## 2 Principal component aggregation in wireless sensor networks
### Tradeoffs  
In this section, we will use the term _default_ to denote the scheme that forwards all the measurements to the sink at each epoch, and the term _PCAg_ to denote the scheme based on the principal component aggregation. Let us first consider the distribution of the network load among sensor nodes, with a particular focus on its upper bound. The upper bound, henceforth referred to as _highest network load_, determines the network capacity, that is the amount of traffic that the network can handle at one time [6]. The highest network load is also related to the network lifetime in terms of time to first failure,2 as the radio is known to be the most energy consuming task of a wireless sensor [5, 25].  
Footnote 2: The time to first failure is the amount of time at which the first node in the network runs out of energy.  
In the _default scheme_, the root of the routing tree is the node that sustains the highest network load as it forwards to the base station the measurements from all other sensors. Its radio throughput therefore causes the root node to have its energy exhausted first, which may lead the rest of the network to be disconnected from the base station.  
In the _PCAg scheme_ (Section 2.1.3), the network load is distributed in a more homogeneous manner. The node \(i^{*}_{\mathcal{C}}=\operatorname*{arg\,max}_{i}C_{i}\) whose number of children is the highest becomes the limiting node in terms of highest network load. The network traffic at this node amounts to \(q(C_{i^{*}_{\mathcal{C}}}+1)\) packets per epoch, where \(q\) is the number of components retained. A PCAg scheme therefore reduces the highest network load in a configuration where  
\[q(C_{i^{*}_{\mathcal{C}}}+1)\leq 2p-1. \tag{7}\]  
We illustrate the tradeoff by considering two extreme configurations. If only the first principal component is retained, the PCAg reduces the highest network load as \(C_{i^{*}_{\mathcal{C}}}<p\). On the contrary, if all the components are retained, i.e. \(q=p\), the equation (7) simplifies into \(pC_{i^{*}_{\mathcal{C}}}\leq p-1\), and the PCAg incurs a higher network load than the default scheme.  
The main parameter of the approach is therefore the number of principal components to retain, which trades network load for sensing task accuracy. The amount of information retained by a set of \(q\) principal components depends on the degree of correlation among the data sources. Whenever a set of sensors collects correlated measurements, a small proportion of principal components is likely to support most of the variations observed by the network. The number of principal components to retain can be specified by the application, for example by means of a cross validation procedure on the accuracy of the sensing task at the base station [14]. It can be also be driven by network constraints or network congestion issues. In the latter case, a policy prioritizing packets containing the component scores of the principal components would allow to retain a maximum amount of information under network load constraints.  
Finally, given a number of principal components, note that the highest network load may be further reduced if the algorithm that builds the routing tree uses \(C_{i^{*}_{\mathcal{C}}}\) as a parameter. The network load indeed depends on the number of children a node has. Although we de not address this aspect, routing strategies that aim at lowering \(C_{i^{*}_{\mathcal{C}}}\) can therefore further improve the efficiency of the PCAg scheme.