Online Parameter Selection for Gas Distribution Mapping

Kernel DM+V, one of the most common and well studied algorithms for gas distribution mapping, relies crucially on a set of parameters. These parameters are often estimated using cross-validation (CV), which is computationally expensive and therefore has to be carried out offline. Here we propose an efficient method for parameter selection, based on Virtual LeaveOne-Out Cross Validation (VLOOCV), which enables online calculation of the optimal set of parameters. VLOOCV approximates the results of CV at greatly reduced computational costs. We validate the proposed method in one indoor experiment where a mobile robot with a Photo Ionization Detector (PID) was collecting gas measurements while moving in the target area. A comparison with the standard measure for model selection, the CV–based NLPD (Negative Log Predictive Density), favors the proposed algorithm that achieves the same model selection performance using just a fraction of the computational resources. ALGORITHM Mapping the distribution of one or multiple gases using a mobile robot is a challenging task. Kernel DM+V [1], probably the most robust approach today, interprets the mapping problem as a kernel regression method. At the core of the algorithm is the well known NadarayaWatson estimator with RBF kernel [2], which is applied twice, once for the estimation of the predictive mean and once for the estimation of the predictive variance. Kernel regression depends crucially on the choice an appropriate bandwidth of the kernel. It is usually selected in a grid search over the kernel bandwidth space using the average NLPD obtained from CV as evaluation criterion. The computational cost of this method is high due to the need for building and evaluating multiple models on different parts of the data for cross-validation. Instead, VLOOCV builds a single model using the whole dataset and estimates the effect of leaving out parts of the data [3]. To this end, VLOOCV calculates a leverage score hi for each data point i indicating the effect of the point on the model. Scores of zero correspond to points that do not affect the model. To estimate the NLPD using VLOOCV, the negative likelihood of each of the training points according to the model built with the whole training set is computed and then weighted by hi according to the following formula: