Autonomous underwater vehicles (AUVs) are robotic platforms that are commonly used to map the sea floor, for example for benthic surveys or for naval mine countermeasures (MCM) operations. AUVs create an acoustic image of the survey area, such that objects on the seabed can be identified and, in the case of MCM, mines can be found and disposed of. The common method for creating such seabed maps is to run a lawnmower survey, which is a standard method in coverage path planning. We are interested in exploring alternate techniques for surveying areas of interest, in order to reduce mission time or assess feasible actions, such as finding a safe path through a hazardous region. In this paper, we use Gaussian Process regression to build models of seabed complexity data, obtained through lawnmower surveys. We evaluate several commonly used kernels to assess their modeling performance, which includes modeling discontinuities in the data. Our results show that an additive Mat 'ern kernel is most suitable for modeling seabed complexity data. On top of the GP model, we use adaptations of two standard path planning methods, A* and RRT*, to find safe paths through the modeled areas. We evaluate the planned paths and also run a vehicle dynamics simulator to assess potential performance by a marine vessel.
GP regression is also known as Kriging in the field of spatial statistics, where it is used primarily as an interpolation technique. Geospatial Kriging assumes there is some spatially related component that represents a trend or a random error term @cite_3 , based on what is informally called Tobler's first law of geography: Everything is related to everything else, but near things are more related than distant things.'' @cite_0 . will explain the basics of GP regression.
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Gaussian process (GP) models are very popular for machine learning and regression and they are widely used to account for spatial or temporal relationships between multivariate random variables. In this paper, we propose a general formulation of underdetermined source separation as a problem involving GP regression. The advantage of the proposed unified view is first to describe the different underdetermined source separation problems as particular cases of a more general framework. Second, it provides a flexible means to include a variety of prior information concerning the sources such as smoothness, local stationarity or periodicity through the use of adequate covariance functions. Third, given the model, it provides an optimal solution in the minimum mean squared error (MMSE) sense to the source separation problem. In order to make the GP models tractable for very large signals, we introduce framing as a GP approximation and we show that computations for regularly sampled and locally stationary GPs can be done very efficiently in the frequency domain. These findings establish a deep connection between GP and nonnegative tensor factorizations (NTF) with the Itakura-Saito distance and lead to effective methods to learn GP hyperparameters for very large and regularly sampled signals. We present a new Gaussian process (GP) regression model whose co-variance is parameterized by the the locations of M pseudo-input points, which we learn by a gradient based optimization. We take M ≪ N, where N is the number of real data points, and hence obtain a sparse regression method which has O(M2N) training cost and O(M2) prediction cost per test case. We also find hyperparameters of the covariance function in the same joint optimization. The method can be viewed as a Bayesian regression model with particular input dependent noise. The method turns out to be closely related to several other sparse GP approaches, and we discuss the relation in detail. We finally demonstrate its performance on some large data sets, and make a direct comparison to other sparse GP methods. We show that our method can match full GP performance with small M, i.e. very sparse solutions, and it significantly outperforms other approaches in this regime. Gaussian processes (GPs) are powerful non-parametric function estimators. However, their applications are largely limited by the expensive computational cost of the inference procedures. Existing stochastic or distributed synchronous variational inferences, although have alleviated this issue by scaling up GPs to millions of samples, are still far from satisfactory for real-world large applications, where the data sizes are often orders of magnitudes larger, say, billions. To solve this problem, we propose ADVGP, the first Asynchronous Distributed Variational Gaussian Process inference for regression, on the recent large-scale machine learning platform, PARAMETERSERVER. ADVGP uses a novel, flexible variational framework based on a weight space augmentation, and implements the highly efficient, asynchronous proximal gradient optimization. While maintaining comparable or better predictive performance, ADVGP greatly improves upon the efficiency of the existing variational methods. With ADVGP, we effortlessly scale up GP regression to a real-world application with billions of samples and demonstrate an excellent, superior prediction accuracy to the popular linear models. In geostatistics it is common practice to assume that the underlying spatial process is stationary and isotropic, i.e. the spatial distribution is unchanged when the origin of the index set is translated and under rotation about the origin. However, in environmental problems, such assumptions are not realistic since local influences in the correlation structure of the spatial process may be found in the data. The paper proposes a Bayesian model to address the anisot- ropy problem. Following Sampson and Guttorp, we define the correlation function of the spatial process by reference to a latent space, denoted by "D", where stationarity and isotropy hold. The space where the gauged monitoring sites lie is denoted by "G". We adopt a Bayesian approach in which the mapping between "G" and "D" is represented by an unknown function d (·). A Gaussian process prior distribution is defined for d (·). Unlike the Sampson-Guttorp approach, the mapping of both gauged and ungauged sites is handled in a single framework, and predictive inferences take explicit account of uncertainty in the mapping. Markov chain Monte Carlo methods are used to obtain samples from the posterior distributions. Two examples are discussed: a simulated data set and the solar radiation data set that also was analysed by Sampson and Guttorp. Copyright 2003 Royal Statistical Society.