  In empirical studies of random walks, continuous trajectories of animals or
individuals are usually sampled over a finite number of points in space and
time. It is however unclear how this partial observation affects the measured
statistical properties of the walk, and we use here analytical and numerical
methods of statistical physics to study the effects of sampling in movements
alternating rests and moves of random durations. We evaluate how the
statistical properties estimated are affected by the way trajectories are
measured and we identify an optimal sampling frequency leading to the best
possible measure. We solve analytically the simplest scenario of a constant
sampling interval and short-tailed distributions of rest and move durations,
which allows us to show that the measured displacement statistics can be
significantly different from the original ones and also to determine the
optimal sampling time. The corresponding optimal fraction of correctly sampled
movements, analytically predicted for this short-tail scenario, is an upper
bound for the quality of a trajectory's sampling. Indeed, we show with
numerical simulations that this fraction is dramatically reduced in any
real-world case where we observe long-tailed distributions of rest duration. We
test our results with high resolution GPS human trajectories, where a constant
sampling interval allows to recover at best $18\%$ of the movements, while
over-evaluating the average trip length by a factor of $2$. If we use a
sampling interval extracted from real communication data, we recover only
$11\%$ of moves, a value that cannot be increased above $16\%$ even with ideal
algorithms. These figures call for a more cautious use of data in all
quantitative studies of individuals' trajectories, casting in particular
serious doubts on the results of previous studies on human mobility based on
mobile phone data.
