Gaussian Variables Research Articles

Data Assimilation in Truncated Plurigaussian Models: Impact of the Truncation Map

Assimilation of production data into reservoir models for which the distribution of porosity and permeability is largely controlled by facies has become increasingly common. When the locations of the facies bodies must be conditioned to observations, the truncated plurigaussian model has been often shown to be a useful method for modeling as it allows gaussian variables to be updated instead of facies types. Previous experience has also shown that ensemble Kalman filter-like methods are particularly effective for assimilation of data into truncated plurigaussian models. In this paper, some limitations are shown of the ensemble-based or gradient-based methods when applied to truncated plurigaussian models of a certain type that is likely to occur for modeling channel facies. It is also shown that it is possible to improve the data match and increase the ensemble spread by modifying the updating step using an approximate derivative of the truncation map.

Quantifying relative importance: computing standardized effects in models with binary outcomes

AbstractScientists commonly ask questions about the relative importance of processes and then turn to statistical models for answers. Standardized coefficients are typically used in such situations, with the goal being to compare effects on a common scale. Traditional approaches to obtaining standardized coefficients were developed with idealized Gaussian variables in mind. When responses are binary, complications arise that impact standardization methods. In this paper, we review, evaluate, and propose new methods for standardizing coefficients from models that contain binary outcomes. We first consider the interpretability of unstandardized coefficients and then examine two main approaches to standardization. One approach, which we refer to as the latent‐theoretical (LT) method, assumes that underlying binary observations, there exists a latent, continuous propensity linearly related to the coefficients. A second approach, which we refer to as the observed‐empirical (OE) method, assumes responses are purely discrete and estimates error variance empirically via reference to a classical R2 estimator. We also evaluate the standard formula for calculating standardized coefficients based on standard deviations. Criticisms of this practice have been persistent, leading us to propose an alternative formula that is based on user‐defined “relevant ranges.” Finally, we implement all of the above in an open‐source package for the statistical software R. Results from simulation studies show that both the LT and OE methods of standardization support a similarly broad range of coefficient comparisons. The LT method estimates effects that reflect underlying latent‐linear propensities, while the OE method computes a linear approximation for the effects of predictors on binary responses. The contrast between assumptions for the two methods is reflected in persistently weaker standardized effects associated with OE standardization. Reliance on standard deviations for standardization (the traditional approach) is critically examined and shown to introduce substantial biases when predictors are non‐Gaussian. The use of relevant ranges in place of standard deviations has the capacity to place LT and OE standardized coefficients on a more comparable scale. As ecologists address increasingly complex hypotheses, especially those that comparing the influences of different controlling factors (e.g., top‐down vs. bottom‐up or biotic vs. abiotic controls), comparable coefficients become necessary for evaluations.

Stochastic tail index model for high frequency financial data with Bayesian analysis

This paper proposes a new dynamic model called Stochastic Tail Index (STI) model to analyze time-varying tail index for financial asset using high frequency return data. Bayesian tools are developed to estimate the model, make related inferences, and perform model selection. To construct efficient posterior sampler for the STI model by an approximation approach, a new algorithm called ALSO (Auxiliary Least Squares Optimization) is introduced, which can quickly make sufficient approximation to a given random variable using Gaussian mixture variables. The posterior sampler takes advantages of the BFGS optimization method to tailor the proposal densities in Metropolis–Hastings chains, and is computationally faster than the existing samplers in literature. Simulation shows that the proposed posterior sampler works well for the STI model. To illustrate the use of the STI model in the real world, we analyze two real high frequency data sets associated with two markets. It is found that the estimated daily tail indexes generally follow a time-varying pattern and tend to fall when large negative events occur. Besides, they significantly drop below 2 during some periods, which implies that the variances of the return distributions during those periods may be infinite, and hence any variance-based risk management for the two markets may be questionable.

Optimization Based Methods for Partially Observed Chaotic Systems

In this paper we consider filtering and smoothing of partially observed chaotic dynamical systems that are discretely observed, with an additive Gaussian noise in the observation. These models are found in a wide variety of real applications and include the Lorenz 96’ model. In the context of a fixed observation interval T, observation time step h and Gaussian observation variance sigma _Z^2, we show under assumptions that the filter and smoother are well approximated by a Gaussian with high probability when h and sigma ^2_Z h are sufficiently small. Based on this result we show that the maximum a posteriori (MAP) estimators are asymptotically optimal in mean square error as sigma ^2_Z h tends to 0. Given these results, we provide a batch algorithm for the smoother and filter, based on Newton’s method, to obtain the MAP. In particular, we show that if the initial point is close enough to the MAP, then Newton’s method converges to it at a fast rate. We also provide a method for computing such an initial point. These results contribute to the theoretical understanding of widely used 4D-Var data assimilation method. Our approach is illustrated numerically on the Lorenz 96’ model with state vector up to 1 million dimensions, with code running in the order of minutes. To our knowledge the results in this paper are the first of their type for this class of models.

Supervised dimension reduction for ordinal predictors

In applications involving ordinal predictors, common approaches to reduce dimensionality are either extensions of unsupervised techniques such as principal component analysis, or variable selection procedures that rely on modeling the regression function. A supervised dimension reduction method tailored to ordered categorical predictors is introduced which uses a model-based dimension reduction approach, inspired by extending sufficient dimension reductions to the context of latent Gaussian variables. The reduction is chosen without modeling the response as a function of the predictors and does not impose any distributional assumption on the response or on the response given the predictors. A likelihood-based estimator of the reduction is derived and an iterative expectation–maximization type algorithm is proposed to alleviate the computational load and thus make the method more practical. A regularized estimator, which simultaneously achieves variable selection and dimension reduction, is also presented. Performance of the proposed method is evaluated through simulations and a real data example for socioeconomic index construction, comparing favorably to widespread use techniques.

Simultaneous Inference for Misaligned Multivariate Functional Data

SummaryWe consider inference for misaligned multivariate functional data that represents the same underlying curve, but where the functional samples have systematic differences in shape. We introduce a class of generally applicable models where warping effects are modelled through non-linear transformation of latent Gaussian variables and systematic shape differences are modelled by Gaussian processes. To model cross-covariance between sample co-ordinates we propose a class of low dimensional cross-covariance structures that are suitable for modelling multivariate functional data. We present a method for doing maximum likelihood estimation in the models and apply the method to three data sets. The first data set is from a motion tracking system where the spatial positions of a large number of body markers are tracked in three dimensions over time. The second data set consists of longitudinal height and weight measurements for Danish boys. The third data set consists of three-dimensional spatial hand paths from a controlled obstacle avoidance experiment. We use the method to estimate the cross-covariance structure and use a classification set-up to demonstrate that the method outperforms state of the art methods for handling misaligned curve data.

VIBNN

Bayesian Neural Networks (BNNs) have been proposed to address the problem of model uncertainty in training and inference. By introducing weights associated with conditioned probability distributions, BNNs are capable of resolving the overfitting issue commonly seen in conventional neural networks and allow for small-data training, through the variational inference process. Frequent usage of Gaussian random variables in this process requires a properly optimized Gaussian Random Number Generator (GRNG). The high hardware cost of conventional GRNG makes the hardware implementation of BNNs challenging. In this paper, we propose VIBNN, an FPGA-based hardware accelerator design for variational inference on BNNs. We explore the design space for massive amount of Gaussian variable sampling tasks in BNNs. Specifically, we introduce two high performance Gaussian (pseudo) random number generators: 1) the RAM-based Linear Feedback Gaussian Random Number Generator (RLF-GRNG), which is inspired by the properties of binomial distribution and linear feedback logics; and 2) the Bayesian Neural Network-oriented Wallace Gaussian Random Number Generator. To achieve high scalability and efficient memory access, we propose a deep pipelined accelerator architecture with fast execution and good hardware utilization. Experimental results demonstrate that the proposed VIBNN implementations on an FPGA can achieve throughput of 321,543.4 Images/s and energy efficiency upto 52,694.8 Images/J while maintaining similar accuracy as its software counterpart.

Phase diagram of restricted Boltzmann machines and generalized Hopfield networks with arbitrary priors.

Restricted Boltzmann machines are described by the Gibbs measure of a bipartite spin glass, which in turn can be seen as a generalized Hopfield network. This equivalence allows us to characterize the state of these systems in terms of their retrieval capabilities, both at low and high load, of pure states. We study the paramagnetic-spin glass and the spin glass-retrieval phase transitions, as the pattern (i.e., weight) distribution and spin (i.e., unit) priors vary smoothly from Gaussian real variables to Boolean discrete variables. Our analysis shows that the presence of a retrieval phase is robust and not peculiar to the standard Hopfield model with Boolean patterns. The retrieval region becomes larger when the pattern entries and retrieval units get more peaked and, conversely, when the hidden units acquire a broader prior and therefore have a stronger response to high fields. Moreover, at low load retrieval always exists below some critical temperature, for every pattern distribution ranging from the Boolean to the Gaussian case.

Bias correction in the realized stochastic volatility model for daily volatility on the Tokyo Stock Exchange

The realized stochastic volatility model has been introduced to estimate more accurate volatility by using both daily returns and realized volatility. The main advantage of the model is that no special bias-correction factor for the realized volatility is required a priori. Instead, the model introduces a bias-correction parameter responsible for the bias hidden in realized volatility. We empirically investigate the bias-correction parameter for realized volatilities calculated at various sampling frequencies for six stocks on the Tokyo Stock Exchange, and then show that the dynamic behavior of the bias-correction parameter as a function of sampling frequency is qualitatively similar to that of the Hansen–Lunde bias-correction factor although their values are substantially different. Under the stochastic diffusion assumption of the return dynamics, we investigate the accuracy of estimated volatilities by examining the standardized returns. We find that while the moments of the standardized returns from low-frequency realized volatilities are consistent with the expectation from the Gaussian variables, the deviation from the expectation becomes considerably large at high frequencies. This indicates that the realized stochastic volatility model itself cannot completely remove bias at high frequencies.

Valid analytical performance specifications for combined analytical bias and imprecision for the use of common reference intervals.

Background Many clinical decisions are based on comparison of patient results with reference intervals. Therefore, an estimation of the analytical performance specifications for the quality that would be required to allow sharing common reference intervals is needed. The International Federation of Clinical Chemistry (IFCC) recommended a minimum of 120 reference individuals to establish reference intervals. This number implies a certain level of quality, which could then be used for defining analytical performance specifications as the maximum combination of analytical bias and imprecision required for sharing common reference intervals, the aim of this investigation. Methods Two methods were investigated for defining the maximum combination of analytical bias and imprecision that would give the same quality of common reference intervals as the IFCC recommendation. Method 1 is based on a formula for the combination of analytical bias and imprecision and Method 2 is based on the Microsoft Excel formula NORMINV including the fractional probability of reference individuals outside each limit and the Gaussian variables of mean and standard deviation. The combinations of normalized bias and imprecision are illustrated for both methods. The formulae are identical for Gaussian and log-Gaussian distributions. Results Method 2 gives the correct results with a constant percentage of 4.4% for all combinations of bias and imprecision. Conclusion The Microsoft Excel formula NORMINV is useful for the estimation of analytical performance specifications for both Gaussian and log-Gaussian distributions of reference intervals.

Low-Dimensional Reconciliation for Continuous-Variable Quantum Key Distribution

We propose an efficient logical layer-based reconciliation method for continuous-variable quantum key distribution (CVQKD) to extract binary information from correlated Gaussian variables. We demonstrate that by operating on the raw-data level, the noise of the quantum channel can be corrected in the low-dimensional (scalar) space, and the reconciliation can be extended to arbitrary dimensions. The CVQKD systems allow an unconditionally secret communication over standard telecommunication networks. To exploit the real potential of CVQKD a robust reconciliation technique is needed. It is currently unavailable, which makes it impossible to reach the real performance of the CVQKD protocols. The reconciliation is a post-processing step separated from the transmission of quantum states, which is aimed to derive the secret key from the raw data. The reconciliation process of correlated Gaussian variables is a complex problem that requires either tomography in the physical layer that is intractable in a practical scenario, or high-cost calculations in the multidimensional spherical space with strict dimensional limitations. To avoid these issues, we define the low-dimensional reconciliation. We prove that the error probability of one-dimensional reconciliation is zero in any practical CVQKD scenario, and provides unconditional security. The results allow for significantly improving the currently available key rates and transmission distances of CVQKD.

Modeling and Fitting of Time Series with Heavy Distribution Tails and Strong Time Dependence by Gaussian Time Series

In the model of Gaussian copula time series with the tails of one-dimensional distributions belonging to the Frechet maximum domain of attraction and the description of dependency based on Gaussian variables (see [A. E. Mazur and V. I. Piterbarg, Moscow Univ. Math. Bull., 70 (2015), pp. 197--201]), an estimator for the copula (which is a nonlinear function that takes Gaussian variables to variables from the Frechet maximum domain of attraction) is built. This opens the way for statistical analysis of data time series with potentially heavy tails using the machinery of asymptotic analysis of Gaussian sequences. The consistency and asymptotic normality for this estimator are proved.

ADMM-Based Robust Beamforming Design for Downlink Cloud Radio Access Networks

The cloud radio access network (C-RAN) has been considered as a promising network architecture to improve both spectrum efficiency and energy efficiency of current wireless networks. In this paper, we aim to address the channel uncertainty issue in the multi-input single-output C-RAN by studying the robust beamforming. We formulate the robust beamforming design problem with an aim to minimize the overall network power and backhaul cost while satisfying the each radio remote head power constraint and guaranteeing individual signal-to-interference-plus-noise ratio (SINR) requirements. The channel state information is assumed to be imperfect, and the additive channel state information error is modeled as Gaussian distributed variables. Formulated problem is hard to solve due to the non-convex $\ell _{0}$ -norm functions and channel uncertainty in the constraints. Two statistical approaches, named average approach and probability approach, are proposed to deal with SINR requirements when including the channel uncertainty. $\ell _{0}$ -norm approximation and a majorization-minimization algorithm are utilized to transform $\ell _{0}$ -norm problem into a series of semidefinite programing (SDP) problems. After that, we propose an alternating direction method of multipliers (ADMM)-based algorithm to solve each SDP problem. We introduce two auxiliary variables to reformulate the SDP problems in an ADMM form, which further ensure that solving the SDP problem can be decomposed to solve three convex subproblems. A subgradient algorithm, Karush–Kuhn–Tucker conditions, and a projected gradient method are applied to solve them, respectively. Simulation results verify that the proposed robust algorithm can significantly enhance the performance compared the non-robust case.

General Semiparametric Shared Frailty Model: Estimation and Simulation with frailtySurv.

The R package frailtySurv for simulating and fitting semi-parametric shared frailty models is introduced. Package frailtySurv implements semi-parametric consistent estimators for a variety of frailty distributions, including gamma, log-normal, inverse Gaussian and power variance function, and provides consistent estimators of the standard errors of the parameters' estimators. The parameters' estimators are asymptotically normally distributed, and therefore statistical inference based on the results of this package, such as hypothesis testing and confidence intervals, can be performed using the normal distribution. Extensive simulations demonstrate the flexibility and correct implementation of the estimator. Two case studies performed with publicly available datasets demonstrate applicability of the package. In the Diabetic Retinopathy Study, the onset of blindness is clustered by patient, and in a large hard drive failure dataset, failure times are thought to be clustered by the hard drive manufacturer and model.

Robust and sparse Gaussian graphical modelling under cell‐wise contamination

Graphical modelling explores dependences among a collection of variables by inferring a graph that encodes pairwise conditional independences. For jointly Gaussian variables, this translates into detecting the support of the precision matrix. Many modern applications feature high‐dimensional and contaminated data that complicate this task. In particular, traditional robust methods that down‐weight entire observation vectors are often inappropriate as high‐dimensional data may feature partial contamination in many observations. We tackle this problem by giving a robust method for sparse precision matrix estimation based on the γ‐divergence under a cell‐wise contamination model. Simulation studies demonstrate that our procedure outperforms existing methods especially for highly contaminated data. Copyright © 2018 John Wiley & Sons, Ltd.

Characterization of non-commutative free Gaussian variables

Characterization of non-commutative free Gaussian variables

Powers of Ginibre eigenvalues

We study the images of the complex Ginibre eigenvalues under the power maps $\pi_M: z \mapsto z^M$, for any integer $M$. We establish the following equality in distribution, $$ {\rm{Gin}}(N)^M \stackrel{d}{=} \bigcup_{k=1}^M {\rm{Gin}} (N,M,k), $$ where the so-called Power-Ginibre distributions ${\rm{Gin}}(N,M,k)$ form $M$ independent determinantal point processes. The decomposition can be extended to any radially symmetric normal matrix ensemble, and generalizes Rains' superposition theorem for the CUE and Kostlan's independence of radii to a wider class of point processes. Our proof technique also allows us to recover a result by Edelman and La Croix for the GUE. Concerning the Power-Ginibre blocks, we prove convergence of fluctuations of their smooth linear statistics to independent gaussian variables, coherent with the link between the complex Ginibre Ensemble and the Gaussian Free Field. Finally, some partial results about two-dimensional beta ensembles with radial symmetry and even parameter $\beta$ are discussed, replacing independence by conditional independence.

Segal–Bargmann transform: the q-deformation

We give identifications of the $q$-deformed Segal-Bargmann transform and define the Segal-Bargmann transform on mixed $q$-Gaussian variables. We prove that, when defined on the random matrix model of \'Sniady for the $q$-Gaussian variable, the classical Segal-Bargmann transform converges to the $q$-deformed Segal-Bargmann transform in the large $N$ limit. We also show that the $q$-deformed Segal-Bargmann transform can be recovered as a limit of a mixture of classical and free Segal-Bargmann transform.

A general algorithm for covariance modeling of discrete data

We propose an algorithm that generalizes to discrete data any given covariance modeling algorithm originally intended for Gaussian responses, via a Gaussian copula approach. Covariance modeling is a powerful tool for extracting meaning from multivariate data, and fast algorithms for Gaussian data, such as factor analysis and Gaussian graphical models, are widely available. Our algorithm makes these tools generally available to analysts of discrete data and can combine any likelihood-based covariance modeling method for Gaussian data with any set of discrete marginal distributions. Previously, tools for discrete data were generally specific to one family of distributions or covariance modeling paradigm, or otherwise did not exist. Our algorithm is more flexible than alternate methods, takes advantage of existing fast algorithms for Gaussian data, and simulations suggest that it outperforms competing graphical modeling and factor analysis procedures for count and binomial data. We additionally show that in a Gaussian copula graphical model with discrete margins, conditional independence relationships in the latent Gaussian variables are inherited by the discrete observations. Our method is illustrated with a graphical model and factor analysis on an overdispersed ecological count dataset of species abundances.

Japanese Janken Recognition by Support Vector Machine Based on Electromyogram of Wrist

We propose a method which can discriminate hand motions in this paper. We measure an electromyogram (EMG) of wrist by using 8 dry type sensors. We focus on four motions, such as “Rock-Scissors-Paper” and “Neutral”. “Neutral” is a state that does not do anything. In the proposed method, we apply fast Fourier transformation (FFT) to measured EMG data, and then remove a hum noise. Next, we combine values of sensors based on a Gaussian function. In this Gaussian function, variance and mean are 0.2 and 0, respectively. We then apply normalization by linear transformation to the values. Subsequently, we resize the values into the range from -1 to 1. Finally, a support vector machine (SVM) conducts learning and discrimination to classify them. We conducted experiments with seven subjects. Average of discrimination accuracy was 89.8%. In the previous method, the discrimination accuracy was 77.1%. Therefore, the proposed method is better in accuracy than the previous method. In future work, we will conduct an experiment which discriminates Japanese Janken of a subject who is not learned.