# Multiple Correspondence Analysis of Emergencies Attended by Integrated Security Services
## 3 MCA Statistical Characterization
### Toy Example of Proposed Modified MCA  
In this section, we present a toy example in order to illustrate the basic elements of the adapted MCA and to give an idea of its theoretical background. Let us assume that we have three categorical variables (FRIs) with two categories each (attended emergencies). In total there are 6 categories that can be present in the same case, either individually or jointly. The generation of a toy example is done by assigning a probability of occurrence to each category, considering both individually and jointly among them in order to observe mutual dependences. For simplicity, the probability of individual occurrence for each variable is the same, and we set it as  
\[\mathrm{P}(V_{i}=cat_{ij})=0.155,\ \ \forall i,j \tag{18}\]  
whereas the joint-occurrence probability between pairs of categories is set to observe higher and lower probabilities, as follows,  
\[\mathrm{P}(V_{2}=cat_{2B},V_{1}=cat_{1B})=0.0275 \tag{19}\] \[\mathrm{P}(V_{2}=cat_{2B},V_{3}=cat_{3B})=0.0275 \tag{20}\]  
and  
\[\mathrm{P}(V_{1}=cat_{1A},V_{1}=cat_{1B})=0.005 \tag{21}\] \[\mathrm{P}(V_{1}=cat_{1A},V_{3}=cat_{3A})=0.005 \tag{22}\]  
In order to show the effect of new occurrences on the obtained eigenvectors, we propose to scrutinize two different situations. In Case 1 we introduce the next occurrence,  
\[\mathrm{P}(V_{1}=cat_{1B},V_{2}=cat_{2B},V_{3}=cat_{3B})=0.005\qquad\text{( Case 1)} \tag{23}\]  
which reinforces the simultaneous occurrence of the three members of the group created in Equations (19) and (20). In Case 2, we introduce the following type of occurrence,  
\[\mathrm{P}(V_{1}=cat_{1A},V_{2}=cat_{2A})=0.005\qquad\qquad\qquad\qquad\text{( Case 2)} \tag{24}\]  
and hence we create the second group of low probability by including one of the categories in the first group. Figure 2 shows an schematic of these tree different configurations and the results when using the proposed bootstrapping and Parzen techniques on the eigenvectors obtained from MCA. It can be observed that the group with larger occurrence probability (categories with points in green, red, and blue) overlap between them. Specifically, \(cat_{2B}\) (red points) is the linkage with the other two categories, and for this reason \(cat_{3B}\) (green points) does not overlap \(cat_{1B}\) (blue points). In Case 1, where the occurrence is joint among these three categories, we can check that they actually overlap, regardless of their low probability. We can also check that despite the categories in blue and green points have the same joint probabilities with the category in red points, the blue one is not so much overlapped as the green one. This is due to the existence of a non-null occurrence between this category and the dark blue. Figure 2: Schemes of the three situations to be scrutinized in the toy example (**a**,**c**,**d**), where \(V_{i}\) with \(i=1,\dots,6\) is the \(i^{th}\) variable and \(cat_{ij}\) with \(i=1,\dots,6\) and \(j\in\{A,B\}\) the \(j^{th}\) category corresponding to the \(i^{th}\) variable. The joint probability of occurrence is shown with a line of different thickness according to its value. The individual probabilities are obviated, as they are equiprobable. In (**b**,**d**,**f**) the projections of the original vectors onto the three most significant eigenvectors are depicted together with their Bootstrap-Parzen confidence regions for each category. In order to further scrutinize this separation effect of the variable in blue points, in Case 2 we can check that the category in blue asterisks tends to be about the center of the newly created group, whose members are the categories with asterisks and with blue points. Moreover, the category with blue points slightly separates apart from the categories with points, and it slightly closes to the category in blue asterisks. Figure 3a shows the analysis of the bootstrapped eigenvectors in terms of their confidence intervals. Vertical red lines correspond to those categories affecting (with a 95% confidence interval) to eigenvector \(v_{i},\forall i=1,\ldots,6\).