Standard Gaussian Research Articles

Sparse Bayesian Learning for non-Gaussian sources

In this paper, we show that in the multiple measurement vector model we can take advantage of having multiple samples to learn the properties of the distributions of the sources as part of the recovery process and demonstrate that this improves the recovery performance. We propose a method to solve the simultaneous sparse approximation problem using a mixture of Gaussians prior, inspired by existing Sparse Bayesian Learning approaches. We justify our proposed prior by showing that there are a number of signals modelled better by a mixture of Gaussians prior than the standard zero-mean Gaussian prior, such as communications signals which often have a multimodal distribution. We further show that this method can be applied to data distributed according to an alpha-stable distribution. We also show that our proposed method can be applied to compressed sensing of ultrasound images and demonstrate an improvement over existing methods.

The maxima and sums of multivariate non-stationary Gaussian sequences

Let {Xk1, …, Xkp, k ≥ 1} be a p-dimensional standard (zero-means, unit-variances) non-stationary Gaussian vector sequence. In this work, the joint limit distribution of the maxima of {Xk1, …, Xkp, k ≥ 1}, the incomplete maxima of those sequences subject to random failure and the partial sums of those sequences are obtained.

Partial stochastic dominance for the multivariate Gaussian distribution

We establish a partial stochastic dominance result for the maximum of a multivariate Gaussian random vector with positive intraclass correlation coefficient and negative expectation. Specifically, we show that the distribution function intersects that of a standard Gaussian exactly once.

Tunable second-order, birefringent interferometric filter for dynamic chirp management

A tunable second-order birefringent interferometric filter for minimizing signal degradation due to chirping in modulated lasers is proposed and demonstrated. Compared with commonly used filters with standard Gaussian and super-Gaussian filter profiles, a high-order loop mirror filter of similar bandwidth has a much sharper filtering slope. This provides more effective, specific filtering of desired components of a signal’s spectrum, minimizing signal power loss experienced during filtering. The steep filtering slope also provides significant improvement in transmission performance even with a change in data rate. Thus, the same filter can be reused for different bit rates. Furthermore, the tuning capability of the loop mirror filter enables good transmission performance when a variation in signal wavelength is present. Moreover, the unique, periodic filter profile potentially enables chirp management for multiple channels at different wavelengths. We experimentally verify the chirp management performance of the second-order birefringent loop mirror filter. Improvements to the transmission performance of both directly modulated lasers and electro-absorption modulated lasers at different data rates are achieved.

Nondiffracting chirped Bessel waves in optical antiguides

Chirped Bessel waves are introduced as stable (nondiffracting) solutions of the paraxial wave equation in optical antiguides with a power-law radial variation in their index of refraction. Through numerical simulations, we investigate the propagation of apodized (finite-energy) versions of such waves, with or without vorticity, in antiguides with practical parameters. The new waves exhibit a remarkable resistance against the defocusing effect of the unstable index potentials, outperforming standard Gaussians with the same full width at half-maximum. The chirped profile persists even under conditions of eccentric launching or antiguide bending and is also capable of self-healing like standard diffraction-free beams in free space.

Analysis, optimization, and implementation of a hybrid DS/FFH spread-spectrum technique for smart grid communications

In recent years, there has been great interest in using hybrid spread-spectrum (HSS) techniques for commercial applications, particularly in the Smart Grid, in addition to their inherent uses in military communications. This is because HSS can accommodate high data rates with high link integrity, even in the presence of significant multipath effects and interfering signals. A highly useful form of this transmission technique for many types of command, control, and sensing applications is the specific code-related combination of standard direct sequence modulation with ‘fast’ frequency hopping, denoted hybrid DS/FFH, wherein multiple frequency hops occur within a single data-bit time. In this paper, error-probability analyses are performed for a hybrid DS/FFH system over standard Gaussian and fading-type channels, progressively including the effects from wide- and partial-band jamming, multi-user interference, and varying degrees of Rayleigh and Rician fading. In addition, an optimization approach is formulated that minimizes the bit-error performance of a hybrid DS/FFH communication system and solves for the resulting system design parameters. The optimization objective function is non-convex and can be solved by applying the Karush-Kuhn-Tucker conditions. We also present our efforts toward exploring the design, implementation, and evaluation of a hybrid DS/FFH radio transceiver using a single field-programmable gate array (FPGA). Numerical and experimental results are presented under widely varying design parameters to demonstrate the adaptability of the waveform for varied harsh smart grid RF signal environments.

Random Triangle Theory with Geometry and Applications

What is the probability that a random triangle is acute? We explore this old question from a modern viewpoint, taking into account linear algebra, shape theory, numerical analysis, random matrix theory, the Hopf fibration, and much more. One of the best distributions of random triangles takes all six vertex coordinates as independent standard Gaussians. Six can be reduced to four by translation of the center to \((0,0)\) or reformulation as a \(2\times 2\) random matrix problem. In this note, we develop shape theory in its historical context for a wide audience. We hope to encourage others to look again (and differently) at triangles. We provide a new constructive proof, using the geometry of parallelians, of a central result of shape theory: triangle shapes naturally fall on a hemisphere. We give several proofs of the key random result: that triangles are uniformly distributed when the normal distribution is transferred to the hemisphere. A new proof connects to the distribution of random condition numbers. Generalizing to higher dimensions, we obtain the “square root ellipticity statistic” of random matrix theory. Another proof connects the Hopf map to the SVD of \(2\times 2\) matrices. A new theorem describes three similar triangles hidden in the hemisphere. Many triangle properties are reformulated as matrix theorems, providing insight into both. This paper argues for a shift of viewpoint to the modern approaches of random matrix theory. As one example, we propose that the smallest singular value is an effective test for uniformity. New software is developed, and applications are proposed.

Anonymisation of address coordinates for microlevel analyses of the built environment: a simulation study

BackgroundData privacy is a major concern in spatial epidemiology because exact residential locations or parts of participants’ addresses such as street or zip codes are used to perform geospatial analyses....

Geostatistical Modelling Using Non‐Gaussian Matérn Fields

AbstractThis work provides a class of non‐Gaussian spatial Matérn fields which are useful for analysing geostatistical data. The models are constructed as solutions to stochastic partial differential equations driven by generalized hyperbolic noise and are incorporated in a standard geostatistical setting with irregularly spaced observations, measurement errors and covariates. A maximum likelihood estimation technique based on the Monte Carlo expectation‐maximization algorithm is presented, and a Monte Carlo method for spatial prediction is derived. Finally, an application to precipitation data is presented, and the performance of the non‐Gaussian models is compared with standard Gaussian and transformed Gaussian models through cross‐validation.

Improving Surface Geostrophic Current from a GOCE-Derived Mean Dynamic Topography Using Edge-Enhancing Diffusion Filtering

With increased geoid resolution provided by the gravity and steady-state ocean circulation explorer (GOCE) mission, the ocean’s mean dynamic topography (MDT) can be now estimated with an accuracy not available prior to using geodetic methods. However, an altimetric-derived MDT still needs filtering in order to remove short wavelength noise unless integrated methods are used in which the three quantities are determined simultaneously using appropriate covariance functions. We studied nonlinear anisotropic diffusive filtering applied to the ocean´s MDT and a new approach based on edge-enhancing diffusion (EED) filtering is presented. EED filters enable controlling the direction and magnitude of the filtering, with subsequent enhancement of computations of the associated surface geostrophic currents (SGCs). Applying this method to a smooth MDT and to a noisy MDT, both for a region in the Northwestern Pacific Ocean, we found that EED filtering provides similar estimation of the current velocities in both cases, whereas a non-linear isotropic filter (the Perona and Malik filter) returns results influenced by local residual noise when a difficult case is tested. We found that EED filtering preserves all the advantages that the Perona and Malik filter have over the standard linear isotropic Gaussian filters. Moreover, EED is shown to be more stable and less influenced by outliers. This suggests that the EED filtering strategy would be preferred given its capabilities in controlling/preserving the SGCs.

Reduced dimensional Gaussian process emulators of parametrized partial differential equations based on Isomap

In this paper, Isomap and kernel Isomap are used to dramatically reduce the dimensionality of the output space to efficiently construct a Gaussian process emulator of parametrized partial differential equations. The output space consists of spatial or spatio-temporal fields that are functions of multiple input variables. For such problems, standard multi-output Gaussian process emulation strategies are computationally impractical and/or make restrictive assumptions regarding the correlation structure. The method we develop can be applied without modification to any problem involving vector-valued targets and vector-valued inputs. It also extends a method based on linear dimensionality reduction to response surfaces that cannot be described accurately by a linear subspace of the high dimensional output space. Comparisons to the linear method are made through examples that clearly demonstrate the advantages of nonlinear dimensionality reduction.

Prediction of mature microRNA and piwi-interacting RNA without a genome reference or precursors.

The discovery of novel microRNA (miRNA) and piwi-interacting RNA (piRNA) is an important task for the understanding of many biological processes. Most of the available miRNA and piRNA identification methods are dependent on the availability of the organism’s genome sequence and the quality of its annotation. Therefore, an efficient prediction method based solely on the short RNA reads and requiring no genomic information is highly desirable. In this study, we propose an approach that relies primarily on the nucleotide composition of the read and does not require reference genomes of related species for prediction. Using an empirical Bayesian kernel method and the error correcting output codes framework, compact models suitable for large-scale analyses are built on databases of known mature miRNAs and piRNAs. We found that the usage of an L1-based Gaussian kernel can double the true positive rate compared to the standard L2-based Gaussian kernel. Our approach can increase the true positive rate by at most 60% compared to the existing piRNA predictor based on the analysis of a hold-out test set. Using experimental data, we also show that our approach can detect about an order of magnitude or more known miRNAs than the mature miRNA predictor, miRPlex.

Estimation in ill-posed linear models with nuisance design

The paper deals with recovering an unknown vector θ ∈ ℝp in two simple linear models: in the first one we observe y = b · θ + ∈ξ and z = b + σξ′, whereas in the second one we have at our disposal y′ = b2 · θ + ∈b · ξ and z = b + σξ′. Here b ∈ ℝp is a nuisance vector with positive components and ξ, ξ′ ∈ ℝp are standard white Gaussian noises in ℝp. It is assumed that p is large and components bk of b are small for large k. In order to get good statistical estimates of θ in this situation, we propose to combine minimax estimates of 1/bk and 1/bk2 with regularization techniques based on the roughness penalty approach. We provide new non-asymptotic upper bounds for the mean square risks of the estimates related to this method.

Increased localization precision by interference fringe analysis.

We report a novel optical single-emitter-localization methodology that uses the phase induced by path length differences in a Mach-Zehnder interferometer to improve localization precision. Using information theory, we demonstrate that the localization capability of a modified Fourier domain signal generated by photon interference enables a more precise localization compared to a standard Gaussian intensity distribution of the corresponding point-spread function. The calculations were verified by numerical simulations and an exemplary experiment, where the centers of metal nanoparticles were localized to a precision of 3 nm.

Raney Distributions and Random Matrix Theory

Recent works have shown that the family of probability distributions with moments given by the Fuss–Catalan numbers permit a simple parameterized form for their density. We extend this result to the Raney distribution which by definition has its moments given by a generalization of the Fuss–Catalan numbers. Such computations begin with an algebraic equation satisfied by the Stieltjes transform, which we show can be derived from the linear differential equation satisfied by the characteristic polynomial of random matrix realizations of the Raney distribution. For the Fuss–Catalan distribution, an equilibrium problem characterizing the density is identified. The Stieltjes transform for the limiting spectral density of the singular values squared of the matrix product formed from $$q$$ inverse standard Gaussian matrices, and $$s$$ standard Gaussian matrices, is shown to satisfy a variant of the algebraic equation relating to the Raney distribution. Supported on $$[0,\infty )$$ , we show that it too permits a simple functional form upon the introduction of an appropriate choice of parameterization. As an application, the leading asymptotic form of the density as the endpoints of the support are approached is computed, and is shown to have some universal features.

Regularization of Non-Normal Matrices by Gaussian Noise

We consider the regularization of matrices |$M^N$| in Jordan form by additive Gaussian noise |$N^{-\gamma }G^N$|⁠, where |$G^N$| is a matrix of i.i.d. standard Gaussians and |$\gamma >{\tfrac {1}{2}}$| so that the operator norm of the additive noise tends to |$0$| with |$N$|⁠. Under mild conditions on the structure of |$M^N,$| we evaluate the limit of the empirical measure of eigenvalues of |$M^N+N^{-\gamma } G^N$| and show that it depends on |$\gamma$|⁠, in contrast with the case of a single Jordan block.

Divisive Gaussian processes for nonstationary regression.

Standard Gaussian process regression (GPR) assumes constant noise power throughout the input space and stationarity when combined with the squared exponential covariance function. This can be unrealistic and too restrictive for many real-world problems. Nonstationarity can be achieved by specific covariance functions, though prior knowledge about this nonstationarity can be difficult to obtain. On the other hand, the homoscedastic assumption is needed to allow GPR inference to be tractable. In this paper, we present a divisive GPR model which performs nonstationary regression under heteroscedastic noise using the pointwise division of two nonparametric latent functions. As the inference on the model is not analytically tractable, we propose a variational posterior approximation using expectation propagation (EP) which allows for accurate inference at reduced cost. We have also made a Markov chain Monte Carlo implementation with elliptical slice sampling to assess the quality of the EP approximation. Experiments support the usefulness of the proposed approach.

A two-sided estimate for the Gaussian noise stability deficit

The Gaussian noise-stability of a set A in R^n is defined by S_rho(A) = P (X in A and Y in A) where X and Y are standard Gaussian vectors whose correlation is rho. Borell's inequality states that for all 0 < rho < 1, among all sets A with a given Gaussian measure, the quantity S_rho(A) is maximized when A is a half-space. We give a novel short proof of this fact, based on stochastic calculus. Moreover, we prove an almost tight, two-sided, dimension-free robustness estimate for this inequality: by introducing a new metric to measure the distance between the set A and its corresponding half-space H (namely the distance between the two centroids), we show that the deficit S_rho(H) - S_rho(A) can be controlled from both below and above by essentially the same function of the distance, up to logarithmic factors. As a consequence, we also establish the conjectured exponent in the robustness estimate proven by Mossel-Neeman, which uses the total-variation distance as a metric. In the limit rho->1, we get an improved dimension free robustness bound for the Gaussian isoperimetric inequality. Our estimates are also valid for a the more general version of stability where more than two correlated vectors are considered.

Probabilistic analysis of consolidation that considers spatial variability using the stochastic response surface method

To obtain more accurate and reasonable results in the analyses of soil consolidation, the spatial variability of the soil properties should be considered. In this study, we analyzed the consolidation by vertical drains for soil improvement considering the spatial variability of the coefficients of consolidation. The coefficients for the variation in the vertical and horizontal coefficients of consolidation in Yeonjongdo, South Korea were evaluated, and the probability density function (PDF) was assumed by the Anderson–Darling goodness-of-fit test. Standard Gaussian random fields were generated based on a Karhunen–Loeve expansion, and then transformed using Hermite polynomials in the random field with the log-Gaussian PDF of the coefficient of consolidation. The average degree of consolidation was subsequently calculated using the finite difference method coupled with log-Gaussian random fields. In addition, the stochastic response surface method (SRSM) was applied for the efficient probabilistic uncertainty propagation. A sensitivity analysis was performed for the input parameters of the random field, and the spatial variability was considered using random variables from the Karhunen–Loeve expansion as the input data for the SRSM. The results indicated that when considering the spatial variability of soil properties, the probability of failure for the target degree of consolidation was smaller when the correlation distance was taken into account than when it was not. Additionally, the probability of failure decreased when the correlation distance decreased. Compared with the Monte Carlo simulation (MCS) results, the SRSM analysis can achieve results of similar accuracy to those obtained using the MCS analysis with a sample size of 100,000 (numerical runs), and a third-order SRSM expansion with only 333 numerical runs is sufficient for obtaining the probability with errors less than 0.01.

Heteroskedasticity Robust Panel Unit Root Testing Under Variance Breaks in Pooled Regressions

Noting that many economic variables display occasional shifts in their second order moments, we investigate the performance of homogenous panel unit root tests in the presence of permanent volatility shifts. It is shown that in this case the test statistic proposed by Herwartz and Siedenburg (2008) is asymptotically standard Gaussian. By means of a simulation study we illustrate the performance of first and second generation panel unit root tests and undertake a more detailed comparison of the test in Herwartz and Siedenburg (2008) and its heteroskedasticity consistent Cauchy counterpart introduced in Demetrescu and Hanck (2012a). As an empirical illustration, we reassess evidence on the Fisher hypothesis with data from nine countries over the period 1961Q2–2011Q2. Empirical evidence supports panel stationarity of the real interest rate for the entire subperiod. With regard to the most recent two decades, the test results cast doubts on market integration, since the real interest rate is diagnosed nonstationary.