## 2 Emergency Representation Model
### Temporal Event Analysis  
From a mathematical point of view, we have a set of \(N\) observed events for all the institutions, and each event has been registered at time \(t_{j}\), \(j=1,\ldots,N\). We denote by \(j\in I_{i}\) the subset of events corresponding to the \(i\)th institution, where \(i=1,\ldots,L\), and \(L\) is the total number of institutions. According to this, and considering that each event occurs at a specific time and in a specific geographical location, we denote an event as \(\delta(t-t_{j},\vec{r}-\vec{r}_{j})\), where \(\delta(t,\vec{r})\) denotes the bivariate Dirac delta function [11]. Its first argument accounts for the time evolution, while the second accounts for the geographical location of each event. We note that \(\vec{r}\) denotes the coordinate vector of an event in general terms, which will be on the earth's surface, specifying the place of the event's occurrence and expressed in some adequate coordinate system. Thus, the original event series is given by the accumulation of shifted bivariate deltas:  
\[f(t,\vec{r})=\sum_{j=1}^{N}\delta(t-t_{j},\vec{r}-\vec{r}_{j}) \tag{1}\]  
whereas the set of events for the \(i\)th institution can be expressed as  
\[f_{i}(t,\vec{r})=\sum_{j\in I_{i}}\delta(t-t_{j},\vec{r}-\vec{r}_{j}) \tag{2}\]  
where \(j\in I_{i}\) denotes the set of indices belonging to the \(i\)th institution in the system. We can obtain an expression for the time evolution of the event series in \(M\) intervals, for the \(i\)th institution, by establishing a time interval width \(T\) and a spatial integration domain \(\Gamma(\vec{r})\). This is  
\[\Delta_{i}(t|T)=\sum_{m=1}^{M}g_{i,T,\vec{r}}(m)\delta(t-mT) \tag{3}\]  
where  
\[g_{i,T,\vec{r}}(m)=\int_{mT-T/2}^{mT+T/2}\iint_{\Gamma(\vec{r})}f_{i}(t,\vec{r })\ d\vec{r}\ dt \tag{4}\]  
and \(g_{i,T,\vec{r}}(m)\) is thus obtained as the number of events available in the dataset in the time period \(m\) (\(m=1,\ldots,M\)) and in the spatial domain \(\Gamma(\vec{r})\) when an observation time width \(T\) is used. With this simple event description in the temporal domain, time averages can be obtained and estimated from the available set of observations through  
\[\bar{\Delta}_{i}(t|T)=E[\Delta_{i}(t|T)] \tag{5}\]  
We note that \(E\) denotes here the theoretical statistical expectation; however, it is not available in our problem. Instead, we constrain ourselves to working with the empirical distribution implicitly when we consider events, and this empirical distribution, as given by the delta summations, is used to give an estimation of this value by obtaining the empirical average of the event sets in convenient time intervals of interest. A CI would be highly desirable for the estimated time averages, but in general these are not easy to calculate from parametric tests, as far as a priori distributions (such as Gaussianity) will likely not always be a reasonable assumption. To overcome this situation, we propose using a nonparametric estimation technique based on bootstrap resampling techniques [12; 13]. A bootstrap resample of a given list of events is obtained by generating a new list of events from sampling with replacement of the original list, yielding:  
\[f^{*}(t,\vec{r},b)=\sum_{j^{*}=1}^{N}\delta(t-t_{j^{*}},\vec{r}-\vec{r}_{j^{* }}) \tag{6}\]  
where the asterisk \({}^{*}\) indicates the usual notation in bootstrap resampling for statistical elements coming from the plug-in principle and resampling process, to distinguish them from the empirical ortheoretical statistical elements; \(t_{j^{*}}\) are the resampled event times, hence appearing zero, one, or several times in the resample; and \(b\) indicates the number of resamples that we are building.