Random Vector Research Articles

Modeling of Nonlinear Systems: Method of Optimal Injections

In this paper, a nonlinear system is interpreted as an operator F transforming random vectors. It is assumed that the operator is unknown and the random vectors are available. It is required to find a model of the system represented by a best constructive operator F approximation. While the theory of operator approximation with any given accuracy has been well elaborated, the theory of best constrained constructive operator approximation is not so well developed. Despite increasing demands from various applications, this subject is minimally tractable because of intrinsic difficulties with associated approximation techniques. This paper concerns the best constrained approximation of a nonlinear operator in probability spaces. The main conceptual novelty of the proposed approach is that, unlike the known techniques, it targets a constructive optimal determination of all 3p+2 ingredients of the approximating operator where p is a nonnegative integer. The solution to the associated problem is represented by a combination of new best approximation techniques with a special iterative procedure. The proposed approximating model of the system has several degrees of freedom to minimize the associated error. In particular, one of the specific features of the developed approximating technique is special random vectors called injections. It is shown that the desired injection is determined from the solution of a special Fredholm integral equation of the second kind. Its solution is called the optimal injection. The determination of optimal injections in this way allows us to further minimize the associated error.

Weight decay regularized adversarial training for attacking angle imbalance

Weight decay regularized adversarial training for attacking angle imbalance

An efficient vision transformer for Alzheimer’s disease classification using magnetic resonance images

An efficient vision transformer for Alzheimer’s disease classification using magnetic resonance images

Improved chance constrained random vector functional link network for prediction interval of wind power forecasting

Improved chance constrained random vector functional link network for prediction interval of wind power forecasting

Short-Term Forecasting of Electricity Price using Ensemble Deep. Kernel Based Random Vector Functional Link Network

Short-Term Forecasting of Electricity Price using Ensemble Deep. Kernel Based Random Vector Functional Link Network

Q‐GEV Based Novel Trainable Clustering Scheme for Reducing Complexity of Data Clustering

ABSTRACTThis paper presents a new data clustering technique aimed at enhancing the performance of the trainable path‐cost algorithm and reducing the computational complexity of data clustering models. The proposed method facilitates the discovery of natural groupings and behaviours, which is crucial for effective coordination in complex environments. It identifies natural groupings within a set of features and detects the best clusters with similar behaviour in the data, overcoming the limitations of traditional state‐of‐the‐art methods. The algorithm utilises a density peak clustering method to determine cluster centers and then extracts features from paths passing through these peak points (centers). These features are used to train the support vector machine (SVM) to predict the labels of other points. The proposed algorithm is enhanced using two key concepts: first, it employs Q‐Generalised Extreme Value (Q‐GEV) under power normalisation instead of traditional generalised extreme value distributions, thereby increasing modelling flexibility; second, it utilises the random vector functional link (RVFL) network rather than the SVM, which helps avoid overfitting and improves label prediction accuracy. The effectiveness of the proposed clustering algorithm is evaluated through various experiments, including those on UCI benchmark datasets and real‐world data, demonstrating significant improvements across multiple performance metrics, including F1 measure, Jaccard index, purity, and accuracy, highlighting its capability in accurately identifying paths between similar clusters. Its average F1 measure, Jaccard index, purity, and accuracy is measured 76.87%, 56.29%, 80.29%, and 79.64%, respectively.

Fast blood flow index reconstruction of diffuse correlation spectroscopy using a back-propagation-free data-driven algorithm

This study introduces a fast and accurate online training method for blood flow index (BFI) and relative BFI (rBFI) reconstruction in diffuse correlation spectroscopy (DCS). We implement rigorous mathematical models to simulate the auto-correlation functions (g 2) for semi-infinite homogeneous and three-layer human brain models. We implemented a fast online training algorithm known as random vector functional link (RVFL) to reconstruct BFI from noisy g 2. We extensively evaluated RVFL regarding both speed and accuracy for training and inference. Moreover, we compared RVFL with extreme learning machine (ELM) architecture, a conventional convolutional neural network (CNN), and three fitting algorithms. Results from semi-infinite and three-layer models indicate that RVFL achieves higher accuracy than the other algorithms, as evidenced by comprehensive metrics. While RVFL offers comparable accuracy to CNNs, it boosts training speeds that are 3900-fold faster and inference speeds that are 19.8-fold faster, enhancing its generalizability across different experimental settings. We also used g 2 from one- and three-layer Monte Carlo (MC)-based in-silico simulations, as well as from analytical models, to compare the accuracy and consistency of the results obtained from RVFL and ELM. Furthermore, we discuss how RVFL is more suitable for embedded hardware due to its lower computational complexity than ELM and CNN for training and inference.

On the use and misuse of time-rescaling to assess the goodness-of-fit of self-exciting temporal point processes

The paper first highlights important drawbacks and biases related to the common use of time-rescaling to assess the goodness-of-fit (Gof) of self-exciting temporal point process (SETPP) models. Then it presents a new predictive time-rescaling approach leading to an asymptotically unbiased Gof framework for general SETPPs in the case of single observed trajectories. The predictive approach focuses on forecasting accuracy and addresses the bias problem resulting from the plugged-in estimated parameters. Dawid's prequential approach is used and the models' checking is mainly based on the forecasting accuracy of arrival times. These times are transformed, using sequentially estimated parameters, into random vectors which are proved to converge in probability under the null hypothesis and standard regulatory conditions to vectors of iid Exponential(1) rv's. Numerical experiments are used to compare the performances of the standard and predictive time-rescaling for Gof assessment of non-homogeneous Poisson and Hawkes self-exciting temporal processes. Data of Japanese seismic events are also used to illustrate the dynamic aspect of the proposed model-checking approach.

Maximum interpoint distance of high-dimensional random vectors

Maximum interpoint distance of high-dimensional random vectors

Complex-valued random vector functional link neural network based on real augmented representation and its applications

Complex-valued random vector functional link neural network based on real augmented representation and its applications

Sequential and simultaneous distance-based dimension reduction

This paper introduces a method called Sequential and Simultaneous Distance-based Dimension Reduction (S2D2R) that performs simultaneous dimension reduction for a pair of random vectors based on distance covariance (dCov). Compared with sufficient dimension reduction (SDR) and canonical correlation analysis (CCA)-based approaches, S2D2R is a model-free approach that does not impose dimensional or distributional restrictions on variables and is more sensitive to nonlinear relationships. Theoretically, we establish a non-asymptotic error bound to guarantee the performance of S2D2R. Numerically, S2D2R performs comparable to or better than other state-of-the-art algorithms and is computationally faster. All codes of our S2D2R method can be found on GitHub https://github.com/Yijin911/S2D2R.git, including an R package named S2D2R.

A new structure of random approach vector space

This paper is explaining the fundamental goal of A-Random approach on nonempty set if it meets the condition. A duo (, R) is dubbed A-Random approach space, the relationship between approach space and A-Random approach space is clarified. We define the R-contraction function and debate some of its properties. We introduce the definition of A-Random approach semi group, A-Random approach group, A-Random approach vector space and we will also discuss solve various problems.

The limiting distribution of a bivariate random vector under univariate truncation

The dependence structure in the tails of bivariate random vectors is studied by means of the copula representation. In particular, asymptotic results for the distribution of a random pair under univariate truncation is proved in the spirit of multivariate extensions of the Pickands-Balkema-de Haan Theorem.

Developing a two stage optimized random vector functional link neural network based predictor model utilizing a swift crow search algorithm

Developing a two stage optimized random vector functional link neural network based predictor model utilizing a swift crow search algorithm

On complete convergence for weighted sums of coordinatewise widely orthant dependent random vectors in Hilbert spaces

. In this article, the concept of coordinatewise widely orthant dependent random vectors is introduced and some results on complete convergence for weighted sums of coordinatewise widely orthant dependent random vectors are established in real separable Hilbert spaces. The results obtained in this article extend the corresponding ones of Wang et al. (2014) from rowwise widely orthant dependent random variables to coordinatewise widely orthant-dependent random vectors and the conclusions stated in Anh and Hien (2022) are also extended from coordinatewise negatively associated random vectors to coordinatewise widely orthant-dependent random vectors.

Inference under multivariate size-biased sampling

The present research deals with statistical inference for the expectation of a function of a random vector based on biased samples. After highlighting with the help of a motivating example the need for conducting this study, using the concept of multivariate weighted distributions, a consistent and asymptotically normally distributed estimator is proposed and utilized for developing statistical inference. A Monte Carlo study is carried out to examine the performance of the estimator proposed. Finally, the analysis of a real-world data set illustrates the benefits of using the proposed methods for statistical inference.

A robust and powerful metric for distributional homogeneity

AbstractAssessing the homogeneity of two random vectors is a fundamental task in statistical inference. In this work, we introduce a weighted multivariate Cramér‐von Mises type metric that transforms each variable through a marginal mixture distribution function and integrates the squared difference in probability functions of these transformed variables. Notably, our metric exhibits scale invariance, rendering it robust against outliers and heterogeneity. The expression for our metric is straightforward and possesses a closed‐form representation. It is non‐negative and attains a value of zero if and only if the two random vectors are identically distributed. Moreover, our metric employs an ‐norm expression, which significantly enhances its effectiveness in high‐dimensional scenarios compared to traditional methods relying on the ‐norm. We validate the efficacy of our proposed approach through extensive simulation studies and empirical data analysis.

Cantelli’s Bounds for Generalized Tail Inequalities

Let X be a centered random vector in a finite-dimensional real inner product space E. For a subset C of the ambient vector space V of E and x,y∈V, write x⪯Cy if y−x∈C. If C is a closed convex cone in E, then ⪯C is a preorder on V, whereas if C is a proper cone in E, then ⪯C is actually a partial order on V. In this paper, we give sharp Cantelli-type inequalities for generalized tail probabilities such as PrX⪰Cb for b∈V. These inequalities are obtained by “scalarizing” X⪰Cb via cone duality and then by minimizing the classical univariate Cantelli’s bound over the scalarized inequalities. Three diverse applications to random matrices, tails of linear images of random vectors, and network homophily are also given.

Improved distance correlation estimation

Distance correlation is a novel class of multivariate dependence measure, taking positive values between 0 and 1, and applicable to random vectors of not necessarily equal arbitrary dimensions. It offers several advantages over the well-known Pearson correlation coefficient, the most important being that distance correlation equals zero if-and-only if- the random vectors are independent. There are two different estimators of the distance correlation available in the literature. The first estimator, proposed by Székely et al. (Ann Stat 35:2769–279 2007), is based on an asymptotically unbiased estimator of the distance covariance, which is a V-statistic. The second builds on an unbiased estimator of the distance covariance proposed in Székely and Rizzo (Stat 42:2382–2412 2014), shown to be a U-statistic by Huo and Székely (Technometrics 58:435–447 2016). This study evaluates their efficiency (mean squared error) and compares computational times for both methods under different dependence structures. Under conditions of independence or near-independence, the V-estimates are biased, while the U-estimator frequently cannot be computed due to negative values. To address this challenge, a convex linear combination of the former estimators is proposed and studied, yielding good results regardless of the level of dependence. Additionally, a medical database is studied and discussed.

A brain-computer interface system for lower-limb exoskeletons based on motor imagery and stacked ensemble approach.

Existing lower limb exoskeletons (LLEs) have demonstrated a lack of sufficient patient involvement during rehabilitation training. To address this issue and better incorporate the patient's motion intentions, this paper proposes an online brain-computer interface (BCI) system for LLE based motor imagery and stacked ensemble. The establishment of this online BCI system enables a comprehensive closed-loop control process, which includes the collection and decoding of brain signals, robotic control, and real-time feedback mechanisms. Additionally, an online experimental protocol that integrates visual and proprioceptive feedback is developed. To enhance decoding precision, we proposed a novel classification algorithm based on the stacking technique, termed weighted random forests-support vector machines (WRF-SVM). In this algorithm, WRFs function as the base learning models, while SVMs act as the meta-learning layer. To assess the efficacy of the BCI system and the classification algorithm, eight subjects were recruited for testing. The outcomes of both online and offline experiments exhibit high classification accuracy, confirming the viability and utility of the BCI system. We are confident that our approach holds significant promise for practical applications in the field of LLE technology.