# Multiple Correspondence Analysis of Emergencies Attended by Integrated Security Services
## 3 MCA Statistical Characterization
### Bootstrap Resampling and Parzen Windowing  
We analyzed the statistical distribution of the projection of each column in the previous MCA formulation, which corresponded to the categories of the events contained in the DB. By usingBootstrap resampling techniques, we can obtain a cloud of points that implicitly contain their empirical distribution in the projected space, and the subsequent generation of confidence regions through Parzen windows allows the compact representation of those regions. In this way, regions of confidence that overlap will indicate significant statistical relationship among those categories analyzed, and regions of confidence not overlapping each will point towards independent event categories. A Bootstrap resample of a given list of statistical measurements [17; 18] is obtained by generating a new list of measurements from sampling with replacement of the original list [19]. In this case, if the operator yielding the coordinates of the projection of an illustrative column from the data matrix is denoted by \(\Gamma\), we can write it as follows:  
\[g_{sup}=\Gamma(\mathbf{X}) \tag{12}\]  
We can also denote its corresponding column eigenvalue as \(\mathbf{v}\), and denote by \(\Theta\) the operator yielding it from the data matrix, i.e.,  
\[\mathbf{v}=\Theta(\mathbf{X}) \tag{13}\]  
Now we denote the resampled data matrix \(\mathbf{X}^{*}\) as the result of sampling with repetitions its rows up to \(I\) times, so that it has the same size as the original data matrix. A bootstrap replication of a statistical measurement is obtained by calculating that measurement on the resampled population, instead of the original one, so that we can resample at a given iteration both the projection of a column and the eigenvector,  
\[g_{sup}^{*}(b)=\Gamma(\mathbf{X}^{*}(b)) \tag{14}\] \[\mathbf{v}^{*}(b)=\Theta(\mathbf{X}^{*}(b)) \tag{15}\]  
where 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 or theoretical statistical elements. Note that the resampled data matrix contains rows of the original ones appearing zero, one, or several times. In the previous equations, \(b\) indicates the number of resamples that we are building. After resample \(b\) is built, a Bootstrap replication can be built for the statistical distribution of any of the described statistical elements [12; 17]. The resampling and replication processes are repeated \(B\) times (typically \(B=50,100,200\), or \(500\) times), which provides an estimation of the empirical distribution function of the statistical element. This procedure can be extended to all the eigenvectors and to all the categories, which allows us to determine their empirical distribution. It is easy to construct confidence intervals for the eigenvectors, as will be seen in the next section. However, the multidimensional distribution of different projected categories is given by a set of point clouds, one cloud per category, which can not be readily visualized in this form. In 1962, [PERSON] introduced a non-parametric method that can be used for estimating probability density functions [20; 21]. In our case, we can consider that a subset of three projected categories (denoted compactly as \(R^{*}(b)=(g_{m,sup}^{*}(b),g_{n,sup}^{*}(b),g_{o,sup}^{*}(b))\)) is an event series of independent and identically distributed observations, then the empirical probability density function is  
\[p_{E}(R^{*})=\sum_{b=1}^{B}\delta(R-R^{*}(b)) \tag{16}\]  
We can use a kernel function to give a continuously supported estimation of the probability density function.