  The widespread adoption of online courses opens opportunities for the
analysis of learner behaviour and for the optimisation of web-based material
adapted to observed usage. Here we introduce a mathematical framework for the
analysis of time series collected from online engagement of learners, which
allows the identification of clusters of learners with similar online behaviour
directly from the data, i.e., the groups of learners are not pre-determined
subjectively but emerge algorithmically from the analysis and the data.The
method uses a dynamic time warping kernel to create a pairwise similarity
between time series of learner actions, and combines it with an unsupervised
multiscale graph clustering algorithm to cluster groups of learners with
similar patterns of behaviour. We showcase our approach on online engagement
data of adult learners taking six web-based courses as part of a post-graduate
degree at Imperial Business School. Our analysis identifies clusters of
learners with statistically distinct patterns of engagement, ranging from
distributed to massed learning, with different levels of adherence to
pre-planned course structure and/or task completion, and also revealing outlier
learners with highly sporadic behaviour. A posteriori comparison with
performance showed that, although the majority of low-performing learners are
part of in the massed learning cluster, the high performing learners are
distributed across clusters with different traits of online engagement. We also
show that our methodology is able to identify low performing learners more
accurately than common classification methods based on raw statistics extracted
from the data.
