Aggregate monitoring over data streams is attracting more and more attention in research community due to its broad potential applications. Existing methods suffer two problems, 1) The aggregate functions which could be monitored are restricted to be first-order statistic or monotonic with respect to the window size. 2) Only a limited number of granularity and time scales could be monitored over a stream, thus some interesting patterns might be neglected, and users might be misled by the incomplete changing profile about current data streams. These two impede the development of online mining techniques over data streams, and some kind of breakthrough is urged. In this paper, we employed the powerful tool of fractal analysis to enable the monitoring of both monotonic and non-monotonic aggregates on time-changing data streams. The monotony property of aggregate monitoring is revealed and monotonic search space is built to decrease the time overhead for accessing the synopsis from O(m) t...