# Resilience and Resilient Systems of Artificial Intelligence: Taxonomy, Models and Methods
## 5 Models and Methods to Ensure and Assess AIS Resilience
### Methods to Assess AIS Resilience  
There are various approaches to the formation of system resilience indicators [9; 12]. Of those, However, most studies are devoted to the analysis of resilience curves constructed in time coordinates and a system performance indicator, which describe the response of a resilient system to a destructive disturbing influence (Figure 10). The following basic indicators of system resilience were proposed in [9; 14; 15]:  
- Response Time, Tres;  
- Recovery Time, Trec;  
- Performance Attenuation, A;  
- Performance Loss, L;  
- Robustness, R;  
- Rapidity, \(\emptyset\);  
- Redundancy;  
- Resourcefulness;  
- Integrated measure of resilience, Re. Response Time (Tres) characterizes the timeliness of the response to a destructive disturbance. Systems with short response times are better at mitigating impacts, reducing performance degradation caused by disturbances. Recovery Time (Trec) is the period required to recover the system functionality to the desired level, at which the system can function in the same way, close to or better than before the disturbance. Performance Attenuation (A) describes the maximum reduction in system performance as a result of a disturbance, while Loss of Performance (L) characterizes the total loss of performance during the response and recovery phases. The loss of productivity is represented by the area highlighted in darker (green) in Figure 10.  
Robustness (Rb) characterizes the ability of a system to withstand a certain level of stress while maintaining functionality without significant deterioration or loss of performance. Robustness allows the system to absorb and resist destructive influences. A system with a high degree of robustness will retain most of its functional characteristics under the influence of destructive factors. Robustness can be defined as the residual functionality after exposure to an extreme destructive disturbance and can be calculated using the following formula  
\[\mathrm{R_{b}}=1-\widetilde{\mathrm{A}}(\mathrm{m_{A}},\mathrm{\sigma_{A}}), \tag{3}\]  
where \(\widetilde{\mathrm{A}}\) is a random variable expressed as a function of the mean value of \(\mathrm{m_{A}}\) and the standard deviation \(\mathrm{\sigma_{A}}\) for Performance Attenuation indicator. In [12; 130; 131], the resilience of AI is defined as robustness, not resilience in the full sense of the word. Rapidity (\(\theta\)) is the ability to recover functionality in a timely manner, limiting losses and avoiding future failures. Mathematically, the recovery rate is the slope of the performance curve during the recovery period (Figure 10), calculated by the formula  
\[\theta=\frac{\mathrm{d}\mathrm{C(t)}}{\mathrm{dt}}, \tag{4}\]  
where \(\mathrm{d}/\mathrm{dt}\) is the differentiation operator;  
\(\mathrm{C(t)}\) is a function that defines the dependence of performance on time. The average estimate of the Rapidity can be determined by the following formula  
\[\theta=\frac{\mathrm{A}}{\mathrm{T_{rec}}}. \tag{5}\]  
Figure 10: Illustration of the resilience curve and its key indicators. Redundancy characterizes the availability of alternative resources at the recovery stage when primary resources are insufficient. Redundancy is also defined as a measure of the availability of alternative paths in the system structure through which supportive forces can be transferred to ensure stability after the failure of any element [26]. Structural redundancy implies the availability of multiple supporting components that can withstand additional loads in the event of a failure of individual main components. That is, if one or more components fail, the remaining structure is able to redistribute the load and prevent the entire system from failing. Resourcefulness of the system is the ability to diagnose problems, prioritize and initiate problem solving by identifying and mobilizing material, financial, information, technological and human resources [11].