Standard Condition Number Research Articles

Introducing the evaluation condition number: A novel assessment of conditioning in radial basis function methods

Background and Objective:RBF methods are crucial for reconstructing functions from data sites, like solving partial differential equations. However, ill-conditioning is common, and the standard condition number may not fully address it as it focuses solely on the Kernel matrix. Methods:The concept of the “largest manageable scale” is introduced, representing the maximum scale where errors are minimized without amplification. To determine this scale, we propose a cheaply available “evaluation condition number” (EVN) that evaluates the conditioning of the linear system arising from kernel methods, as well as “reliable digits” that indicate how many digits in the function evaluation are accurate. The results are implemented across various changes in scale, types of RBFs, and sizes of kernel matrices. Results and Conclusions:The evaluation condition number enhances solution quality estimation in kernel method linear systems over standard methods. We establish the connection between three conditioning criteria: the standard, effective, and our proposed evaluation condition numbers. Furthermore, equipped with the EVN criterion, we demonstrate the advantage of normalization techniques, such as truncated SVD over full SVD and MATLAB backslash. Our results confirm the proposed approach’s effectiveness in maintaining stability and accuracy despite rank loss. We provide the code implementation at: https://doi.org/10.24433/CO.4444506.v2.

Conditioning of linear systems arising from penalty methods

Penalizing incompressibility in the Stokes problem leads, under mildassumptions, to matrices with condition numbers $\kappa =\mathcal{O}(\varepsilon ^{-1}h^{-2})$, with $\varepsilon =$ penalty parameter $\ll1$ and$h= $ meshwidth $<1$. Although $\kappa =\mathcal{O}(\varepsilon^{-1}h^{-2}) $ is large, practical tests seldom report difficulty in solvingthese systems. In the SPD case, using the conjugate gradient method, thisis usually explained by spectral gaps occurring in the penalized coefficientmatrix. Herein we point out a second contributing factor. Since the solutionis approximately incompressible, solution components in the eigenspacesassociated with the penalty terms can be small. As a result, the <i>effective</i> condition number can be much smaller than the standard condition number.

On the selection of a better radial basis function and its shape parameter in interpolation problems

A traditional criterion to calculate the numerical stability of the interpolation matrix is its standard condition number. In this paper, it is observed that the effective condition number (κeff) is more informative than the standard condition number (κ) in investigating the numerical stability of the interpolation problem. While the κeff considers the function to be interpolated, the standard condition number only depends on the interpolation matrix. We propose using the shape parameter corresponding to the maximum κeff to obtain a small error in RBF interpolation. It is also observed that the κeff helps to predict the error behavior with respect to the type of the RBF, where the basis function with a higher effective condition number yields a smaller error. In the end, we conclude that the effective condition number links to the error with respect to the selection of a radial basis function, choosing its shape parameter, number of collocation points, and test function. To this end, ten test functions are interpolated using the multiquadric, Matern family, and Gaussian basis functions to show the advantage of the proposed method.

A variational Bayesian approach to multiantenna spectrum sensing under correlated noise

In this paper, we address the problem of spectrum sensing when the secondary user (SU) is equipped with a multiantenna receiver and noise across antenna elements are correlated. Covariance matrix of correlated noise is non-diagonal and its standard-condition-number (SCN) is much greater than one, which results in significantly degraded detection performance. Many of the existing techniques either whiten the covariance matrix or utilize geometrical properties of noise covariance matrix for the design of decision statistic. However, these techniques have low detection performance in uncorrelated noise or when the correlation is low. To address this problem and further improve the detection performance, a novel spectrum sensing algorithm based on variational Bayesian Gaussian mixture model (VBGMM) is proposed. Specifically, a new decision vector is proposed which is amenable to VBGMM and incorporates signal features suitable for spectrum sensing in both correlated and uncorrelated noise. Decision vectors extracted from the received signal are clustered using variational expectation-maximization (VEM) algorithm to detect the presence of primary user (PU). Compared to existing techniques, the proposed method is shown by simulation to achieve higher detection rates for the entire range of correlation.

Application of random matrix model in multiple abnormal sources detection and location based on PMU monitoring data in distribution network

With the conversion of the global power economy and energy structure, access to a large amount of renewable energy has led to a decrease in power system inertia. The slight abnormal disturbance in the distribution network may have a significant impact on social and economic development. Aim at enhancing power stability and system resiliency; this study focuses on the detection and location of multiple abnormal sources in the distribution network. Most traditional methods use models relying on precise line parameters, subject to poor adaptability to the distribution network with a large number of nodes, and rapidly changing topology. Therefore, this study proposes a novel random matrix model, driven by monitoring data from phasor measurement units distributed on the overhead transmission lines. In this model, linear shrinkage (LS) theory, and Marchenko–Pastur law are combined for noise reduction to ensure the dynamic character and anti-noise ability. Moreover, data dimensions and sample points may be at the same level in an extensive scale network. The LS and standard condition number rule (SCN) are used for estimating the number of abnormal sources. Finally, the effectiveness of this paper's model is verified in PSCAD. The results indicate that the method has specific dynamic performance and anti-noise ability.

Paraunitary matrices, entropy, algebraic condition number and Fourier computation

The Fourier Transform is one of the most important linear transformations used in science and engineering. Cooley and Tukey's Fast Fourier Transform (FFT) from 1964 is a method for computing this transformation in time O(nlog⁡n). From a lower bound perspective, relatively little is known. Ailon shows in 2013 an Ω(nlog⁡n) bound for computing the normalized Fourier Transform assuming only unitary operations on two coordinates are allowed at each step, and no extra memory is allowed. In 2014, Ailon then improved the result to show that, in a κ-well conditioned computation, Fourier computation can be sped up by no more than O(κ). The main conjecture is that Ailon's result can be exponentially improved, in the sense that κ-well condition cannot admit ω(log⁡κ) speedup.The main result here is that ‘algebraic’ κ-well condition cannot admit ω(κ) speedup. One equivalent definition of algebraic condition number is related to the degree of polynomials naturally arising as the computation evolves. Using the maximum modulus theorem from complex analysis, we show that algebraic condition number upper bounds standard condition number, and equals it in certain cases. Algebraic condition number is an interesting measure of numerical computation stability in its own right, and provides a novel computational lens. Moreover, based on evidence from other recent related work, we believe that the approach of algebraic condition number has a good chance of establishing an algebraic version of the main conjecture.

Standard Condition Number Based Spectrum Sensing Under Asynchronous Primary User Activity

Cognitive radio (CR) is a promising technology which enables the secondary user (SU) to sense and detect the presence or absence of the primary user (PU) in the frequency band of interest. Therefore, high detection probability is needed to ensure that primary user is adequately protected, while low false-alarm probability provides the opportunity of using free channel and a better throughput performance of secondary user. Most of the studies on spectrum sensing techniques were conducted assuming a perfect synchronization between the secondary and the primary users. However, in practice, the synchronous assumption is very hard to satisfy in the context of real cognitive radio networks environment. In this paper, we present a theoretical formulation of the standard condition number (SCN) based spectrum sensing technique under an asynchronous cognitive radio networks. We assume the case where the primary users generate asynchronous slotted traffic. The standard condition number distributions under null and alternative hypotheses are derived where the secondary user is equipped with two receive antennas. Under asynchronous primary user traffic, we establish the existence of a lower limit of the false-alarm probability below which we are unable to derive a decision threshold for the SCN-based detector.

ELASTIC- Enabling Massive-Antenna for Joint Spectrum Sensing and Sharing: How Many Antennas Do We Need?

Massive antenna and cognitive radio (CR) technologies have attracted many research interests due to the additional resources offered in striving against the spectrum crisis. In this paper, we propose a general framework to enlable massive antenna exploitation for spectrum sensing and sharing in CR. Using random matrix theory and moment matching method, we derived a simple approximation of the distributions of three eigen-value-based detectors namely the largest eigenvalue (LE), the scaled LE, and standard condition number. This has led to a simple analytical formulation and optimization of the number of antennas and the number of samples to reach a target performance. To exploit the large number of antennas, we proposed two sensing scenarios. The first is based on full antenna exploitation and guarantees an optimal performance despite the transmission conditions. The second is based on partial antenna exploitation, which determines the exact number of antennas and samples required to reach the target performance. We have shown that the framework offers an additional degree of freedom in the selection of the optimal system parameters, namely the number of antennas, while the remaining antennas are exploited for sharing in other dimensions of the spectrum hypercube.

Standard Condition Number Distributions of Finite Wishart Matrices for Cognitive Radio Networks

The standard condition number (SCN) in random matrix theory (RMT) has widely been employed in cognitive radio networks (CRNs). The paper studies the SCN distributions of central and complex Wishart matrices with finite dimensions. The exact and closed-form expressions of SCN distributions are first proposed in sum of multiple polynomials. The proposed formulations provide an efficient way to determine probability density function and cumulative distribution function of the SCN in finite RMT. In particular, the specific SCN distributions of triple Wishart matrix are initially presented. The theoretical and precise sensing thresholds for cooperative spectrum sensing (CSS) systems in CRNs are calculated using the proposed SCN distributions. Numerical results indicate that the proposed theoretical SCN distributions match the empirical SCN distributions very well. Furthermore, the exact SCN-based CSS systems are able to obtain higher sensing performance than the asymptotic SCN-based schemes because precise sensing thresholds can be determined.

Asymptotic Approximation of the Standard Condition Number Detector for Large Multi-Antenna Cognitive Radio Systems

Standard condition number (SCN) detector is a promising detector that can work efficiently in uncertain environments. In this paper, we consider a Cognitive Radio (CR) system with large number of antennas (eg. Massive MIMO) and we provide an accurate and simple closed form approximation for the SCN distribution using the generalized extreme value (GEV) distribution. The approximation framework is based on the moment-matching method where the expressions of the moments are approximated using bi-variate Taylor expansion and results from random matrix theory. In addition, the performance probabilities and the decision threshold are considered. Since the number of antennas and/or the number of samples used in the sensing process may frequently change, this paper provides simple form decision threshold and performance probabilities offering dynamic and real-time computations. Simulation results show that the provided approximations are tightly matched to relative empirical ones.

Approximating the standard condition number for cognitive radio spectrum sensing with finite number of sensors

In this study, the authors consider the standard condition number (SCN) detector for a cognitive radio with finite number of cooperative sensors. They derive an exact nested form of the distribution of the SCN for the central uncorrelated, non-central uncorrelated and central semi-correlated Wishart matrices under H 0 and H 1 hypotheses. Due to the complexity of these expressions, the authors approximate the distribution of the SCN by the generalised extreme value distribution using moment matching. They derive the exact form of the pth moment of the SCN for these cases. Consequently, the performance probabilities are approximated and a simple decision threshold formula is provided. In addition, a similar approximation for the detection probability is provided using non-central/central approximation. They show that the proposed analytical approximations provide high accuracy using Monte-Carlo simulations.

A new joint eigenvalue distribution of finite random matrix for cognitive radio networks

A new joint eigenvalue distribution (JED) based on dual extreme eigenvalues of finite random matrix is proposed in this study. Different from conventional JED based on K(K ≥ 2) variables, only two variables are included in the proposed formulation. The upper and lower bounds of the new JED are determined. The new JED provides a simple and efficient way to deduce the distributions of key characteristics of finite random matrix, such as the extreme (largest and smallest) eigenvalues, standard condition number, and scaled largest eigenvalue. Moreover, a novel cooperative spectrum sensing (CSS) scheme based on the new JED is proposed for cognitive radio networks. The simulation results verify the proposed JED and the proposed CSS scheme can improve sensing performance.

Exact Distributions of Finite Random Matrices and Their Applications to Spectrum Sensing.

The exact and simple distributions of finite random matrix theory (FRMT) are critically important for cognitive radio networks (CRNs). In this paper, we unify some existing distributions of the FRMT with the proposed coefficient matrices (vectors) and represent the distributions with the coefficient-based formulations. A coefficient reuse mechanism is studied, i.e., the same coefficient matrices (vectors) can be exploited to formulate different distributions. For instance, the same coefficient matrices can be used by the largest eigenvalue (LE) and the scaled largest eigenvalue (SLE); the same coefficient vectors can be used by the smallest eigenvalue (SE) and the Demmel condition number (DCN). A new and simple cumulative distribution function (CDF) of the DCN is also deduced. In particular, the dimension boundary between the infinite random matrix theory (IRMT) and the FRMT is initially defined. The dimension boundary provides a theoretical way to divide random matrices into infinite random matrices and finite random matrices. The FRMT-based spectrum sensing (SS) schemes are studied for CRNs. The SLE-based scheme can be considered as an asymptotically-optimal SS scheme when the dimension K is larger than two. Moreover, the standard condition number (SCN)-based scheme achieves the same sensing performance as the SLE-based scheme for dual covariance matrix . The simulation results verify that the coefficient-based distributions can fit the empirical results very well, and the FRMT-based schemes outperform the IRMT-based schemes and the conventional SS schemes.

On the detection probability of the standard condition number detector in finite-dimensional cognitive radio context

Standard condition number (SCN) detector is an efficient detector in multi-dimensional cognitive radio systems since no a priori knowledge is needed. The earlier studies usually assume a large number of dimensions and a large number of samples per dimension and use random matrix theory (RMT) to derive asymptotic distributions of the SCN metric. In practice, the number of dimensions may not be large enough for the SCN distribution to be well approximated by the asymptotic ones. In this context, the false alarm probability is considered in literature and formulas for 2D, 3D, and infinite-dimensional systems have been derived. However, the detection probability, which is of great importance in cognitive radio, has not been well discussed in literature. In this paper, we discuss, analytically, the detection probability of the SCN detector. Since the probability of detection is totally related to the SCN distribution, we derive new results on the joint ordered eigenvalues and SCN distributions for central semi-correlated Wishart matrices. These results are used to approximate the detection probability by the non-central/central approximation. We consider systems with three or more dimensions, and we give an approximated form of the detection probability. The analytical results of this paper on probability of detection along with those on probability of false alarm present a complete performance analysis and are validated through simulations. We show that the proposed analytical expressions provide high accuracy and that the SCN detector outperforms the well-known energy detector and the largest eigenvalue detector even with a small number of dimensions and low noise uncertainty environments.

Signal‐to‐noise ratio estimation for DVB‐S2 based on eigenvalue decomposition

Signal-to-noise ratio (SNR) estimation is required in many receivers and systems of communications. In this study, the authors propose a novel SNR estimation method for DVB-S2 over additive white Gaussian noise channel. This is a non-data-aided, eigenvalue decomposition-based method. First, they use the even and odd component vectors of received signal to construct a novel 2 × 2 matrix, and propose an estimation variable using the standard condition number of this matrix. And then, they derive the relationship between this variable and SNR, and give the detailed expressions for SNR estimation on different modulations used in DVB-S2, including quadrature phase shift keying, 8PSK, 16 amplitude PSK (APSK) and 32APSK. Monte Carlo simulation results show the good estimation performance for different modulations. The proposed method is asymptotic unbiased and can be used for both constant and non-constant modulus constellations such as M-ary PSK, MAPSK and M-ary quadrature amplitude modulation.

Eigenvalue-Based Spectrum Sensing for Multiple Received Signals Under the Non-Reconstruction Framework of Compressed Sensing

Eigenvalue-based algorithm is generally acknowledged to be a promising method for spectrum sensing. However, it possesses high computational complexity, because the sampled covariance matrix and the corresponding eigenvalues are calculated. Furthermore, the detection performance of eigenvalue-based algorithms experiences a huge decline when the received signals are uncorrelated. Therefore, compressed sensing is adopted to reduce the computational complexity and introduce the relevance for multiple received signals. First, the sampled covariance matrix and its eigenvalues are calculated under the non-reconstruction framework of compressed sensing. The corresponding standard condition number of the eigenvalues is employed as test statistic to perform spectrum sensing. Then, the impact of the measurement matrix on the detection performance is discussed, and the computational complexity is analyzed. Next, the measurement matrix is optimized to improve the detection performance. In addition, a novel method of setting decision threshold is proposed to maintain a stable false alarm probability for the proposed spectrum sensing algorithm, and its computational complexity is compared with some existing methods. Finally, the corresponding simulations are performed to testify the theoretical results. Theoretical analysis and simulation results certify the effectiveness and validity of the proposed method.

Relative Perturbation Theory for Diagonally Dominant Matrices

In this paper, strong relative perturbation bounds are developed for a number of linear algebra problems involving diagonally dominant matrices. The key point is to parameterize diagonally dominant matrices using their off-diagonal entries and diagonally dominant parts and to consider small relative componentwise perturbations of these parameters. This allows us to obtain new relative perturbation bounds for the inverse, the solution to linear systems, the symmetric indefinite eigenvalue problem, the singular value problem, and the nonsymmetric eigenvalue problem. These bounds are much stronger than traditional perturbation results, since they are independent of either the standard condition number or the magnitude of eigenvalues/singular values. Together with previously derived perturbation bounds for the LDU factorization and the symmetric positive definite eigenvalue problem, this paper presents a complete and detailed account of relative structured perturbation theory for diagonally dominant matrices.

Eigenvalue-Based Sensing and SNR Estimation for Cognitive Radio in Presence of Noise Correlation

Herein, we present a detailed analysis of an eigenvalue-based sensing technique in the presence of correlated noise in the context of a cognitive radio (CR). We use standard-condition-number (SCN)-based decision statistics based on asymptotic random matrix theory (RMT) for the decision process. First, the effect of noise correlation on eigenvalue-based spectrum sensing (SS) is analytically studied under both the noise-only and signal-plus-noise hypotheses. Second, new bounds for the SCN are proposed to achieve improved sensing in correlated noise scenarios. Third, the performance of fractional-sampling (FS)-based SS is studied, and a method to determine the operating point for the FS rate in terms of sensing performance and complexity is suggested. Finally, a signal-to-noise ratio (SNR) estimation technique based on the maximum eigenvalue of the covariance matrix of the received signal is proposed. It is shown that the proposed SCN-based threshold improves sensing performance in correlated noise scenarios, and SNRs up to 0 dB can be reliably estimated with a normalized mean square error (MSE) of less than <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$\hbox{1}\%$</tex></formula> in the presence of correlated noise without the knowledge of noise variance.

Spectrum Sensing Algorithms via Finite Random Matrices

We address the Primary User (PU) detection (spectrum sensing) problem, relevant to cognitive radio, from a finite random matrix theoretical (RMT) perspective. Specifically, we employ recently-derived closed-form and exact expressions for the distribution of the standard condition number (SCN) of uncorrelated and semi-correlated random dual central Wishart matrices of finite sizes in the design Hypothesis-Testing algorithms to detect the presence of PU signals. In particular, two algorithms are designed, with basis on the SCN distribution in the absence (H_0) and in the presence (H_1) of PU signals, respectively. Due to an inherent property of the SCN's, the H_0 test requires no estimation of SNR or any other information on the PU signal, while the H_1 test requires SNR only. Further attractive advantages of the new techniques are: a) due to the accuracy of the finite SCN distributions, superior performance is achieved under a finite number of samples, compared to asymptotic RMT-based alternatives; b) since expressions to model the SCN statistics both in the absence and presence of PU signal are used, the statistics of the spectrum sensing problem in question is completely characterized; and c) as a consequence of a) and b), accurate and simple analytical expressions for the receiver operating characteristic (ROC) — both in terms of the probability of detection as a function of the probability of false alarm (P_D versus P_F) and in terms of the probability of acquisition as a function of the probability of miss detection (P_A versus P_M) — are yielded. It is also shown that the proposed finite RMT-based algorithms outperform all similar alternatives currently known in the literature, at a substantially lower complexity. In the process, several new results on the distributions of eigenvalues and SCNs of random Wishart Matrices are offered, including a closed-form of the Marchenko-Pastur's Cumulative Density Function (CDF) and extensions of the latter, as well as variations of asymptotic the distributions of extreme eigenvalues (Tracy-Widom) and their ratio (Tracy-Widom-Curtiss), which are simpler than those obtained with the "spiked population model".

On the condition number distribution of complex wishart matrices

This paper investigates the distribution of the condition number of complex Wishart matrices. Two closely related measures are considered: the standard condition number (SCN) and the Demmel condition number (DCN), both of which have important applications in the context of multiple-input multiple-output (MIMO) communication systems, as well as in various branches of mathematics. We first present a novel generic framework for the SCN distribution which accounts for both central and non-central Wishart matrices of arbitrary dimension. This result is a simple unified expression which involves only a single scalar integral, and therefore allows for fast and efficient computation. For the case of dual Wishart matrices, we derive new exact polynomial expressions for both the SCN and DCN distributions. We also formulate a new closed-form expression for the tail SCN distribution which applies for correlated central Wishart matrices of arbitrary dimension and demonstrates an interesting connection to the maximum eigenvalue moments of Wishart matrices of smaller dimension. Based on our analytical results, we gain valuable insights into the statistical behavior of the channel conditioning for various MIMO fading scenarios, such as uncorrelated/semi-correlated Rayleigh fading and Ricean fading.