  Hate speech is ubiquitous on the Web. Recently, the offline causes that
contribute to online hate speech have received increasing attention. A
recurring question is whether the occurrence of extreme events offline
systematically triggers bursts of hate speech online, indicated by peaks in the
volume of hateful social media posts. Formally, this question translates into
measuring the association between a sparse event series and a time series. We
propose a novel statistical methodology to measure, test and visualize the
systematic association between rare events and peaks in a time series. In
contrast to previous methods for causal inference or independence tests on time
series, our approach focuses only on the timing of events and peaks, and no
other distributional characteristics. We follow the framework of event
coincidence analysis (ECA) that was originally developed to correlate point
processes. We formulate a discrete-time variant of ECA and derive all required
distributions to enable analyses of peaks in time series, with a special focus
on serial dependencies and peaks over multiple thresholds. The analysis gives
rise to a novel visualization of the association via quantile-trigger rate
plots. We demonstrate the utility of our approach by analyzing whether Islamist
terrorist attacks in Western Europe and North America systematically trigger
bursts of hate speech and counter-hate speech on Twitter.
