  In this article, we discuss various implementation of L1 filtering in order
to detect some properties of noisy signals. This filter consists of using a L1
penalty condition in order to obtain the filtered signal composed by a set of
straight trends or steps. This penalty condition, which determines the number
of breaks, is implemented in a constrained least square problem and is
represented by a regularization parameter ? which is estimated by a
cross-validation procedure. Financial time series are usually characterized by
a long-term trend (called the global trend) and some short-term trends (which
are named local trends). A combination of these two time scales can form a
simple model describing the process of a global trend process with some
mean-reverting properties. Explicit applications to momentum strategies are
also discussed in detail with appropriate uses of the trend configurations.
