# On IoT-Friendly Skewness Monitoring for Skewness-Aware Online Edge Learning
## Appendix A Opening the Developed Methods  
Following the spirit of open methods that act \"as a higher-level strategy over open-source tools and open-access data\" to facilitate scientific work [20], we also share the proofs of the developed methods in this paper. In fact, the formula derivation details are particularly crucial in the context of running statistics, as discussed in Section 3.1.2. Proof. : _Proof of Equation (2) for Calculating Cumulative Sample Mean (CSM)_  
Proof. : Given a series of \(n\) observation values \(x_{1},x_{2},\ldots,x_{n}\), their arithmetic average \(\bar{x}_{n}\) can be calculated by Equation (11):  
\[\bar{x}_{n}=\frac{\sum_{i=1}^{n}x_{i}}{n} \tag{12}\]When a new observation value \(x_{n+1}\) arrives, the sum of the first \(n\) values can be substituted by \(n\cdot\bar{x}_{n}\) for calculating the arithmetic average of the \(n+1\) values, i.e.,  
\[\bar{x}_{n+1}=\frac{\sum_{i=1}^{n+1}x_{i}}{n+1}=\frac{\sum_{i=1}^{n}x_{i}+x_{n+1 }}{n+1}=\frac{n\cdot\bar{x}_{n}+x_{n+1}}{n+1}=\bar{x}_{n}+\frac{x_{n+1}-\bar{x} _{n}}{n+1} \tag{11}\]  
It is clear that \(\bar{x}_{0}\) is zero by default when no value is involved in the average calculation. This completes the proof of Equation (2). Proof of Equation System (3) for Calculating Cumulative Sample Variance (CSV): Given a series of \(n\) discrete random values \(x_{1},x_{2},\ldots,x_{n}\) as the sample dataset, the unbiased estimator for the population variance is:  
\[s_{n}^{2}=\frac{\sum_{i=1}^{n}(x_{i}-\bar{x}_{n})^{2}}{n-1} \tag{12}\]  
By introducing the auxiliary notation \(\mathbb{V}\), this can be rewritten as:  
\[\mathbb{V}_{n}=(n-1)\cdot s_{n}^{2}=\sum_{i=1}^{n}(x_{i}-\bar{x}_{n})^{2} \tag{13}\]  
When a new value \(x_{n+1}\) arrives, a straightforward modification of the unbiased estimator can be:  
\[\mathbb{V}_{n+1}=n\cdot s_{n+1}^{2}=\sum_{i=1}^{n+1}(x_{i}-\bar{x}_{n+1})^{2} \tag{14}\]  
Then, we calculate the differences between \(\mathbb{V}_{n+1}\) and \(\mathbb{V}_{n}\) via Equation (10). \[\mathbb{V}_{n+1}-\mathbb{V}_{n}=\sum_{i=1}^{n+1}(x_{i}-\bar{x}_{n +1})^{2}-\sum_{i=1}^{n}(x_{i}-\bar{x}_{n})^{2} =(x_{n+1}-\bar{x}_{n+1})^{2}+\sum_{i=1}^{n}[(x_{i}-\bar{x}_{n+1}) ^{2}-(x_{i}-\bar{x}_{n})^{2}] \tag{15}\] \[=(x_{n+1}-\bar{x}_{n+1})^{2}+\sum_{i=1}^{n}(2x_{i}-\bar{x}_{n+1}- \bar{x}_{n})(\bar{x}_{n}-\bar{x}_{n+1})\] \[=(x_{n+1}-\bar{x}_{n+1})^{2}+(\bar{x}_{n}-\bar{x}_{n+1})\sum_{i=1 }^{n}(2x_{i}-\bar{x}_{n+1}-\bar{x}_{n})\]  
Recall that \(\sum_{i=1}^{n}x_{i}=n\cdot\bar{x}_{n}\); we have:  
\[ \begin{split}\sum_{i=1}^{n}(2x_{i}-\bar{x}_{n+1}-\bar{x}_{n})=2 \sum_{i=1}^{n}x_{i}-n\cdot\bar{x}_{n+1}-n\cdot\bar{x}_{n}&=\sum _{i=1}^{n}x_{i}-n\cdot\bar{x}_{n+1}\\ &=(n+1)\cdot\bar{x}_{n+1}-x_{n+1}-n\cdot\bar{x}_{n+1}\\ &=\bar{x}_{n+1}-x_{n+1}\end{split}  \tag{16}\]  
Eventually,  
\[\mathbb{V}_{n+1}-\mathbb{V}_{n}=(x_{n+1}-\bar{x}_{n+1})^{2}+( \bar{x}_{n}-\bar{x}_{n+1})(\bar{x}_{n+1}-x_{n+1}) =(x_{n+1}-\bar{x}_{n+1})(x_{n+1}-\bar{x}_{n+1}-\bar{x}_{n}+\bar{x }_{n+1}) \tag{17}\] \[=(x_{n+1}-\bar{x}_{n+1})(x_{n+1}-\bar{x}_{n})\]  
i.e.,  
\[\mathbb{V}_{n+1}=\mathbb{V}_{n}+(x_{n+1}-\bar{x}_{n+1})(x_{n+1}-\bar{x}_{n}) \tag{18}\]  
The sample standard deviation can conveniently be obtained by taking the square root of the unbiased estimator \(s_{n+1}^{2}\). In addition, \(\bar{x}_{0}\) and \(\bar{s}_{0}^{2}\) are initialised as zero by default when no value is involved in the calculation. This completes the proof of Equation System (3). Proof. : Given \(n\) observations, the arithmetic average \(\tilde{x}_{\lceil n\rceil_{k}}\) of the most recent \(k\) values is:  
\[\tilde{x}_{\lceil n\rceil_{k}}=\frac{\sum_{i=n-k+1}^{n}x_{i}}{k} \tag{15}\]After observing a new value \(x_{n+1}\), the sample mean within the same size of rolling window can be updated as \(\bar{x}_{\lceil n+1\rceil_{k}}\) in a recursive manner.