cumulative_ETAS_workbook.ipynb

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    "### Cumulative nETAS Workbook\n",
    "Following the example by John Wilson (2016-17 or so), we add an option to compute, rather than the instantaneous expected nETAS rate, the cumulative number of events (aftershocks) between $t_1 < t < t_2$. In practice, we add a second time parameter; if the second parameter is provided, we compute the definite integral of the Omori functiono, rather than its instantaneous value.\n",
    "\n",
    "One trick here is to correctly generalize the behaviors for $t_1$, $t_{cat}$, {t_2} -- the starting time, ending time of the catalog, and the final forecast time. We want, for example, to:\n",
    "- Forecast the expected number of events, over some finite period (say one week, one month) after today\n",
    "- Compute the cumulative nETAS count between $t_1 < t < t_2$, for which we have a catalog complete to $t = t_2$\n",
    "   - NOTE: This is a sort of funny number. We *know* the number of earthquakes, exactly for this time-space period. We can, however, use this a few different ways:\n",
    "       - A fast geo-spatial normalization (aka, estimate the partial number of events in the finite space wtihout actually computing the spatial distributions -- we assume that most of the events occur nearby).\n",
    "       - Generally, for normalization. The above can be used/interpreted as a normalization constant. We expect the preliminary estimate (with no catalog inside $t_1 < t < t_2$). We can use this computation to estimate the bias.\n",
    "    - Compute the evolving distribution: The expected number of events between $t_1 < t < t_2$ for a catalog complete to some time $t_1 < t' t_2$, perhaps to compare it in real-time to the $t<t_1$ estimate or retrospectively to the $t<t_2$ distribution.\n",
    "\n",
    "The last scenario is of particular interest, as we are considering it for a prediction/forecasting metric, sort of an advanced version of the RI/PI methods developed by Rundle, Holliday, Tiampo, et al.\n",
    "\n"
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      "The autoreload extension is already loaded. To reload it, use:\n",
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