## 2 Emergency Representation Model
### Geographical Event Analysis
#### 2.3.1 Parzen Windows  
In 1962, [PERSON] introduced a nonparametric method that can be used for estimating PDFs [14]. In our case, if we consider that the set of events is an event series of independent and identically distributed observations, then the estimated probability by Parzen windows is denoted \(p(\vec{r})\), and by constraining it to \(N\) observed samples, we have that the empirical probability distribution function is given by  
\[\hat{p}_{\varepsilon}(\vec{r}|mT)=\frac{1}{N}\Delta(\vec{r}|mT) \tag{9}\]  
Now, we can use a kernel function to give a continuously supported estimation of the PDF. For instance, the use of the Gaussian kernel often yields a good-quality estimator by virtue of the central limit theorem [15], and in this case, the estimated distribution is expressed as  
\[\hat{p}_{C}(\vec{r}|mT)=\frac{1}{N(2\pi)^{d/2}\sigma^{d}}\sum_{j\in I_{i,mT}} e^{-\frac{|\vec{r}_{j}|^{2}}{2\sigma^{2}}} \tag{10}\]where \(d\) is the dimensionality of the Gaussian kernel feature space using a common variance \(\sigma^{2}\); this is a free parameter to be chosen according to the observation scale and the number of events.  
At this point, we can define the space slices for a specific time. However, we give an example in one dimension in order to better understand this proposal. For a one-dimensional and single-mode random variable \(x\), with known distribution \(f_{X}(x)\), we can define its CI, with confidence level (CL) \(1-\alpha\), in terms of the area integrated by the portion of \(f_{X}(x)\) larger than a given threshold \(u\), as follows. The CI \((x_{t},x_{h})\) is the set of points fulfilling  
\[\int_{x_{\alpha}}^{x_{h}}f(x|f\geq u)\ dx=1-\alpha \tag{11}\]  
which can be readily extended to multimodal distributions. We define the limiting points as those accounting for the boundary in the \(x\) domain that ensure that the CL is \(1-\alpha\) when integrating the distribution function when it is larger than a threshold. Thus, we can define the space slice \(\xi(T,\alpha)\) as the set of points in the spatial domain for which the estimated density function integrates \(1-\alpha\) of its area, for the events in a given period of time; that is,  
\[\xi(T,\alpha)=\vec{r}_{\xi}\left|\int_{D(\vec{r}_{\xi})}\hat{\rho}_{G}(\vec{r},T|\hat{\rho}_{G}\geq u)\ d\vec{r}\ =1-\alpha\right. \tag{12}\]  
where \(D(\vec{r}_{\xi})\) denotes the bidimensional domain enclosed by the boundary points in a path given by \(\vec{r}_{\xi}\)[16].  
We can think of a Parzen window density estimation in our problem as a data interpolation technique that uses the emergency's coordinates (i.e., latitude and longitude) to estimate a PDF [14]. This way, it allows us to graph a bidimensional surface with its higher amplitudes over the densest area of the represented events. We can define a CL, for example, 40% or 20%, as shown in Figure 3, and visualize the resulting contour where the plane \(P\) intercepts the PDF's generated surface.  
Figure 3: A confidence level (CL) visualization shows the contour that results from the intersection of the Parzen solid representation and the settled horizontal plane, as seen in (**a**); (**b**,**c**) contour visualization for CL = 40% and CL = 20%.