Figure 6

Parameters Sweep. The shock indicator function is relatively insensitive to functional forms \(\mathcal{K}^{(\cdot )}\) and values of the kernel’s parameter vector θ so long as the kernel functions are qualitatively similar (e.g., for cusp-like dynamics—as considered in this figure and in Eq. (10)—\(\mathcal{K}^{(C)}\) displaying increasing rates of increase followed by decreasing rates of decrease). Here we have computed the shock indicator function \(\mathrm{C}_{\mathcal{K}^{(S)}}(\tau |\theta )\) (Eq. (12)) for three different time series: two sociotechnical and one null example. From left to right, the top row of figures displays the rank usage time series of the word “bling” on Twitter, the price of the cryptocurrency Bitcoin, and a simple Gaussian random walk. Below each time series we display parameter sweeps over combinations of \((\theta , W_{\max })\) for two kernel functions: one kernel given by the function of Eq. (10) and another of the identical form but constructed by setting \(\mathcal{K}^{(S)}(\tau |W, \theta )\) to the function given in Eq. (1). The \(\ell _{1}\) norms of the shock indicator function are nearly invariant across the values of the parameters θ for which we evaluated the kernels. However, the shock indicator function does display dependence on the maximum window size \(W_{\max }\), with large \(W_{\max }\) associated with larger \(\ell _{1}\) norm. This is because a larger window size allows the DST to detect shock-like behavior over longer periods of time