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This paper introduces a new estimation-theoretic framework for experiment design in the context of MR image reconstruction under sparsity constraints. The new framework is called OEDIPUS (Oracle-based Experiment Design for Imaging Parsimoniously Under Sparsity constraints) and is based on combining the constrained Cramér-Rao bound with classical experiment design techniques. Compared to popular random sampling approaches, OEDIPUS is fully deterministic and automatically tailors the sampling pattern to the specific imaging context of interest (i.e., accounting for coil geometry, anatomy, image contrast, etc.). OEDIPUS-based experiment designs are evaluated using retrospectively subsampled in vivo MRI data in several different contexts. The results demonstrate that OEDIPUS-based experiment designs have some desirable characteristics relative to conventional MRI sampling approaches. [ABSTRACT FROM AUTHOR]

To advance the utility of polarimetric synthetic aperture radar observations for extracting biophysical information about vegetation canopies, it is important to develop good understanding of the scattering mechanisms that give rise to the polarimetric scattering response and to apply it toward the development of effective decomposition models of the polarimetric covariance matrices. In this paper, we introduce a more rigorous approach to characterizing the volume scattering component of the three-component scattering model developed by Freeman and Durden. The improved rigor has two aspects: 1) the use of a T-matrix model for computing the polarimetric scattering by an inclined dielectric cylinder and 2) the incorporation of the change in local incidence angle as a function of the orientation of the vegetation cylinders. The improvements address some of the current limitations of the Freeman–Durden model. This paper also provides a sensitivity analysis of the various components of the covariance matrix as a function of several physical parameters including cylinder size, its dielectric constant, and the orientation distribution of cylinders. Such an analysis is a precursor to the development of improved inversion algorithms. [ABSTRACT FROM AUTHOR]

In this paper, we proposed to evaluate the optimal phasor measurement unit (PMU) placement based on the sum of variance (SV) of the robust estimators. Variance is traditionally used as a measure of the quality of estimates. In this paper, after satisfying the requirements of the minimum number of PMUs, maximum measurement redundancy, and observability, the placements obtained are further distinguished on the basis of the variance of the estimated states. Both the weighted least squares (WLS) and robust estimators are considered. Examples on the IEEE 30-bus system using least absolute value (LAV) estimation are given. The covariance of the state estimates from the WLS can be calculated using the existing formulas. However, there is no equivalent formula for the robust LAV estimator. A formula is derived in this paper using influence function to approximately calculate the covariance matrix of the state estimates from the robust LAV estimator. The optimum placement, i.e., the placement with the smallest SV, can be selected. [ABSTRACT FROM AUTHOR]

Three main characteristics of massive multiple-input-multiple-output (MIMO) are studied. The widespread use of these characteristics and their lack of validation motivates this paper based in measurements. First, we study the channel hardening when the number of antennas in the base station (BS) increases. Second, we focus on the channel vector orthogonality between two users. Third, we investigate the rank of the spatial covariance matrix. The data used for this paper have been obtained in two measurement campaigns with all real antennas (i.e., neither virtual arrays nor virtual users). The first one has 64 BS elements arranged in 3 configurations, and it serves 8 users with 2 antennas each. The second campaign has 128 BS elements, serving 2 users with 2 antennas each. Both campaigns include line-of-sight and nonline-of-sight scenarios, designed according to the future 5G deployment scenarios. We show the rate of channel hardening when the number of BS elements increases. We evaluate the sum rate of two users at specific distances between them. We observe the large angular spread occupied by the user. [ABSTRACT FROM AUTHOR]

This paper considers a massive random access problem in which a large number of sporadically active devices wish to communicate with a base station (BS) equipped with massive multiple-input multiple-output (MIMO) antennas. Each device is preassigned a unique signature sequence, and the BS identifies the active devices by detecting which sequences are transmitted. This device activity detection problem can be formulated as a maximum likelihood estimation (MLE) problem for which the sample covariance matrix of the received signal is a sufficient statistic. The goal of this paper is to characterize the feasible set of problem parameters under which this covariance based approach is able to successfully recover the device activities in the massive MIMO regime. Through an analysis of the asymptotic behaviors of MLE via its associated Fisher information matrix, this paper derives a necessary and sufficient condition on the Fisher information matrix to ensure a vanishing probability of detection error as the number of antennas goes to infinity, based on which a numerical phase transition analysis is obtained. This condition is also examined from a perspective of covariance matching, which relates the phase transition analysis to a recently derived scaling law. Further, we provide a characterization of the distribution of the estimation error in MLE, based on which the error probabilities in device activity detection can be accurately predicted. Finally, this paper studies a random access scheme with joint device activity and data detection and analyzes its performance in a similar way. [ABSTRACT FROM AUTHOR]

Reconfigurable intelligent surfaces (RISs) appear as one of the most promising paradigms for future wireless communications, because of their high adjustability for diverse communication demands and the additional information-carrying capability by reflecting patterns. This paper investigates the capacity of RIS-assisted multiple-input multiple-output (MIMO) symbiotic communications utilizing multiple reflecting patterns, where each reflecting pattern is non-uniformly activated to carry additional information. To enhance transmission performance, the reflecting patterns, reflecting activation probability, and the transmit covariance matrix are jointly designed. Since the exact expression of the system capacity is intractable, the lower and upper bounds on the capacity are derived and used for optimization in this paper. Based on the lower bound on the capacity, a gradient ascent algorithm is developed to find the optimal reflecting patterns, reflecting activation probability, and the transmit covariance matrix. By taking advantage of the concise-form upper bound on the capacity, closed-form solutions of the reflecting activation probability and transmit covariance matrix can be derived after optimizing the reflecting patterns. The superiority of the proposed design is investigated and verified by computer simulations. Some selected numerical results demonstrate that the proposed design can achieve a higher capacity than the benchmark adopting only one reflecting pattern. [ABSTRACT FROM AUTHOR]

This paper investigates the compress-and-forward scheme for an uplink cloud radio access network (C-RAN) model, where multi-antenna base stations (BSs) are connected to a cloud-computing-based central processor (CP) via capacity-limited fronthaul links. The BSs compress the received signals with Wyner-Ziv coding and send the representation bits to the CP; the CP performs the decoding of all the users’ messages. Under this setup, this paper makes progress toward the optimal structure of the fronthaul compression and CP decoding strategies for the compress-and-forward scheme in the C-RAN. On the CP decoding strategy design, this paper shows that under a sum fronthaul capacity constraint, a generalized successive decoding strategy of the quantization and user message codewords that allows arbitrary interleaved order at the CP achieves the same rate region as the optimal joint decoding. Furthermore, it is shown that a practical strategy of successively decoding the quantization codewords first, then the user messages, achieves the same maximum sum rate as joint decoding under individual fronthaul constraints. On the joint optimization of user transmission and BS quantization strategies, this paper shows that if the input distributions are assumed to be Gaussian, then under joint decoding, the optimal quantization scheme for maximizing the achievable rate region is Gaussian. Moreover, Gaussian input and Gaussian quantization with joint decoding achieve to within a constant gap of the capacity region of the Gaussian multiple-input multiple-output (MIMO) uplink C-RAN model. Finally, this paper addresses the computational aspect of optimizing uplink MIMO C-RAN by showing that under fixed Gaussian input, the sum rate maximization problem over the Gaussian quantization noise covariance matrices can be formulated as convex optimization problems, thereby facilitating its efficient solution. [ABSTRACT FROM PUBLISHER]

A review of turbulence measurements using ground-based wind lidars is carried out. Works performed in the last 30 yr, i.e. from 1972–2012 are analyzed. More than 80% of the work has been carried out in the last 15 yr, i.e. from 1997–2012. New algorithms to process the raw lidar data were pioneered in the first 15 yr, i.e. from 1972–1997, where standard techniques could not be used to measure turbulence. Obtaining unfiltered turbulence statistics from the large probe volume of the lidars has been and still remains the most challenging aspect. Until now, most of the processing algorithms that have been developed have shown that by combining an isotropic turbulence model with raw lidar measurements, we can obtain unfiltered statistics. We believe that an anisotropic turbulence model will provide a more realistic measure of a turbulence statistic. Future development in algorithms will depend on whether the unfiltered statistics can be obtained without the aid of any turbulence model. With the tremendous growth of the wind energy sector, we expect that lidars will be used for turbulence measurements much more than ever before. [ABSTRACT FROM AUTHOR]

For any machine learning model, finding the optimal hyperparameter setting has a direct and significant impact on the model's performance. In this paper, we discuss different types of hyperparameter optimization techniques. We compare the performance of some of the hyperparameter optimization techniques on image classification datasets with the help of AutoML models. In particular, the paper studies Bayesian optimization in depth and proposes the use of genetic algorithm, differential evolution and covariance matrix adaptation—evolutionary strategy for acquisition function optimization. Moreover, we compare these variants of Bayesian optimization with conventional Bayesian optimization and observe that the use of covariance matrix adaptation—evolutionary strategy and differential evolution improves the performance of standard Bayesian optimization. We also notice that Bayesian optimization tends to perform poorly when genetic algorithm is used for acquisition function optimization. [ABSTRACT FROM AUTHOR]

This paper proposes a joint estimation method for source number and DOA based on an improved convolutional neural network for unknown source number and undetermined DOA estimation. By analyzing the signal model, the paper designs a convolutional neural network model based on the existence of a mapping relationship between the covariance matrix and both the source number and DOA estimation. The model, which discards the pooling layer to avoid data loss and introduces the dropout method to improve generalization, takes the signal covariance matrix as input and the two branches of source number estimation and DOA estimation as outputs, and achieves the unfixed number of DOA estimation by filling in invalid values. Simulation experiments and analysis of the results show that the algorithm can effectively achieve the joint estimation of source number and DOA. Under the conditions of high SNR and a large snapshot number, both the proposed algorithm and the traditional algorithm have high estimation accuracy, while under the conditions of low SNR and a small snapshot, the algorithm is better than the traditional algorithm, and under the underdetermined conditions, where the traditional algorithm often fails, the algorithm can still achieve the joint estimation. [ABSTRACT FROM AUTHOR]

This paper introduces a power-law subordinator and a power-law Lévy process whose Laplace transform and characteristic function are simply made up of power functions or the ratio of power functions, respectively, and proposes a power-law vector random field whose finite-dimensional characteristic functions consist merely of a power function or the ratio of two power functions. They may or may not have first-order moment, and contain Linnik, variance Gamma, and Laplace Lévy processes (vector random fields) as special cases. For a second-order power-law vector random field, it is fully characterized by its mean vector function and its covariance matrix function, just like a Gaussian vector random field. An important feature of the power-law Lévy processes (random fields) is that they can be used as the building blocks to construct other Lévy processes (random fields), such as hyperbolic secant, cosine ratio, and sine ratio Lévy processes (random fields). [ABSTRACT FROM AUTHOR]

This paper presents a robust method for robust estimation of quadratic discriminant analysis. The mean and covariance matrix for estimating quadratic discriminant rule is computed using a robust estimation method called Sn method established from a robust covariance estimator S n C o v. The performance of the proposed method is evaluated using the results of simulated samples. This outlier detection method is compared with some well-known methods available in the current literature. The application of the proposed method in real-life data is also executed in this paper. [ABSTRACT FROM AUTHOR]

This paper describes the effect of analysis window functions on the performance of Mel Frequency Cepstral Coefficient (MFCC) based speaker recognition (SR). The MFCCs of speech signal are extracted from the fixed length frames using Short Time Fourier Analysis (STFA) technique where an appropriate analysis window function is required to extract frames from the complete speech signal of a speaker prior to STFA. The number of frames are consider as the number of MFCC feature vectors of a speaker which uniquely represents the speaker in feature space (domain). For the recognition purpose Vector Quantization (VQ) and/or Gaussian Mixture Model (GMM) and/or Universal Background Model GMM (UBM-GMM) based classifiers are used and a comparative study is made. Generally in state-of-the-art MFCC feature vector extraction, Hamming (in some places abbreviated as Ham in this paper) window function is used, but here we also examine the effect of other window functions like rectangular window, Hann window, B-spline windows, polynomial windows, adjustable windows, hybrid windows and Lanczos window in SR. In the present paper, we briefly describe the analysis window functions and try to evaluate text-independent speaker identification (SI). We also use voice activity detector (VAD) to discard the silence frames before STFA. Indeed, silence frames removal leads to the better performance of SR because MFCC of silent frames make the MFCC feature space intrinsic (MFCC with impurity). Here IITG MV SR database contains speech signal of speakers recorded by different devices, namely, D01, H01, T01, M01 and M02, in different environment, different language, different session. This is the reason for calling the database multi variability. It is observed that VQ classifier performs better than other GMM based classifiers for this database and the classifiers VQ-GMM, VQ-UGM-GMM and the combination of them suffers from singularity problem of covariance matrix. So we evaluate the performance of device D01 for all the classifiers and the three classifiers namely, GMM, UBM-GMM and VQ are used for the remaining four recording devices, H01, T01, M01, M02 because except these three classifiers, all other classifiers suffer from singularity problem of covariance matrix in SI. It is observed that VQ provide the highest accuracy for all the devices. [ABSTRACT FROM AUTHOR]

Balancing a bicycle is typical for the balance control humans perform as a part of a whole range of behaviors (walking, running, skating, skiing, etc.). This paper presents a general model of balance control and applies it to the balancing of a bicycle. Balance control has both a physics (mechanics) and a neurobiological component. The physics component pertains to the laws that govern the movements of the rider and his bicycle, and the neurobiological component pertains to the mechanisms via which the central nervous system (CNS) uses these laws for balance control. This paper presents a computational model of this neurobiological component, based on the theory of stochastic optimal feedback control (OFC). The central concept in this model is a computational system, implemented in the CNS, that controls a mechanical system outside the CNS. This computational system uses an internal model to calculate optimal control actions as specified by the theory of stochastic OFC. For the computational model to be plausible, it must be robust to at least two inevitable inaccuracies: (1) model parameters that the CNS learns slowly from interactions with the CNS-attached body and bicycle (i.e., the internal noise covariance matrices), and (2) model parameters that depend on unreliable sensory input (i.e., movement speed). By means of simulations, I demonstrate that this model can balance a bicycle under realistic conditions and is robust to inaccuracies in the learned sensorimotor noise characteristics. However, the model is not robust to inaccuracies in the movement speed estimates. This has important implications for the plausibility of stochastic OFC as a model for motor control. [ABSTRACT FROM AUTHOR]

The reasonable power quality assessment model of microgrid is significant to the planning and management for a microgrid. In the power quality assessment, how to extract and integrate the implicative information in different evaluation indicators, to realize the comprehensive power quality assessment, are needs to be further studied. To solve the above problems, a microgrid power quality assessment model based on multivariate Gaussian distribution and local sensitivity analysis is proposed in this paper. Firstly, the probability density function is used to integrate the individual assessment indicators, and the correlation between the assessment indicators is reflected by the covariance matrix, and then the above characteristics of evaluation indicators are quantized to be as the indicator weights by using the local sensitivity. Comparing with the traditional power quality assessment method, the proposed evaluation model is more suitable for microgrid. Finally, an example is developed to verify that the overall situation of microgrid power quality can be reasonably reflected by the proposed evaluation model. [ABSTRACT FROM AUTHOR]

This paper mainly discusses the demand bidding and risk management of user aggregators by considering profit and risk. The covariance matrix of demand price was used to analyze the risk model under an uncertain demand price. By considering revenue and cost, the demand bidding strategy of user aggregators was derived to obtain the maximum profit. By using a risk-tolerance parameter β, a new demand bidding model for the user aggregators that takes both risk and profit into consideration was formulated. We simulated the risk posed by fluctuating demand prices for user aggregators using this model. Finally, this paper proposes Feasible Particle Swarm Optimization (FPSO) to solve the demand bidding model of user aggregators. Through the dynamic adjustment of control factor parameters in the FPSO, we changed the behavioral characteristics of various types of particles, improved the search efficiency and stability of particles in high-dimensional space, and sought the optimal solution for the system as a whole. This paper provides a parameter adjustment mechanism, improves the capability of algorithm implementation, and increases the probability of finding the optimal solution. The simulation results suggest that a tradeoff between profit and risk needs to be considered in the search process. By doing so, enterprises' abilities in terms of operation and management control can be enhanced, and effective demand management can be achieved. [ABSTRACT FROM AUTHOR]

Manifold fitting is a manifold verification technique for data with noise and manifold structures. By extracting the expected manifold structure, the reliability of the data manifold hypothesis can be determined, and the true structure of the data without noise can conform to a manifold. This paper proposes a manifold fitting algorithm for the variable-scale spectral graph theory and estimates the deviation of the manifold structure caused by noise. Considering the scale variations in frequency-domain analysis based on spectral graph theory, details of the data under the effect of small scale and characteristics of data shape under the effect of large scale are highlighted. This study uses the average calculation and meanshift method to obtain two types of mean vectors from each neighborhood, which are essential in suppressing noise and maintaining shape, respectively. Therefore, manifold fitting is carried out from two aspects, specifically weakening noise and characterizing the manifold shape. To obtain a closer estimate of the deviation caused by noise to the manifold structure, this study also estimates the neighborhood distribution of data under the effect of medium scale, obtains the covariance information of each neighborhood, and uses the variance information to estimate the manifold structure deviations. [ABSTRACT FROM AUTHOR]

This paper proposes a lightweight bidirectional scattering distribution function (BSDF) model for layered materials with anisotropic reflection and refraction properties. In our method, each layer of the materials can be described by a microfacet BSDF using an anisotropic normal distribution function (NDF). Furthermore, the NDFs of layers can be defined on tangent vector fields, which differ from layer to layer. Our method is based on a previous study in which isotropic BSDFs are approximated by projecting them onto base planes. However, the adequateness of this previous work has not been well investigated for anisotropic BSDFs. In this paper, we demonstrate that the projection is also applicable to anisotropic BSDFs and that the BSDFs are approximated by elliptical distributions using covariance matrices. [ABSTRACT FROM AUTHOR]

This paper proposes a novel algorithm to determine the optimal orientation of sensing axes of redundant inertial sensors such as accelerometers and gyroscopes (gyros) for increasing the sensing accuracy. In this paper, we have proposed a novel iterative algorithm to find the optimal sensor configuration. The proposed algorithm utilizes the majorization-minimization (MM) algorithm and the duality principle to find the optimal configuration. Unlike the state-of-the-art approaches which are mainly geometrical in nature and restricted to sensors’ noise being uncorrelated, the proposed algorithm gives the exact orientations of the sensors and can easily deal with the cases of correlated noise. The proposed algorithm has been implemented and tested via numerical simulation in the MATLAB. The simulation results show that the algorithm converges to the optimal configurations and shows the effectiveness of the proposed algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the downlink ergodic sum rate of a single-cell large-scale multiuser MIMO system in which the base station employs $N$ antennas to communicate with $K$ single-antenna user equipments (UEs). A regularized zero-forcing (RZF) scheme is used for precoding under the assumption that each UE uses a specific power and each link forms a spatially correlated MIMO Rician fading channel. The analysis is conducted assuming that $N$ and $K$ grow large with a given ratio and perfect channel state information is available at the base station. New results from random matrix theory and large system analysis are used to compute an asymptotic expression of the signal-to-interference-plus-noise ratio as a function of system parameters, spatial correlation matrix, and Rician factor. Numerical results are used to validate the accuracy of asymptotic approximations in the finite system regime and to evaluate the performance under different operating conditions. It turns out that the asymptotic expressions provide accurate approximations even for relatively small values of $N$ and $K$. [ABSTRACT FROM AUTHOR]

Time variation is a fundamental problem in statistical and econometric analysis of macroeconomic and financial data. Recently, there has been considerable focus on developing econometric modelling that enables stochastic structural change in model parameters and on model estimation by Bayesian or nonparametric kernel methods. In the context of the estimation of covariance matrices of large dimensional panels, such data requires taking into account time variation, possible dependence and heavy-tailed distributions. In this paper, we introduce a nonparametric version of regularization techniques for sparse large covariance matrices, developed by Bickel and Levina (2008) and others. We focus on the robustness of such a procedure to time variation, dependence and heavy-tailedness of distributions. The paper includes a set of results on Bernstein type inequalities for dependent unbounded variables which are expected to be applicable in econometric analysis beyond estimation of large covariance matrices. We discuss the utility of the robust thresholding method, comparing it with other estimators in simulations and an empirical application on the design of minimum variance portfolios. [ABSTRACT FROM AUTHOR]

Multiple signal classification (MUSIC) has been widely applied in multiple-input multiple-output (MIMO) receivers for direction-of-arrival (DOA) estimation. To reduce the cost of radio frequency (RF) chains at millimeter-wave bands, hybrid analog-digital structure has been adopted in massive MIMO transceivers. In this situation, received signals at the antennas are unavailable to the digital receiver, and as a consequence, the spatial covariance matrix (SCM), which is essential in MUSIC algorithm, cannot be obtained using traditional sample average approach. Based on our previous work, we propose a novel algorithm for SCM reconstruction in hybrid massive MIMO systems with multiple RF chains in this paper. By switching the analog beamformers to a group of predetermined DOAs, SCM can be reconstructed through the solutions of a set of linear equations. Furthermore, a low-complexity algorithm, as well as a careful selection of the predetermined DOAs, will be also presented in this paper. Simulation results show that the proposed algorithms can reconstruct the SCM accurately so that MUSIC algorithm can be well used in hybrid massive MIMO systems with multiple RF chains. [ABSTRACT FROM AUTHOR]

This paper studies the device activity detection problem in a multi-cell massive multiple-input multiple-output (MIMO) system, in which the active devices transmit signature sequences to multiple base stations (BSs) that are connected to a central unit (CU), and the BSs cooperate across multiple cells to detect the active devices based on the sample covariance matrices at the BSs. This paper demonstrates the importance of exploiting the knowledge of channel large-scale fadings in this cooperative detection setting through a phase transition analysis, which characterizes the length of signature sequences needed for successful device activity detection in the massive MIMO regime. It is shown that when the large-scale fadings are known, the phase transition for the multi-cell scenario is approximately the same as that of a single-cell system. In this case, the length of the signature sequences required for reliable activity detection in the multi-cell system can be made to be independent of the number of cells through cooperation, in contrast to the case where the large-scale fadings are not known. Further, this paper considers the case in which the fronthaul links between the BSs and the CU have capacity constraints and proposes a novel cooperation scheme based on the quantization of preliminary detection results at the BSs and the reconstruction of the sample covariance matrices at the CU. Simulations show that the proposed method significantly outperforms the scheme of directly quantizing the sample covariance matrices. [ABSTRACT FROM AUTHOR]

In order to improve the effect of agricultural economic structure optimisation, this paper builds an agricultural economic structure optimisation analysis system based on factor analysis and intelligent algorithms, and uses a set of low-dimensional hidden factor vectors to describe the covariance matrix structure of high-dimensional observation vectors. Moreover, this paper combines the algorithm with the actual situation to obtain the optimisation analysis process of the agricultural economic structure. On this basis, this paper analyses the optimisation of agricultural economic structure in practice with a case analysis method. Moreover, this paper uses mathematical analysis to process agricultural economic structure data, and uses mathematical statistics to visualise the process of data processing. In addition, this paper combines the weighted average to perform parameter processing, combines with the comparative analysis method to obtain the agricultural economic results parameters, and analyses these parameters. Finally, on the basis of the above, this paper proposes the corresponding optimisation strategy of agricultural economic structure. The research results show that the method proposed in this paper has a certain effect. [ABSTRACT FROM AUTHOR]

This paper presents novel numerical approaches to finding the secrecy capacity of the multiple-input multiple-output (MIMO) wiretap channel subject to multiple linear transmit covariance constraints, including sum power constraint, per antenna power constraints and interference power constraint. An analytical solution to this problem is not known and existing numerical solutions suffer from slow convergence rate and/or high per-iteration complexity. Deriving computationally efficient solutions to the secrecy capacity problem is challenging since the secrecy rate is expressed as a difference of convex functions (DC) of the transmit covariance matrix, for which its convexity is only known for some special cases. In this paper we propose two low-complexity methods to compute the secrecy capacity along with a convex reformulation for degraded channels. In the first method we capitalize on the accelerated DC algorithm which requires solving a sequence of convex subproblems, for which we propose an efficient iterative algorithm where each iteration admits a closed-form solution. In the second method, we rely on the concave-convex equivalent reformulation of the secrecy capacity problem which allows us to derive the so-called partial best response algorithm to obtain an optimal solution. Notably, each iteration of the second method can also be done in closed form. The simulation results demonstrate a faster convergence rate of our methods compared to other known solutions. We carry out extensive numerical experiments to evaluate the impact of various parameters on the achieved secrecy capacity. [ABSTRACT FROM AUTHOR]

Support vector regression is a promising method for time-series prediction, as it has good generalisability and an overall stable behaviour. Recent studies have shown that it can describe the dynamic characteristics of financial processes and make more accurate forecasts than other machine learning techniques. The first main contribution of this paper is to propose a methodology for dynamic modelling and forecasting covariance matrices based on support vector regression using the Cholesky decomposition. The procedure is applied to range-based covariance matrices of returns, which are estimated on the basis of low and high prices. Such prices are most often available with closing prices for many financial series and contain more information about volatility and relationships between returns. The methodology guarantees the positive definiteness of the forecasted covariance matrices and is flexible, as it can be applied to different dependence patterns. The second contribution of the paper is to show with an example of the exchange rates from the forex market that the covariance matrix forecasts calculated using the proposed approach are more accurate than the forecasts from the benchmark dynamic conditional correlation model. The advantage of the suggested procedure is higher during turbulent periods, i.e., when forecasting is the most difficult and accurate forecasts matter most. [ABSTRACT FROM AUTHOR]

In this paper, we develop a maintenance model for systems subjected to multiple correlated degradation processes, where a multivariate stochastic process is used to model the degradation processes, and the covariance matrix is employed to describe the interactions among the processes. The system is considered failed when any of its degradation features hits the pre-specified threshold. Due to the dormancy of degradation-based failures, inspection is implemented to detect the hidden failures. The failed systems are replaced upon inspection. We assume an imperfect inspection, in such a way that a failure can only be detected with a specific probability. Based on the degradation processes, system reliability is evaluated to serve as the foundation, followed by a maintenance model to reduce the economic losses. We provide theoretical boundaries of the cost-optimal inspection intervals, which are then integrated into the optimisation algorithm to relieve the computational burden. Finally, a fatigue crack propagation process is employed as an example to illustrate the effectiveness and robustness of the developed maintenance policy. Numerical results imply that the inspection inaccuracy contributes significantly to the operating cost and it is suggested that more effort should be paid to improve the inspection accuracy. [ABSTRACT FROM AUTHOR]

Steering is a type of nonseparable quantum correlation, where its inherent asymmetric feature makes it distinct from Bell nonlocality and entanglement. In this paper, we investigate quantum steering in a two-mode Gaussian state ϱ ̂ c 1 c 2 coupled to a two-mode vacuum reservoir. The mode c 1  ( c 2 ) is emitted during the first (second) transition of a nondegenerate three-level cascade laser. By means of the master equation of the state ϱ ̂ c 1 c 2 , we derive the analytical expression of the steady-state covariance matrix of the modes c 1 and c 2 . Using realistic experimental parameters, we show that the state ϱ ̂ c 1 c 2 can exhibit asymmetric steering. Furthermore, by an appropriate choice of the physical parameters of the state ϱ ̂ c 1 c 2 , we show that one-way steering can be achieved. Essentially, we demonstrate that one-way steering can, in general, occur only from c 1 → c 2 . Besides, we perform a comparative study between the steering of the two laser modes and their Gaussian Rényi-2 entanglement. As a result, we found that the entanglement and steering behave similarly in the same circumstances, i.e. both of them decay under dissipation effect; moreover, they can be well enhanced by inducing more and more quantum coherence in the state ϱ ̂ c 1 c 2 . In particular, we found that the steering remains always less than the Gaussian Rényi-2 entanglement. [ABSTRACT FROM AUTHOR]

This paper proposes a multi-objective optimization method for permanent magnet motors using a fast optimization algorithm, Covariance Matrix Adaptation Evolution Strategy (CMA-ES), and deep learning. Multi-objective optimization with topology optimization is effective in the design of permanent magnet motors. Although CMA-ES needs fewer population size than genetic algorithm for single objective problems, this is not evident for multi-objective problems. For this reason, the proposed method generates training data by solving the single-objective optimization multiple times using CMA-ES, and constructs a deep neural network (NN) based on the data to predict performance from motor images at high speed. The deep NN is then used for fast solution of multi-objective optimization problems. Numerical examples demonstrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we present an unsupervised, completely blind, no-reference (NR) stereoscopic (S3D) image quality prediction model to assess the perceptual quality of natural S3D images. We study the joint dependencies between color and depth features of S3D images and empirically model these dependencies by using a bivariate generalized Gaussian distribution (BGGD). We compute the parameters of BGGD, and we also obtain the determinant and the coherence values from the covariance matrix of the proposed BGGD model. We extract the features of BGGD model and covariance matrix from the reference S3D image, followed by multivariate Gaussian (MVG) distribution modeling on the predicted features of the reference. We estimate the joint color and depth quality of the S3D images by computing the likelihood of the image features with respect to the reference MVG model. We apply the popular 2D unsupervised NIQE model on individual stereo views to estimate the overall spatial quality of the S3D images. Finally, we pool the likelihood scores and the spatial NIQE scores to achieve the estimation for the overall perceived quality of the S3D images. The performance of the proposed model is evaluated on the MICT, LIVE Phase I and II S3D image datasets. The results indicate consistent and robust performance for all datasets. Our proposed estimator is completely blind, as it requires neither training on subjective scores nor reference S3D images. [ABSTRACT FROM AUTHOR]

Let H denote the quaternion algebra. This paper investigates the generalized complementary covariance, which is the ϕ -Hermitian quaternion matrix. We give the properties of the generalized complementary covariance matrices. In addition, we explore the unitary diagonalization of the covariance and generalized complementary covariance. Moreover, we give the generalized quaternion unitary transform algorithm and test the performance by numerical simulation. [ABSTRACT FROM AUTHOR]

Peak electron density data derived from GOLD measurements during 2018–2022 are used to analyze the magnitudes and correlations of ionospheric diurnal variability at low latitudes. The correlation distance describes the correlations between spatial locations and is defined in this paper as the angular separation at which the correlation coefficient decreases from 1 to 0.7. Variations in correlation distance with local time, season, magnetic latitude, solar activity, and geomagnetic activity are discussed in this study. The average value of the zonal correlation distance is approximately 8.55° and 3.56° for the meridional direction at low geomagnetic latitudes (magnetic latitudes <30°). The statistical results indicate that both zonal and meridional correlation distances vary little with local time premidnight, while they show pronounced seasonal and latitudinal variations. Both zonal and meridional correlation distances increase with increasing solar activity and decrease with enhancing geomagnetic activity. The EIA strength and gradient mainly modulate the distributions of correlation distances related to magnetic latitude, season, and solar flux level. An empirical model is constructed to describe the temporal and spatial variations in the correlation distance at low latitudes. The study of correlation distances would contribute to a better understanding of ionospheric variability and improvements in data assimilation. Plain Language Summary: The correlation distance is an important parameter in determining the background covariance matrix in data assimilations, which helps to transfer information from observations to the model‐driven background. The study of correlation distances can not only promote a better understanding of ionospheric variability but also improve ionospheric forecasts. More than 1,400 days of GOLD Nmax (peak electron density) data are applied for the statistical analysis of the correlation distance. In general, the correlation distance presents pronounced seasonal and latitudinal variations, varies little with local time premidnight, increases with increasing solar activity, and decreases with enhancing geomagnetic activity. The morphology of the EIA has a direct control on the correlation distance. The EIA strength and gradient mainly modulate the distributions of correlation distances related to magnetic latitude, season, and solar flux level. An empirical model is constructed, and it can effectively describe the temporal and spatial variations in the correlation distance at low latitudes. The results of correlation distances could further improve the accuracy of data assimilation. Key Points: EIA structures directly control the variation of correlation distancesHorizontal correlation distances increase with solar activity and vary with seasonAn empirical model is constructed to describe the variation in correlation distances with LT, mlat, season, solar and geomagnetic activities [ABSTRACT FROM AUTHOR]

Introduction: The paper presents a hybrid generative/discriminative classification method aimed at identifying abnormalities, such as cancer, in lung X-ray images.Methods: The proposed method involves a generative model that performs generative embedding in Probabilistic Component Analysis (PrCA). The primary goal of PrCA is to model co-existing information within a probabilistic framework, with the intent to locate the feature vector space for X-ray data based on a defined kernel structure. A kernel-based classifier, grounded in information-theoretic principles, was employed in this study.Results: The performance of the proposed method is evaluated against nearest neighbour (NN) classifiers and support vector machine (SVM) classifiers, which use a diagonal covariance matrix and incorporate normal linear and non-linear kernels, respectively.Discussion: The method is found to achieve superior accuracy, offering a viable solution to the class of problems presented. Accuracy rates achieved by the kernels in the NN and SVM models were 95.02% and 92.45%, respectively, suggesting the method's competitiveness with state-of-the-art approaches. [ABSTRACT FROM AUTHOR]

This paper is devoted to study of a class of conformally at (α, β)- metrics that have of the form F = α exp(2s)=s; where s:= βα. They are called Kropina change of exponential (α, β)-metrics. We prove that if F has relatively isotropic mean Landsberg curvature or almost vanishing Ξ-curvature then it is a Riemannian metric or a locally Minkowski metric. Also, we prove that, if F be a weak Einstein metric, then it is either a Riemannian metric or a locally Minkowski metric. [ABSTRACT FROM AUTHOR]

Adaptive EWMA (AEWMA) control charts have gained remarkable recognition by monitoring productions over a wide range of shifts. The adaptation of computational statistic as per system shift is the main aspect behind the proficiency of these charts. In this paper, a function-based AEWMA multivariate control chart is suggested to monitor the stability of the variance–covariance matrix for normally distributed process control. Our approach involves utilizing an unbiased estimator applying the EWMA statistic to estimate the process shift in real-time and adapt the smoothing or weighting constant using a suggested continuous function. Preferably, the Monte Carlo simulation method is utilized to determine the characteristics of the suggested AEWMA chart in terms of proficient detection of process shifts. The underlying computed results are compared with existing EWMA and existing AEWMA charts and proved to outperform in providing quick detection for different sizes of shifts. To illustrate its real-life application, the authors employed the concept in the bimetal thermostat industry dataset. The proposed research contributes to statistical process control and provides a practical tool for the solution while monitoring covariance matrix changes. [ABSTRACT FROM AUTHOR]

A novel online critical neural network weight adjustment algorithm is proposed in this paper by combining policy iteration and recursive least squares (RLSs) to address the problem of optimal control of players in nonlinear systems with nonzero-sum games. The interaction between players and the nonlinearity of the system make it difficult to solve the Hamiltonian function directly. From a linear regression perspective, this paper regards any admissible control and its corresponding value function as the input and output affected by perturbations. By using RLS to process the current data and store the historical data's covariance matrix to adjust the weights, the calculation is greatly simplified by avoiding the space waste caused by a lot of historical data storage and the time waste caused by data collection. When calculating the cumulative error, a discount factor is introduced to avoid the total error value tending to infinity and the effect of historical policy evaluation, where the covariance matrix tends to zero and loses its adjustment effect. When the error of the Hamiltonian equation is zero, the proposed adjustment law will also tend to zero. After that, the stability of the covariance matrix was analysed as well as the convergence of the weight errors was proved. Finally, two simulations are conducted to verify the effectiveness of the proposed algorithm based on the RLS method for solving the Nash equilibrium solution online. [ABSTRACT FROM AUTHOR]

In this paper, we address the problem of covariance matrix estimation for radar adaptive detection under non-Gaussian clutter. Traditional model-based estimators may suffer from performance loss due to the mismatch between real data and assumed models. Therefore, we resort to a data-driven deep-learning method and propose a covariance matrix estimation method based on a complex-valued convolutional neural network (CV-CNN). Moreover, a real-valued (RV) network with the same framework as the proposed CV network is also constructed to serve as a natural competitor. The obtained clutter covariance matrix estimation based on the network is applied to the adaptive normalized matched filter (ANMF) detector for performance assessment. The detection results via both simulated and real sea clutter illustrate that the estimator based on CV-CNN outperforms other traditional model-based estimators as well as its RV competitor in terms of probability of detection (PD). [ABSTRACT FROM AUTHOR]

To improve the performance of underdetermined direction of arrival (DOA) estimation in the scenario of strong mutual coupling and unknown nonuniform noise, this paper presents a novel coprime polarization sensitive array (PSA) with limited antenna number and proposes a low-complexity DOA estimation method via iterative covariance matrix reconstruction. The proposed synthetic array is composed of two single-polarized subarrays with dipole or loop antennas. Owing to the polarization domain, it can efficiently reduce the mutual coupling between subarrays, which dominates the effect of mutual coupling in the existing coprime arrays. For effectively eliminating the nonuniform noise, the cross-covariance matrix between subarrays is utilized. Based on it, the full aperture and all degrees of freedoms of the difference coarray with holes are obtained, and then, initial DOAs are estimated by using orthogonal matching pursuit algorithm. The oblique projection operators depending on initial DOAs are constructed to fill holes in difference coarray, in order to generate a larger covariance matrix for refined angle estimation. Then, a fast iterative procedure associated with covariance matrix reconstruction is given to achieve final DOA estimation, for further enhancing estimation performance. Theoretical analysis and simulation results verify the effectiveness of the proposed array and method. [ABSTRACT FROM AUTHOR]

Monte Carlo simulation (MCS) has been widely applied in the preflight evaluation of flight systems because of advantages such as its ability to evaluate nonlinear systems with uncertain parameters, which are incorporated into MCS randomly and simultaneously. Some aerodynamic and control derivatives can have significant effects on flight control and vehicle performance, so it is important to evaluate the influences of their uncertainties. However, it has not been easy to incorporate uncertainties of derivatives into MCS appropriately. A derivative is the slope of a curve, such as the Cm-α curve. To randomly vary it for use in MCS, the rotation point on the curve must be determined. However, the deviation from the nominal curve becomes greater as the flight condition, such as α, moves further from the rotation point, which can result in excessive variance of the aerodynamic coefficient. This paper presents a new method to generate derivative and bias uncertainties randomly using a covariance matrix of uncertain parameters. The method is applied to the MCS of an existing experimental flight system, and the MCS results are compared with a result that excludes derivative uncertainties to show how to apply the presented method. [ABSTRACT FROM AUTHOR]

The traditional coprime parallel array (TrCPA) is prone to suffer from the problem of angle ambiguity, due to its larger inter-element spacing and limited array aperture, for two-dimensional direction of arrival estimation. And the classic multiple signal classification (MUSIC) algorithm has heavy complexity and has poor estimation ability in some unsatisfactory situations. Both restrict the improvement of angle estimation performance. In this paper, we design a novel displaced coprime parallel array termed NDCPA, by rearranging and misplacing all subarrays of TrCPA along the coordinate axis, which brings about larger array aperture making contributions to superior angle estimation accuracy. Besides, we propose an improved two-step reduced-dimensional MUSIC algorithm to further enhance angle estimation accuracy. The proposed method reconstructs the covariance matrix, which can fully exploit the information contained in uniform linear subarrays. Meanwhile, the transformation of angle domain divides the original two-dimensional search into two one-dimensional searches, reducing complexity to a certain extent. Finally, simulation results verify the effectiveness and applicability of the proposed configuration and method compared with the other structures and methods. [ABSTRACT FROM AUTHOR]

This paper proposes a fast direction of arrival (DOA) estimation method based on positive incremental modified Cholesky decomposition atomic norm minimization (PI-CANM) for augmented coprime array sensors. The approach incorporates coprime sampling on the augmented array to generate a non-uniform, discontinuous virtual array. It then utilizes interpolation to convert this into a uniform, continuous virtual array. Based on this, the problem of DOA estimation is equivalently formulated as a gridless optimization problem, which is solved via atomic norm minimization to reconstruct a Hermitian Toeplitz covariance matrix. Furthermore, by positive incremental modified Cholesky decomposition, the covariance matrix is transformed from positive semi-definite to positive definite, which simplifies the constraint of optimization problem and reduces the complexity of the solution. Finally, the Multiple Signal Classification method is utilized to carry out statistical signal processing on the reconstructed covariance matrix, yielding initial DOA angle estimates. Experimental outcomes highlight that the PI-CANM algorithm surpasses other algorithms in estimation accuracy, demonstrating stability in difficult circumstances such as low signal-to-noise ratios and limited snapshots. Additionally, it boasts an impressive computational speed. This method enhances both the accuracy and computational efficiency of DOA estimation, showing potential for broad applicability. [ABSTRACT FROM AUTHOR]

Robust adaptive beamforming (RAB) can be used to suppress interference signals while retaining the desired signals received by a sensor array. However, desired signal self-cancellation and model mismatch can affect RAB performance. In this paper, a novel interference-plus-noise covariance matrix (INCM) reconstruction method is proposed for RAB to solve these problems. The proposed method divides the desired signal into two ranges according to the input signal-to-noise ratio (SNR), namely low SNR and high SNR. In the low SNR range, INCM reconstruction directly uses the same sample covariance matrix as the sample matrix inversion (SMI) beamformer to retain the advantages of the traditional SMI algorithm. In the high SNR range, the eigenvalues of the sample covariance matrix are used to estimate the interference power and noise power. The optimized interference steering vector (SV) is obtained by solving a quadratic convex optimization problem in an interference subspace. The INCM is reconstructed from the interference SVs, interference power, and noise power. The reconstructed INCM is then used to correct the desired signal SV via maximizing the beamformer output power. This is achieved by solving a quadratically constrained quadratic programming (QCQP) problem. Analysis and simulation results are presented which demonstrate that the proposed method performs well under a variety of mismatch conditions. [ABSTRACT FROM AUTHOR]

In this paper, we study the entanglement structure of a system of N quantum oscillators with distinctive coupling strengths, all linearly coupled to a common massless scalar quantum field. This study is helpful in characterizing the notion of an entanglement domain and its symmetry features, which is useful for understanding the interplay between different levels of structure in many-body quantum systems. The effect of the quantum field on the system is derived via the influence functional and the correlation functions are obtained from the solutions of the evolutionary operator of the reduced density matrix. They are then used to construct the covariance matrix, which forms the basis for our analysis of the structure of quantum entanglement in this open system. To make the physical features explicit, we consider a system of three quantum coupled oscillators placed at the vertices of an equilateral triangle with disparate pairwise couplings. We analyze the entanglement between one oscillator and the other two with equal (symmetric) and unequal (asymmetric) coupling strengths. As a physical illustration, we apply the results for these two different configurations to address some basic issues in macroscopic quantum phenomena from the quantum entanglement perspective. [ABSTRACT FROM AUTHOR]

Recently, many robust adaptive beamforming (RAB) algorithms have been proposed to improve beamforming performance when model mismatches occur. For a uniform linear array, a larger aperture array can obtain higher array gain for beamforming when the inter-sensor spacing is fixed. However, only the small aperture array can be used in the equipment limited by platform installation space, significantly weakening beamforming output performance. This paper proposes two beamforming methods for improving beamforming output in small aperture sensor arrays. The first method employs an integration algorithm that combines angular sector and gradient vector search to reconstruct the interference covariance matrix (ICM). Then, the interference-plus-noise covariance matrix (INCM) is reconstructed combined with the estimated noise power. The INCM and ICM are used to optimize the desired signal steering vector (SV) by solving a quadratically constrained quadratic programming (QCQP) problem. The second method proposes a beamforming algorithm based on a virtual extended array to increase the degree of freedom of the beamformer. First, the virtual conjugated array element is designed based on the structural characteristics of a uniform linear array, and received data at the virtual array element are obtained using a linear prediction method. Then, the extended INCM is reconstructed, and the desired signal SV is optimized using an algorithm similar to the actual array. The simulation results demonstrate the effectiveness of the proposed methods under different conditions. [ABSTRACT FROM AUTHOR]

This paper proposes a class of algorithms for analyzing event count time series, based on state space modeling and Kalman filtering. While the dynamics of the state space model is kept Gaussian and linear, a nonlinear observation function is chosen. In order to estimate the states, an iterated extended Kalman filter is employed. Positive definiteness of covariance matrices is preserved by a square-root filtering approach, based on singular value decomposition. Non-negativity of the count data is ensured, either by an exponential observation function, or by a newly introduced "affinely distorted hyperbolic" observation function. The resulting algorithm is applied to time series of the daily number of seizures of drug-resistant epilepsy patients. This number may depend on dosages of simultaneously administered anti-epileptic drugs, their superposition effects, delay effects, and unknown factors, making the objective analysis of seizure counts time series arduous. For the purpose of validation, a simulation study is performed. The results of the time series analysis by state space modeling, using the dosages of the anti-epileptic drugs as external control inputs, provide a decision on the effect of the drugs in a particular patient, with respect to reducing or increasing the number of seizures. [ABSTRACT FROM AUTHOR]

Approaching space target tracking is a typical and challenging mission in the space situational awareness (SSA) field. As the space-based radar is able to monitor the space targets of interest full-weather all-time, the space-based radar system is utilized in this paper. However, most multi-target tracking (MTT) filters in target tracking studies merely utilize the location or narrow measurements, and many potentially valuable electromagnetic scattering characteristics are missed, which leads to space target false tracking problems. The space-based radar transmits a wide-band signal, and the measured high-resolution range profile (HRRP) information is an effective characteristic for different target discrimination. Therefore, the HRRP characteristics of space targets are implemented into the update recursion of the MTT filter, which can be utilized to improve the tracking performance. Then, to predict the target HRRP sequence, the geometrical theory of diffraction (GTD) model is utilized. Additionally, a modified spatial spectrum method with a novel covariance matrix is designed to improve the scattering parameter estimation accuracy. Finally, an adapting threshold is devised for merging the Gaussian mixture (GM) components weights. The proposed threshold is on the basis of the proposed HRRP characteristic-aided probability hypothesis density (PHD) filter, and it can tackle the problem of space target discrimination. Simulation results validate the effectiveness and robustness of the proposed probability hypothesis density (HGI-PHD) filter aided by HRRP information and improved with GM weights. [ABSTRACT FROM AUTHOR]

In this paper, we consider the dynamics of Rényi negativities after a quantum quench in the free-boson chain with homogeneous dissipation. Initially we prepare the system in the squeezed thermal state, and then let it evolves under the tight-binding bosonic Hamiltonian with local linear dissipation. We use the Lindblad equation to solve the time evolution of the covariance matrix, from which one can obtain the time dependence of Rényi negativities. We are interested in the weak dissipation hydrodynamic limit where a quasi-particle picture emerges. In this limit, exact results of non-equilibrium dynamics of Rényi negativities can be obtained using the stationary phase method. We consider the Rényi negativities between both adjacent and disjoint regions in a infinite chain. We numerically test our analytical predictions and perfect matches have found. [ABSTRACT FROM AUTHOR]

This paper makes an empirical comparison of prominent methods in portfolio optimisation, such as nodewise regression, the sample covariance matrix, observable factor model-based covariance matrices, linear and nonlinear shrinkage methods, and principal orthogonal complement thresholding (POET) estimators. Empirically, we find that the nodewise regression approach that uses a direct estimator of the sparse inverse covariance matrix improves portfolio performance among existing methods in mean-variance portfolio optimisation when the number of stocks is greater than the number of observations. [ABSTRACT FROM AUTHOR]

Accurate online parameter identification and state of charge (SOC) estimation are both very crucial for ensuring the operating safety of lithium-ion batteries and usually the former is a base of the latter. To achieve accurate and stable SOC estimation results, this paper proposes a model-based method, which incorporates a vector forgetting factor least square (VFFLS) algorithm and an improved adaptive cubature Kalman filter (IACKF). Firstly, considering it is difficult for the traditional forgetting factor recursive least square (FFRLS) algorithm to balance the accuracy, convergence, and stability for multiple parameters with different time-varying periods, an improved VFFLS method is employed to determine the multiple parameters of the first-order RC battery model online. It supersedes the single forgetting factor in the FFRLS with multiple forgetting factors in a vector form for improving adaptive capability to multiple time-varying parameters. Secondly, aiming at the fact that the standard cubature Kalman filter (CKF) cannot operate properly when the error covariance matrix is non-positive definite, which is caused by disturbance, initial error, and the limit of the computer word length, the UR decomposition rather than the Cholesky decomposition is applied, thus improving the algorithm stability. In addition, an adaptive update strategy is added to the CKF to enhance accuracy and convergence speed. Finally, comparative experiments with different operating patterns, positive and non-positive definite error covariance matrices, and temperatures are carried out. Experimental results showed that the proposed method can estimate the SOC accurately and stably. [ABSTRACT FROM AUTHOR]

The paper proposes a fast method for the multidimensional parameter estimation of a polarization-sensitive array. Compared with conventional methods (e.g., MUSIC algorithm), the proposed method applies an iterative approach based on Newton's method to obtain joint estimation results instead of a spectral search and dimension reduction. It also extends the original Newton method to the 4D scale using the Hessian matrix. To reduce the complexity of establishing the aim function, Nystrom's method is applied to process the covariance matrix. A new threshold is also proposed to select the results, which can accomplish the parameter estimation with a small number of iterations while guaranteeing a high estimation accuracy. Finally, the proposed algorithm is analyzed in detail and the numerical simulations of various algorithms are compared to verify its effectiveness. [ABSTRACT FROM AUTHOR]