Mathematical comparison of risk-score transformations for dynamic risk trajectories in mathematics learning

Abstract

Learning Analytics has become an important approach for supporting mathematics learning evaluation through Early Warning Systems (EWS). However, most existing studies directly employ prediction probabilities generated by machine learning models without examining how different mathematical probability transformations influence the representation of students' risk evolution over time. This study presents a mathematical analysis of probability transformation functions within a Dynamic Learning Analytics framework using the Open University Learning Analytics Dataset (OULAD). Student risk scores were estimated using LightGBM at five Learning Progress Checkpoints (LPC20–LPC100) and subsequently transformed using five nonlinear probability functions, namely Sigmoid, Tanh, Probit, Arctan, and Softsign. The transformed trajectories were quantitatively evaluated using variance, mean slope, total variation, and risk accumulation, while Dynamic Time Warping and Agglomerative Clustering were employed to identify temporal risk patterns. The results demonstrate that different transformation functions produce distinct mathematical characteristics despite preserving similar cumulative risk distributions. Under the selected variance, mean absolute slope, and total variation criteria, Softsign produced the least variable and smoothest transformed trajectories, with the lowest variance (0.1048) and total variation (0.3320).  Based upon the types of trajectories identified, there were four representative types of trajectories identified: Persistence, Escalation, Recovery, and Instability. These results provide a descriptive mathematical comparison of the five risk-score transformations and show that the choice of transformation affects the stability and temporal characteristics of trajectory-based representations.