## 3 QoS Correlation Analysis
#### Correlation Analysis of Empirical Measurements  
The correlation between the metrics was determined using both the Pearson Correlation Coefficient (PCC) and the Spearman Correlation Coefficient (SCC). The results indicated that the PCC between the QoS metrics was stronger than the SCC, indicating that the data had a stronger linear relationship than a monotonic one. As a result, for ease of analysis, only PCC was considered in the cross-correlation analysis [16]. Figure 1 is a heat-map displaying the Pearson correlation of the four QoS measures. All network operators were shown to generate similar correlation values between the QoS measures. The \"Delay-Jitter\" set of parameters has the highest correlation (dark red), followed by the \"Delay-Packet Loss\" and \"Jitter-Packet Loss\" combinations. The correlation between Delay and Jitter ranges from 69% to 79%, which is logical given that Jitter is the variability in delay between packets. One of the most important QoS pairs is \"Delay-Loss\", which showed correlation ranging from 33% to 50%. This correlation was expected to be higher based on the model  
Figure 1: Correlation Heat-Map between QoS Metric. of queuing theory, since in a simple queuing system, as packet delay increases, the buffer fills up and packets are lost once the buffer is full. This is explained by the method used to measure packet loss. As previously mentioned, probes were inserted into the network traffic in order to evaluate QoS. The server was unable to calculate the average delay when all 20 packets were lost because the packets never reached their destination. This created a data gap that was not accounted for in a scenario where probes were lost due to extreme delays. As a result, the correlation analysis produced a lower value than anticipated. Regardless, correlations of up to 50% may still be significant in QoE analysis, given that packet loss has a significant impact on the perception of any real-time applications [29]. The range of the \"Delay-Loss\" obtained in this analysis was much higher than the results obtained in [8], where the correlation of the same pair was determined to be 23%. Finally, none of the operators other than operator 4 demonstrates any significant correlation between Signal Strength and other QoS metrics. Even if it sounds counter-intuitive, the mobile signal intensity can be explained by comparing it to a switch or step function. Signal Strength is registered as 0 when a phone cannot send packets to a cell tower owing to network congestion or interference, regardless of the signal's strength (Off). When it can make a connection, the 'Signal Strength' is 1 (On) (On). Therefore, in the majority of the cases, the pinging of packet performance characteristics, such as delay and loss, is independent of the device's signal strength. This was the same conclusion reached by the authors in studies [30; 31] and explains this discrepancy due to the way RSSI is measured. According to these studies, RSSI in commercial Network Interface Controllers (NICs) is measured during the Physical Layer Protocol where before any data packets are sent, the headers and preamble are sent at low data rate. If the transmission of header and preamble fails, then the data packets are not sent giving RSSI a value of 0. But in case it does reach, regardless of the RSSI values, the data packets are sent at a stable rate. Hence, signal strength reading (RSSI) shows no variation in performance for different values. Next the delay distributions of each operator were generated from the measured data. First we obtained the probability density function (PDF) of delay distributions obtained from four operators' QoS data. This was then best fitted to an existing distribution based on Maximum Likelihood Estimation (MLE) and Information Criterion. It was discovered that the Log-Normal distribution best fits the data for all operators. This is the expected result given that packet latency over the internet is generally long [32].