We show how to find regions of interest (ROIs) in trajectory databases. ROIs are regions where a large number of moving objects remain for at least a given time interval. Previous techniques use somewhat restrictive definitions for ROIs, and are parameter-dependent. They require sequential scanning of the entire dataset to find ROIs when the ROI parameters change. Our approach is parameter independent, so that the user can quickly identify ROIs under different parametric definitions without rescanning the whole database. We also generalize ROIs to be regions of arbitrary shape of some predefined density. We have tested our methods with large real and synthetic datasets to test the scalability and verify the output of our methods. Our methods give meaningful output and scale very well.

The current paper studies equity markets for the contagion of squared index returns as a proxy for stock market volatility, which has not been studied earlier. The study examines squared stock index returns of equity in 35 markets, including the US, UK, Euro Zone and BRICS (Brazil, Russia, India, China and South Africa) countries, as a proxy for the measurement of volatility. Results from the conditional heteroskedasticity long memory model show the evidence of long memory in the squared stock returns of all 35 stock indices studied. Empirical findings show the evidence of contagion during the global financial crisis (GFC) and Euro Zone crisis (EZC). The intensity of contagion varies depending on its sources. This implies that the effects of shocks are not symmetric and may have led to some structural changes. The effect of contagion is also studied by decomposing the level series into explained and unexplained behaviors.

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A region \(\mathcal {R} \) is a dwell region for a moving object O if, given a threshold distance r q and duration τ q , every point of \(\mathcal {R} \) remains within distance r q from O for at least time τ q . Points within \(\mathcal {R} \) are likely to be of interest to O , so identification of dwell regions has applications such as monitoring and surveillance. We first present a logarithmic-time online algorithm to find dwell regions in an incoming stream of object positions. Our method maintains the upper and lower bounds for the radius of the smallest circle enclosing the object positions, thereby greatly reducing the number of trajectory points needed to evaluate the query. It approximates the radius of the smallest circle enclosing a given subtrajectory within an arbitrarily small user-defined factor, and is also able to efficiently answer decision queries asking whether or not a dwell region exists. For the offline version of the dwell region problem, we first extend our online approach to develop the ρ -Index, which indexes subtrajectories using query radius ranges. We then refine this approach to obtain the τ -Index, which indexes subtrajectories using both query radius ranges and dwell durations. Our experiments using both real-world and synthetic datasets show that the online approach can scale up to hundreds of thousands of moving objects. For archived trajectories, our indexing approaches speed up queries by many orders of magnitude.