Showing papers on "Gaussian published in 2013"

Estimating Continuous Distributions in Bayesian Classifiers

Abstract: When modeling a probability distribution with a Bayesian network, we are faced with the problem of how to handle continuous variables. Most previous work has either solved the problem by discretizing, or assumed that the data are generated by a single Gaussian. In this paper we abandon the normality assumption and instead use statistical methods for nonparametric density estimation. For a naive Bayesian classifier, we present experimental results on a variety of natural and artificial domains, comparing two methods of density estimation: assuming normality and modeling each conditional distribution with a single Gaussian; and using nonparametric kernel density estimation. We observe large reductions in error on several natural and artificial data sets, which suggests that kernel estimation is a useful tool for learning Bayesian models.

On asymptotically optimal confidence regions and tests for high-dimensional models

Abstract: We propose a general method for constructing confidence intervals and statistical tests for single or low-dimensional components of a large parameter vector in a high-dimensional model. It can be easily adjusted for multiplicity taking dependence among tests into account. For linear models, our method is essentially the same as in Zhang and Zhang [J. R. Stat. Soc. Ser. B Stat. Methodol. 76 (2014) 217-242]: we analyze its asymptotic properties and establish its asymptotic optimality in terms of semiparametric efficiency. Our method naturally extends to generalized linear models with convex loss functions. We develop the corresponding theory which includes a careful analysis for Gaussian, sub-Gaussian and bounded correlated designs.

MCMC Methods for Functions: Modifying Old Algorithms to Make Them Faster

Abstract: Many problems arising in applications result in the need to probe a probability distribution for functions. Examples include Bayesian nonparametric statistics and conditioned diffusion processes. Standard MCMC algorithms typically become arbitrarily slow under the mesh refinement dictated by nonparametric description of the unknown function. We describe an approach to modifying a whole range of MCMC methods, applicable whenever the target measure has density with respect to a Gaussian process or Gaussian random field reference measure, which ensures that their speed of convergence is robust under mesh refinement. Gaussian processes or random fields are fields whose marginal distributions, when evaluated at any finite set of NNpoints, are ℝ^N-valued Gaussians. The algorithmic approach that we describe is applicable not only when the desired probability measure has density with respect to a Gaussian process or Gaussian random field reference measure, but also to some useful non-Gaussian reference measures constructed through random truncation. In the applications of interest the data is often sparse and the prior specification is an essential part of the overall modelling strategy. These Gaussian-based reference measures are a very flexible modelling tool, finding wide-ranging application. Examples are shown in density estimation, data assimilation in fluid mechanics, subsurface geophysics and image registration. The key design principle is to formulate the MCMC method so that it is, in principle, applicable for functions; this may be achieved by use of proposals based on carefully chosen time-discretizations of stochastic dynamical systems which exactly preserve the Gaussian reference measure. Taking this approach leads to many new algorithms which can be implemented via minor modification of existing algorithms, yet which show enormous speed-up on a wide range of applied problems.

Lattice Signatures and Bimodal Gaussians

Abstract: Our main result is a construction of a lattice-based digital signature scheme that represents an improvement, both in theory and in practice, over today’s most efficient lattice schemes. The novel scheme is obtained as a result of a modification of the rejection sampling algorithm that is at the heart of Lyubashevsky’s signature scheme (Eurocrypt, 2012) and several other lattice primitives. Our new rejection sampling algorithm which samples from a bimodal Gaussian distribution, combined with a modified scheme instantiation, ends up reducing the standard deviation of the resulting signatures by a factor that is asymptotically square root in the security parameter. The implementations of our signature scheme for security levels of 128, 160, and 192 bits compare very favorably to existing schemes such as RSA and ECDSA in terms of efficiency. In addition, the new scheme has shorter signature and public key sizes than all previously proposed lattice signature schemes.

Planck 2018 results. VII. Isotropy and Statistics of the CMB

Abstract: Analysis of the Planck 2018 data set indicates that the statistical properties of the cosmic microwave background (CMB) temperature anisotropies are in excellent agreement with previous studies using the 2013 and 2015 data releases. In particular, they are consistent with the Gaussian predictions of the $\Lambda$CDM cosmological model, yet also confirm the presence of several so-called "anomalies" on large angular scales. The novelty of the current study, however, lies in being a first attempt at a comprehensive analysis of the statistics of the polarization signal over all angular scales, using either maps of the Stokes parameters, $Q$ and $U$, or the $E$-mode signal derived from these using a new methodology (which we describe in an appendix). Although remarkable progress has been made in reducing the systematic effects that contaminated the 2015 polarization maps on large angular scales, it is still the case that residual systematics (and our ability to simulate them) can limit some tests of non-Gaussianity and isotropy. However, a detailed set of null tests applied to the maps indicates that these issues do not dominate the analysis on intermediate and large angular scales (i.e., $\ell \lesssim 400$). In this regime, no unambiguous detections of cosmological non-Gaussianity, or of anomalies corresponding to those seen in temperature, are claimed. Notably, the stacking of CMB polarization signals centred on the positions of temperature hot and cold spots exhibits excellent agreement with the $\Lambda$CDM cosmological model, and also gives a clear indication of how Planck provides state-of-the-art measurements of CMB temperature and polarization on degree scales.

Fastfood: approximating kernel expansions in loglinear time

Abstract: Despite their successes, what makes kernel methods difficult to use in many large scale problems is the fact that computing the decision function is typically expensive, especially at prediction time. In this paper, we overcome this difficulty by proposing Fastfood, an approximation that accelerates such computation significantly. Key to Fastfood is the observation that Hadamard matrices when combined with diagonal Gaussian matrices exhibit properties similar to dense Gaussian random matrices. Yet unlike the latter, Hadamard and diagonal matrices are inexpensive to multiply and store. These two matrices can be used in lieu of Gaussian matrices in Random Kitchen Sinks (Rahimi & Recht, 2007) and thereby speeding up the computation for a large range of kernel functions. Specifically, Fastfood requires O(n log d) time and O(n) storage to compute n non-linear basis functions in d dimensions, a significant improvement from O(nd) computation and storage, without sacrificing accuracy. We prove that the approximation is unbiased and has low variance. Extensive experiments show that we achieve similar accuracy to full kernel expansions and Random Kitchen Sinks while being 100x faster and using 1000x less memory. These improvements, especially in terms of memory usage, make kernel methods more practical for applications that have large training sets and/or require real-time prediction.

Gaussian approximations and multiplier bootstrap for maxima of sums of high-dimensional random vectors

Abstract: We derive a Gaussian approximation result for the maximum of a sum of high-dimensional random vectors. Specifically, we establish conditions under which the distribution of the maximum is approximated by that of the maximum of a sum of the Gaussian random vectors with the same covariance matrices as the original vectors. This result applies when the dimension of random vectors ($p$) is large compared to the sample size ($n$); in fact, $p$ can be much larger than $n$, without restricting correlations of the coordinates of these vectors. We also show that the distribution of the maximum of a sum of the random vectors with unknown covariance matrices can be consistently estimated by the distribution of the maximum of a sum of the conditional Gaussian random vectors obtained by multiplying the original vectors with i.i.d. Gaussian multipliers. This is the Gaussian multiplier (or wild) bootstrap procedure. Here too, $p$ can be large or even much larger than $n$. These distributional approximations, either Gaussian or conditional Gaussian, yield a high-quality approximation to the distribution of the original maximum, often with approximation error decreasing polynomially in the sample size, and hence are of interest in many applications. We demonstrate how our Gaussian approximations and the multiplier bootstrap can be used for modern high-dimensional estimation, multiple hypothesis testing, and adaptive specification testing. All these results contain nonasymptotic bounds on approximation errors.

Expectation-Maximization Gaussian-Mixture Approximate Message Passing

Abstract: When recovering a sparse signal from noisy compressive linear measurements, the distribution of the signal's non-zero coefficients can have a profound effect on recovery mean-squared error (MSE). If this distribution was a priori known, then one could use computationally efficient approximate message passing (AMP) techniques for nearly minimum MSE (MMSE) recovery. In practice, however, the distribution is unknown, motivating the use of robust algorithms like LASSO-which is nearly minimax optimal-at the cost of significantly larger MSE for non-least-favorable distributions. As an alternative, we propose an empirical-Bayesian technique that simultaneously learns the signal distribution while MMSE-recovering the signal-according to the learned distribution-using AMP. In particular, we model the non-zero distribution as a Gaussian mixture and learn its parameters through expectation maximization, using AMP to implement the expectation step. Numerical experiments on a wide range of signal classes confirm the state-of-the-art performance of our approach, in both reconstruction error and runtime, in the high-dimensional regime, for most (but not all) sensing operators.

Gaussian Process Kernels for Pattern Discovery and Extrapolation

Abstract: Gaussian processes are rich distributions over functions, which provide a Bayesian nonparametric approach to smoothing and interpolation. We introduce simple closed form kernels that can be used with Gaussian processes to discover patterns and enable extrapolation. These kernels are derived by modelling a spectral density -- the Fourier transform of a kernel -- with a Gaussian mixture. The proposed kernels support a broad class of stationary covariances, but Gaussian process inference remains simple and analytic. We demonstrate the proposed kernels by discovering patterns and performing long range extrapolation on synthetic examples, as well as atmospheric CO2 trends and airline passenger data. We also show that we can reconstruct standard covariances within our framework.

Computing the Jacobian in Gaussian Spatial Autoregressive Models: An Illustrated Comparison of Available Methods

Abstract: This is the accepted version of the following article:Computing the Jacobian in Gaussian Spatial Autoregressive Models: An Illustrated Comparison of Available Methods,Geographical Analysis 2013, 45(2):150-179, which has been published in final form at http://onlinelibrary.wiley.com/doi/10.1111/gean.12008/abstract. © 2013 The Ohio State University

MCMC Methods for Functions: ModifyingOld Algorithms to Make Them Faster

Abstract: Many problems arising in applications result in the need to probe a probability distribution for functions. Examples include Bayesian nonparametric statistics and conditioned diffusion processes. Standard MCMC algorithms typically become arbitrarily slow under the mesh refinement dictated by nonparametric description of the un- known function. We describe an approach to modifying a whole range of MCMC methods, applicable whenever the target measure has density with respect to a Gaussian process or Gaussian random field reference measure, which ensures that their speed of convergence is robust under mesh refinement. Gaussian processes or random fields are fields whose marginal distri- butions, when evaluated at any finite set of N points, are RN-valued Gaussians. The algorithmic approach that we describe is applicable not only when the desired probability measure has density with respect to a Gaussian process or Gaussian random field reference measure, but also to some useful non-Gaussian reference measures constructed through random truncation. In the applications of interest the data is often sparse and the prior specification is an essential part of the over- all modelling strategy. These Gaussian-based reference measures are a very flexible modelling tool, finding wide-ranging application. Examples are shown in density estimation, data assimilation in fluid mechanics, subsurface geophysics and image registration. The key design principle is to formulate the MCMC method so that it is, in principle, applicable for functions; this may be achieved by use of proposals based on carefully chosen time-discretizations of stochas- tic dynamical systems which exactly preserve the Gaussian reference measure. Taking this approach leads to many new algorithms which can be implemented via minor modification of existing algorithms, yet which show enormous speed-up on a wide range of applied problems.

Spatio-temporal modeling of particulate matter concentration through the SPDE approach

Abstract: In this work, we consider a hierarchical spatio-temporal model for particulate matter (PM) concentration in the North-Italian region Piemonte. The model involves a Gaussian Field (GF), affected by a measurement error, and a state process characterized by a first order autoregressive dynamic model and spatially correlated innovations. This kind of model is well discussed and widely used in the air quality literature thanks to its flexibility in modelling the effect of relevant covariates (i.e. meteorological and geographical variables) as well as time and space dependence. However, Bayesian inference—through Markov chain Monte Carlo (MCMC) techniques—can be a challenge due to convergence problems and heavy computational loads. In particular, the computational issue refers to the infeasibility of linear algebra operations involving the big dense covariance matrices which occur when large spatio-temporal datasets are present. The main goal of this work is to present an effective estimating and spatial prediction strategy for the considered spatio-temporal model. This proposal consists in representing a GF with Matern covariance function as a Gaussian Markov Random Field (GMRF) through the Stochastic Partial Differential Equations (SPDE) approach. The main advantage of moving from a GF to a GMRF stems from the good computational properties that the latter enjoys. In fact, GMRFs are defined by sparse matrices that allow for computationally effective numerical methods. Moreover, when dealing with Bayesian inference for GMRFs, it is possible to adopt the Integrated Nested Laplace Approximation (INLA) algorithm as an alternative to MCMC methods giving rise to additional computational advantages. The implementation of the SPDE approach through the R-library INLA ( www.r-inla.org ) is illustrated with reference to the Piemonte PM data. In particular, providing the step-by-step R-code, we show how it is easy to get prediction and probability of exceedance maps in a reasonable computing time.

A Nonlocal Bayesian Image Denoising Algorithm

Abstract: Recent state-of-the-art image denoising methods use nonparametric estimation processes for $8 \times 8$ patches and obtain surprisingly good denoising results. The mathematical and experimental evidence of two recent articles suggests that we might even be close to the best attainable performance in image denoising ever. This suspicion is supported by a remarkable convergence of all analyzed methods. Still more interestingly, most patch-based image denoising methods can be summarized in one paradigm, which unites the transform thresholding method and a Markovian Bayesian estimation. As the present paper shows, this unification is complete when the patch space is assumed to be a Gaussian mixture. Each Gaussian distribution is associated with its orthonormal basis of patch eigenvectors. Thus, transform thresholding (or a Wiener filter) is made on these local orthogonal bases. In this paper a simple patch-based Bayesian method is proposed, which on the one hand keeps most interesting features of former metho...

Optimal Inversion of the Generalized Anscombe Transformation for Poisson-Gaussian Noise

Abstract: Many digital imaging devices operate by successive photon-to-electron, electron-to-voltage, and voltage-to-digit conversions. These processes are subject to various signal-dependent errors, which are typically modeled as Poisson-Gaussian noise. The removal of such noise can be effected indirectly by applying a variance-stabilizing transformation (VST) to the noisy data, denoising the stabilized data with a Gaussian denoising algorithm, and finally applying an inverse VST to the denoised data. The generalized Anscombe transformation (GAT) is often used for variance stabilization, but its unbiased inverse transformation has not been rigorously studied in the past. We introduce the exact unbiased inverse of the GAT and show that it plays an integral part in ensuring accurate denoising results. We demonstrate that this exact inverse leads to state-of-the-art results without any notable increase in the computational complexity compared to the other inverses. We also show that this inverse is optimal in the sense that it can be interpreted as a maximum likelihood inverse. Moreover, we thoroughly analyze the behavior of the proposed inverse, which also enables us to derive a closed-form approximation for it. This paper generalizes our work on the exact unbiased inverse of the Anscombe transformation, which we have presented earlier for the removal of pure Poisson noise.

Properties of nonlinear noise in long, dispersion-uncompensated fiber links.

Abstract: We study the properties of nonlinear interference noise (NLIN) in fiber-optic communications systems with large accumulated dispersion. Our focus is on settling the discrepancy between the results of the Gaussian noise (GN) model (according to which NLIN is additive Gaussian) and a recently published time-domain analysis, which attributes drastically different properties to the NLIN. Upon reviewing the two approaches we identify several unjustified assumptions that are key in the derivation of the GN model, and that are responsible for the discrepancy. We derive the true NLIN power and verify that the NLIN is not additive Gaussian, but rather it depends strongly on the data transmitted in the channel of interest. In addition we validate the time-domain model numerically and demonstrate the strong dependence of the NLIN on the interfering channels' modulation format.

Generating Efficient Quantum Chemistry Codes for Novel Architectures

Abstract: We describe an extension of our graphics processing unit (GPU) electronic structure program TeraChem to include atom-centered Gaussian basis sets with d angular momentum functions. This was made possible by a "meta-programming" strategy that leverages computer algebra systems for the derivation of equations and their transformation to correct code. We generate a multitude of code fragments that are formally mathematically equivalent, but differ in their memory and floating-point operation footprints. We then select between different code fragments using empirical testing to find the highest performing code variant. This leads to an optimal balance of floating-point operations and memory bandwidth for a given target architecture without laborious manual tuning. We show that this approach is capable of similar performance compared to our hand-tuned GPU kernels for basis sets with s and p angular momenta. We also demonstrate that mixed precision schemes (using both single and double precision) remain stable and accurate for molecules with d functions. We provide benchmarks of the execution time of entire self-consistent field (SCF) calculations using our GPU code and compare to mature CPU based codes, showing the benefits of the GPU architecture for electronic structure theory with appropriately redesigned algorithms. We suggest that the meta-programming and empirical performance optimization approach may be important in future computational chemistry applications, especially in the face of quickly evolving computer architectures.

Random matrices and complexity of spin glasses

Abstract: CERN ´ Y Abstract. We give an asymptotic evaluation of the complexity of spherical p-spin spin- glass models via random matrix theory. This study enables us to obtain detailed infor- mation about the bottom of the energy landscape, including the absolute minimum (the ground state), the other local minima, and describe an interesting layered structure of the low critical values for the Hamiltonians of these models. We also show that our ap- proach allows us to compute the related TAP-complexity and extend the results known in the physics literature. As an independent tool, we prove a LDP for the k-th largest eigenvalue of the GOE, extending the results of (BDG01). How many critical values of given index and below a given level does a typical random Morse function have on a high dimensional manifold? Our work addresses this question in a very special case. We look at certain natural random Gaussian functions on the N- dimensional sphere known as p-spin spherical spin glass models. We cannot yet answer the question above about the typical number, but we can study thoroughly the mean number, which we show is exponentially large in N. We introduce a new identity, based on the classical Kac-Rice formula, relating random matrix theory and the problem of counting these critical values. Using this identity and tools from random matrix theory, we give an asymptotic evaluation of the complexity of these spherical spin-glass models. The complexity mentioned here is defined as the mean number of critical points of given index whose value is below (or above) a given level. This includes the important question of counting the mean number of local minima below a given level, and in particular the question of finding the ground state energy (the minimal value of the Hamiltonian). We show that this question is directly related to the study of the edge of the spectrum of the Gaussian Orthogonal Ensemble (GOE). The question of computing the complexity of mean-field spin glass models has recently been thoroughly studied in the physics literature (see for example (CLR03) and the refer- ences therein), mainly for a different measure of the complexity, i.e. the mean number of solutions to the Thouless-Anderson-Palmer equations, or TAP-complexity. Our approach to the complexity enables us to recover known results in the physics literature about TAP-complexity, to compute the ground state energy (when p is even), and to describe an interesting layered structure of the low energy levels of the Hamiltonians of these models, which might prove useful for the study of the metastability of Langevin dynamics for these models (in longer time scales than those studied in (BDG01)). The paper is organised as follows. In Section 2, we give our main results. In Section 3, we prove two main formulas (Theorem 2.1 and 2.2), relating random matrix theory (specif- ically the GOE) and spherical spin glasses. These formulas are direct consequences of the Kac-Rice formula (we learned the version needed here in the book (AT07), for another modern account see (AW09)). The main new ingredient is the fact that, for spherical spin- glass models, the Hessian of the Hamiltonian at a critical point, conditioned on the value of the Hamiltonian, is a symmetric Gaussian random matrix with independent entries (up to symmetry) plus a diagonal matrix. This implies, in particular, that it is possible to

Properties of nonlinear noise in long, dispersion-uncompensated fiber links

Abstract: We study the properties of nonlinear interference noise (NLIN) in fiber-optic communications systems with large accumulated dispersion. Our focus is on settling the discrepancy between the results of the Gaussian noise (GN) model (according to which NLIN is additive Gaussian) and a recently published time-domain analysis, which attributes drastically different properties to the NLIN. Upon reviewing the two approaches we identify several unjustified assumptions that are key in the derivation of the GN model, and that are responsible for the discrepancy. We derive the true NLIN power and verify that the NLIN is not additive Gaussian, but rather it depends strongly on the data transmitted in the channel of interest. In addition we validate the time-domain model numerically and demonstrate the strong dependence of the NLIN on the interfering channels' modulation format.

Gaussian Process Kernels for Pattern Discovery and Extrapolation

Abstract: Gaussian processes are rich distributions over functions, which provide a Bayesian nonparametric approach to smoothing and interpolation. We introduce simple closed form kernels that can be used with Gaussian processes to discover patterns and enable extrapolation. These kernels are derived by modelling a spectral density - the Fourier transform of a kernel - with a Gaussian mixture. The proposed kernels support a broad class of stationary covariances, but Gaussian process inference remains simple and analytic. We demonstrate the proposed kernels by discovering patterns and performing long range extrapolation on synthetic examples, as well as atmospheric CO2 trends and airline passenger data. We also show that it is possible to reconstruct several popular standard covariances within our framework.

GLIMMPSE: Online Power Computation for Linear Models with and without a Baseline Covariate.

Abstract: GLIMMPSE is a free, web-based software tool that calculates power and sample size for the general linear multivariate model with Gaussian errors (http://glimmpse.SampleSizeShop.org/). GLIMMPSE provides a user-friendly interface for the computation of power and sample size. We consider models with fixed predictors, and models with fixed predictors and a single Gaussian covariate. Validation experiments demonstrate that GLIMMPSE matches the accuracy of previously published results, and performs well against simulations. We provide several online tutorials based on research in head and neck cancer. The tutorials demonstrate the use of GLIMMPSE to calculate power and sample size.

Gaussian Bare-Bones Differential Evolution

Abstract: Differential evolution (DE) is a well-known algorithm for global optimization over continuous search spaces. However, choosing the optimal control parameters is a challenging task because they are problem oriented. In order to minimize the effects of the control parameters, a Gaussian bare-bones DE (GBDE) and its modified version (MGBDE) are proposed which are almost parameter free. To verify the performance of our approaches, 30 benchmark functions and two real-world problems are utilized. Conducted experiments indicate that the MGBDE performs significantly better than, or at least comparable to, several state-of-the-art DE variants and some existing bare-bones algorithms.

Benchmarking the Starting Points of the GW Approximation for Molecules

Abstract: The GW approximation is nowadays being used to obtain accurate quasiparticle energies of atoms and molecules. In practice, the GW approximation is generally evaluated perturbatively, based on a prior self-consistent calculation within a simpler approximation. The final result thus depends on the choice of the self-consistent mean-field chosen as a starting point. Using a recently developed GW code based on Gaussian basis functions, we benchmark a wide range of starting points for perturbative GW, including Hartree–Fock, LDA, PBE, PBE0, B3LYP, HSE06, BH&HLYP, CAM-B3LYP, and tuned CAM-B3LYP. In the evaluation of the ionization energy, the hybrid functionals are clearly superior results starting points when compared to Hartree–Fock, to LDA, or to the semilocal approximations. Furthermore, among the hybrid functionals, the ones with the highest proportion of exact-exchange usually perform best. Finally, the reliability of the frozen-core approximation, that allows for a considerable speed-up of the calculatio...

A Versatile Probability Distribution Model for Wind Power Forecast Errors and Its Application in Economic Dispatch

Abstract: The existence of wind power forecast errors is one of the most challenging issues for wind power system operation. It is difficult to find a reasonable method for the representation of forecast errors and apply it in scheduling. In this paper, a probability distribution model named “versatile distribution” is formulated and developed along with its properties and applications. The model can well represent forecast errors for all forecast timescales and magnitudes. The incorporation of the model in economic dispatch (ED) problems can simplify the wind-induced uncertainties via a few analytical terms in the problem formulation. The ED problem with wind power could hence be solved by the classical optimization methods, such as sequential linear programming which has been widely accepted by industry for solving ED problems. Discussions are also extended on the incorporation of the proposed versatile distribution into unit commitment problems. The results show that the new distribution is more effective than other commonly used distributions (i.e., Gaussian and Beta) with more accurate representation of forecast errors and better formulation and solution of ED problems.

BIG & QUIC: Sparse Inverse Covariance Estimation for a Million Variables

Abstract: The l1-regularized Gaussian maximum likelihood estimator (MLE) has been shown to have strong statistical guarantees in recovering a sparse inverse covariance matrix even under high-dimensional settings. However, it requires solving a difficult non-smooth log-determinant program with number of parameters scaling quadratically with the number of Gaussian variables. State-of-the-art methods thus do not scale to problems with more than 20,000 variables. In this paper, we develop an algorithm BIGQUIC, which can solve 1 million dimensional l1-regularized Gaussian MLE problems (which would thus have 1000 billion parameters) using a single machine, with bounded memory. In order to do so, we carefully exploit the underlying structure of the problem. Our innovations include a novel block-coordinate descent method with the blocks chosen via a clustering scheme to minimize repeated computations; and allowing for inexact computation of specific components. In spite of these modifications, we are able to theoretically analyze our procedure and show that BIGQUIC can achieve super-linear or even quadratic convergence rates.

Stable calculation of Gaussian-based RBF-FD stencils

Abstract: Traditional finite difference (FD) methods are designed to be exact for low degree polynomials. They can be highly effective on Cartesian-type grids, but may fail for unstructured node layouts. Radial basis function-generated finite difference (RBF-FD) methods overcome this problem and, as a result, provide a much improved geometric flexibility. The calculation of RBF-FD weights involves a shape parameter @e. Small values of @e (corresponding to near-flat RBFs) often lead to particularly accurate RBF-FD formulas. However, the most straightforward way to calculate the weights (RBF-Direct) becomes then numerically highly ill-conditioned. In contrast, the present algorithm remains numerically stable all the way into the @e->0 limit. Like the RBF-QR algorithm, it uses the idea of finding a numerically well-conditioned basis function set in the same function space as is spanned by the ill-conditioned near-flat original Gaussian RBFs. By exploiting some properties of the incomplete gamma function, it transpires that the change of basis can be achieved without dealing with any infinite expansions. Its strengths and weaknesses compared with the Contour-Pade, RBF-RA, and RBF-QR algorithms are discussed.

Pairwise likelihood ratios for estimation of non-Gaussian structural equation models

Abstract: We present new measures of the causal direction, or direction of effect, between two non-Gaussian random variables. They are based on the likelihood ratio under the linear non-Gaussian acyclic model (LiNGAM). We also develop simple first-order approximations of the likelihood ratio and analyze them based on related cumulant-based measures, which can be shown to find the correct causal directions. We show how to apply these measures to estimate LiNGAM for more than two variables, and even in the case of more variables than observations. We further extend the method to cyclic and nonlinear models. The proposed framework is statistically at least as good as existing ones in the cases of few data points or noisy data, and it is computationally and conceptually very simple. Results on simulated fMRI data indicate that the method may be useful in neuroimaging where the number of time points is typically quite small.

Atomic orbital basis sets

Abstract: Electronic structure methods for molecular systems rely heavily on using basis sets composed of Gaussian functions for representing the molecular orbitals. A number of hierarchical basis sets have been proposed over the last two decades, and they have enabled systematic approaches to assessing and controlling the errors due to incomplete basis sets. We outline some of the principles for constructing basis sets, and compare the compositions of eight families of basis sets that are available in several different qualities and for a reasonable number of elements in the periodic table. © 2012 John Wiley & Sons, Ltd.

Quantum parameter estimation using general single-mode Gaussian states

Abstract: We calculate the quantum Cramer--Rao bound for the sensitivity with which one or several parameters, encoded in a general single-mode Gaussian state, can be estimated. This includes in particular the interesting case of mixed Gaussian states. We apply the formula to the problems of estimating phase, purity, loss, amplitude, and squeezing. In the case of the simultaneous measurement of several parameters, we provide the full quantum Fisher information matrix. Our results unify previously known partial results, and constitute a complete solution to the problem of knowing the best possible sensitivity of measurements based on a single-mode Gaussian state.

MAMPOSSt: Modelling Anisotropy and Mass Profiles of Observed Spherical Systems – I. Gaussian 3D velocities

Abstract: Mass modelling of spherical systems through internal motions is hampered by the mass/velocity anisotropy (VA) degeneracy inherent in the Jeans equation, as well as the lack of techniques that are both fast and adaptable to realistic systems. A new fast method, called MAMPOSSt, which performs a maximum likelihood fit of the distribution of observed tracers in projected phase space, is developed and thoroughly tested. MAMPOSSt assumes a shape for the gravitational potential, but instead of postulating a shape for the distribution function in terms of energy and angular momentum, or supposing Gaussian line-of-sight velocity distributions, MAMPOSSt assumes a VA profile and a shape for the 3D velocity distribution, here Gaussian. MAMPOSSt requires no binning, differentiation, nor extrapolation of the observables. Tests on cluster-mass haloes from LambdaCDM cosmological simulations show that, with 500 tracers, MAMPOSSt is able to jointly recover the virial radius, tracer scale radius, dark matter scale radius and outer or constant VA with small bias (<10% on scale radii and <2% on the two other quantities) and inefficiencies of 10%, 27%, 48% and 20%, respectively. MAMPOSSt does not perform better when some parameters are frozen, and even worse when the virial radius is set to its true value, which appears to be the consequence of halo triaxiality. The accuracy of MAMPOSSt depends weakly on the adopted interloper removal scheme, including an efficient iterative Bayesian scheme that we introduce here, which can directly obtain the virial radius with as good precision as MAMPOSSt. Our tests show that MAMPOSSt with Gaussian 3D velocities is very competitive with, and up to 1000x faster than other methods. Hence, MAMPOSSt is a very powerful and rapid tool for the mass and anisotropy modeling of systems such as clusters and groups of galaxies, elliptical and dwarf spheroidal galaxies.