  We consider the problem of detecting multiple changes in multiple independent
time series. The search for the best segmentation can be expressed as a
minimization problem over a given cost function. We focus on dynamic
programming algorithms that solve this problem exactly. When the number of
changes is proportional to data length, an inequality-based pruning rule
encoded in the PELT algorithm leads to a linear time complexity. Another type
of pruning, called functional pruning, gives a close-to-linear time complexity
whatever the number of changes, but only for the analysis of univariate time
series.
  We propose a few extensions of functional pruning for multiple independent
time series based on the use of simple geometric shapes (balls and
hyperrectangles). We focus on the Gaussian case, but some of our rules can be
easily extended to the exponential family. In a simulation study we compare the
computational efficiency of different geometric-based pruning rules. We show
that for small dimensions (2, 3, 4) some of them ran significantly faster than
inequality-based approaches in particular when the underlying number of changes
is small compared to the data length.
