Gaussian likelihood inference on data from trans-Gaussian random fields with Matérn covariance function
KAUST DepartmentApplied Mathematics and Computational Science Program
Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
Online Publication Date2017-07-12
Print Publication Date2018-08
Permanent link to this recordhttp://hdl.handle.net/10754/625682
MetadataShow full item record
AbstractGaussian likelihood inference has been studied and used extensively in both statistical theory and applications due to its simplicity. However, in practice, the assumption of Gaussianity is rarely met in the analysis of spatial data. In this paper, we study the effect of non-Gaussianity on Gaussian likelihood inference for the parameters of the Matérn covariance model. By using Monte Carlo simulations, we generate spatial data from a Tukey g-and-h random field, a flexible trans-Gaussian random field, with the Matérn covariance function, where g controls skewness and h controls tail heaviness. We use maximum likelihood based on the multivariate Gaussian distribution to estimate the parameters of the Matérn covariance function. We illustrate the effects of non-Gaussianity of the data on the estimated covariance function by means of functional boxplots. Thanks to our tailored simulation design, a comparison of the maximum likelihood estimator under both the increasing and fixed domain asymptotics for spatial data is performed. We find that the maximum likelihood estimator based on Gaussian likelihood is overall satisfying and preferable than the non-distribution-based weighted least squares estimator for data from the Tukey g-and-h random field. We also present the result for Gaussian kriging based on Matérn covariance estimates with data from the Tukey g-and-h random field and observe an overall satisfactory performance.
CitationYan Y, Genton MG (2017) Gaussian likelihood inference on data from trans-Gaussian random fields with Matérn covariance function. Environmetrics: e2458. Available: http://dx.doi.org/10.1002/env.2458.
SponsorsThis research was supported by the King Abdullah University of Science and Technology (KAUST).
CollectionsArticles; Applied Mathematics and Computational Science Program; Applied Mathematics and Computational Science Program; Statistics Program; Statistics Program; Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division; Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
Showing items related by title, author, creator and subject.
A non-Gaussian multivariate distribution with all lower-dimensional Gaussians and related familiesDutta, Subhajit; Genton, Marc G. (Journal of Multivariate Analysis, Elsevier BV, 2014-07-28) [Article]Several fascinating examples of non-Gaussian bivariate distributions which have marginal distribution functions to be Gaussian have been proposed in the literature. These examples often clarify several properties associated with the normal distribution. In this paper, we generalize this result in the sense that we construct a pp-dimensional distribution for which any proper subset of its components has the Gaussian distribution. However, the jointpp-dimensional distribution is inconsistent with the distribution of these subsets because it is not Gaussian. We study the probabilistic properties of this non-Gaussian multivariate distribution in detail. Interestingly, several popular tests of multivariate normality fail to identify this pp-dimensional distribution as non-Gaussian. We further extend our construction to a class of elliptically contoured distributions as well as skewed distributions arising from selections, for instance the multivariate skew-normal distribution.
Analytic Treatment of Deep Neural Networks Under Additive Gaussian NoiseAlfadly, Modar (2018-04-12) [Thesis]
Advisor: Ghanem, Bernard
Committee members: Heidrich, Wolfgang; Wonka, PeterDespite the impressive performance of deep neural networks (DNNs) on numerous vision tasks, they still exhibit yet-to-understand uncouth behaviours. One puzzling behaviour is the reaction of DNNs to various noise attacks, where it has been shown that there exist small adversarial noise that can result in a severe degradation in the performance of DNNs. To rigorously treat this, we derive exact analytic expressions for the first and second moments (mean and variance) of a small piecewise linear (PL) network with a single rectified linear unit (ReLU) layer subject to general Gaussian input. We experimentally show that these expressions are tight under simple linearizations of deeper PL-DNNs, especially popular architectures in the literature (e.g. LeNet and AlexNet). Extensive experiments on image classification show that these expressions can be used to study the behaviour of the output mean of the logits for each class, the inter-class confusion and the pixel-level spatial noise sensitivity of the network. Moreover, we show how these expressions can be used to systematically construct targeted and non-targeted adversarial attacks. Then, we proposed a special estimator DNN, named mixture of linearizations (MoL), and derived the analytic expressions for its output mean and variance, as well. We employed these expressions to train the model to be particularly robust against Gaussian attacks without the need for data augmentation. Upon training this network on a loss that is consolidated with the derived output probabilistic moments, the network is not only robust under very high variance Gaussian attacks but is also as robust as networks that are trained with 20 fold data augmentation.
Generating Correlated QPSK Waveforms By Exploiting Real Gaussian Random VariablesJardak, Seifallah; Ahmed, Sajid; Alouini, Mohamed-Slim (2012 Conference Record of the Forty Sixth Asilomar Conference on Signals, Systems and Computers (ASILOMAR), Institute of Electrical and Electronics Engineers (IEEE), 2012-11) [Conference Paper]The design of waveforms with specified auto- and cross-correlation properties has a number of applications in multiple-input multiple-output (MIMO) radar, one of them is the desired transmit beampattern design. In this work, an algorithm is proposed to generate quadrature phase shift- keying (QPSK) waveforms with required cross-correlation properties using real Gaussian random-variables (RV’s). This work can be considered as the extension of what was presented in  to generate BPSK waveforms. This work will be extended for the generation of correlated higher-order phase shift-keying (PSK) and quadrature amplitude modulation (QAM) schemes that can better approximate the desired beampattern.