Finding critical thresholds for defining bursts in event logs

A burst, i.e., an unusally high frequency of occurrence of an event in a time-window, is interesting in many monitoring applications that give rise to temporal data as it often indicates an abnormal activity. While the problem of detecting bursts from time-series data has been well addressed, the question of what choice of thresholds, on the number of events as well as on the window size, makes a window "unusally bursty" remains a relevant one. We consider the problem of finding critical values of both these thresholds. Since for most applications, we hardly have any apriori idea of what combination of thresholds is critical, the range of possible values for either threshold can be very large. We formulate finding the combination of critical thresholds as a two-dimensional search problem and design efficient deteministic and randomized divide-and-conquer heuristics. For the deterministic heuristic, we show that under some weak assumptions, the computational overhead is logarithmic in the sizes of the ranges. Under identical assumptions, the expected computational overhead of the randomized heuristic in the worst case is also logarithmic. Using data obtained from logs of medical equipment, we conduct extensive simulations that reinforce our theoretical results, and show that on average, the randomized heuristic beats its deteministic counterpart in practice.