  We study the problem of reconstructing and predicting the future of a
dynamical system by the use of time-delay measurements of typical observables.
Considering the case of too few measurements, we prove that for Lipschitz
systems on compact sets in Euclidean spaces, equipped with an invariant Borel
probability measure $\mu$ of Hausdorff dimension $d$, one needs at least $d$
measurements of a typical (prevalent) Lipschitz observable for $\mu$-almost
sure reconstruction and prediction. Consequently, the Hausdorff dimension of
$\mu$ is the precise threshold for the minimal delay (embedding) dimension for
such systems in a probabilistic setting. Furthermore, we establish a lower
bound postulated in the Schroer--Sauer--Ott--Yorke prediction error conjecture
from 1998, after necessary modifications (whereas the upper estimates were
obtained in our previous work). To this aim, we prove a general theorem on the
dimensions of conditional measures of $\mu$ with respect to time-delay
coordinate maps.
