  In the time-decay model for data streams, elements of an underlying data set
arrive sequentially with the recently arrived elements being more important. A
common approach for handling large data sets is to maintain a \emph{coreset}, a
succinct summary of the processed data that allows approximate recovery of a
predetermined query. We provide a general framework that takes any
offline-coreset and gives a time-decay coreset for polynomial time decay
functions.
  We also consider the exponential time decay model for $k$-median clustering,
where we provide a constant factor approximation algorithm that utilizes the
online facility location algorithm. Our algorithm stores
$\mathcal{O}(k\log(h\Delta)+h)$ points where $h$ is the half-life of the decay
function and $\Delta$ is the aspect ratio of the dataset. Our techniques extend
to $k$-means clustering and $M$-estimators as well.
