  The time series cluster kernel (TCK) provides a powerful tool for analysing
multivariate time series subject to missing data. TCK is designed using an
ensemble learning approach in which Bayesian mixture models form the base
models. Because of the Bayesian approach, TCK can naturally deal with missing
values without resorting to imputation and the ensemble strategy ensures
robustness to hyperparameters, making it particularly well suited for
unsupervised learning.
  However, TCK assumes missing at random and that the underlying missingness
mechanism is ignorable, i.e. uninformative, an assumption that does not hold in
many real-world applications, such as e.g. medicine. To overcome this
limitation, we present a kernel capable of exploiting the potentially rich
information in the missing values and patterns, as well as the information from
the observed data. In our approach, we create a representation of the missing
pattern, which is incorporated into mixed mode mixture models in such a way
that the information provided by the missing patterns is effectively exploited.
Moreover, we also propose a semi-supervised kernel, capable of taking advantage
of incomplete label information to learn more accurate similarities.
  Experiments on benchmark data, as well as a real-world case study of patients
described by longitudinal electronic health record data who potentially suffer
from hospital-acquired infections, demonstrate the effectiveness of the
proposed methods.
