# On IoT-Friendly Skewness Monitoring for Skewness-Aware Online Edge Learning
## Appendix A Opening the Developed Methods
\[\mathbb{V}_{n+1}=\mathbb{V}_{n}+(x_{n+1}-\bar{x}_{n+1})(x_{n+1}-\bar{x}_{n}) \tag{18}\]1 \[\mathbb{V}_{n+1}=\mathbb{V}_{n}+(x_{n+1}-\bar{x}_{n+1})(x_{n+1}-\bar{x}_{n}) \tag{18}\]2 \[\mathbb{V}_{n+1}=\mathbb{V}_{n}+(x_{n+1}-\bar{x}_{n+1})(x_{n+1}-\bar{x}_{n}) \tag{18}\]3 \[\mathbb{V}_{n+1}=\mathbb{V}_{n}+(x_{n+1}-\bar{x}_{n+1})(x_{n+1}-\bar{x}_{n}) \tag{18}\]4 \[\mathbb{V}_{n+1}=\mathbb{V}_{n}+(x_{n+1}-\bar{x}_{n+1})(x_{n+1}-\bar{x}_{n}) \tag{18}\]5 \[\mathbb{V}_{n+1}=\mathbb{V}_{n}+(x_{n+1}-\bar{x}_{n+1})(x_{n+1}-\bar{x}_{n}) \tag{18}\]6 On the other hand, \[\mathbb{V}_{n+1}=\mathbb{V}_{n}+(x_{n+1}-\bar{x}_{n+1})(x_{n+1}-\bar{x}_{n}) \tag{18}\]7 (27) \[\mathbb{V}_{n+1}=\mathbb{V}_{n}+(x_{n+1}-\bar{x}_{n+1})(x_{n+1}-\bar{x}_{n}) \tag{18}\]8 \[\mathbb{V}_{n+1}=\mathbb{V}_{n}+(x_{n+1}-\bar{x}_{n+1})(x_{n+1}-\bar{x}_{n}) \tag{18}\]9 Then, \[\tilde{x}_{\lceil n\rceil_{k}}=\frac{\sum_{i=n-k+1}^{n}x_{i}}{k} \tag{15}\]0 \[\tilde{x}_{\lceil n\rceil_{k}}=\frac{\sum_{i=n-k+1}^{n}x_{i}}{k} \tag{15}\]1 (28) \[\tilde{x}_{\lceil n\rceil_{k}}=\frac{\sum_{i=n-k+1}^{n}x_{i}}{k} \tag{15}\]2 \[\tilde{x}_{\lceil n\rceil_{k}}=\frac{\sum_{i=n-k+1}^{n}x_{i}}{k} \tag{15}\]3To initialise the sample skewness \(\mathbb{S}_{[\bar{k}]_{k}}^{2}\) of the first \(k\) values before rolling the observation window, the total number of observations \(n\) must be at least equal to \(k\), while the rolling window \(k\) must be bigger than two to avoid the error of division by zero. This completes the proof of Equation System (7). Proof of Equation (8) for Calculating Exponentially Weighted SampleMean (EWSM): We refer to [36] to briefly rephrase the derivation of the EWSM formula. Similarly, the derivation starts from the discussion about CSM by rewriting (2) as:  
\[\tilde{x}_{\lceil n\rceil_{k}}=\frac{\sum_{i=n-k+1}^{n}x_{i}}{k} \tag{15}\]4  
It is clear that when the observation amount \(n\) approaches infinity, the new observation value \(x_{n+1}\) will barely play a part in the calculation of the sample mean, which violates the EWSM scenario where the more recent data are more valuable/significant. In contrast, by replacing the item \(\frac{1}{n+1}\) with a constant coefficient \(\alpha\) between 0 and 1, and correspondingly replacing \(\frac{n}{n+1}\) with \((1-\alpha)\), the contribution of older observations will become progressively smaller. Eventually, we obtain the equation for calculating EWSM as follows. \[\tilde{x}_{\lceil n\rceil_{k}}=\frac{\sum_{i=n-k+1}^{n}x_{i}}{k} \tag{15}\]5  
In particular, to avoid confusion with the notation \(\bar{x}_{n}\), we use \(\bar{x}_{w(n)}\) to denote the sample mean that involves all the existing weighting effects. Moreover, in this form, a bigger \(\alpha\) discounts older data faster. This completes the proof of Equation (8). Proof of Equation (9) for Calculating Exponentially Weighted SampleVariance (EWSV): Following the same strategy of drawing the EWSM formula, we start the derivation from rewriting the CSV formula (3) into:  
\[\tilde{x}_{\lceil n\rceil_{k}}=\frac{\sum_{i=n-k+1}^{n}x_{i}}{k} \tag{15}\]6  
and then:  
\[\tilde{x}_{\lceil n\rceil_{k}}=\frac{\sum_{i=n-k+1}^{n}x_{i}}{k} \tag{15}\]7  
There is no doubt that increasing observation amount \(n\) will decrease the contribution of new value \(x_{n+1}\) to the variance calculation. By fixing a constant coefficient \(\beta\) between 0 and 1 to replace \(\frac{1}{n}\), we obtain Equation (10) for calculating EWSV:  
\[\tilde{x}_{\lceil n\rceil_{k}}=\frac{\sum_{i=n-k+1}^{n}x_{i}}{k} \tag{15}\]8  
We use the notation \(s_{w(n)}^{2}\) to indicate the sample variance that involves the past weighting effects. Similar to \(\alpha\), a bigger \(\beta\) also discounts older data faster in this case. This completes the proof of Equation (9). Proof of Equation System (10) for Calculating Exponentially Weighted SampleSkewness (EWSS): Given the previous proofs for EWSM and EWSV, a natural deduction is that the EWSS formula can be obtained from the CSS formula by replacing its \(n\)-related parameters with suitable constant coefficients.