## 4 Experimental results
### Distributed principal component computation  
We finally discuss the ability of the power iteration method to properly identify the principal components. The main parameter for the estimation of the eigenvectors is the convergence criteria used. We illustrate in Fig. 13 the difference in accuracy obtained on the test set, for different convergence criteria,between the set of exact eigenvectors (Computed in a centralized manner with the QR method), and the set of approximated eigenvectors obtained by means of the power iteration method. The convergence threshold \(\delta\) was set to \(10^{-3}\), and we tested the accuracy obtained for \(5\), \(10\), \(20\), \(30\), \(40\) and \(50\) iterations. Results reported are averages of ten-fold cross validations, and the confidence level of the results was about \(2\%\).  
This Figure allows to experimentally illustrate a set of possible behaviors, which are understood by keeping in mind that the rate of convergence of the power method depends on the ratio of the two dominant eigenvalues. Typically, in correlated data, the ratio of subsequent eigenvalues decreases exponentially (which can be seen qualitatively in Fig. 7), making the convergence speed lower as the number of principal components computed increases. As a result, few iterations are usually enough to converge to the first eigenvectors. This is seen in Fig. 13 for the first PC, for which as little as five iterations are enough to nicely converge. In the computation of the subsequent PCs, we observe that the number of iterations required to properly converge is higher (about 20 iterations led to accuracy similar to the centralized approach). If the number of iterations is not high enough, the power iteration leads to a PC that may not represent the data as well as the centralized approach, as is observed from the second PC for a maximum number of five iterations. Note that, given the fact that PCs are estimated on the basis of past measurements whose distribution is not exactly the same as upcoming measurements, it may also happen that approximations provide better accuracies than the centralized approach. This explains the gains in accuracy obtained around the \(8\)-th component where the difference between subsequent eigenvalues gets very low.  
Additional experiments were also run to observe the consequence of the local covariance hypothesis on the non positive semi-definiteness of the matrix. We observed that negative eigenvalues could lead the algorithm to stop at stages as early as between \(5\) to \(10\) components (this was observed for radio rangesof \(30\) and \(40\) meters). It is however important to note that despite the early stopping of the algorithm, the set of principal components identified was enough to retain more than \(90\%\) of the variance (cf. Fig. 11).  
Finally, we report in Fig. 14 estimates of the communication costs entailed by the computation of the principal components. The radio range was of \(10\) meters, and the costs were computed following the analysis of Sections 2.1.3 and 3.5. While about two hundred packets per node suffice for the computation of the first eigenvector, this quantity can reach a significant network load of \(6000\) packets on average for the computation of \(15\) components. As discussed in Section 3.5, the main advantage of the distributed computation of the principal components lies in the distribution of the computational and memory costs, and the approach does not scale well to the computation of a large number of principal components.