Gaussian Data Research Articles

Additive noise model structure learning based on rank correlation

To examine the structural learning of the additive noise model in causal discovery, a new algorithm, i.e., RCB (Rank-Correlation-Based), is proposed in combination with the method of Rank Correlation. This algorithm can effectively process multivariate linear Gaussian, non-Gaussian and multivariate nonlinear non-Gaussian data. In this article we have made three contributions. First, it is proven that rank correlation can be used as the criterion of the independence test. Second, through a series of experiments, the optimal threshold of rank correlation is found to select the potential neighbors of the target node . Thus, the RCB algorithm greatly reduces the search space and achieves good time performance. The third contribution is the improvement of the RCB algorithm in combination with the hypothesis testing method, and the RCS (Rank-Correlation-Statistics) algorithm is proposed to solve the theoretical basis for the threshold selection. Compared with the existing technology on 7 networks, the RCS algorithm is superior to existing algorithms in terms of both accuracy and time performance. In addition, simulations show that the RCS algorithm can achieve a good time performance and accuracy on low-dimensional large samples, high-dimensional large samples and real data.

Relaxation factor optimization based on cuckoo search algorithm for limited – Angle CT

X-ray Computed Tomography (CT), is a medical imaging technique used in non-invasive exploration by obtaining cross-sectional information of objects. This technique is based on scanning an object at different angles to obtain projections used later to reconstruct the object. Computed Tomography has wide applications in medical science and industry as non-destructive testing. Iterative reconstruction algorithms are very attractive due to their ability of reducing radiation dose while being of high image quality. To achieve high image quality when using these algorithms, relaxation factor plays a very important role in image reconstruction. The present work focuses on relaxation factor estimation using a heuristic algorithm: the ”cuckoo search optimization” for the reconstruction of CT images. We used our heuristic-based relaxation factor estimation with Gaussian noisy data: our method showed high robustness in preserving image features (edges and contours) and obtained improved image quality with a reduced number of projections views.

Actual error rates in linear discrimination of spatial Gaussian data in terms of semivariograms

The novel approach for classifying spatial Gaussian data based on Bayes discriminant functions in terms of semivariograms has been developed. We derived the closed-form expressions for the Maximum Likelihood estimator of regression parameters and the actual error rate (AER). Results are illustrated through simulation study of spatial Gaussian data sampled on a regular two-dimensional unit spacing lattice endowed with neighborhood structure based on Euclidean distance between sites. The accuracy of the derived AER for various values of spatial dependence parameters, Mahalanobis distances and neighborhood sizes are evaluated and compared.

A Computationally Efficient, High-Dimensional Multiple Changepoint Procedure with Application to Global Terrorism Incidence

Abstract Detecting changepoints in data sets with many variates is a data science challenge of increasing importance. Motivated by the problem of detecting changes in the incidence of terrorism from a global terrorism database, we propose a novel approach to multiple changepoint detection in multivariate time series. Our method, which we call SUBSET, is a model-based approach which uses a penalised likelihood to detect changes for a wide class of parametric settings. We provide theory that guides the choice of penalties to use for SUBSET, and that shows it has high power to detect changes regardless of whether only a few variates or many variates change. Empirical results show that SUBSET out-performs many existing approaches for detecting changes in mean in Gaussian data; additionally, unlike these alternative methods, it can be easily extended to non-Gaussian settings such as are appropriate for modelling counts of terrorist events.

Feature-Enhanced Data Classification Using B-center Function

Linear-kernel Support Vector Machine (SVM) classification algorithm needs to constantly calculate the similarity between the data to be tested and the sample during classification. If samples in the dataset are linearly indivisible, the accuracy of the algorithm will get worse. The feature weight calculation based on category information in Multi-layer Perceptron (MLP) algorithm is not accurate enough to express the relationship between features and categories, i.e. it cannot distinguish categories precisely when the features with the same category frequency. In order to improve the performance of MLP and linear-kernel SVM algorithms in classification, an improved function: the Base function, which is based on clustering, is proposed in this paper. Before running the classification algorithm, a series of possible centers is predicted, after which the Base function is used for data clustering to increase the dimension of data, and then samples with more features are passed to classification models. In the high-position Gaussian data space, the Base function can obviously improve the spatial distribution of samples. Experimental comparison in the thesis also proved that the Base function can improve the performance of MLP or SVM in classification effectively. The proposed feature-enhanced MLP is superior to the baseline with 32.6% accuracy promotion on the Gaussian dataset.

Research on image reconstruction algorithms based on autoencoder neural network of Restricted Boltzmann Machine (RBM)

Aiming at the problem of low quality in image reconstruction of traditional image reconstruction algorithm of electromagnetic tomography(EMT), an EMT image reconstruction algorithm based on autoencoder neural network of Restricted Boltzmann Machine (RBM) is proposed. Firstly, the basic principles of EMT system and autoencoder neural network are analyzed. Autoencoder neural network is a deep learning model, which contains two parts: encoder and decoder. The encoding process of the encoder is equivalent to the object field detection process in the EMT system; the decoding process of the decoder is equivalent to the image reconstruction process. On this basis, an autoencoder neural network model is built. In this model, the RBM is used for layer by layer pre-training to obtain the initial weight and offset, and the global weight and offset are adjusted by BP algorithm. The parameter file generated in the trained autoencoder neural network is used to construct a decoder. Finally, the detected voltage value output by the EMT system is input into the decoder network to obtain the reconstructed image of the EMT. Furthermore, data with Gaussian noise and data regarding flow pattern not in training dataset are used to test the generalization ability and practicability of the network, respectively. The experimental results show that the method in this paper is a kind of EMT image reconstruction method with higher accuracy, which also provides a new means for EMT image reconstruction.

Fast Univariate Inference for Longitudinal Functional Models

We propose fast univariate inferential approaches for longitudinal Gaussian and non-Gaussian functional data. The approach consists of three steps: (i) fit massively univariate pointwise mixed-effects models; (ii) apply any smoother along the functional domain; and (iii) obtain joint confidence bands using analytic approaches for Gaussian data or a bootstrap of study participants for non-Gaussian data. Methods are motivated by two applications: (i) Diffusion tensor imaging measured at multiple visits along the corpus callosum of multiple sclerosis patients; and (ii) physical activity (PA) data measured by body-worn accelerometers for multiple days. An extensive simulation study indicates that model fitting and inference are accurate and much faster than existing approaches. Moreover, the proposed approach was the only one that was computationally feasible for the PA data application. Methods are accompanied by R software, though the method is “read-and-use,” as it can be implemented by any analyst who is familiar with mixed-effects model software. Supplementary files for this article are available online.

Geostatistical Learning: Challenges and Opportunities

Statistical learning theory provides the foundation to applied machine learning, and its various successful applications in computer vision, natural language processing and other scientific domains. The theory, however, does not take into account the unique challenges of performing statistical learning in geospatial settings. For instance, it is well known that model errors cannot be assumed to be independent and identically distributed in geospatial (a.k.a. regionalized) variables due to spatial correlation; and trends caused by geophysical processes lead to covariate shifts between the domain where the model was trained and the domain where it will be applied, which in turn harm the use of classical learning methodologies that rely on random samples of the data. In this work, we introduce thegeostatistical (transfer) learningproblem, and illustrate the challenges of learning from geospatial data by assessing widely-used methods for estimating generalization error of learning models, under covariate shift and spatial correlation. Experiments with synthetic Gaussian process data as well as with real data from geophysical surveys in New Zealand indicate that none of the methods are adequate for model selection in a geospatial context. We provide general guidelines regarding the choice of these methods in practice while new methods are being actively researched.

A game-theoretic approach to adversarial linear Gaussian classification

We employ a game-theoretic model to analyse the interaction between an adversary and a classifier. There are two (i.e., positive and negative) classes to which data points can belong. The adversary wants to maximize the probability of miss-detection for the positive class (i.e., false negative probability) while it does not want to significantly modify the data point so that it still maintains favourable traits of the original class. The classifier, on the other hand, wants maximize the probability of correct detection for the positive class (i.e., true positive probability) subject to a lower-bound on the probability of correct detection for the negative class (i.e., true negative probability). For conditionally Gaussian data points (conditioned on the class) and linear support vector machine classifiers, we rewrite the optimization problems of the adversary and the classifier as convex problems and use best response dynamics to learn an equilibrium of the game. This results in computing a linear support vector machine classifier that is robust against adversarial input manipulations.

Machine-learning interatomic potential for W–Mo alloys

In this work, we develop a machine-learning interatomic potential for W x Mo1−x random alloys. The potential is trained using the Gaussian approximation potential framework and density functional theory data produced by the Vienna ab initio simulation package. The potential focuses on properties such as elastic properties, melting, and point defects for the whole range of W x Mo1−x compositions. Moreover, we use all-electron density functional theory data to fit an adjusted Ziegler–Biersack–Littmarck potential for the short-range repulsive interaction. We use the potential to investigate the effect of alloying on the threshold displacement energies and find a significant dependence on the local chemical environment and element of the primary recoiling atom.

Equivalence class selection of categorical graphical models

Learning the structure of dependence relations between variables is a pervasive issue in the statistical literature. A directed acyclic graph (DAG) can represent a set of conditional independencies, but different DAGs may encode the same set of relations and are indistinguishable using observational data. Equivalent DAGs can be collected into classes, each represented by a partially directed graph known as essential graph (EG). Structure learning directly conducted on the EG space, rather than on the allied space of DAGs, leads to theoretical and computational benefits. Still, the majority of efforts has been dedicated to Gaussian data, with less attention to methods designed for multivariate categorical data. A Bayesian methodology for structure learning of categorical EGs is then proposed. Combining a constructive parameter prior elicitation with a graph-driven likelihood decomposition, a closed-form expression for the marginal likelihood of a categorical EG model is derived. Asymptotic properties are studied, and an MCMC sampler scheme developed for approximate posterior inference. The methodology is evaluated on both simulated scenarios and real data, with appreciable performance in comparison with state-of-the-art methods.

Distributionally-robust machine learning using locally differentially-private data

We consider machine learning, particularly regression, using locally-differentially private datasets. The Wasserstein distance is used to define an ambiguity set centered at the empirical distribution of the dataset corrupted by local differential privacy noise. The radius of the ambiguity set is selected based on privacy budget, spread of data, and size of the problem. Machine learning with private dataset is rewritten as a distributionally-robust optimization. For general distributions, the distributionally-robust optimization problem can be relaxed as a regularized machine learning problem with the Lipschitz constant of the machine learning model as a regularizer. For Gaussian data, the distributionally-robust optimization problem can be solved exactly to find an optimal regularizer. Training with this regularizer can be posed as a semi-definite program.

An integrated approach based on Gaussian noises-based data augmentation method and AdaBoost model to predict faecal coliforms in rivers with small dataset

Machine Learning (ML) techniques can be valuable for modelling the faecal contamination in the rivers to overcome the limitations of the process-based models. However, this approach requires large sufficient data for training and validation processes to avoid the over-fitting problem. This study attempts to overcome the small dataset limitation by relying on the data augmentation techniques. To that end, Adaptive boosting (AdaBoost) models were trained and integrated into the data augmentation method to generate 600 virtual samples based on 40 original datasets. The results revealed that the proposed method significantly improved the accuracy (RMSE = 0.716ln(Colony Forming Unit (CFU)/100 ml)) and generalization ability of the AdaBoost model for predicting the faecal coliform in the rivers compared to the baseline model developed only with a small dataset (RMSE = 2.348ln(CFU/100 ml)). However, the study showed that generating and using too many virtual data could deteriorate the generalization ability of the ML model and the optimal virtual datasets are about (337–415) virtual samples. Globally, the results of this study provide new insights to improve the prediction accuracy of the health risk related to the faecal coliforms in raw water used for drinking purposes under a small dataset. The developed method can broaden the application of ML to water resources and environmental sciences when it is impossible to get a large dataset required by ML models.

Effect of polydispersity on bubble characteristics of Geldart Group B particles

In order to enhance the understanding of the influences of bubble characteristics in bubbling fluidized beds of Gaussian and lognormal particle size distributions (PSDs) of Geldart Group B particles, machine learning tools were harnessed. The PSDs had the same Sauter-mean diameter and widths (i.e., ratio of standard deviation to Sauter-mean diameter) of between 10 and 30% and 10 – 70%, respectively. Self-organizing maps (SOMs) analysis of more than a thousand data rows each of Gaussian and lognormal PSD data indicate that bubble velocity, frequency, length, and probability are highly correlated. The optimal number of data assemblies that either the Gaussian or lognormal dataset can be divided into per the Calinski-Harabasz criterion was determined to be three, and it was found that the critical parameter that underlies the demarcation of the datasets was PSD width. This agrees with an earlier study on clusters, wherein the non-monodispersity of the particle systems was also responsible for the division of the dataset into distinct data assemblies. Furthermore, the number-based frequency of the particle species was better correlated with the bubbles than the mass-based one. The key highlight is the predominant influence of PSD width in demarcating the datasets into distinct data assemblies, which underscores the need to account for the polydispersity of particle systems and provides valuable insights towards model development.

Instability results for Euclidean distance, nearest neighbor search on high dimensional Gaussian data

In 1998, Beyer et al. described a nearest neighbor query as unstable if the query point has nearly identical distance from all points in the dataset. Subsequently, researchers have proven that, as data dimensionality goes to infinity, the probability of query instability approaches one for various kinds of data distributions, dataset size functions, and distance metrics. This paper addresses the problem of characterizing query instability behavior over centered Gaussian data generation distributions and Euclidean distance. Sufficient conditions are established on the covariance matrices and dataset size function under which the probability of query instability approaches one. Furthermore, conditions are also established under which the query instability probability is strictly bounded away from one for a non-vanishing set of query points.

On the diffusion NLMS algorithm applied to adaptive networks: Stochastic modeling and performance comparisons

This paper aims to develop an accurate stochastic model for the diffusion normalized least-mean-square (dNLMS) algorithm operating with both combine-then-adapt (CTA) and adapt-then-combine (ATC) strategies, aiming to provide a theoretical basis for supporting the study of this algorithm. In particular, considering uncorrelated and correlated Gaussian input data, model expressions are derived for predicting the mean and mean-square behavior of either an individual node or the whole adaptive network for both transient and steady-state phases. Based on these expressions, the impact of the diffusion strategy along with a combination rule on the algorithm performance is assessed and discussed. In addition, examples are presented to demonstrate how model expressions can help the designer in the adjustment of the algorithm parameters without the need of extensive trial-and-error procedures, making performance comparisons less laborious. The effectiveness of the proposed model is assessed through simulation results covering different operating conditions, network topologies, combination rules, step-size values, as well as for a wide range of eigenvalue spreads of the input data correlation matrix and signal-to-noise ratio (SNR) values.

10 OAM × 16 Wavelengths Two-Layer Switch Based on an Integrated Mode Multiplexer for 19.2 Tb/s Data Traffic

A two-layer switch exploiting orbital angular momentum (OAM) and wavelength of the light as switching domains is presented, aiming to increase the scalability with respect to the single-layer switches. The switch is able to accept 160 optical Gaussian data inputs on a 16-channel wavelength division multiplexing (WDM) grid and direct each input signals to different output ports exploiting 10 OAMs. The optical switch is based on an integrated OAM multiplexer followed by a compact OAM demultiplexer consisting of two refractive elements. Its experimental characterization confirmed a total enabled throughput of 19.2 Tb/s, thanks to the 30 GHz bandwidth available for each port. The switching time can be lower than 1 μs. The OAM switch power consumption, solely due to the thermal tuning of the OAM emitters, since the OAM demux is passive, is 1.35 mW/Gb/s. In the proposed switching architecture the number of active components, i.e., the power consumption, scales linear with the number of ports. This is favorable in comparison with single-layer switches that cascade e.g., 2 × 2 elementary blocks to obtain large port counts, which scale with the square of the number of ports. The switch accepts input and output signals with Gaussian phase profile that propagate through optical fibers and waveguides, thus making it compatible with standard telecom devices. The suitability of the switch to support real data-traffic is proved by successfully testing it with 10G Ethernet and fiber channel over Ethernet (FCoE) data and video traffic. A possible application scenario is represented by a data-center network where the switch can be used to create a low-power consumption network parallel to the network based on standard electronic routers, to manage large traffic flows.

Filling in Cosmic Microwave Background map missing regions via Generative Adversarial Networks

In this work, we propose a new method to fill in (or in-paint) the CMB signal in regions masked out following a point source extraction process. We adopt a modified Generative Adversarial Network (GAN) and compare different combinations of internal (hyper-)parameters and training strategies. We study the performance using a suitable \U0001d49er variable in order to estimate the performance regarding the CMB power spectrum recovery. We consider a test set where one point source is masked out in each sky patch with a 1.83 × 1.83 squared degree extension, which, in our gridding, corresponds to 64 × 64 pixels. The GAN is optimized for estimating performance on Planck 2018 total intensity simulations. The training makes the GAN effective in reconstructing a masking corresponding to about 1500 pixels with 1% error down to angular scales corresponding to about 5 arcminutes. Finally, we show that GAN is able to mimic PDF and number density of peaks for both Gaussian and non-Gaussian data with less than 0.5% relative error.

Classification of Gaussian spatio-temporal data with stationary separable covariances

The novel approach to classification of spatio-temporal data based on Bayes discriminant functions is developed. We focus on the problem of supervised classifying of the spatiotemporal Gaussian random field (GRF) observation into one of two classes specified by different drift parameters, separable nonlinear covariance functions and nonstationary label field. The performance of proposed classification rule is validated by the values of local Bayes and empirical error rates realized by leave one out procedure. A simulation study for spatial covariance functions belonging to powered-exponential family and temporal covariance functions of AR(1) models is carried out. The influence of the values of spatial and temporal covariance parameters to error rates for several label field models are studied. The results showed that the proposed classification methodology can be applied successfully in practice with small error rates and can be a useful tool for discriminant analysis of spatio-temporal data.

Relation between the phase-lag index and lagged coherence for assessing interactions in EEG and MEG data

Over the last two decades, a large number of estimators have been proposed to assess brain connectivity from electroencephalography (EEG) and magnetoencephalography (MEG) data. The statistical theory underlying these estimators, however, is relatively underdeveloped. In particular, the theoretical relationships between different estimators were unknown until very recently. In a recent preprint, Nolte et al. derived formulas for such relationships under the assumption that the data are Gaussian. One of the results was that the phase-lag index and the lagged coherence concentrate around identical values for large sample sizes, i.e. have identical asymptotic limits. A proof of this statement, however, was only sketched, the model assumptions were not checked in experimental data, and sampling properties of the estimators were not considered. We derive the probability density of the relative-phase in the Gaussian model and use it to provide an alternative proof of the asymptotic equality of the phase-lag index and the lagged coherence. The proof is based on power series expansions of the Fourier coefficients of the phase-lag index and the lagged coherence, which are demonstrated to be hypergeometric functions. We also assess the sampling properties of the phase-lag index and the lagged coherence through numerical simulations from the Gaussian model. These demonstrate that throughout the entire parameter space of the model and for all sample sizes, the standard error of the phase-lag index is higher than that of the lagged coherence and thus establish that the lagged coherence is a uniformly better estimator than the phase-lag index. We use experimental EEG and MEG data to verify to what extent the Gaussian assumption is appropriate. Deviations from normality were observed precisely at frequencies for which EEG/MEG power spectra had local maxima, i.e. at oscillatory resonances. Depending on the data-set, the resonances were located in the delta, alpha, and beta frequency bands and correspond to the respective brain rhythms. Based on these observations, we propose to model EEG/MEG data with exponential power densities, which include the Gaussian and Laplace densities as special cases. Lastly, we demonstrate that the asymptotic equality of the phase-lag index and the lagged coherence, as well as the large relative standard errors of the phase lag index, also hold in experimental EEG/MEG data. This establishes that the lagged coherence is not only a better estimator for Gaussian data, but for experimental EEG/MEG data as well.