## 4 Experimental results
### Principal component aggregation  
We study in this section the amount of variance that the PCA can retain in the measurements for different training periods and number of components. We rely on a 10-fold cross validation technique to simulate the fact that the measurements used to compute the principal components are not used for assessing the accuracy of the approach. More precisely, the dataset is split in ten consecutive blocks (1440 observations - i.e., half a day of measurements). Each of the ten blocks is used in turn as a _training_ set to compute the covariance matrix and its eigenvectors, while the remaining observations, referred to as _test_ set, are used to estimate the percentage of retained variance. This provides ten estimates of the percentage of retained variance on measurements not used for computing the eigenvectors, which we average to obtain an overall estimate of the method.  
Fig. 7 provides the average retained variance on the 10 test sets for the first 25 principal components. The upper line gives the average amount of variance retained when the principal components are computed with the test sets, and provides an upper bound on the compression efficiency that the PCA can achieve on this dataset. The lower curve gives the average amount of variance retained on the test set when the components are computed with the training set. This figure shows that the first principal com  
Figure 6: Map of sensors and the routing obtained for a connectivity of 10 meters. The root node connecting the sink is the top right sensor. The routing tree has a depth of seven.  
ponent accounts on average for almost \(80\%\) of the variance, while \(90\%\) and \(95\%\) of variance are retained with \(4\) and \(10\) components, respectively. The confidence level of these estimates was about \(\pm 5\%\). Additional experiments were run using \(K\)-fold cross validation with \(K\) ranging from \(2\) to \(30\). The percentages of retained variance on the test data blocks tended to decrease with \(K\). Losses of a few percents were observed for \(K\) higher than \(15\) (less than nine hours of data).  
The amount of retained variance increases very fast with the first principal component, and becomes almost linear after about ten components. A linear increase of retained variance with the number of principal components reflects the fact that the components obtained by the PCA are actually no better than random components [28]. From Fig. 7, it therefore seems that from \(10\) or \(15\) components onwards, the remaining variations can be considered as white noise.  
Fig. 8 illustrates the approximations obtained during the first round of the cross validation (i.e., principal components are computed from the first 12 hours of measurements) for the sensor 49, using one, five and ten principal components. A single principal component provides rough approximations, which cannot account for fine-grained details of some of the sensor measurements. For example, the stabilization of the temperature around \(20^{\circ}\mathrm{C}\) around noon during the second, third and fourth day (probably due to the activation of an air conditioning system at a location close to sensor 49) are not captured by the approximations. Increasing the number of principal components allows to better approximate the local variations, and passing to five components provides for example a much better approximation of the sensor measurements.  
Figure 7: Capacity of principal components to retain the measurement variance.