  This paper investigates the benefits of the side information on the universal
compression of sequences from a mixture of $K$ parametric sources. The output
sequence of the mixture source is chosen from the source $i \in \{1,\ldots
,K\}$ with a $d_i$-dimensional parameter vector at random according to
probability vector $\mathbf{w} = (w_1,\ldots,w_K)$. The average minimax
redundancy of the universal compression of a new random sequence of length $n$
is derived when the encoder and the decoder have a common side information of
$T$ sequences generated independently by the mixture source. Necessary and
sufficient conditions on the distribution $\mathbf{w}$ and the mixture
parameter dimensions $\mathbf{d} = (d_1,\ldots,d_K)$ are determined such that
the side information provided by the previous sequences results in a reduction
in the first-order term of the average codeword length compared with the
universal compression without side information. Further, it is proved that the
optimal compression with side information corresponds to the clustering of the
side information sequences from the mixture source. Then, a clustering
technique is presented to better utilize the side information by classifying
the data sequences from a mixture source. Finally, the performance of the
clustering on the universal compression with side information is validated
using computer simulations on real network data traces.
