Statistics Online Computational Resource for Education

Interactive Visualization and Computation of 2D and 3D Probability Distributions

Purpose

Mathematical modeling, probability estimation, and statistical inference represent core elements of modern artificial intelligence (AI) approaches for data-driven prediction, forecasting, classification, risk-estimation, and prognosis. Currently there are many tools that help calculate and visualize univariate probability distributions, however, very few resources venture beyond into multivariate distributions, which are commonly used in advanced statistical inference and AI decision-making. This article presents a new web-calculator that enables some calculation and visualization of bivariate and trivariate probability distributions.

Methods

Several methods are explored to compute the joint bivariate and trivariate probability densities, including the optimal multivariate modeling using Gaussian copula. We developed an interactive webapp to visually illustrate the parallels between the mathematical formulation, computational implementation, and graphical depiction of multivariate probability density and cumulative distribution functions. To ensure the interface and functionality are hardware platform independent, scalable, and functional, the app and its component widgets are implemented using HTML5 and JavaScript.

Results

We validated the webapp by testing the multivariate copula models under different experimental conditions and inspecting the performance in terms of accuracy and reliability of the estimated multivariate probability densities and distribution function values.

Conclusion

This article demonstrates the construction, implementation, and utilization of multivariate probability calculators. The proposed webapp implementation is freely available online (https://socr.umich.edu/HTML5/BivariateNormal/BVN2/) and can be used to assist with education and research of a diverse array of data scientists, STEM instructors, and AI learners.

Model-based and Model-free Machine Learning Techniques for Diagnostic Prediction and Classification of Clinical Outcomes in Parkinson's Disease

In this study, we apply a multidisciplinary approach to investigate falls in PD patients using clinical, demographic and neuroimaging data from two independent initiatives (University of Michigan and Tel Aviv Sourasky Medical Center). Using machine learning techniques, we construct predictive models to discriminate fallers and non-fallers. Through controlled feature selection, we identified the most salient predictors of patient falls including gait speed, Hoehn and Yahr stage, postural instability and gait difficulty-related measurements. The model-based and model-free analytical methods we employed included logistic regression, random forests, support vector machines, and XGboost. The reliability of the forecasts was assessed by internal statistical (5-fold) cross validation as well as by external out-of-bag validation. Four specific challenges were addressed in the study: Challenge 1, develop a protocol for harmonizing and aggregating complex, multisource, and multi-site Parkinson's disease data; Challenge 2, identify salient predictive features associated with specific clinical traits, e.g., patient falls; Challenge 3, forecast patient falls and evaluate the classification performance; and Challenge 4, predict tremor dominance (TD) vs. posture instability and gait difficulty (PIGD). Our findings suggest that, compared to other approaches, model-free machine learning based techniques provide a more reliable clinical outcome forecasting of falls in Parkinson's patients, for example, with a classification accuracy of about 70-80%.

Technology-enhanced Interactive Teaching of Marginal, Joint and Conditional Probabilities: The Special Case of Bivariate Normal Distribution

Data analysis requires subtle probability reasoning to answer questions like What is the chance of event A occurring, given that event B was observed? This generic question arises in discussions of many intriguing scientific questions such as What is the probability that an adolescent weighs between 120 and 140 pounds given that they are of average height? and What is the probability of (monetary) inflation exceeding 4% and housing price index below 110? To address such problems, learning some applied, theoretical or cross-disciplinary probability concepts is necessary. Teaching such courses can be improved by utilizing modern information technology resources. Students' understanding of multivariate distributions, conditional probabilities, correlation and causation can be significantly strengthened by employing interactive web-based science educational resources. Independent of the type of a probability course (e.g. majors, minors or service probability course, rigorous measure-theoretic, applied or statistics course) student motivation, learning experiences and knowledge retention may be enhanced by blending modern technological tools within the classical conceptual pedagogical models. We have designed, implemented and disseminated a portable open-source web-application for teaching multivariate distributions, marginal, joint and conditional probabilities using the special case of bivariate Normal distribution. A real adolescent height and weight dataset is used to demonstrate the classroom utilization of the new web-application to address problems of parameter estimation, univariate and multivariate inference.

Web-based tools for modelling and analysis of multivariate data: California ozone pollution activity

This article presents a hands-on web-based activity motivated by the relation between human health and ozone pollution in California. This case study is based on multivariate data collected monthly at 20 locations in California between 1980 and 2006. Several strategies and tools for data interrogation and exploratory data analysis, model fitting and statistical inference on these data are presented. All components of this case study (data, tools, activity) are freely available online at: http://wiki.stat.ucla.edu/socr/index.php/SOCR_MotionCharts_CAOzoneData. Several types of exploratory (motion charts, box-and-whisker plots, spider charts) and quantitative (inference, regression, analysis of variance (ANOVA)) data analyses tools are demonstrated. Two specific human health related questions (temporal and geographic effects of ozone pollution) are discussed as motivational challenges.

A Study of Students' Learning Styles, Discipline Attitudes and Knowledge Acquisition in Technology-Enhanced Probability and Statistics Education

Many modern technological advances have direct impact on the format, style and efficacy of delivery and consumption of educational content. For example, various novel communication and information technology tools and resources enable efficient, timely, interactive and graphical demonstrations of diverse scientific concepts. In this manuscript, we report on a meta-study of 3 controlled experiments of using the Statistics Online Computational Resources in probability and statistics courses. Web-accessible SOCR applets, demonstrations, simulations and virtual experiments were used in different courses as treatment and compared to matched control classes utilizing traditional pedagogical approaches. Qualitative and quantitative data we collected for all courses included Felder-Silverman-Soloman index of learning styles, background assessment, pre and post surveys of attitude towards the subject, end-point satisfaction survey, and varieties of quiz, laboratory and test scores. Our findings indicate that students' learning styles and attitudes towards a discipline may be important confounds of their final quantitative performance. The observed positive effects of integrating information technology with established pedagogical techniques may be valid across disciplines within the broader spectrum courses in the science education curriculum. The two critical components of improving science education via blended instruction include instructor training, and development of appropriate activities, simulations and interactive resources.

SOCR Motion Charts: An Efficient, Open-Source, Interactive and Dynamic Applet for Visualizing Longitudinal Multivariate Data

The amount, complexity and provenance of data have dramatically increased in the past five years. Visualization of observed and simulated data is a critical component of any social, environmental, biomedical or scientific quest. Dynamic, exploratory and interactive visualization of multivariate data, without preprocessing by dimensionality reduction, remains a nearly insurmountable challenge. The Statistics Online Computational Resource (www.SOCR.ucla.edu) provides portable online aids for probability and statistics education, technology-based instruction and statistical computing. We have developed a new Java-based infrastructure, SOCR Motion Charts, for discovery-based exploratory analysis of multivariate data. This interactive data visualization tool enables the visualization of high-dimensional longitudinal data. SOCR Motion Charts allows mapping of ordinal, nominal and quantitative variables onto time, 2D axes, size, colors, glyphs and appearance characteristics, which facilitates the interactive display of multidimensional data. We validated this new visualization paradigm using several publicly available multivariate datasets including Ice-Thickness, Housing Prices, Consumer Price Index, and California Ozone Data. SOCR Motion Charts is designed using object-oriented programming, implemented as a Java Web-applet and is available to the entire community on the web at www.socr.ucla.edu/SOCR_MotionCharts. It can be used as an instructional tool for rendering and interrogating high-dimensional data in the classroom, as well as a research tool for exploratory data analysis.