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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 ...

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). ...

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). ...

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: ...

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). ...

In this paper, we introduce a new locally multivariate procedure to quantitatively extract voxel-wise patterns of abnormal perfusion in individual whitepatients . This a contrario approach uses a multivariate metric from the computer vision community whitethat is suitable to detect abnormalities even in the presence of closeby hypo- and hyper-perfusions . This method whitetakes into account local information without whiteapplying Gaussian smoothing to the data. Furthermore, to improve on the standard a contrario approach, which assumes white noise, we introduce an updated a contrario approach whitethat takes into account the spatial coherency of the noise in the probability estimation. Validation is undertaken on a dataset of 25 patients whitediagnosed with brain tumors and 61 healthy volunteers. whiteWe show how the a contrario approach outperforms the massively univariate General Linear Model usually employed for this type of analysis.

We use a Bayesian method, optimal interpolation, to improve satellite derived irradiance estimates at city-scales using ground sensor data. Optimal interpolation requires error covariances in the satellite estimates and ground data, which define how information from the sensor locations is distributed across a large area. We describe three methods to choose such covariances, including a covariance parameterization that depends on the relative cloudiness between locations. Results are computed with ground data from 22 sensors over a 75×80 km area centered on Tucson, AZ, using two satellite derived irradiance models. The improvements in standard error metrics for both satellite models indicate that our approach is applicable to additional satellite derived irradiance models. We also show that optimal interpolation can nearly eliminate mean bias error and improve the root mean squared error by 50%.

The Wenzel roughness parameter of isotropic Gaussian surfaces is analytically described in terms of the Power Spectral Density function without the smooth surface approximation. This Wenzel roughness parameter- Power Spectral Density link was examined for distinct roughnesses of Aluminum-oxide thin films. The Power Spectral Density functions of the surfaces were determined in a wide spatial frequency range by combining different scan areas of Atomic Force Microscopy measurements. The calculated results presented a good agreement with the Wenzel roughness parameter values obtained directly from the topography measured by Atomic Force Microscopy. Finally, wetting behavior was ascertained through determination of water contact angles, including superhydrophobic behavior. This approach, together with an empirical procedure based on a structural parameter, can predict the wetting properties of a surface by taking all its relevant roughness components into account.

Many simulation-intensive tasks in the applied sciences, such as sensitivity analysis, parameter inference or real time control, are hampered by slow simulators. Emulators provide the opportunity of speeding up simulations at the cost of introducing some inaccuracy. An emulator is a fast approximation to a simulator that interpolates between design input-output pairs of the simulator. Increasing the number of design data sets is a computationally demanding way of improving the accuracy of emulation. We investigate the complementary approach of increasing emulation accuracy by including knowledge about the mechanisms of the simulator into the formulation of the emulator. To approximately reproduce the output of dynamic simulators, we consider emulators that are based on a system of linear, ordinary or partial stochastic differential equations with a noise term formulated as a Gaussian process of the parameters to be emulated. This stochastic model is then conditioned to the design data so that it mimics the behavior of the nonlinear simulator as a function of the parameters. The drift terms of the linear model are designed to provide a simplified description of the simulator as a function of its key parameters so that the required corrections by the conditioned Gaussian process noise are as small as possible. The goal of this paper is to compare the gain in accuracy of these emulators by enlarging the design data set and by varying the degree of simplification of the linear model. We apply this framework to a simulator for the shallow water equations in a channel and compare emulation accuracy for emulators based on different spatial discretization levels of the channel and for a standard non-mechanistic emulator. Our results indicate that we have a large gain in accuracy already when using the simplest mechanistic description by a single linear reservoir to formulate the drift term of the linear model. Adding some more reservoirs does not lead to a significant improvement in accuracy. However, the transition to a spatially continuous linear model leads again to a similarly large gain in accuracy as the transition from the non-mechanistic emulator to that based on one reservoir.

The important place of images in the modern world is undeniable. They are intimately integrated into our organic life ("visual perception" is particularly well developed in human beings). They are frequently involved in our daily life (magazines, newspapers, telephones, televisions and video games, etc.), personal life (medical imaging, biological imaging and photographs, etc.), professional life (plant control, office automation, remote monitoring, scanners and video conferencing), etc. They are not confined to the various technological sectors, but they are vectors of observations and investigations of matter at very small scales (electron microscopes and scanning probe microscopes, etc.), or of the universe at very large scales (telescopes and space probes, etc.), sometimes leading major scientific discoveries. Mankind is now able to see images of other worlds without going there (e.g. distant planets, stars and galaxies, or the surface terrain of the Earth) and worlds within (e.g. human organs, geological imaging, or atomic and molecular structures at the nanoscale level). From a technological point of view, this importance is enhanced by the performance of the systems of investigation by imaging and the powers of calculation of computers, which expanded considerably in the second half of the 20th Century, and that are still progressing, with both hardware and software advances.

A novel technique for the high-resolution interpolation of in situ sea surface salinity (SSS) observations is developed and tested. The method is based on an optimal interpolation (OI) algorithm that includes satellite sea surface temperature (SST) in the covariance estimation. The covariance function parameters (i.e., spatial, temporal, and thermal decorrelation scales) and the noise-to-signal ratio are determined empirically, by minimizing the root-mean-square error and mean error with respect to fully independent validation datasets. Both in situ observations and simulated data extracted from a numerical model output are used to run these tests. Different filters are applied to sea surface temperature data in order to remove the large-scale variability associated with air–sea interaction, because a high correlation between SST and SSS is expected only at small scales. In the tests performed on in situ observations, the lowest errors are obtained by selecting covariance decorrelation scales of 400 km, 6 days, and 2.758C, respectively, a noise-to-signal ratio of 0.01 and filtering the scales longer than 1000 km in the SST time series. This results in a root-mean-square error of ;0.11 g kg 21 and a mean error of ;0.01 g kg 21 , that is, reducing the errors by ;25% and ;60%, respectively, with respect to the first guess.

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