Temporal overlap, in the context of event intensities like those found in Hawkes processes, refers to the degree to which two or more events' influence periods coincide in time. Specifically, it can be quantified as the ratio of the shared area under their respective intensity functions to the total combined area under those functions.
Here's a breakdown:
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Hawkes Processes: These are point processes where the occurrence of an event increases the probability of future events. The intensity function represents this probability over time.
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Intensity Function: Represents the expected rate of events occurring at a particular time. In Hawkes processes, this rate increases after an event, then typically decays over time.
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Area under the Intensity Function: This area represents the cumulative influence or contribution of an event to the overall process.
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Temporal Overlap (Calculation):
- Imagine two events, each with its own intensity function.
- Identify the region where the intensity functions of these two events are active at the same time. This is the "overlap" region.
- Calculate the area of this overlap region.
- Calculate the total area under both intensity functions individually.
- The temporal overlap is the ratio: (Area of Overlap) / (Total Area under both Intensity Functions).
In simpler terms, temporal overlap assesses how much two events "influence" each other's timing by quantifying how much their respective periods of increased event probability coincide. A high temporal overlap suggests a strong interdependency, while a low overlap suggests more independent event occurrences.
