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Covariance functions

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... We choose joint Gaussian distribution on value function -more specifically, Gaussian Process (GP) -because GP provides a principled, practical, probabilistic approach to learn in kernel machines (Rasmussen & Williams, 2006). ...
... Theorem 1 When a set (X , f ) is used to estimate f (x * ) in GP, the expectation of variance on test points x * with distribution p(x) conditioned on all possible training set (X', f) set would not be less than what conditioned on the training set X sampled from distribution p(x), if the size of sample set is large enough to ignore the approximation error. (Rasmussen & Williams, 2006). ...
... For each i, focus on φ * i (X )K(X , X ) −1 φ i (X ). Using numerical approximation of eigenfunctions (Rasmussen & Williams, 2006), when each x l is sampled from ...
Preprint
Efficient Reinforcement Learning usually takes advantage of demonstration or good exploration strategy. By applying posterior sampling in model-free RL under the hypothesis of GP, we propose Gaussian Process Posterior Sampling Reinforcement Learning(GPPSTD) algorithm in continuous state space, giving theoretical justifications and empirical results. We also provide theoretical and empirical results that various demonstration could lower expected uncertainty and benefit posterior sampling exploration. In this way, we combined the demonstration and exploration process together to achieve a more efficient reinforcement learning.
... This kernel is smooth enough to avoid the GP becoming too rough whilst not being excessively smooth, which is appropriate for modelling physical relationships. Examples of other kernels are exponential, squared exponential, rational quadratic, and piecewise polynomial ( [106]). The kernels have parameters (also called length scales) that are solved along with other hyperparameters via non-linear optimization in a maximum likelihood estimation (MLE) scheme (other approaches such as a Bayesian procedure are possible in MOGP). ...
Thesis
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The work presented in this thesis focuses on the development of fast computational methods for modelling tsunamis. A large emphasis is placed on the newly redeveloped tsunami code, Volna-OP2, which is optimised to utilise the latest high performance computing architectures. The code is validated/verified against various benchmark tests. An extensive error analysis of this redeveloped code has been completed, where the occurrence and relative importance of numerical errors is presented. The performance of the GPU version of the code is investigated by simulating a submarine landslide event. A first of its kind tsunami hazard assessment of the Irish coastline has been carried out with Volna-OP2. The hazard is captured on various levels of refinement. The efficiency of the redeveloped version of the code is demonstrated by its ability to complete an ensemble of simulations in a faster than real time setting. The code also forms an integral part of a newly developed workflow which would allow for tsunami warning centres to capture the uncertainty on the tsunami hazard within warning time constraints. The uncertainties are captured by coupling Volna-OP2 with a computationally cheap statistical emulator. The steps of the proposed workflow are outlined by simulating a test case, the Makran 1945 event. The code is further utilised to validate and expand upon a new analytical theory which quantifies the energy of a tsunami generated by a submarine landslide. Some preliminary work on capturing the scaling relationships between the parameters of the set up and the tsunami energy has been completed. Transfer functions, which are based upon extensions to Green's Law, and machine learning techniques which quantify the local response to an incoming tsunami are presented. The response, if captured ahead of time, would allow a warning centre to rapidly forecast the local tsunami impact. This work is the only chapter in the thesis which doesn't draw upon Volna-OP2, but nevertheless showcases another fast computational method for modelling tsunamis.
... Here, we use the Matern 5/2 kernel that is smooth enough to avoid a rough GP, but not extremely smooth thus being suitable for modelling the physics. The piecewise polynomial, rational quadratic, exponential, and squared exponential functions are other candidates [56]. The parameters (or length scales) in the kernels and other hyperparameters are found via non-linear optimization (L-BFGS-B) using maximum likelihood estimation (MLE). ...
Article
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In this paper, statistical emulation is shown to be an essential tool for the end-to-end physical and numerical modelling of local tsunami impact, i.e. from the earthquake source to tsunami velocities and heights. In order to surmount the prohibitive computational cost of running a large number of simulations, the emulator, constructed using 300 training simulations from a validated tsunami code, yields 1 million predictions. This constitutes a record for any realistic tsunami code to date, and is a leap in tsunami science since high risk but low probability hazard thresholds can be quantified. For illustrating the efficacy of emulation, we map probabilistic representations of maximum tsunami velocities and heights at around 200 locations about Karachi port. The 1 million predictions comprehensively sweep through a range of possible future tsunamis originating from the Makran Subduction Zone (MSZ). We rigorously model each step in the tsunami life cycle: first use of the three-dimensional subduction geometry Slab2 in MSZ, most refined fault segmentation in MSZ, first sediment enhancements of seabed deformation (up to 60% locally) and bespoke unstructured meshing algorithm. Owing to the synthesis of emulation and meticulous numerical modelling, we also discover substantial local variations of currents and heights.
... Only a function that satisfies the positive semi-definiteness can be a valid covariance function. Let us give some examples of covariance functions [151]. The squared exponential covariance function is one of the most commonly used covariance functions: ...
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A robotic system can be characterized by its interactions with environments. With growing demand for robots deployed in various scenarios, the ability to perform physical interaction in uncontrolled environments has become of great interest. While a robot performs interactive tasks, its visual and spatial sensing plays a critical role. Being a major source of learning, vision not only guides immediate actions, but also indirectly improves future actions and decisions. How visual information is gathered and represented will significantly influence how a robot can plan and act. Although recent advances in machine perception have presented unprecedented performance in some areas, there still exist challenges in various aspects. In this dissertation, I will address two such issues and suggest an online probabilistic approach to each problem. Most successful approaches in visual learning depend on fragments of exemplars prepared by humans. It is simply unaffordable to provide constant human supervision to a robotic system that would receive tens of new image frames per second. Ideally, a robotic system is required to gather information from its unique experience and keep growing knowledge on the fly without such external aids. One way to implement the self-learning is to take advantage of the naturally correlated sensations of different sensory modalities. The first part of this talk presents a probabilistic online self-learning framework to alleviate the dependency in robotic visual learning by leveraging structural priors. Another challenge in robotics is its spatial understanding. Aside from planning and performing actions, spatial representation itself still largely requires more research. While point or grid-based representations are currently being employed for practical conveniences, these methods suffer from discretization and disconnected spatial information. On the other hand, Gaussian Processes (GP) have recently gained attention as an alternative to represent the distance field of structures continuously and probabilistically. It is not only the seamless expression of structures, but also direct access to the distance and direction to obstacles that make the representation invaluable. The second part of the talk presents an online framework for continuous spatial mapping using GP.
... Given this estimate of the autocorrelation, σ 2 (i,j) is easily computed (cf. for example (Rasmussen and Williams, 2006) p.84). ...
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