# Gaussian Processes for Signal Processing and Representation in Control Engineering
## 5 Gaussian Process Usage in Terms of Electrical Energy
### Prediction of Battery Condition Parameters
The authors focused on a special GP variant known as the special Gaussian Process (SGP). SGP solves two problems of GP Regression: GPR accuracy dependence on the locations of training data points (its highly accurate near those datapoints) and computational complexity (of inversed kernel matrix). In SGP it is assumed that the training dataset is augmented with pseudo-inputs \(Z\) and the associated outputs \(u=f(K)\). SGP has been also extended by [PERSON] and [PERSON] [28] to deep Gaussian Process Regression. The main idea is to use the \(l\) layer as on input to the \(l+1\) layer. Both SGP and DGPR concepts can be seen in Figure 1. DGPR in fact is created by stacking some SGPs on top of each other. Layers between first input and last output are called \"hidden\". The last reviewed approach in [27] focuses on searching and estimating a special moment in the battery life cycle called \"knee point\". The algorithm is also different as it consists not only of the Gaussian Process but is also augmented with feature extraction and selection modules. The authors point out that not all batteries behave just like those from the provided datasets so the knee point can happen in different points that could be assumed. The proper indication of knee points is important in that it informs that the cell will soon need a replacement [27]. There also exist some other estimation methods (for instance approaching 80% of capacity), but they are mostly based on other searching principals such as point estimations instead of whole transients. The whole idea presentedin Figure 2 takes a special algorithm to extract features from the datasets (voltage, current etc. ), select a relevant subset of key features and finally feed the Gaussian Process regression model with it. The output should be a capacity fade trajectory estimation [27]. GP was poorly conditioned by initial settings. The hyperparameters were fitted to the training data in the standard way using maximum marginal likelihood estimation. The kernel was set to the Matern 2.5 function, because of previous research in [24; 25], where it has been shown that it works well. Moreover, contrary to previous examples, the authors used different datasets in their research. The general conclusion is that the whole idea makes practical sense. The accuracy is about half a day of time for achieving knee point (or 2.6% across 600 predictions). They also used different subsets of data, but it turned out that the more the better. Overall, the information on how all different features correlate with each other is extremely viable with the use of the accurate forecast of capacity, end of life and the knee point. Moreover, they suggest that the problem should still be developed for other approaches and deeper research of data correlation. Figure 1: Particular problem of predicting battery parameters such as State of Health and Remaining Useful Life was not solved but is a simple use of GP. In order to receive desirable solution, the authors enhanced GP into Deep GP (DGP), which uses principals of spatial GP (SGP). The figure shows the major differences in such approaches. To create an SGP, a standard GP dataset is augmented with pseudo-inputs \(z\) and the outputs \(u\) associated with them. DGP uses the SGP idea and stacks layers of outputs and pseudo-inputs on each other, which creates hidden layers \(h\). This solves problems of accuracy dependence on the locations of data points and computational complexity [28]. Figure 2: In the problem of predicting the knee point (point of drastic acceleration of battery condition degradation), the authors developed a strict algorithm which used a large amount of data as an input. To receive useful information from those datasets, they created extraction and selection algorithms for features. First, it has to only obtain the information necessary to build the model (like voltage, current etc.) and then specific features of it had to be selected in order to pass them to GP.