# Gaussian Processes for Signal Processing and Representation in Control Engineering
## 6 Decision Support for Bike-Sharing System
The censoring problem was implemented, referring to the Tobias model [34], which can be simplified to: for each observation \(i\), the model assumes that there is a latent (e.g., unobservable) variable \(y_{i}^{*}\) and an observable censored realization \(y_{i}\). The real demand is represented by \(y_{i}^{*}\) while the available historical observations are represented by \(y_{i}\). The simple form of the Tobit model is a dependency of variables on input data \(x\) in a linear relationship with parameters \(\beta\) with the addition of error \(\varepsilon_{i}\), which we assume to be normally distributed (Formula (27)) [33]. \[y_{i}^{*}=\beta\mathbf{x}_{i}+\varepsilon_{i},\quad\varepsilon_{i}\sim N\Big{(}0, \sigma^{2}\Big{)}. \tag{27}\]  
The Gaussian Process, as mentioned before, is used for the prediction of demand. The authors note that, one of the most important values which comes from GP is not only the prediction on its own, but also its confidence interval. They also focus on three different kernels: Squared Exponential Kernel (SE), Periodic Kernel, Matern Kernel, which are used in the project. Moreover, the optimization methods based on the maximizing likelihood, which was mentioned earlier in this review, were used to optimize hyperparameters [33]. Figure 4: Prediction framework used for solving the bike-sharing system problem consists of various algorithms, which work together. The point of such a complex algorithm is to inform us ahead of time about the possibility of crowding the station and allow us to reposition bikes in advance. Bike data gathered from the whole city are firstly divided into clusters (by city segments, stations etc.) by AdaTC Clustering method. Based on this, a 3-level hierarchy is created and then meteorology data are added in order to create the Hierarchical Consistency Prediction (HCP) model. This with the combination of data about meteorology, time and particular events led to the creation of GP, the main purpose of which is to predict the check-outs of bike stations (count and time of renting a bike). With Transition based Inference (TINF) and ride time data, check-in times are inferred (count and time of returning a bike). Offline processes have a white background while online processes have greyed one [32]. What distinguishes this approach is combinations of GP and censoring problem, which gives us Censored Gaussian Processes (CGP). CGP have to deal with a censored regression problem and simultaneously be able to exploit GP flexibility to fit complex patters. It can be done by defining a censoring-aware likelihood within a GP framework, using it instead of standard GP likelihood. At the end, experiments on datasets from Donkey Republic (bike-sharing provider in Copenhagen) were conducted. Results showed that taking into account the limiting effect of supply on demand is very important for creating an unbiased predictive model of user demand behavior. Moreover, GP is considered a great way to solve a problem, because of its capability of representing an elegant framework which is able to work in a fully probabilistic way and to deal with uncertainty in predictions [33].