We consider the problem of estimating the distance between any two large data streams in small-space constraint. This problem is of utmost importance in data intensive monitoring applications where input streams are generated rapidly. These streams need to be processed on the fly and accurately to quickly determine any deviance from nominal behavior. We present a new metric, the \emph{Sketch $\star$-metric}, which allows to define a distance between updatable summaries (or sketches) of large data streams. An important feature of the \emph{Sketch $\star$-metric} is that, given a measure on the entire initial data streams, the \emph{Sketch $\star$-metric} preserves the axioms of the latter measure on the sketch (such as the non-negativity, the identity, the symmetry, the triangle inequality but also specific properties of the $f$-divergence or the Bregman one). Extensive experiments conducted on both synthetic traces and real data sets allow us to validate the robustness and accuracy of the \emph{Sketch $\star$-metric}.