## 5 Results and Discussion
### The Comparative Performance of PRDFS and BRDFS in Detecting Malicious Nodes
#### 5.3.1 Major Findings  
* The demand forecast resilience is directly correlated to the accuracy in estimating current aggregate demand \(\left(\overline{D(t)}\right)\) that acts as the input signal while forecasting the demand time series (Figures 7 and 12).
* Since PRDFS detects and recovers all compromised nodes at the very early stages of games (within first 30-time steps) (Figure 10), the mean accuracy of estimating \(\left(\overline{D(t)}\right)\) in PRDFS is almost constant to around 98% over the time frame of the game (i.e., 1000-time steps) regardless of the percentage of the nodes compromised by the adversary (Figure 12).
* During the first 40-time steps of games, while the attack is still occurring or just ended, the mean \(\left(\overline{D(t)}\right)\) estimation accuracy in PRDFS decreases from around 98% to 83% as the percentage of malicious nodes increases. In contrast, the mean \(\left(\overline{D(t)}\right)\) estimation accuracy in BRDFS falls sharply over the simulation period (i.e., 1000 time-steps) as the percentage of malicious nodes increases.  
Figure 12: A comparison of accuracy between PRDFS and BRDFS for estimating current aggregate demand. Figure 12 depicts the prediction accuracy of current aggregate demands in PDRFS and BRDFS under the varying percentage of malicious sensor nodes. The accuracy is estimated based on the difference between the most trustworthy measurement of the current aggregate demand predicted by the TMS and the actual current aggregate demand (which is the simple average of the actual current measurements of the sensors). The X-axis represents the percentages of sensor nodes that were malicious in the simulation trial. The Y-axis represents the accuracy. The accuracy is estimated based on MAAPE. The green and blue lines plot the mean accuracy of PRDFS in respect to the mean current aggregate demand over 1000-time steps and over the first 40-time steps, respectively, under different percentages of malicious nodes. The red line plots the mean accuracy of BRDFS in terms of the mean current aggregate demand over 1000-time steps under different percentages of malicious nodes. The  
green line is almost flat to X-axis, indicating that the mean accuracy of mean current aggregate demand over 1000-time steps predicted by (the TMS of) PRDFS is not sensitive to the percentage of malicious nodes in the system. The blue line drops gradually as the percentage of almost flat to X-axis, indicating that the mean accuracy of mean current aggregate demand over 1000-time steps predicted by (the TMS of) PRDFS is not sensitive to the percentage of malicious nodes in the system. The blue line drops gradually as the percentage of malicious nodes increases, indicating that the mean accuracy of mean current aggregate demand over the first 40-time steps predicted by PRDFS is mildly sensitive to the percentage of malicious nodes in the system. The red line falls at an accelerated rate indicating that the mean accuracy of mean current aggregate demand over 1000-time steps predicted by BRDFS is highly sensitive to the percentage of malicious nodes in the system.