## 5 Results and Discussion
### The Comparative Performance of PRDFS and BRDFS in Detecting Malicious Nodes
#### 5.2.2 Further Observations  
We attribute the above observations to the following characteristics of PRDFS and BRDFS:  
* PRDFS uses a supervised intrusion detection system (SIDS) for detecting malicious nodes.
* SIDS learns from the input and output data sets that are contamination-free since they are generated through the game theoretic approach. It employs a supervised learning algorithm (GRNN) to distinguish between malicious and normal nodes.
* It can detect malicious nodes even when all nodes are malicious because it compares the present behaviors of the nodes in respect to dynamic spatio-temporal patterns of trustworthiness and the skewness of residual distribution with previous correct behaviors.
* BRDFS uses an unsupervised anomaly-based intrusion detection system (AIDS) for detecting malicious nodes.
* AIDS learn from unlabeled real data that are contaminated with many undetected malicious nodes. Hence, its learning is not very effective.
* AIDS learn under the assumption that statistically rare or atypical behaviors are abnormal and are possible signs of maliciousness, whereas typical or common behaviors are normal. Hence, AIDS can only detect malicious nodes when the percentage of malicious nodes is low, and the malicious nodes are operating for a short period of time where it is not possible to inject a significant number of malicious instances in the data set. However, stealthy malicious nodes are highly difficult to detect at very early stage.
* The skewness detector and the temporal collusion detection scheme of CAD uses confidence interval-based anomaly detection. If a node's behavioral traits fall outside the confidence interval, the node is detected as malicious. However, such a strategy is ineffective against collusion attacks. In collusion attacks, the adversary takes control of multiple sensors and changes the readings of these sensors to the desired upward or downward direction in such a way that the malicious nodes always lie within the confidence intervals of correlation coefficients and skewness coefficients. One of the major challenges in time series forecasting is that the distribution of the time series changes over time. Hence, the forecasting model needs to be adapted to the most recent data--the adversaries take advantage of this situation. Returning the upper or lower extreme of the acceptable range as the readings of the compromised sensors, they can gradually move the forecasting model in the wrong direction without getting caught. Since the forecasting model is evolving towards the false data, over time, the non-compromised sensors are marked as compromised one after another and will lose all influence over the model's evolution, thereby accelerating the compromise rate of the entire forecasting system.
* The spatial collusion detector of the CAD in BRDFS uses unsupervised clustering to separate malicious nodes. These clustering algorithms are based on arbitrary rules. Thus, it is natural that its accuracy is not as good as PRDFS.  
The Comparative Performance of PRDFS and BRDFS in Estimating the Most Trustworthy Measurement of the Current Aggregate Demand \(\left(\overline{D(t)}\right)\)