## 4 A New Approach
### An Example  
In this section, we present an example of the profiling function defined by Equation (3) works. Table 2 presents data about four users extracted from the case study presented in Section 6, grouped on the basis of their role in the calculus of Equations (5), (8), and (10) respectively. The same boundaries proposed [53] have been maintained:  
* a user CU is considered _capable_ iff \(index_{TOT}^{CU}\geq 0.70\);
* a user CU is considered _slow but gradual_ iff \(0.60\geq index_{TOT}^{CU}<0.70\);
* a user CU is considered _complicated_ iff \(0.50\geq index_{TOT}^{CU}<0.60\);
* a user CU is considered _static_ iff \(index_{TOT}^{CU}<0.50\).  
First of all, let's make some considerations common to all the users A, B, C, and D:  
* The denominator of Equation (5) is equal to 3600 MET;
* the denominator of Equation (8) is equal to 600 MET;
* the denominator of Equation (10) is equal to 48, being \(V_{SE_{MAX}}=8\times 8=64\) and \(V_{SE_{MIN}}=8\times 2=16\), given that in our case study \(n=8\) and \(j=2\), thus the maximum and minimum value for weekly \(SE\) are 8 and 2, respectively. About user A, we obtain the following value for \(index_{TOT}\):  
\[index_{TOT}^{A}=\frac{1-\frac{480}{3600}+\frac{421.25}{640}+1-\frac{22}{48}}{ 3}=0.70347222\]  
User A can be classified as _capable_, according to the clusters created by the CBR method. They demonstrated excellent physical performance, while their personal motivations showed some problems. The overall index seems able to capture these peculiarities, posing user A at the lower bound of the capable region proposed in Figure 1.  
The calculus of \(index_{TOT}\) for user B is as follows:  
\[index_{TOT}^{B}=\frac{1-\frac{840}{3600}+\frac{369}{600}+1-\frac{37}{48}}{3} =0.60638889\]  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & & & & & & & & \\ \hline \multicolumn{1}{c}{**Index Color Characterization based on Equation (5)**} & & & & & & & \\ \multicolumn{1}{c}{\(MET_{OR_{MIN}}\)**} & & & & & & & & \\ \multicolumn{1}{c}{\(MET_{OR_{MIN}}\)**} & & & & & & & & \\ \hline A & 120 & 240 & 360 & 480 & 360 & 480 & 480 & 600 \\ B & 120 & 240 & 360 & 360 & 360 & 480 & 480 & 360 \\ C & 120 & 240 & 240 & 240 & 360 & 480 & 360 & 240 \\ D & 120 & 120 & 240 & 120 & 120 & 240 & 120 & 120 \\ \hline \hline \multicolumn{1}{c}{**Physical Activity Characterization based on Equation (8)**} & & & & & & & \\ \multicolumn{1}{c}{\(MET_{HITE_{HITE_{HITE_{HITE_{HITE_{HITE_{HITE_{HITE_{HITE_{HITE_HITE_HITE_HITE_HITE_HITE_HITE User B presents very good physical attitudes, lacking in motivation: this is main reason why the app suggests them to maintain of reduce training goals when needed. Thus, the profile evaluation for this person is _slow but gradual_.  
User C is characterized on the basis of the following calculus:  
\[index^{C}_{TOT}=\frac{1-\frac{13200}{3600}+\frac{301.5}{600}+1-\frac{38}{48}}{3 }=0.531388889\]  
Physical performance decreases significantly with respect to previous users, both form the goal and the effective met points of view. From the psychological standpoint, the self-efficacy estimation maintains good standards, which is slightly higher than user B. This profile can be clustered as _complicated_, and the final value of the overall index confirms this opinion.  
Finally, user D is characterized by the following formula:  
\[index^{D}_{TOT}=\frac{1-\frac{2400}{3600}+\frac{122.5}{600}+1-\frac{29}{48}}{3 }=0.269444444\]  
User D shows significant physical and psychological problems; the overall performance is characterized by many failures and few successes, as pointed out by the very low value of the overall index variable. Moreover, in this case, we can conclude that user D has been correctly profiled as _static_.