## 2 Materials and Methods
### Information Fusion in Chain-Based IOT Platform using Proposed Method  
#### 2.8.1 Background  
We first consider a chain-based information fusion technique where all nodes are organized into a linear chain. Each node fuses its own measurement and received information from the nearest neighbor as shown in Figure 1.  
The resultant information is then quantized and transmitted to the nearest neighbor in downstream. The information fusion process continues, until the sink node receives the information. We assume that nodes are placed in equal distances. Consider the \(k\)th node in the chain, it has only one child and that child node is represented by \(k-1\). The linearly scaled received information from the child node is denoted by \(r^{{}^{\prime}}_{k-1}\), which is given using Equation (23),  
\[r^{{}^{\prime}}_{k-1}=(\theta+n_{k-1}+w_{k-1})+\gamma_{k-1}/(1-2p_{k-1}) \tag{23}\]  
Figure 1: Chain-based network topology.  
Here, \(w_{k-1}\) is the quantization noise and the mean of the quantization noise is equal to zeros and its variance is upper bounded by \(\delta^{2}_{k-1}\), which is equivalent to \(W^{2}/(2^{l_{k-1}}-1)^{2}\). \(\gamma_{k-1}\) has zero mean and variance \(\epsilon_{k-1}^{2}=(4W^{2}(2^{L_{k-1}}+1)p_{k-1})/3(2^{L_{k-1}}-1)\). The \(k\)th node in the chain fuse \(\gamma_{k-1}^{{}^{\prime}}\) and its own measurement \(s_{k}\) together. As the mean and the approximated variances of the \(\gamma_{k-1}^{{}^{\prime}}\) and \(s_{k}\) are available, the weighting property of the BLUE estimation [52] can be used to fuse the data to provide the fusion estimate given by Equation (24) as,  
\[\hat{\theta}=(1/\sigma_{r_{k-1}}^{2}+1/\sigma_{s_{k}}^{2})^{-1}(r_{k-1}^{{}^{ \prime}}/\sigma_{r_{k-1}}^{2}+s_{k}/\sigma_{s_{k}}^{2}) \tag{24}\]  
\(\sigma_{r_{k-1}}^{2}\) is the approximated variance of the \(r_{k-1}^{{}^{\prime}}\). The variance of the quantized message at \(k\)th node is obtained using the inequality Equation (25) as,  
\[\sigma_{\tilde{\theta}_{k}}^{2}\leq\delta_{k}^{2}+(1/\sigma_{r_{k-1}}^{2}+1/ \sigma_{s_{k}}^{2})^{-1} \tag{25}\]  
Inequality Equation (25) can be further simplified to Equation (26) and the details are given in Appendix A (\(k=2\) scenario).  
\[\sigma_{\tilde{\theta}_{k}}^{2}\leq\delta_{k}^{2}+(\sigma_{r_{k-1}}^{2}/4)+( \sigma_{s_{k}}^{2}/4) \tag{26}\]  
Equation (26) can be further expanded using a similar approach in [53] as given in inequality Equation (27).  
\[\begin{split}\sigma_{\tilde{\theta}_{k}}^{2}&\leq( 1/4)^{k-1}\sigma_{s_{1}}^{2}+\sum_{j=2}^{k}{(1/4)^{k-j+1}\sigma_{s_{j}}^{2}}\\ &+\sum_{j=1}^{k}{(1/4)^{k-j}\delta_{j}^{2}}+\sum_{j=1}^{k-1}{(1/4 )^{k-j}(\epsilon_{j}^{2}/(1-2p_{j})^{2})}\end{split} \tag{27}\]  
Equation (27) is an analytical upper bound for the variance of the quantized resultant information at any sensor node k in the chain. Substituting k by K, an analytical expression for variance at the sink node can be obtained.  
Only the knowledge of the mean and the approximated variances of the \(\gamma_{k-1}^{{}^{\prime}}\) and \(s_{k}\) is required for this fusion technique. The knowledge on Probability Density Functions (PDFs), which was used to obtain such noise variances, is not required in generating the fusion function. For instance, the output of the fusion function for same mean and variance of \(\gamma_{k-1}^{{}^{\prime}}\) and \(s_{k}\) which can be generated using any distribution such as Gaussian, Normal, Poisson, Chi-Square, etc. will have the same fusion result. Therefore, the proposed method does not require knowledge on PDF for information fusion.