Noncentral Chi-square Distribution Research Articles

Streaming variational probabilistic principal component analysis for monitoring of nonstationary process

Modern industrial processes are characteristic of nonstationary and uncertainty. To address these issues, this paper proposes a probabilistic principal component analysis based model that utilizes streaming variational inference for online monitoring of nonstationary process. The model parameters are updated in a streaming/batch-wise manner to track the nonstationary behavior in the process. In the updating procedures, the posterior distributions obtained from the previous data batch are used as priors for the current model, whose parameters can be estimated using the variational Bayesian inference. The streaming variational inference enables recursive parameter estimation over time. Additionally, in order to improve computational efficiency, a distributed update strategy is also applied to the proposed model. For online process monitoring, the noncentral chi-square distributions are derived to estimate the statistical control limits for each data batch, resulting in better adaptation to the nonstationary characteristics. The effectiveness and superiority of the proposed method are validated through simulation example and real-world industrial application.

Identification of the Damages and Abnormal Objects in Tibetan Stone Walls Based on GPR Data Analysis

ABSTRACT Based on the amplitude attribute analysis of ground penetrating radar (GPR) data, this paper applies GPR to the identification of internal damages and abnormal objects in stone walls of Tibetan ruins. The corresponding relationship between the characteristics of internal damages and abnormal objects and the radar data is explored through the Tibetan stone walls (TSWs) simulation test. The identification of location and size of the damages and abnormal objects in TSWs is realized based on the Mahalanobis distance abnormal data discrimination method. Using the normal distribution and the noncentral chi-square distribution, the identification law of types of the damages and abnormal objects is constructed, which takes root mean square (RMS) amplitude and interface reflection coefficient as the characteristic values. The field detection results of TSWs at three Tibetan sites are taken as an extension and supplement to the simulation results, and a set of identification atlas for the location, size and type of the internal damages and abnormal objects in TSWs is established. The application results of the identification atlas were verified by the field detection and the results show that the atlas has a high accuracy, which can significantly improve the efficiency of the identification, and can provide a basis for the performance research, protection and maintenance of TSWs.

Analytically Computing the Moments of a Conic Combination of Independent Noncentral Chi-Square Random Variables and Its Application for the Extended Cox–Ingersoll–Ross Process with Time-Varying Dimension

This paper focuses mainly on the problem of computing the γth, γ>0, moment of a random variable Yn:=∑i=1nαiXi in which the αi’s are positive real numbers and the Xi’s are independent and distributed according to noncentral chi-square distributions. Finding an analytical approach for solving such a problem has remained a challenge due to the lack of understanding of the probability distribution of Yn, especially when not all αi’s are equal. We analytically solve this problem by showing that the γth moment of Yn can be expressed in terms of generalized hypergeometric functions. Additionally, we extend our result to computing the γth moment of Yn when Xi is a combination of statistically independent Zi2 and Gi in which the Zi’s are distributed according to normal or Maxwell–Boltzmann distributions and the Gi’s are distributed according to gamma, Erlang, or exponential distributions. Our paper has an immediate application in interest rate modeling, where we can explicitly provide the exact transition probability density function of the extended Cox–Ingersoll–Ross (ECIR) process with time-varying dimension as well as the corresponding γth conditional moment. Finally, we conduct Monte Carlo simulations to demonstrate the accuracy and efficiency of our explicit formulas through several numerical tests.

Enhancing UAV-Enabled Communications via Multiple Intelligent Omni-Surfaces

This letter investigates an air-ground communication system where an unmanned aerial vehicle (UAV) communicates with multiple users with the assistance of multiple intelligent omni-surfaces (IOSs). Our goal is to maximize the minimum average expectation of the achievable rate through the joint optimization of the users’ scheduling, IOSs’ phase shifts, and UAV’s trajectory. We first show the random channel power gains follow the more tractable noncentral Chi-square distribution. Then, the resulting optimization problem is a mixed-integer non-convex problem with tightly coupled optimization variables, which is difficult to solve optimally. We propose an efficient algorithm to obtain the suboptimal solution based on block coordinate descent (BCD) and successive convex approximation (SCA) methods. Simulation results show that our proposed scheme significantly outperforms other benchmark schemes.

Performance Analysis of Intelligent Reflecting Surface Assisted Wireless Communication System

In this paper, we investigate the end-to-end performance of intelligent reflecting surface (IRS)-assisted wireless communication systems. We consider a system in which an IRS is deployed on a uniform planar array (UPA) configuration, including a large number of reflecting elements, where the transmitters and receivers are only equipped with a single antenna. Our objective is to analytically obtain the achievable ergodic rate, outage probability, and bit error rate (BER) of the system. Furthermore, to maximize the system’s signal-to-noise ratio (SNR), we design the phase shift of each reflecting element and derive the optimal reflection phase of the IRS based on the channel state information (CSI). We also derive the exact expression of the SNR probability density function (p.d.f.) and show that it follows a non-central Chi-square distribution. Using the p.d.f., we then derive the theoretical results of the achievable rate, outage probability, and BER. The accuracy of the obtained theoretical results is also verified through numerical simulation. It was shown that the achievable rate, outage probability, and BER could be improved by increasing the number of reflecting elements and choosing an appropriate SNR regime. Furthermore, we also find that the IRS-assisted communication system achieves better performance than the existing end-to-end wireless communication.

IRS-Assisted Full Duplex Systems Over Rician and Nakagami Fading Channels

Intelligent reflecting surface (IRS) has been deemed as an energy and spectral-efficient technology, that can potentially enhance network coverage and transmission reliability, with minimum impact on transceivers' complexity. Motivated by this, we develop a comprehensive analysis on the performance of integrating IRS into full-duplex (FD) cellular or Internet of Things (IoT) networks in both realistic Rician and Nakagami fadings. Firstly, in the context of reciprocal channels in Rician fadings, we derive the closed-form approximations of the users' outage probability (OP) and ergodic capacity (EC), under the non-central Chi-square distribution assumption on the signal-to-interference-plus-noise ratio (SINR). Further following by the Gamma distribution assumption on the SINR, we derive the cumulative distribution function (CDF) expression of the user's SINR, which is then leveraged to obtain simple yet effective closed-form expressions in terms of OP and EC. Subsequently, in Nakagami fading scenarios with the reciprocal and non-reciprocal channels, the closed forms of both users' OP and EC are obtained. Finally, the correctness of all the theoretical expressions is verified through substantial Monte Carlo simulations. The results indicate that the OP and EC deduced from Gamma distribution exhibit the fairly precise results for the arbitrary number of IRS elements, especially in Nakagami fadings.

Performance of a Bayes factor (BF) based broadband active acoustic system

We explore the performance of a Bayes Factor (BF) active acoustic inference approach under various refraction and rough surface scattering regimes that typify diverse ocean waveguides. The BF processor, a time varying quadratic form over beam-angle observations, enables various user risk minimization strategies. The nature of the BF quadratic form is shaped by the expected ambient noise, reverberation and target angle-delay spectra. The structure of the BF processor provides intuition regarding the role of prior environmental and bathymetric information on detection in shallow water ocean waveguides with rough boundary scattering. The distribution of the BF under the composite null hypothesis when target and reverberation are well approximated as uncorrelated is shown to be a superposition of heteroskedastic gamma densities with the degree of freedom being the dimension of the target hypothesis scattering sub space. The distribution of the BF under the composite alternative is a superposition of heteroskedastic scaled non-central chi-square distributions. These distributions do not admit simple closed forms but can be well approximated. We consider a number of refractive and scattering regimes and relate the probability of detection as a function of probability of false alarm under conventional fixed threshold and cost rules. [This work supported by the Office of Naval Research.]

Computation of cumulative density of noncentral chi-square distribution in Excel

Computation of cumulative density of noncentral chi-square distribution in Excel

Performance Analysis of IRS-Assisted Full-Duplex Wireless Communication Systems With Interference

This letter considers a full-duplex wireless communication system, where two users communicate to each other through an intelligent reflecting surface (IRS). Each user is assumed to be equipped with two antennas, one each for reception and transmission. The system suffers from loop-interference (LI) between the transmit and receive antennas, self-interference (SI) through the IRS, and external interferers (EI) available at the IRS and users. The fading channels are modelled by Weibull distribution as it is a generalized and accurate model for both indoor and outdoor communications. By taking into account the IRS random reflection coefficient (RRC), and by applying the Laguerre expansion approximation, we derive closed-form expressions for outage probability and ergodic capacity. Numerical and simulation results show that LI, SI, EI and RRC have sever impacts on the system performance. The results also show the robustness of the Laguerre expansion approximation compared to non-central Chi-squared distribution, in particular, for a small number of IRS elements.

Refined normal approximations for the central and noncentral chi-square distributions and some applications

In this paper, we prove a local limit theorem for the chi-square distribution with r>0 degrees of freedom and noncentrality parameter . We use it to develop refined normal approximations for the survival function. Our maximal errors go down to an order of , which is significantly smaller than the maximal error bounds of order recently found by Horgan and Murphy [On the convergence of the chi square and noncentral chi square distributions to the normal distribution. IEEE Commun Lett. 2013;17(12):2233–2236. DOI:10.1109/LCOMM.2013.111113.131879] and Seri [A tight bound on the distance between a noncentral chi square and a normal distribution. IEEE Commun Lett. 2015;19(11):1877–1880. DOI:10.1109/LCOMM.2015.2461681]. Our results allow us to drastically reduce the number of observations required to obtain negligible errors in the energy detection problem, from 250, as recommended in the seminal work of Urkowitz [Energy detection of unknown deterministic signals. Proc IEEE. 1967;55(4):523–531. DOI:10.1109/PROC.1967.5573], to only 8 here with our new approximations. We also obtain an upper bound on several probability metrics between the central and noncentral chi-square distributions and the standard normal distribution, and we obtain an approximation for the median that improves the lower bound previously obtained by Robert [On some accurate bounds for the quantiles of a noncentral chi squared distribution. Stat Probab Lett. 1990;10(2):101–106. Available from: https://www.ams.org/mathscinet-getitem?mr=MR1072495].

On the measure of anchored Gaussian simplices, with applications to multivariate medians

We consider anchored Gaussian ℓ-simplices in the d-dimensional Euclidean space, that is, simplices with one fixed vertex y∈Rd and the remaining vertices X1,…,Xℓ randomly sampled from the d-variate standard normal distribution. We determine the distribution of the measure of such simplices for any d, any ℓ, and any anchor point y, which is of interest, e.g., when studying the asymptotic behaviour of U-statistics based on such simplex measures. We provide two proofs of the results. The first one is short but is not self-contained as it crucially relies on a technical result for non-central Wishart distributions. The second one is a simple and self-contained proof, that also provides some geometric insight on the results. Quite nicely, variations on this second argument reveal intriguing distributional identities on products of central and non-central chi-square distributions with Beta-distributed non-centrality parameters. We independently establish these distributional identities by making use of Mellin transforms. Beyond the aforementioned use to study the asymptotic behaviour of some U-statistics, our results do find natural applications in the context of robust location estimation, as we illustrate by considering a class of simplex-based multivariate medians that contains the celebrated spatial median and Oja median as special cases. Throughout, our results are confirmed by numerical experiments.

Simple: Statistical Inference on Membership Profiles in Large Networks

AbstractNetwork data are prevalent in many contemporary big data applications in which a common interest is to unveil important latent links between different pairs of nodes. Yet a simple fundamental question of how to precisely quantify the statistical uncertainty associated with the identification of latent links still remains largely unexplored. In this paper, we propose the method of statistical inference on membership profiles in large networks (SIMPLE) in the setting of degree-corrected mixed membership model, where the null hypothesis assumes that the pair of nodes share the same profile of community memberships. In the simpler case of no degree heterogeneity, the model reduces to the mixed membership model for which an alternative more robust test is also proposed. Both tests are of the Hotelling-type statistics based on the rows of empirical eigenvectors or their ratios, whose asymptotic covariance matrices are very challenging to derive and estimate. Nevertheless, their analytical expressions are unveiled and the unknown covariance matrices are consistently estimated. Under some mild regularity conditions, we establish the exact limiting distributions of the two forms of SIMPLE test statistics under the null hypothesis and contiguous alternative hypothesis. They are the chi-square distributions and the noncentral chi-square distributions, respectively, with degrees of freedom depending on whether the degrees are corrected or not. We also address the important issue of estimating the unknown number of communities and establish the asymptotic properties of the associated test statistics. The advantages and practical utility of our new procedures in terms of both size and power are demonstrated through several simulation examples and real network applications.

Approximating the Mode of the Non-Central Chi-Squared Distribution

In this paper we consider the probability density function (pdf) of the non-central χ2 distribution with arbitrary number of degrees of freedom and non-centrality. For this function we find the approximate location of the maximum and discuss related edge cases of 1 and 2 degrees of freedom. We also use this expression to demonstrate the improved performance of the C++ Boost’s implementation of the non-central χ2 and extend the domain of its applicability.

A theoretical minimal solution for heuristics: The case of the spatial harvest timber problem

Heuristic methods are widely used to address the spatial optimization of timber harvests at the forest level. These methods have been shown to solve the timber harvest problem in a timely and computationally efficient manner. However, solutions provided by any heuristic are often suboptimal, and inquiries often arise regarding the quality of those solutions. One way to assess the quality of the solutions is to compare them against a minimum solution estimated using probability functions associated with the theoretical distribution of the solution space. A thorough theoretical framework is proposed to estimate the parameters of the probability distribution of the solution space based on the noncentral chi-square (χ2) distribution as an underlying distribution for the instances. A case study using the Lincoln Tract dataset suggests that the best objective function out of six thousand solutions was 0.26, whereas the theoretical minimal solution was 2×10−6.

Intelligent Reflecting Surface Aided Wireless Systems with Imperfect Hardware

In this article, we investigate the design of reconfigurable intelligent surface (RIS)-aided transmission as a smart method to reflect signals received from access points to users and, hence, improving users’ performance. To implement smart Internet of Things (IoT) networks, massive connectivity and low-cost deployment are essential in designing such systems. In particular, we consider two practical scenarios (dual-hop and single-hop transmissions). These scenarios highlight the potential of RIS in enhancing the system’s outage probability performance. Furthermore, to characterize channel conditions in practice, we pay particular attention to two-channel distributions that are non-central chi-square (NCCS) distributions that approximate the channel distribution of the RIS-aided wireless system and the squared KG distribution. In addition, the RIS-aided system may face imperfect hardware-related issues in practice. Therefore, we need to consider the degraded performance of practical RIS-aided systems by considering the detrimental impact of in-phase and quadrature-phase imbalance (IQI). To characterize the main system performance metric, we provide closed-form formulas of outage probability and ergodic capacity. We then evaluate system performance under the impacts of signal-to-noise ratio (SNR), the number of meta-surfaces, and channel parameters. All closed-form outage expressions are validated via Monte Carlo simulations. Simulation results indicate that the considered RIS scheme at dual-hop and single hop under the impact of IQI and RIS hardware impairment achieves significant improvements in terms of outage probability at high SNR and high meta-surface number N. Additionally, the simulation results demonstrate that the impact of IQI on the proposed system is limited. It is worth noting that, in terms of ergodic capacity, ergodic capacity faces an upper limit. Despite this limitation, the proposed system can still work well once some parameters are controlled well, such as the transmit SNR, levels of IQI, and the number of RIS components.

Bayesian‐MLE signal detection for multi‐antenna ambient backscatter communication

Multi-antenna signal detection is one of the most critical and challenging issues for ambient backscatter communication (AmBC) systems. This paper proposes an efficient multi-antenna AmBC signal detection method, called Bayesian-MLE (maximum likelihood estimation). It shows good performance on high transmission rate, detection accuracy and low energy consumption. Particularly, a practical multi-antenna AmBC system model is developed to offer transmit-receive diversity, and then an efficient multi-antenna AmBC signal detection method is presented using Bayesian optimization and MLE theory. Furthermore, to maximize the detection performance, an optimal detection threshold selection scheme is developed. Particularly, a non-central chi-square distribution conditional probability density function (PDF) is considered instead of the conventional Gaussian PDF. Extensive qualitative and quantitative experiments are performed, showing that the proposed Bayesian-MLE detector achieves the state-of-the-art signal detection performance.

New Developments on the Non-Central Chi-Squared and Beta Distributions

New formulas for the moments about zero of the Non-central Chi-Squared and the Non-central Beta distributions are achieved by means of novel approaches. The mixture representation of the former model and a new expansion of the ascending factorial of a binomial are the main ingredients of the first approach, whereas the second one hinges on an interesting relationship of conditional independence and a simple conditional density of the latter model. Then, a simulation study is carried out in order to pursue a twofold purpose: providing numerical validations of the derived moment formulas on one side and discussing the advantages of the new formulas over the existing ones on the other.

Detection of radar signals scattered by acoustic disturbances generated by UAVs

The problem of UAV radar monitoring by its acoustic radiation is considered. It is shown that in a number of practical cases such an approach is preferable to observation by radar methods directly from the UAV airframe. It is noted that the radio signals scattered by acoustic packets from the UAV are characterized by an unknown in advance complex envelope, which does not allow the use of optimal filtering methods for their detection and estimation. It is shown that to solve these problems, it is advisable to use the principle of accumulation over the observation interval of the energy of a narrow-band random process reduced to the noise dispersion, using the statistical differences between noise fluctuations and the additive "signal-plus-noise" mixture. It is shown that the energy estimate reduced to noise has either a central or an off-center "chi-square" distribution with a certain number of degrees of freedom and an off-center parameter greater than or equal to zero. As a result of comparing the current value of the no centrality parameter with the threshold value, a decision is made on the presence or absence of a useful signal in the observation interval with minimal a priori information about its parameters. It is noted that the well-known expressions for the differential probability densities of the central and non-central chi-square distributions allow one to obtain qualitative estimates of the synthesized detector. A practical structural diagram of a detector using processing of received oscillations in quadrature channels is proposed.

Cooperative fault detection and recovery in the GNSS positioning of mobile agent swarms based on relative distance measurements

Relative measurements are exploited to cooperatively detect and recover faults in the positioning of Mobile Agent (MA) Swarms (MASs). First, a network vertex fault detection method based on edge testing is proposed. For each edge, a property that has a functional relationship with the properties of its two vertices is measured and tested. Based on the edge testing results of the network, the maximum likelihood principle is used to identify the vertex fault sources. Second, an edge distance testing method based on the noncentral chi-square distribution is developed for detecting faults in the Global Navigation Satellite System (GNSS) positioning of MASs. Third, a recovery strategy for faults in the positioning of MASs based on distance measurement is provided. The effectiveness of the proposed methods is validated by a simulation case in which an MAS passes through a GNSS spoofing zone. The proposed methods are conducive to increasing the robustness of the positioning of MASs in complex environments. The main novelties include the following: (A) network vertex fault detection is based on concrete probability analysis rather than simple majority voting, and (B) the relation of detectability and recoverability of MAS positioning faults with the structure of the relative measurement network is first disclosed.

Analytical prediction of low-frequency fluctuations inside a one-dimensional shock

Linear instability of high-speed boundary layers is routinely examined assuming quiescent edge conditions, without reference to the internal structure of shocks or to instabilities potentially generated in them. Our recent work has shown that the kinetically modeled internal nonequilibrium zone of straight shocks away from solid boundaries exhibits low-frequency molecular fluctuations. The presence of the dominant low frequencies observed using the direct simulation Monte Carlo (DSMC) method has been explained as a consequence of the well-known bimodal probability density function (PDF) of the energy of particles inside a shock. Here, PDFs of particle energies are derived in the upstream and downstream equilibrium regions, as well as inside shocks, and it is shown for the first time that they have the form of the noncentral Chi-squared (NCCS) distributions. A linear correlation is proposed to relate the change in the shape of the analytical PDFs at a specified upstream number density and temperature as a function of Mach number, within the range \(3 \le M \le 10\), with the DSMC-derived average characteristic low-frequency of shocks, as computed in our earlier work. At a given Mach number \(M=7.2\) and upstream number density \(n_1=10^{22}\,\hbox {m}^{-3}\), it is shown that the variation in DSMC-derived low frequencies is correlated with the change in most-probable-speed inside shocks at the location of maximum bulk velocity gradient for upstream translational temperature in the range \(\sim 90 \le T_{tr,1}/(K) \le 1420\). Using the proposed linear functions, average low frequencies are estimated within the examined ranges of Mach number and input temperature and a semi-empirical relationship is derived to predict low-frequency oscillations in shocks. Our model can be used to provide realistic physics-based boundary conditions in receptivity and linear stability analysis studies of laminar-turbulent transition in high-speed flows.