Student-t processes as alternatives to Gaussian processes

We investigate the Student-t process as an alternative to the Gaussian process as a non-parametric prior over functions. We derive closed form expressions for the marginal likelihood and predictive distribution of a Student-t process, by integrating away an inverse Wishart process prior over the co-variance kernel of a Gaussian process model. We show surprising equivalences between different hierarchical Gaussian process models leading to Student-t processes, and derive a new sampling scheme for the inverse Wishart process, which helps elucidate these equivalences. Overall, we show that a Student-t process can retain the attractive properties of a Gaussian process - a nonparamet-ric representation, analytic marginal and predictive distributions, and easy model selection through covariance kernels - but has enhanced flexibility, and predictive covariances that, unlike a Gaussian process, explicitly depend on the values of training observations. We verify empirically that a Student-t process is especially useful in situations where there are changes in covariance structure, or in applications such as Bayesian optimization, where accurate predictive covariances are critical for good performance. These advantages come at no additional computational cost over Gaussian processes.

Avoiding pathologies in very deep networks

Choosing appropriate architectures and regularization strategies of deep networks is crucial to good predictive performance. To shed light on this problem, we analyze the analogous problem of constructing useful priors on compositions of functions. Specifically, we study the deep Gaussian process, a type of infinitely-wide, deep neural network. We show that in standard architectures, the representational capacity of the network tends to capture fewer degrees of freedom as the number of layers increases, retaining only a single degree of freedom in the limit. We propose an alternate network architecture which does not suffer from this pathology. We also examine deep covariance functions, obtained by composing infinitely many feature transforms. Lastly, we characterize the class of models obtained by performing dropout on Gaussian processes.

First principles interatomic potential for tungsten based on Gaussian process regression

An accurate description of atomic interactions, such as that provided by first principles quantum mechanics, is fundamental to realistic prediction of the properties that govern plasticity, fracture or crack propagation in metals. However, the computational complexity associated with modern schemes explicitly based on quantum mechanics limits their applications to systems of a few hundreds of atoms at most. This thesis investigates the application of the Gaussian Approximation Potential (GAP) scheme to atomistic modelling of tungsten - a bcc transition metal which exhibits a brittle-to-ductile transition and whose plasticity behaviour is controlled by the properties of $\frac{1}{2} \langle 111 \rangle$ screw dislocations. We apply Gaussian process regression to interpolate the quantum-mechanical (QM) potential energy surface from a set of points in atomic configuration space. Our training data is based on QM information that is computed directly using density functional theory (DFT). To perform the fitting, we represent atomic environments using a set of rotationally, permutationally and reflection invariant parameters which act as the independent variables in our equations of non-parametric, non-linear regression. We develop a protocol for generating GAP models capable of describing lattice defects in metals by building a series of interatomic potentials for tungsten. We then demonstrate that a GAP potential based on a Smooth Overlap of Atomic Positions (SOAP) covariance function provides a description of the $\frac{1}{2} \langle 111 \rangle$ screw dislocation that is in agreement with the DFT model. We use this potential to simulate the mobility of $\frac{1}{2} \langle 111 \rangle$ screw dislocations by computing the Peierls barrier and model dislocation-vacancy interactions to QM accuracy in a system containing more than 100,000 atoms.

Differential equation based length scales to improve DES and RANS simulations

Surprisingly expensive to compute wall distances are still used in a range of key turbulence and peripheral physics models. Potentially economical, accuracy improving differential equation based distance algorithms are considered. These involve elliptic Poisson and hyperbolic natured Eikonal equation approaches. Numerical issues relating to non-orthogonal curvilinear grid solution of the latter are addressed. Eikonal extension to a Hamilton-Jacobi (HJ) equation is discussed. Use of this extension to improve turbulence model accuracy and, along with the Eikonal, enhance Detached Eddy Simulation (DES) techniques is considered. Application of the distance approaches is studied for various geometries. These include a plane channel flow with a wire at the centre, a wing-flap system, a jet with co-flow and a supersonic double-delta configuration. Although less accurate than the Eikonal, Poisson method based flow solutions are extremely close to those using a search procedure. For a moving grid case the Poisson method is found especially efficient. Results show the Eikonal equation can be solved on highly stretched, non-orthogonal, curvilinear grids. A key accuracy aspect is that metrics must be upwinded in the propagating front direction. The HJ equation is found to have qualitative turbulence model improving properties. © 2003 by P. G. Tucker.

Linear dimensionality reduction: Survey, insights, and generalizations

© 2015 John P. Cunningham and Zoubin Ghahramani. Linear dimensionality reduction methods are a cornerstone of analyzing high dimensional data, due to their simple geometric interpretations and typically attractive computational properties. These methods capture many data features of interest, such as covariance, dynamical structure, correlation between data sets, input-output relationships, and margin between data classes. Methods have been developed with a variety of names and motivations in many fields, and perhaps as a result the connections between all these methods have not been highlighted. Here we survey methods from this disparate literature as optimization programs over matrix manifolds. We discuss principal component analysis, factor analysis, linear multidimensional scaling, Fisher's linear discriminant analysis, canonical correlations analysis, maximum autocorrelation factors, slow feature analysis, sufficient dimensionality reduction, undercomplete independent component analysis, linear regression, distance metric learning, and more. This optimization framework gives insight to some rarely discussed shortcomings of well-known methods, such as the suboptimality of certain eigenvector solutions. Modern techniques for optimization over matrix manifolds enable a generic linear dimensionality reduction solver, which accepts as input data and an objective to be optimized, and returns, as output, an optimal low-dimensional projection of the data. This simple optimization framework further allows straightforward generalizations and novel variants of classical methods, which we demonstrate here by creating an orthogonal-projection canonical correlations analysis. More broadly, this survey and generic solver suggest that linear dimensionality reduction can move toward becoming a blackbox, objective-agnostic numerical technology.

Conditions for the local manipulation of tripartite Gaussian states

We present the normal form of the covariance matrix for three-mode tripartite Gaussian states. By means of this result, the general form of a necessary and sufficient criterion for the possibility of a state transformation from one tripartite entangled Gaussian state to another with three modes is found. Moreover, we show that the conditions presented include not only inequalities but equalities as well.

Modern sedimentary environments and dynamic depositional systems in the southern Yellow Sea

Based on analyses of more than 600 surface sediment samples together with large amounts of previous sedimentologic and hydrologic data, the characteristics of modern sedimentary environments and dynamic depositional systems in the southern Yellow Sea (SYS) are expounded, and the controversial formation mechanism of muddy sediments is also discussed. The southern Yellow Sea shelf can be divided into low-energy sedimentary environment and high-energy sedimentary environment; the low-energy sedimentary environment can be further divided into cyclonic and anticyclonic ones, and the high-energy environment is subdivided into high-energy depositional and eroded environments. In the shelf low-energy environments, there developed muddy depositional system. In the central part of the southern Yellow Sea, there deposited the cold eddy sediments under the actions of a meso-scale cyclonic eddy (cold eddy), and in the southeast of the southern Yellow Sea, an anticyclonic eddy muddy depositional system (warm eddy sediment) was formed. These two types of sediments showed evident differences in grain size, sedimentation rate, sediment thickness and mineralogical characteristics. The high-energy environments were covered with sandy sediments on seabed; they appeared mainly in the west, south and northeast of the southern Yellow Sea. In the high-energy eroded environment, large amounts of sandstone gravels were distributed on seabed. In the high-energy depositional environment, the originally deposited fine materials (including clay and fine silt) were gradually re-suspended and then transported to a low-energy area to deposit again. In this paper, the sedimentation model of cyclonic and anticyclonic types of muddy sediments is established, and a systematic interpretation for the formation cause of muddy depositional systems in the southern Yellow Sea is given.

The effects of random surface waves on the steady Ekman current solutions

The response of near-surface current profiles to wind and random surface waves are studied based on the approach of Jenkins [1989. The use of a wave prediction model for driving a near surface current model. Dtsch. Hydrogr. Z. 42,134-149] and Tang et al. [2007. Observation and modeling of surface currents on the Grand Banks: a study of the wave effects on surface currents. J. Geophys. Res. 112, C10025, doi:10.1029/2006JC004028]. Analytic steady solutions are presented for wave-modified Ekman equations resulting from Stokes drift, wind input and wave dissipation for a depth-independent constant eddy viscosity coefficient and one that varies linearly with depth. The parameters involved in the solutions can be determined by the two-dimensional wavenumber spectrum of ocean waves, wind speed, the Coriolis parameter and the densities of air and water, and the solutions reduce to those of Lewis and Belcher [2004. Time-dependent, coupled, Ekman boundary layer solutions incorporating Stokes drift. Dyn. Atmos. Oceans. 37, 313-351] when only the effects of Stokes drift are included. As illustrative examples, for a fully developed wind-generated sea with different wind speeds, wave-modified current profiles are calculated and compared with the classical Ekman theory and Lewis and Belcher's [2004. Time-dependent, coupled, Ekman boundary layer solutions incorporating Stokes drift. Dyn. Atmos. Oceans 37, 313-351] modification by using the Donelan and Pierson [1987. Radar scattering and equilibrium ranges in wind-generated waves with application to scatterometry. J. Geophys. Res. 92, 4971-5029] wavenumber spectrum, the WAM wave model formulation for wind input energy to waves, and wave energy dissipation converted to currents. Illustrative examples for a fully developed sea and the comparisons between observations and the theoretical predictions demonstrate that the effects of the random surface waves on the classical Ekman current are important, as they change qualitatively the nature of the Ekman layer. But the effects of the wind input and wave dissipation on surface current are small, relative to the impact of the Stokes drift. (C) 2008 Elsevier Ltd. All rights reserved.

Interannual horizontal heat advection in the surface mixed layer over the equatorial Pacific Ocean: assimilation versus TAO analysis

The interannual anomalies of horizontal heat advection in the surface mixed layer over the equatorial Pacific Ocean in an assimilation experiment are studied and compared with existing observational analyses. The assimilation builds upon a hindcast study that has produced a good simulation of the observed equatorial currents and optimizes the simulation of the Reynolds sea surface temperature (SST) data. The comparison suggests that the assimilation has improved the simulation of the interannual horizontal heat advection of the surface mixed layer significantly. During periods of interrupted current measurements, the assimilation is shown to produce more meaningful anomalies of the heat advection than the interpolation of the observational data does. The assimilation also shows that the eddy heat flux due to the correlation between high-frequency current and SST variations, which is largely overlooked by the existing observational analyses, is important for the interannual SST balance over the equatorial Pacific. The interannual horizontal heat advection anomalies are found to be sensitive to SST errors where oceanic currents are strong, which is a challenge for ENSO prediction. The study further suggests that the observational analyses of the tropical SST balance based on the TAO and the Reynolds SST data contain significant errors due to the large gradient errors in the Reynolds SST data, which are amplified into the advection anomalies by the large equatorial currents.

Ship-board measurements and estimations of air-sea fluxes in the western tropical Pacific during TOGA COARE

Direct air-sea flux measurements were made on RN Kexue #1 at 40 degrees S, 156 degrees E during the Tropical Ocean Global Atmosphere (TOGA) Coupled Ocean-Atmospheric Response Experiment (COARE) Intensive Observation Period (IOP). An array of six accelerometers was used to measure the motion of the anchored ship, and a sonic anemometer and Lyman-alpha hygrometer were used to measure the turbulent wind vector and specific humidity. The contamination of the turbulent wind components by ship motion was largely removed by an improvement of a procedure due to Shao based on the acceleration signals. The scheme of the wind correction for ship motion is briefly outlined. Results are presented from data for the best wind direction relative to the ship to minimize flow distortion effects. Both the time series and the power spectra of the sonic-measured wind components show swell-induced ship motion contamination, which is largely removed by the accelerometer correction scheme, There was less contamination in the longitudinal wind component than in the vertical and transverse components. The spectral characteristics of the surface-layer turbulence properties are compared with those from previous land and ocean results, Momentum and latent heat fluxes were calculated by eddy correlation and compared to those estimated by the inertial dissipation method and the TOGA COARE bulk formula. The estimations of wind stress determined by eddy correlation are smaller than those from the TOGA COARE bulk formula, especially for higher wind speeds, while those from the bulk formula and inertial dissipation technique are generally in agreement. The estimations of latent heal flux from the three different methods are in reasonable agreement. The effect of the correction for ship motion on latent heat fluxes is not as large as on momentum fluxes.

THE VARIATION AND MOVEMENT OF A KIND OF EDDIES IN SHELF SLOPE WATER

Considering the characteristics of the time and space scales of the eddies we established a quasi-static and quasi-geostrophic model to describe their variation and movement in shelf slope water. The analytical solution revealed the main properties of the variation: slow expansion and fast stagnation processes and the law of the eddy motion affected under the background field. All theoretical results are proved by satellite image measurements.

Variability of surface circulation in the South China Sea from satellite altimeter data

11-year satellite altimeter sea surface height (SSH) anomaly data from January 1993 to December 2003 are used to present the dominant spatial patterns and temporal variations of the South China Sea (SCS) surface circulation through Empirical Orthogonal Function (EOF) analysis. The first three EOF modes show the obvious seasonal variations of SSH in the SCS. EOF mode one is generally characterized by a basin-wide circulation. Mode two describes the double-cell basin scale circulation structure. The two cells were located off west of the Luzon Island and southeast of Vietnam, respectively. EOF mode three presents the mesoscale eddy structure in the western SCS, which develops into a strong cyclonic eddy rapidly from July to September. EOF mode one and mode three are also embedded with interannual signals, indicating that the SCS surface circulation variation is influenced by El Nino events prominently. The strong El Nino of 1997/98 obviously changed the SCS circulation structure. This study also shows that there existed a series of mesoscale eddies in the western SCS, and their temporal variation indicates intra-seasonal and interannual signals.

Anti-cyclonic eddies northwest of Luzon in summer-fall observed by satellite altimeters

Anti-cyclonic eddies northwest of Luzon of the Philippines in summer-fall are identified in the merged data products of satellite altimeters of Topex/Poseidon, Jason-1 and European Research Satellites. The generation and propagation of the anti-cyclonic eddies, which are confirmed by satellite ocean color data, are found to be a seasonal phenomenon that is phase-locked to the onset of the southwesterly monsoon and the relaxation of the cyclonic wind curl in the northeastern South China Sea. The eddies originate from northwest of Luzon in summer, move across the northeastern South China Sea to reach the China continental slope in fall, and propagate southwestward along the continental slope in fall-winter, inducing shelfbreak current variations in the western South China Sea in fall-winter. The anti-cyclonic eddy discovered by Li et al. (1998) in the northern South China Sea is found to originate from northwest of Luzon and carry primarily the South China Sea waters. It does not appear to be an eddy shed from the Kuroshio in the Luzon Strait area as alluded by Li et al. (1998) and others.

The main phosphorous sources in the Changjiang estuary

Analysis using historical data on the phosphate sources in Changjiang (Yangtze River) estuary show that phosphate was supplied equally from the east, south, west and north of the estuary. These sources include the Changjiang River, the Taiwan Warm Current (TWC), a cyclone-type eddy, and the 32A degrees N Upwelling, supplying different phosphates in different times, ways and intensities. The magnitude of their supplying phosphate concentration was related with the size in the order of the Changjiang River < the TWC < the 32A degrees N Upwelling < the cyclone-type eddy, and the duration of the supplying was: the Changjiang River > the TWC > the cyclone-type eddy > the 32A degrees N Upwelling. The four sources supplied a great deal of phosphate so that the phosphate concentration in the estuary was kept above 0.2 mu mol/L in previous years, satisfying the phytoplankton growth. The horizontal and vertical distribution of the phosphate concentration showed that near shallow marine areas at 122A degrees E/31A degrees N, the TWC in low nutrient concentration became an upwelling through sea bottom and brought up nutrients from sea bottom to marine surface. In addition, horizontal distribution of phosphate concentration was consistent with that of algae: Rhizosolenia robusta, Rhizosolenia calcaravis and Skeletonema, which showed that no matter during high water or low water of Changjiang River, these species brought by the TWC became predominant species. Therefore, the authors believe that the TWC flowed from south to north along the coast and played a role in deflecting the Changjiang River flow from the southern side.

Application of altimetry data assimilation on mesoscale eddies simulation

Mesoscale eddy plays an important role in the ocean circulation. In order to improve the simulation accuracy of the mesoscale eddies, a three-dimensional variation (3DVAR) data assimilation system called Ocean Variational Analysis System (OVALS) is coupled with a POM model to simulate the mesoscale eddies in the Northwest Pacific Ocean. In this system, the sea surface height anomaly (SSHA) data by satellite altimeters are assimilated and translated into pseudo temperature and salinity (T-S) profile data. Then, these profile data are taken as observation data to be assimilated again and produce the three-dimensional analysis T-S field. According to the characteristics of mesoscale eddy, the most appropriate assimilation parameters are set up and testified in this system. A ten years mesoscale eddies simulation and comparison experiment is made, which includes two schemes: assimilation and non-assimilation. The results of comparison between two schemes and the observation show that the simulation accuracy of the assimilation scheme is much better than that of non-assimilation, which verified that the altimetry data assimilation method can improve the simulation accuracy of the mesoscale dramatically and indicates that it is possible to use this system on the forecast of mesoscale eddies in the future.