## 4 Data Aggregation
### Spatial
As a hierarchical approach, CH performs the first level of data aggregation as Local Aggregators (LA) and a subset of LAs forms the next level of data aggregation, called Master Aggregators (MA). The authors provide two versions of the problem: a two-level scheme and a multi-level scheme. The only difference of two schemes is regarding the data aggregation at MAs level. While in two-level scheme, MAs are not allowed to aggregate the received messages from other MAs, in multi-level scheme MAs are able to do so. GRASS tries to find out the optimal route with selection of the minimum data aggregation points to minimize the maximum power consumption at each LA node by integer linear program formulation and genetic heuristic approach. According to the analytical calculation and simulation results presented by the paper, the network achieves its maximum lifetime extension factor under two-level scheme when the number of MAs reaches the half of the number of selected LAs. In terms of energy-delay tradeoffs a two-level aggregation scheme imposes less reporting delay than a multi-level aggregation scheme, however, it decreases network lifetime contrarily. To improve the spatial credibility of the aggregated data at CHs, [PERSON] _et al._[110] introduces fault probability to map dependence weight of each sensor node. According to the protocol, the fault probability of each sensor node is estimated by using a Bayesian Belief Network (BBN) [111], while higher fault probability maps lower dependency weight and _vice versa_. Therefore, a CH by having the fault probability information of its cluster members generates a mapping function of the dependency weight of the cluster nodes, and forwards the aggregated data to the BS only if the accumulated dependency weight of an event collected from sensor nodes in a cluster exceeds a threshold value. This threshold value is calculated by each CH according to the average weight in the cluster, the average number of nodes that sense the event and the average hop counts in a cluster. To answer the special event situations that the accumulated weight of an event is below the threshold value and there is no exactly same event received, the paper also proposes another threshold, called adaptive threshold. Using the adaptive threshold, CH exchanges the average number of sensor nodes that sensed the event into the number of source nodes that the CH received so far. Although the proposed algorithm improves the credibility of the aggregated data, it increases the delay time and memory storage due to the fault information. In this respect, a waiting time is proposed, which starts to count down by receiving of the first event and is formulated via the maximum hop counts in cluster, average computing time, average sensing time and average transmission time. In Unbalanced clustering (UBC) [112], the significance of the optimal cluster radius according to the aggregation characteristic in a correlated data field is investigated. Given the decrease of the spatial correlation of the generated data of environmental factor in real world along the space field, the main idea of the paper is to find the optimal cluster radius to achieve the most efficient data aggregation ratio. To exploit correlation as an independent coefficient, UBC uses the Discrete Cosine Transform (DCT) [113] as a linear transform, in which statistically spatial-dependent data are mapped into a set of more independent coefficients. Using the statistical results presented by the paper, the trend of the aggregation ratio _versus_ cluster radius R is a typical exponential graph. Therefore, the authors approximate the aggregation ratio \(Ag_{a}\) in Equation (20):  
\[Ag_{a}=(1-r\_{min})e^{-\pi R^{2}\rho_{d}\ /\ \delta_{c}\ }+r\_{min} \tag{20}\]  
where \(\rho_{d}\) is the node density in the field, \(\delta_{c}\) is a constant factor that manages the dropping speed of aggregation ratio and \(r\_{min}\) is the minimum aggregation convergence point obtained from the statistical results.