Linear Random Fields Research Articles

Limit theorems for linear random fields with innovations in the domain of attraction of a stable law

In this paper we study the convergence in distribution and the local limit theorem for the partial sums of linear random fields with i.i.d. innovations that have infinite second moment and belong to the domain of attraction of a stable law with index 0<α≤2 under the condition that the innovations are centered if 1<α≤2 and are symmetric if α=1. We establish these two types of limit theorems as long as the linear random fields are well-defined, the coefficients are either absolutely summable or not absolutely summable.

Estimating space–time wave statistics using a sequential sampling method and Gaussian process regression

The traditional Monte Carlo sampling of waves requires generating tens of thousands of random waves to achieve stable statistics in the tail of the distribution, which is computationally costly for numerical simulations and time-consuming and often impractical for experiments. To improve the sampling efficiency, we present a sequential sampling method, which predicts the wave statistics based on the nonlinear response of deterministic wave groups. This data-driven method starts with parameterising the linear random wave field with a series of Gaussian wave groups. The nonlinear system response from the water wave system is then approximated by observations through the nonlinear evolution of wave groups with carefully designed initial conditions. A sequential sampling method is introduced during this sampling procedure, which determines the next best sampling point based on the previous observations and the distance metric. We examine the performance of the proposed method with Monte Carlo simulation of random wave fields as well as a grid search sampling strategy. The results show the proposed method can achieve computational cost savings over several orders of magnitude for numerical simulations. The results also suggest such a sequential sampling method can be used to determine the test matrix of the wave experiments with better coverage over the parameter space.

A stratified approach to function fingerprinting in program binaries using diverse features

Fingerprinting individual functions in binary code is useful in many security applications ranging from digital forensic analysis of malware corpora to the detection of critical security vulnerabilities. However, existing approaches for fingerprinting functions are typically not resilient to code transformation methods or the use of different compilers. Moreover, another common weakness with these approaches is that when they report a similarity, they do not provide reverse engineers with any insight into the underlying evidence. In order to bridge this gap, our paper presents Plumeria, an obfuscation-resilient and scalable approach based on a stratified architecture comprised of three layers. The first layer retrieves as many candidates as possible by capturing statistical characteristics, function behavior, and function neighborhood relationships. The second layer then trains a linear conditional random field to learn the correlations between the features of the function and its semantics. This layer is designed to reduce the number of false positives. Finally, the third layer is designed to provide insights into the underlying evidence by collecting the side effects exhibited from the candidates selected by the previous layer. Our study evaluates Plumeria in the context of several scenarios: fingerprinting functions in obfuscated/de-obfuscated binaries; fingerprinting functions across different compilers; fingerprinting various vulnerabilities across compilers and versions; and fingerprinting standard library functions. We then benchmark Plumeria on real-world projects and malware binaries, comparing it with existing state-of-the-art solutions. Our results show that Plumeria outperforms existing solutions, with an average precision of over 89%

On the weak invariance principle for non-adapted stationary random fields under projective criteria

In this paper, we study the central limit theorem (CLT) and its weak invariance principle (WIP) for sums of stationary random fields non-necessarily adapted, under different normalizations. To do so, we first state sufficient conditions for the validity of a suitable ortho-martingale approximation. Then, with the help of this approximation, we derive projective criteria under which the CLT as well as the WIP hold. These projective criteria are in the spirit of Hannan’s condition and are well adapted to linear random fields with ortho-martingale innovations and which exhibit long memory.

Limit theorems for linear random fields with tapered innovations. II: The stable case

Limit theorems for linear random fields with tapered innovations. II: The stable case

Bound on the maximal function associated to the law of the iterated logarithms for Bernoulli random fields

We provide a sufficient condition for the bounded law of the iterated logarithms for strictly stationary random fields expressable as a functional of i.i.d. random fields when the summation is done on rectangles. The study is done via the control of the moments of an appropriated maximal function. Applications to functionals of linear random fields, functions of a Gaussian linear random field and Volterra process are given.

Limit Theorems for Linear Random Fields with Tapered Innovations. I: The Gaussian case

We consider the limit behavior of partial-sum random field $$ {S}_{\mathrm{n}}\left({t}_1,{t}_2;X\left({b}_{\mathrm{n}}\right)\right)={\sum}_{k=1}^{\left[{n}_1{t}_1\right]}{\sum}_{l=1}^{\left[{n}_2{t}_2\right]}{X}_{k,l}^{\left(\mathrm{n}\right)} $$ , where $$ \left\{{X}_{k,l}^{\left(\mathrm{n}\right)}={\sum}_{i=0}^{\infty }{\sum}_{j=0}^{\infty }{c}_i{,}_j{\upxi}_{k-i,l-j}\left({b}_{\mathrm{n}}\right),k,l\in \mathrm{\mathbb{Z}}\right\},n\ge 1 $$ , is a family (indexed by n = (n1, n2), ni ≥ 1) of linear random fields with filter ci,j = aibj and innovations ξk,l(bn) having heavy-tailed tapered distributions with tapering parameter bn growing to infinity as n → ∞. We consider the so-called hard tapering as bn grows relatively slowly and the limit random fields for appropriately normalized Sn(t1, t2;X(bn)) are Gaussian random fields. We consider all cases where sequences {ai} and {bj} are long-range, short-range, and negative dependent. In the second part (as a separate paper), we will consider the case of soft tapering, where bn grows more rapidly, and limit random fields are stable.

Frequency domain bootstrap methods for random fields

This paper develops a frequency domain bootstrap method for random fields on Z2. Three frequency domain bootstrap schemes are proposed to bootstrap Fourier coefficients of observations. Then, inverse-transformations are applied to obtain resamples in the spatial domain. As a main result, we establish the invariance principle of the bootstrap samples, from which it follows that the bootstrap samples preserve the correct second-order moment structure for a large class of random fields. The frequency domain bootstrap method is simple to apply and is demonstrated to be effective in various applications including constructing confidence intervals of correlograms for linear random fields, testing for signal presence using scan statistics, and testing for spatial isotropy in Gaussian random fields. Simulation studies are conducted to illustrate the finite sample performance of the proposed method and to compare with the existing spatial block bootstrap and subsampling methods.

On Lamperti type limit theorem and scaling transition for random fields

In the paper we consider a real-valued stationary random field of discrete argument and summation of its values over rectangles. Taking into account Lamperti type theorem for random fields we introduce notions of stable and unstable behavior with respect to summation over rectangles. We present simple examples of linear random fields defined on Z2 and Z3 having unstable behavior with respect to summation over rectangles, in particular, exhibiting the scaling transition phenomenon.

Scaling transition and edge effects for negatively dependent linear random fields on [formula omitted

We obtain a complete description of anisotropic scaling limits and the existence of scaling transition for a class of negatively dependent linear random fields X on Z2 with moving-average coefficients a(t,s) decaying as |t|−q1 and |s|−q2 in the horizontal and vertical directions, q1−1+q2−1<1 and satisfying ∑(t,s)∈Z2a(t,s)=0. The scaling limits are taken over rectangles whose sides increase as λ and λγ when λ→∞, for any γ>0. The scaling transition occurs at γ0X>0 if the scaling limits of X are different and do not depend on γ for γ>γ0X and γ<γ0X. We prove that the scaling transition in this model is closely related to the presence or absence of the edge effects. The paper extends the results in Pilipauskaitė and Surgailis (2017) on the scaling transition for a related class of random fields with long-range dependence.

Guiding Attention in Sequence-to-Sequence Models for Dialogue Act Prediction

The task of predicting dialog acts (DA) based on conversational dialog is a key component in the development of conversational agents. Accurately predicting DAs requires a precise modeling of both the conversation and the global tag dependencies. We leverage seq2seq approaches widely adopted in Neural Machine Translation (NMT) to improve the modelling of tag sequentiality. Seq2seq models are known to learn complex global dependencies while currently proposed approaches using linear conditional random fields (CRF) only model local tag dependencies. In this work, we introduce a seq2seq model tailored for DA classification using: a hierarchical encoder, a novel guided attention mechanism and beam search applied to both training and inference. Compared to the state of the art our model does not require handcrafted features and is trained end-to-end. Furthermore, the proposed approach achieves an unmatched accuracy score of 85% on SwDA, and state-of-the-art accuracy score of 91.6% on MRDA.

Correction to: A note on linear processes with tapered innovations

Investigating the same problems for linear random fields with tapered innovations, I realized that the Hurst index of the limit FBM process and conditions on tapering parameter γ in Theorem 1 were incorrectly calculated.

On the Quenched Central Limit Theorem for Stationary Random Fields Under Projective Criteria

Motivated by random evolutions which do not start from equilibrium, in a recent work, Peligrad and Volný (J Theor Probab, 2018. arXiv:1802.09106 ) showed that the central limit theorem (CLT) holds for stationary ortho-martingale random fields when they are started from a fixed past trajectory. In this paper, we study this type of behavior, also known under the name of quenched CLT, for a class of random fields larger than the ortho-martingales. We impose sufficient conditions in terms of projective criteria under which the partial sums of a stationary random field admit an ortho-martingale approximation. More precisely, the sufficient conditions are of the Hannan’s projective type. We also discuss some aspects of the functional form of the quenched CLT. As applications, we establish new quenched CLTs and their functional form for linear and nonlinear random fields with independent innovations.

Quenched Invariance Principles for Orthomartingale-Like Sequences

In this paper, we study the central limit theorem and its functional form for random fields which are started not from their equilibrium, but rather under the measure conditioned by the past sigma field. The initial class considered is that of orthomartingales and then the result is extended to a more general class of random fields by approximating them, in some sense, with an orthomartingale. We construct an example which shows that there are orthomartingales which satisfy the CLT but not its quenched form. This example also clarifies the optimality of the moment conditions used for the validity of our results. Finally, by using the so-called orthomartingale-coboundary decomposition, we apply our results to linear and nonlinear random fields.

Application of machine learning in BGP anomaly detection

Aiming at the problem of BGP anomalies, a support vector machine-based BGP anomaly detection method (SVM-BGPAD) is proposed. We first employ feature selection algorithm based on Fisher linear analysis and Markov random field technology to select features that can maximize the distance among classes and minimize the distance within the class, and then use grid search and cross-validation methods to optimize the parameters of SVM model. We evaluate the performance of classification model with linear, polynomial, RBF and Sigmoid kernels. The results are compared based on accuracy and F1-Score. Experimental results show that the model based on RBF kernel function can achieve the best classification accuracy of 91.36% and F1-Score of 96.03%.

Cramér type moderate deviations for random fields

AbstractWe study the Cramér type moderate deviation for partial sums of random fields by applying the conjugate method. The results are applicable to the partial sums of linear random fields with short or long memory and to nonparametric regression with random field errors.

Central limit theorem for Fourier transform and periodogram of random fields

In this paper, we show that the limiting distribution of the real and the imaginary part of the Fourier transform of a stationary random field is almost surely an independent vector with Gaussian marginal distributions, whose variance is, up to a constant, the field’s spectral density. The dependence structure of the random field is general and we do not impose any restrictions on the speed of convergence to zero of the covariances, or smoothness of the spectral density. The only condition required is that the variables are adapted to a commuting filtration and are regular in some sense. The results go beyond the Bernoulli fields and apply to both short range and long range dependence. They can be easily applied to derive the asymptotic behavior of the periodogram associated to the random field. The method of proof is based on new probabilistic methods involving martingale approximations and also on borrowed and new tools from harmonic analysis. Several examples to linear, Volterra and Gaussian random fields will be presented.

Anisotropic scaling limits of long-range dependent linear random fields on Z3

We provide a complete description of anisotropic scaling limits of stationary linear random field (RF) on Z 3 with long-range dependence and moving average coefficients decaying as O ( | t i | − q i ) in the i th direction, i = 1 , 2 , 3 . The scaling limits are taken over rectangles in Z 3 whose sides increase as O ( λ γ i ) , i = 1 , 2 , 3 when λ → ∞ , for any fixed γ i > 0 , i = 1 , 2 , 3 . We prove that all these limits are Gaussian RFs whose covariance structure is determined by the fulfillment or violation of the balance conditions γ i q i = γ j q j , 1 ≤ i < j ≤ 3 . The paper extends recent results in Puplinskaitė and Surgailis (2015, 2016) [27] , [28] , Pilipauskaitė and Surgailis (2016, 2017) [31] , [29] on anisotropic scaling of long-range dependent RFs from dimension 2 to dimension 3.

Exact moderate and large deviations for linear random fields

Abstract By extending the methods of Peligrad et al. (2014), we establish exact moderate and large deviation asymptotics for linear random fields with independent innovations. These results are useful for studying nonparametric regression with random field errors and strong limit theorems.

Martingale approximations for random fields

In this paper we provide necessary and sufficient conditions for the mean square approximation of a random field by an ortho-martingale. The conditions are formulated in terms of projective criteria. Applications are given to linear and nonlinear random fields with independent innovations.