  In this paper we investigate the problem of detecting dynamically evolving
signals. We model the signal as an $n$ dimensional vector that is either zero
or has $s$ non-zero components. At each time step $t\in \mathbb{N}$ the
non-zero components change their location independently with probability $p$.
The statistical problem is to decide whether the signal is a zero vector or in
fact it has non-zero components. This decision is based on $m$ noisy
observations of individual signal components collected at times $t=1,\ldots,m$.
We consider two different sensing paradigms, namely adaptive and non-adaptive
sensing. For non-adaptive sensing the choice of components to measure has to be
decided before the data collection process started, while for adaptive sensing
one can adjust the sensing process based on observations collected earlier. We
characterize the difficulty of this detection problem in both sensing paradigms
in terms of the aforementioned parameters, with special interest to the speed
of change of the active components. In addition we provide an adaptive sensing
algorithm for this problem and contrast its performance to that of non-adaptive
detection algorithms.
