# Decomposition Methods for Machine Learning with Small, Incomplete or Noisy Datasets
## 3 Results
### Scada Data Completion in Water Networks  
In industrial applications, usually having a Supervisory Control And Data Acquisition (SCADA) system collecting data, is habitual to have missing values due to several problems (sensor failures, communication loss, etc.). In this scenario, tensor completion algorithms can be used to reconstruct missing data providing a better alternative compared to classical interpolation methods. Because of the cyclic behaviour of data consumption, a tensor structure can be defined considering time scales (days and weeks). The redundancy present into the data is exploited by the tensor completion algorithms, allowing to recover missing data with greater accuracy outperforming classical interpolation or filtering methods. Drinking water network distribution enterprises usually have a SCADA system which manages the information collected by flow-meters, manometers, level sensors, valves, pumps, etc. By accessing to this centralized dataset, they are able to manage and optimize the operation of the water network. One of the major problems involved in managing long term SCADA data is to deal with the loss of bursts of data due to sensor failures, sensor re-calibrations, or communication failures that take time to be repaired and therefore cause the loss of entire bursts of data. Completing the data lost in bursts remains a difficult task, and most data completion methods that work fairly well when data are lost more or less evenly distributed over time, collapse in that situation. Water network data completion was investigated in [61; 62]. For this application, authors used data coming from the drinking water network distribution enterprise named Aigues de Vic S.A. (AVSA) which is the responsible for the water supply of the city of Vic, in Catalonia, where the heterogeneous data from the SCADA system are stored every 5 min in Structured Query Language (SQL) databases. To explore how tensor completion methods would behave in this scenario, a block of 77 consecutive weeks of data was selected. Then, some parts of the data were deleted to simulate missing burst. The Mean Square Error (MSE) per sample was used as a measure to check the accuracy of the reconstructed data. In [61], besides of taking advantage of the classical methods, an additional improvement was achieved by performing a tensorization of the data and applying tensor decomposition to recover missing data. This tensorization operation is illustrated in Figure 6.  
In [62], a new approach was developed that improved the performance of the previous method by performing two concatenated tensor decompositions. The new approach has several steps. The first step, consisted of applying a smoothing process to the signal to avoid oscillations between adjacent discrete values eventually produced around the point of quantification. The second step consisted of using a very rough imputation method (linear interpolation). After this operation, no empty values were present. Then tensorization that places the original burst positions precisely in the central positions, the burst centered tensorization, was performed (Figure 6a). The tensor obtained was used to make a low-rank tensor model to capture the background trend of the missing burst. The data of the low-rank model in the burst positions obtained was offset corrected and burst-centred again in a new tensor which was used to find a most refined model by employing more modes in a second tensor approximation. The samples \(\hat{x}_{i}\) occupying the positions of the burst were retrieved, offset corrected and became the output of the algorithm. The concatenated decompositions that optimize the results independently of the length of the burst to recover were the Tuker(4,6,1) followed by the Tuker(4,7,7) when using the Tucker model, and the CP(1) followed by the CP(15) when using the CANDECOMP/PARAFAC model.