability than existing alternatives, thereby affording new possibilities for analysing multivariate categorical data. However, since categorical variables can occur within more complex data structures, we also consider their presence in networks and hypergraphs, which require specialised methods.

To demonstrate the application of these techniques, we draw on two linguistic case studies that focus on languages of special significance in Aotearoa New Zealand. Addressing the low-resource status of Māori, the country’s Indigenous language, we first contribute two related Twitter datasets—a monolingual Māori corpus and a mixed–language Māori–English corpus—together with an architecture for differentiating Māori and English words. Our initial case study uses the monolingual Māori corpus and proposed visualisation techniques to investigate grammatical possession in Māori, offering fresh insights into the linguistic practices of contemporary speakers. The second case study uses networks and hypergraphs with categorical attributes to explore Māori loanword co-occurrence in New Zealand English newspaper articles. We find that loanwords tend not to occur in isolation and that New Zealanders are still importing new (unlisted) borrowings from Māori.

Ultimately, the techniques developed in this thesis have broad applications both within and beyond the corpus linguistics community. By enabling more effective visualisation and analysis of multivariate categorical data, this research has the potential to facilitate deeper insights into domains as diverse as education, healthcare, business and science.

text: Visualising categorical data: Linguistic case studies from te Reo Māori and New Zealand English Categorical variables are prevalent in real-world datasets across numerous domains, yet few visualisation techniques accommodate them effectively. This is especially true of datasets comprising three or more categorical variables, termed multivariate categorical data. Visualising such data is challenging due to the lack of inherent ordering of nominal categories, the so-called ‘curse of dimensionality’, and the potential variability in the number of categories per variable. Corpus linguistics, which involves the study of large digital collections of naturally occurring language, serves as the primary application domain in this thesis. This domain was chosen because it is rich in multivariate categorical data and, at the same time, is often visualised using only basic techniques.

This thesis contributes to the area of categorical data visualisation in several ways. First, we propose a taxonomy of techniques for visualising categorical data, highlight limitations of existing solutions, and identify relevant analysis tasks. Building on this foundation, the thesis introduces novel techniques and enhancements for visualising datasets involving multiple categorical variables. We focus on adapting the layout and interactive capabilities of an existing technique that uses a matrix of heatmaps to represent pairwise category intersections. These modifications show that directly visualising statistical test results for categorical data can be beneficial for exploring bivariate patterns and associations. Furthermore, we contribute the design, implementation and evaluation of a novel technique called MultiCat, which is not restricted to pairwise intersections but rather facilitates analysis of relationships among multiple variables simultaneously. Both these techniques are interactive and offer greater scalability than existing alternatives, thereby affording new possibilities for analysing multivariate categorical data. However, since categorical variables can occur within more complex data structures, we also consider their presence in networks and hypergraphs, which require specialised methods.

To demonstrate the application of these techniques, we draw on two linguistic case studies that focus on languages of special significance in Aotearoa New Zealand. Addressing the low-resource status of Māori, the country’s Indigenous language, we first contribute two related Twitter datasets—a monolingual Māori corpus and a mixed–language Māori–English corpus—together with an architecture for differentiating Māori and English words. Our initial case study uses the monolingual Māori corpus and proposed visualisation techniques to investigate grammatical possession in Māori, offering fresh insights into the linguistic practices of contemporary speakers. The second case study uses networks and hypergraphs with categorical attributes to explore Māori loanword co-occurrence in New Zealand English newspaper articles. We find that loanwords tend not to occur in isolation and that New Zealanders are still importing new (unlisted) borrowings from Māori.

Ultimately, the techniques developed in this thesis have broad applications both within and beyond the corpus linguistics community. By enabling more effective visualisation and analysis of multivariate categorical data, this research has the potential to facilitate deeper insights into domains as diverse as education, healthcare, business and science.

year: 2024