## 4 Redundant Transmission Control Clustering Approach
### Data Collection at Cluster Heads
#### 4.4.1 Variance Study  
We perform some statistical tests to find the similarity in between the collected data. To perform these statistical tests, we assume that the variance is not substantial in all the correlated data sets. Consequently, the \(S_{out}\) is calculated using the statistical tests, while \(S_{out}\) is a ratio in variances which is dependent on the computed measurements. The correlated data sets are replicated each time if the value of \(S_{out}\) is found lesser than the threshold \(T_{DOF}\) values. Assumption and Definitions for Variance Test  
* Presume \(N=\{N_{1},N_{2},...,N_{n}\}\) express a set of FN generating a data set \(S=\{S_{1},S_{2},...,S_{n}\}\) in each slot. * Presume \(CHs=\{CH_{1},CH_{2},...,CH_{l}\}\) express a set of CHs, where \(l\leq n\) and the RH gathers \(n\) number of data sets from the MNs in its specified region. * Every time the collected data comprehends \(T\) number of measures. * We similarly assume that the collected data sets \(|S_{j}|\) are independent of number of measures the mean \(\overline{X_{i}}\) though \(\sigma_{n}^{2}=\sigma\). **Definition 1**. : _When two functions having the same measures are found in a set of FN generating a data set \(S=\{S_{1},S_{2},...,S_{n}\}\) can be defined as similar function and expressed as:_  
\[Similar(s_{i},s_{j})=\left\{ \begin{array}{ll}1&\quad\text{if }\|s_{i},s_{j}\|\leq\delta.\\ 0&\quad\text{Otherwise.}\end{array} \right. \tag{16}\]  
_where \(s_{i},s_{j}\in S\) and \(\delta\) are threshold values._  
**Definition 2**. : _The measurement weight \(s_{i}\) is co-occurrence of an alike function in a similar set._  
**Definition 3**. : _The cardinality of a data set \(S_{n}\) is equal to the number of elements in that data set._Definition 4: Weighted cardinality \(W_{card}\) of set \(S_{n}\) is equivalent to the measure's weight in the set \(W_{card}(S_{n})\) and the measure's variable can be expressed [9; 18] as:  
\[s_{ji}=\overline{X_{i}}+\epsilon_{ji};\qquad j=1,...,n\text{ and }i=1,...,|S_{i}|. \tag{17}\]  
where, \(\epsilon_{ji}\) is a residual, which follows the Normal distribution \(N(0,\sigma^{2})\). For the collected data sets \(|S_{j}|\), we symbolize \(\overline{X_{i}}\) as its mean, \(\sigma_{j}^{2}\) as its variance, and \(\overline{X}\) as its mean of the available data sets, respectively. \[\overline{X_{j}}=\frac{1}{W_{card}(S_{j})}\sum_{k=1}^{|S_{j}|}\big{(}s_{jk} \times W(s_{jk})\big{)}. \tag{18}\]  
\[\sigma_{j}^{2}=\frac{1}{W_{card}(S_{j})}\sum_{k=1}^{|S_{j}|}\bigg{(}\big{(}s_{ jk}-\overline{X_{j}}\big{)}^{2}\times W(s_{jk})\bigg{)}. \tag{19}\]  
\[\overline{X}=\sum_{j=1}^{n}\sum_{k=1}^{|S_{j}|}\frac{\big{(}s_{jk}\times W(s_{ jk})\big{)}}{W_{card}(s_{j})}. \tag{20}\]  
where, \(s_{jk}\in S_{j}\) and \(W(s_{jk})\) is measure's weight. Since, \(W_{card}(S_{1})=...=W_{card}(S_{i})=...=W_{card}(S_{n})=T\). \[\overline{X_{j}}=\frac{1}{T}\sum_{k=1}^{|S_{j}|}\big{(}s_{jk}\times W(s_{jk}) \big{)}. \tag{21}\]  
\[\sigma_{i}^{2}=\frac{1}{T}\sum_{k=1}^{|S_{j}|}\bigg{(}\big{(}s_{jk}-\overline{ X_{j}}\big{)}^{2}\times W(s_{jk})\bigg{)}. \tag{22}\]  
\[\overline{X}=\frac{1}{T}\sum_{j=1}^{n}\sum_{k=1}^{|S_{j}|}\big{(}s_{jk}\times W (s_{jk})\big{)}. \tag{23}\]  
Honestly Significant Difference (HSD) Test  
To find the similarity in the available data sets, we perform the HSD test [19] to compute the variances and means of the available data sets. Then, we are able to choose the similarity in the available data sets and later which can completely eliminated.