## 4 Experimental results
### Communication costs  
We compare in the following the communication costs entailed by the default and PCAg schemes for different types of routing trees. Fig. 9 reports the total number of packets processed by nodes (received and transmitted) during an epoch, for the default and PCAg scheme, as the radio communication range increases. The subfigure on the left reports the overall sensor network load, while subfigures in the middle and on the right detail with boxplots the distributions of the network load per node in the network, for the default and PCAg schemes, respectively. Given that sensors choose as their parent the sensor within radio range that is the closest to the base station, increasing the radio communication range typically leads the routing tree to have a smaller depth, and its nodes to have a higher number of children. As was pointed out in Section 2.5, this is detrimental to the PCAg scheme, as this may lead the numbers of packets received by a node to be higher than in the default scheme. For the default scheme, increasing the radio range reduces the overall sensor network load (Fig. 9, left), and eventually leads all the nodes but the root to only transmit one packet (Fig. 9, middle, radio range of 50m). In this extreme case, the routing tree has depth one, and all nodes but the root are leaf nodes. Note that the highest network load does not depend on the tree topology. This highest load is sustained by the root node which, whatever the depth of the tree, is required to forward other node's packets (i.e., 51 measurements to receive and send), and to send its own measurement to send. Its network load is therefore of 103 packets per epoch. Figure 8: Approximations obtained on the test set for the sensor 21, using one, five and fifteen principal components. The covariance matrix was computed from the first twelve hours of measurements (before the vertical dashed line). For the PCAg scheme, we first report results for the extraction of one component. A nice feature of aggregation is that the overall network load (Fig. 9, left) does not depend on the topology, thanks to the fact that forwarding does not increase the number of transmissions. Looking at the details of the distribution of the network load per node (Fig. 9, right), it is interesting to see that increasing the radio range has a reverse effect for the PCAg scheme, as it tends to increase the network load. This is a direct consequence of the increased number of children induced by routing trees with smaller depths. Eventually, for a fully interconnected network, we observe the same effect than for the default scheme, where all the sensors send only one packet, while the root node sustains the higher network load due to the forwarding task. Note however than thanks to the aggregation process, it only sends one packet, which reduces to \(52\) packets per epoch its network load (one packet transmitted and \(51\) packets received). The extraction of one component therefore decreases the network load supported by the root node. The network load incurred by \(k\) components is obtained by multiplying the values found in Fig. 9 by \(k\). The PCAg scheme may therefore be less efficient than the the default scheme if many components are extracted. This is illustrated in Fig. 10 where the number of packets processed (received and sent) is reported as a function of the number of principal components extracted for a radio range of \(10\) (Illustrated in Fig. 6). In this routing tree, the highest number of children is \(6\). For the extraction of one PC, the highest  
Figure 9: Communication costs entailed by D and A operations. The overall network load (left) is in all cases higher for a D operation than for an A operation. Details of the distribution of the network load per node reported as boxplots (middle and right) reveal that the distributions of the load are more balanced using aggregation. Aggregation also leads to reduce the highest network load.