## 3 Target Functions
As mentioned earlier, a more accurate model could also include higher-order powers in the Xi variables or derivative terms. However, considering the almost complete absence of linear models similar to the one considered, we prefer not to further complicate the modelling in this context. The STF f1 function, by construction, does not address the second objective, i.e., the temporal and geospatial localization of information. To meet this requirement, a generalization of the function is necessary. In this regard, in (4), we introduce the following function, STF_2 = f2, defined as follows:  
\[f_{2}:T\times P\times A\ldots\times A\subset\mathbb{R}^{10}\to T\times S \times B\times C\subset\mathbb{R}^{5} \tag{4}\]  
where T represents the temporal domain and **P** the spatial domain defined using the two coordinates, latitude and longitude, provided using GPS. This function allows temporal and spatial resolution of events. From a computational perspective, we can find the function of 10 real variables in a five-dimensional space over a set of real numbers, i.e., a vector-valued function as follows:  
\[f_{2}=(f_{2,1},f_{2,2},f_{2,3}) \tag{5}\]  
where  
\[f_{2,1}=t \tag{6}\]with \(t\) being the temporal parameter belonging to the domain T,  
\[f_{2,2}=(long,\,lat) \tag{7}\]  
with (_long, lat_) being the GPS georeferencing coordinates,  
\[f_{2,3}=f_{1}. \tag{8}\]  
Through construction, the STF f2 function in (4) addresses the first two objectives. It could happen that the decision-maker does not provide complete information, i.e., information about all seven pointers. This could happen for various reasons, such as  
* There is a need for rapid intervention, so only the most significant alterations are reported;
* There are significant alterations in only CSFs, and the others are not reported;
* Etc. The results will be overly complex for the system, which will have to managed incompletely and using partial information. Therefore, to address this complexity, the model needs to account for it. Specifically, the system will associate a value of zero with the missing CSFs, which will automatically imply necessary freezing of the CSFs at the previously introduced values in the subsequent historical data. While this solves the problem of managing input, it does not provide an exhaustive response in terms of services to be prepared for monitoring and analysis. In other words, there is a need to address the third of the three themes considered in the three objectives to which the STFs must be subordinated. To control such aspects of information incompleteness, two output parameters are introduced into the target function, defined as proximity and completeness. Proximity represents the belonging of an output to a given scenario (e.g., equilibrium, explosive, implosive, etc. ), so  
\[V:A\times\ldots\times A\subset\mathbb{R}^{7}\to S_{c}\subset\mathbb{R} \tag{9}\]  
where \(S_{c}\subset\mathbb{N}\) is the space of admissible scenarios that we will describe later. Completeness provides a level of informational completeness regarding the seven CSFs, and it is defined as  
\[C:A\subset\mathbb{R}\to B\subset\mathbb{R} \tag{10}\]  
which computationally can be expressed as  
\[f_{1}=(E,S)=\bigg{(}\frac{1}{7}\sum_{i=1}^{7}\alpha_{i}X_{i},\frac{1}{7}\sum_ {i=1}^{7}\alpha_{i}\bigg{|}X_{i}-\widetilde{X}\bigg{|}\bigg{)} \tag{2}\]0  
where \(n\) is the number of CSFs which assume values other than zero. In this way, an answer to the third of the objectives identified at the beginning of the section is also provided. In conclusion, the complete target function to be considered will be STF_3 = f3 as formally described below in (12):  
\[f_{1}=(E,S)=\bigg{(}\frac{1}{7}\sum_{i=1}^{7}\alpha_{i}X_{i},\frac{1}{7}\sum_ {i=1}^{7}\alpha_{i}\bigg{|}X_{i}-\widetilde{X}\bigg{|}\bigg{)} \tag{2}\]1  
In other words, the achievement of the monitoring goal for alterations in the system is obtained through a vector-valued function, i.e., a function of 10 real variables (inputs) that returns a real vector with six components (outputs).