Cold atomic clouds have been shown to have some similarities with plasmas. Previous studies showed that such clouds exhibit instabilities induced by long-range interactions. However, they did not describe the spatial properties of the dynamics. In this paper, we study experimentally the spatial nature of stochastic instabilities, and we find out that the dynamics is localized. Data are analyzed both in the spectral domain and in the spatial domain (principal component analysis). Both methods fail to describe the dynamics in terms of eigenmodes, showing that space and time are not separable.