  Modern time series analysis requires the ability to handle datasets that are
inherently high-dimensional; examples include applications in climatology,
where measurements from numerous sensors must be taken into account, or
inventory tracking of large shops, where the dimension is defined by the number
of tracked items. The standard way to mitigate computational issues arising
from the high dimensionality of the data is by applying some dimension
reduction technique that preserves the structural properties of the ambient
space. The dissimilarity between two time series is often measured by
``discrete'' notions of distance, e.g. the dynamic time warping or the discrete
Fr\'echet distance. Since all these distance functions are computed directly on
the points of a time series, they are sensitive to different sampling rates or
gaps. The continuous Fr\'echet distance offers a popular alternative which aims
to alleviate this by taking into account all points on the polygonal curve
obtained by linearly interpolating between any two consecutive points in a
sequence.
  We study the ability of random projections \`a la Johnson and Lindenstrauss
to preserve the continuous Fr\'echet distance of polygonal curves by
effectively reducing the dimension. In particular, we show that one can reduce
the dimension to $O(\epsilon^{-2} \log N)$, where $N$ is the total number of
input points while preserving the continuous Fr\'echet distance between any two
determined polygonal curves within a factor of $1\pm \epsilon$. We conclude
with applications on clustering.
