  Supervised learning by extreme learning machines resp. neural networks with
random weights is studied under a non-stationary spatial-temporal sampling
design which especially addresses settings where an autonomous object moving in
a non-stationary spatial environment collects and analyzes data. The stochastic
model especially allows for spatial heterogeneity and weak dependence. As
efficient and computationally cheap learning methods (unconstrained) least
squares, ridge regression and $\ell_s$-penalized least squares (including the
LASSO) are studied. Consistency and asymptotic normality of the least squares
and ridge regression estimates as well as corresponding consistency results for
the $\ell_s$-penalty are shown under weak conditions. The results also cover
bounds for the sample squared predicition error.
