  Despite being extensively studied, time series classification and forecasting
on noisy data remain highly difficult. The main challenges lie in finding
suitable mathematical concepts to describe time series and effectively
separating noise from the true signals. Instead of treating time series as a
static vector or a data sequence as often seen in previous methods, we
introduce a novel framework that considers each time series, not necessarily of
fixed length, as a sample realization of a continuous-time stochastic process.
Such mathematical model explicitly captures the data dependence across several
timestamps and detects the hidden time-dependent signals from noise. However,
since the underlying data is often composed of several distinct dynamics,
modeling using a single stochastic process is not sufficient. To handle such
settings, we first assign each dynamics a signature vector. We then propose the
abstract concept of the most informative timestamps to infer a sparse
approximation of the individual dynamics based on their assigned vectors. The
final model, referred to as Motion Code, contains parameters that can fully
capture different underlying dynamics in an integrated manner. This allows
unmixing classification and generation of specific sub-type forecasting
simultaneously. Extensive experiments on sensors and devices noisy time series
data demonstrate Motion Code's competitiveness against time series
classification and forecasting benchmarks.
