Eigenvalues Of Correlation Matrix Research Articles

Complexity of fermionic states

How much information does a fermionic state contain? To address this fundamental question, we define the complexity of a particle-conserving many-fermion state as the entropy of its Fock space probability distribution, minimized over all Fock representations. The complexity characterizes the minimum computational and physical resources required to represent the state and store the information obtained from it by measurements. Alternatively, the complexity can be regarded as a Fock space entanglement measure describing the intrinsic many-particle entanglement in the state. We establish a universal lower bound for the complexity in terms of the single-particle correlation matrix eigenvalues and formulate a finite-size complexity scaling hypothesis. Remarkably, numerical studies on interacting lattice models suggest a general model-independent complexity hierarchy: ground states are exponentially less complex than average excited states, which, in turn, are exponentially less complex than generic states in the Fock space. Our work has fundamental implications on how much information is encoded in fermionic states. Published by the American Physical Society 2024

Characterizing nonclassical correlations of tensorizing states in a bilocal scenario

In the present paper, we attempt to address the question of “can tensorizing states $$(\rho \otimes \rho \; \text {or}\;\rho \otimes \rho ')$$ have quantum advantages?”. To answer this question, we exploit the notion of measurement-induced nonlocality (MIN) and advocate a fidelity-based nonbilocal measure to capture the nonlocal effects of tensorizing states due to locally invariant von Neumann projective measurements. We show that the properties of the fidelity-based nonbilocal measures are retrieved from that of MIN. Analytically, we evaluate the nonbilocal measure for any arbitrary pure state. The upper bounds of the nonbilocal measure based on fidelity are also obtained in terms of eigenvalues of correlation matrix. As an illustration, we have computed the nonbilocality for some popular input states.

Few-body nature of Kondo correlated ground states

The quenching of degenerate impurity states in metals generally induces a long-range correlated quantum state known as the Kondo screening cloud. While a macroscopic number of particles clearly take part in forming this extended structure, assessing the number of truly entangled degrees of freedom requires a careful analysis of the relevant many-body wavefunction. For this purpose, we examine the natural single-particle orbitals that are eigenstates of the single-particle density (correlation) matrix for the ground state of two quantum impurity problems: the interacting resonant level model (IRLM) and the single impurity Anderson model (SIAM). As a simple and general probe for few-body versus many-body character we consider the rate of exponential decay of the correlation matrix eigenvalues towards inactive (fully empty or filled) orbitals. We find that this rate remains large in the physically most relevant region of parameter space, implying a few-body character. Genuine many-body correlations emerge only when the Kondo temperature becomes exponentially small, for instance near a quantum critical point. In addition, we demonstrate that a simple numerical diagonalization of the few-body problem restricted to the Fock space of the most correlated orbitals converges exponentially fast with respect to the number of orbitals, to the true ground state of the IRLM. We also show that finite size effects drastically affect the correlation spectrum, shedding light on an apparent paradox arising from previous studies on short chains.

The Effect of the Eigenvalues of the Zero-Forcing Detector on Its Performance in the Space-Division Multiplexing System

Zero-Forcing (ZF) detector is used in Space-Division Multiplexing (SDM) receiver to remove interference among the received symbols. Previous works showed that the power of channel noise is enhanced in the output of the ZF detector. They recommend using the ZF detector when the received Signal-to-Noise Ratio (SNR) is high. This work proves that the performance of the ZF detector depends on the eigenvalues of the channel correlation matrix. The paper shows that if the sum of the eigenvalues of this correlation matrix is equal to the rank of the channel matrix, the ZF detector will not enhance noise power at its outputs. Moreover, if the sum of the eigenvalues is smaller than the rank of the channel matrix, the ZF detector will reduce noise power at its outputs. In this work, a theorem, which demonstrates the performance of the ZF detector in SDM receiver, is introduced and proved. The proposed work uses smart antennas in the transmitter and receiver to control the elements and eigenvalues of the channel matrix. The introduced theorem and a complete SDM receiver with ZF detector are simulated and evaluated at different conditions with different criteria. A real-time SDM receiver with ZF detector is also implemented and evaluated. The simulation and implementation results are shown at the end of this study. The results of the proposed systems show that a ZF detector can be used to remove interference in the SDM system without enhancing the channel noise.

Characterizing Human Postural Stability by Using Features Extracted From the Correlation Structure of Ground Reaction Force Components

A force plate is one of the most popular tools for postural stability assessment. It decomposes the ground reaction force (GRF) applied to the human body into multiple force components to determine the center of pressure (COP), which is the point at which the GRF is applied. Many COP measures have been proposed to characterize postural stability. Despite postural stability being closely related to GRF components, the interactions between them have rarely been studied. By studying the correlation structure of these GRF components, we developed a set of features to assess postural stability. We determined the correlation matrix of these GRF components and subsequently solved the corresponding eigenvalue problem; we used the resulting eigenvalues to characterize postural stability. The effectiveness of the proposed features was demonstrated by using them to differentiate between individuals in two age groups: 18–24 and 65–73 years. Statistical test results showed that the correlation matrix eigenvalues of the two age groups differed significantly. The classification results demonstrated that most of the correlation matrix eigenvalues were more sensitive to age variations than one of the most reliable and accurate conventional COP features. Furthermore, by reducing the force sensing requirement from three-dimensional to one-dimensional by considering only the vertical GRF components, a simplified version of the proposed approach can be obtained, which could reduce the cost of the force plate system substantially.

Analyzing financial correlation matrix based on the eigenvector–eigenvalue identity

Previous empirical studies have shown that correlation matrices in financial markets usually have dominant eigenvalues. This paper applies a classic eigenvector–eigenvalue identity to analyze the properties of financial correlation matrices with super-dominant eigenvalues. Empirical analysis shows that there is an approximate relationship between the maximum eigenvalue and the eigenvector component. If the correlation matrix has a super eigenvalue, we can estimate the maximum eigenvalue of the sub-matrix from the maximum eigenvalue of the large-dimensional correlation matrix. Conversely, we can also estimate the maximum eigenvalue of the correlation matrix of a large number of stocks from the maximum eigenvalues corresponding to a few stocks. In addition, we find that different stock sets constructed based on the components of the eigenvector generate different predicted values, and the most accurate estimates can be obtained by selecting stocks at equal intervals. This paper reveals that eigenvector–eigenvalue identity helps to analyze the spectrum of financial correlation matrix in depth.

Optimal Power Allocation and Training Duration for Uplink Multiuser Massive MIMO Systems With MMSE Receivers

This paper investigates the optimal training duration and the optimal power allocation for the training and the data transmission that maximize the ergodic sum rate in single-cell uplink massive MIMO with MMSE receivers. Our channel model assumes that each user experiences the same spatial channel correlation. The expression for the ergodic sum rate is obtained in the large system regime where the number of antennas ( $N$ ) at the base station and the number of users ( $K$ ) tend to infinity with a fixed ratio. Interestingly, we show that the optimal training duration is equal to $K$ and independent of the spatial correlation. We also derive the optimal power allocation that in facts depends on the spatial correlation, the channel coherence interval, and the uplink SNR. We show that more energy should be allocated for the training if the data transmission duration ( $t_{d}$ ) is less than $K$ , and vice versa. Moreover, equal power allocation is optimal when $t_{d}=K$ . We also obtain an approximation for the optimal power allocation that depends on the mean of the correlation matrix eigenvalues. Numerical simulations show that our results based on the large system approximation are accurate and applicable for finite-size systems. The simulations also show that (1) the resulting ergodic sum rates obtained by employing the optimal power allocation and its approximation are indistinguishable, and (2) the optimal power allocation obtained from the uncorrelated channel model can be applied to the cases involving the correlated channels with indiscernible penalties on the ergodic sum rates.

Research and methodological approach to scoring the impact of banks performing operations involving transnational capital on the banking system of Ukraine

The article examines the impact of banks performing operations involving transnational banking capital on the banking system of Ukraine through the use of the principal components analysis. This approach is related to the method of multidimensional statistical analysis and is widely used to reduce the dimension of an analyzed set of attributes, and to select the most informative indicators and classification features.From the research, it is found that the principal components analysis is based on the idea according to which structural properties of relations between variables can be determined by their dependence on fewer non-measureable (concealed, latent, hypothetical) general factors.The direct estimation of the impact of banks performing operations involving transnational capital on the banking system of Ukraine is carried out by means of the application package named Statistica. For this analysis, the following relative indicators showing how Ukraine’s banks perform operations involving transnational capital are used: 1) percentage of funds raised from legal entities; 2) percentage of funds raised from individuals; 3) percentage of owned capital; 4) percentage of profit / loss after tax; 5) percentage of assets; 6) percentage of loans and debts of legal entities; 7) percentage of loans and debts of individuals.Based on the research findings, it is shown that there are three principal components which take into account more than 5% of total variance. However, there is only one component among them the value of which exceeds 1. The second component equals 0.954179, meaning that it approaches 1. The scree plot shows that the breaking point can be viewed as the eigen-value of correlation matrix which equals 23.85%. Consequently, the two principal factors have an essential impact on the processes of foreign capital penetration into Ukraine.It is claimed that the expansion of transnational banking capital into the banking system of Ukraine is controversial. On the one hand, these trends are quite obvious. On the other hand, financial, economic, military, and political instability hampers such processes. In conclusion, it should be stressed that unlike other approaches, the research and methodological approach to scoring the impact of banks performing operations involving transnational capital on the banking system of Ukraine, is based on the principal components analysis. This makes it possible to develop an aggregate index which characterizes the impact of banks performing operations involving transnational capital on the banking system of Ukraine, and to use it for monitoring of structural changes in the banking system.

Power and sample size requirements for GEE analyses of cluster randomized crossover trials.

The cluster randomized crossover design has been proposed to improve efficiency over the traditional parallel cluster randomized design, which often involves a limited number of clusters. In recent years, the cluster randomized crossover design has been increasingly used to evaluate the effectiveness of health care policy or programs, and the interest often lies in quantifying the population-averaged intervention effect. In this paper, we consider the two-treatment two-period crossover design, and develop sample size procedures for continuous and binary outcomes corresponding to a population-averaged model estimated by generalized estimating equations, accounting for both within-period and interperiod correlations. In particular, we show that the required sample size depends on the correlation parameters through an eigenvalue of the within-cluster correlation matrix for continuous outcomes and through two distinct eigenvalues of the correlation matrix for binary outcomes. We demonstrate that the empirical power corresponds well with the predicted power by the proposed formulae for as few as eight clusters, when outcomes are analyzed using the matrix-adjusted estimating equations for the correlation parameters concurrently with a suitable bias-corrected sandwich variance estimator.

Fidelity based measurement induced nonlocality and its dynamics in quantum noisy channels

Measurement induced nonlocality (MIN) captures global nonlocal effect of bipartite quantum state due to locally invariant projective measurements. In this paper, we propose a new version of MIN using fidelity induced metric, and the same is calculated for pure and mixed states. For mixed state, the upper bound is obtained from eigenvalues of correlation matrix. Further, dynamics of MIN and fidelity based MIN under various noisy quantum channels show that they are more robust than entanglement.

Cluster structure in the correlation coefficient matrix can be characterized by abnormal eigenvalues

In a large number of previous studies, the researchers found that some of the eigenvalues of the financial correlation matrix were greater than the predicted values of the random matrix theory (RMT). Here, we call these eigenvalues as abnormal eigenvalues. In order to reveal the hidden meaning of these abnormal eigenvalues, we study the toy model with cluster structure and find that these eigenvalues are related to the cluster structure of the correlation coefficient matrix. In this paper, model-based experiments show that in most cases, the number of abnormal eigenvalues of the correlation matrix is equal to the number of clusters. In addition, empirical studies show that the sum of the abnormal eigenvalues is related to the clarity of the cluster structure and is negatively correlated with the correlation dimension.

Determining of spatial distribution patterns and temporal trends of an air pollutant using proper orthogonal decomposition basis functions

Abstract This study aims to determine spatial patterns of an air pollutant dispersion and its temporal trends using proper orthogonal decomposition (POD) basis functions. The POD method is a model reduction technique for complex nonlinear problems and POD basis functions contain essential dynamics and physics of original problem. In the present work, the POD basis functions are applied to identify the dominant modes of carbon monoxide (CO) concentration in ambient air. For this purpose, CO concentration data for 15 monitoring stations over mega city of Tehran for 1339 days (1 Jan. 2006 to 31 Aug. 2009) are used. Data of monitoring stations are interpolated to generate 100 × 100 grid point network. Generated grid based data for all days create a time series of data that is the basic for constructing the POD basis functions. POD basis functions are obtained using eigenvectors of correlation matrix that is obtained using correlation of time series of data. The few number of the POD basis functions corresponding to the few first largest eigenvalues of correlation matrix are dominant modes. The results indicate that the first 7 largest eigenvalues of correlation matrix are 99 percent of the first 100 largest eigenvalues. This indicates that the first 7 POD basis functions out of 1339 capture the essential physics of CO distribution over region. Distribution of the first POD basis function over the city shows that the central and west-central parts of the city are more affected by CO pollutant. In addition, using the recorded data and the POD basis functions the temporal variation of each POD basis function is obtained. Results for the temporal variations of the POD basis functions show that the largest temporal trend belongs to the first POD basis function.

The Scheme and Performance Analysis of Combined Beamforming with Space-Time Block Coding

Alamouti空时块码与波束形成相结合的方案只使用了两个功率均分的波束,在很多情况下并非最优。该文提出了新的空时块码与波束形成相结合的方案,新方案能够根据信道互相关矩阵的特征值,展开不超过发射端天线数量的波束,在满足空时块码发射天线数量等于波束数量的条件下,与任意码率空时块码结合。该文还利用Gauss-Chebyshev积分对线性调制下新方案的误符号率进行了性能分析,得出了简单的数值计算表达式。新方案由于同时获得了分集增益和波束形成增益,显著提高了系统的性能。

On Statistical Distribution of Eigenvalues of Channel Correlation Matrix in MIMO Multi-Keyhole Environment

Y MIMO leads to dramatic improvement in channel capacity and/or link reliability of wireless systems. However, a MIMO channel has only one degree of freedom in a keyhole environment. As a result, this environment reduces achievable channel capacity and link quality. This paper proposes a MIMO repeater system, which can realize a multi-stream transmission. Although the averaged channel capacity in the MIMO repeater system is discussed in several published papers, the probability density functions of eigenvalues of correlation matrix are not analyzed. MIMO transmission performance can basically be estimated from eigenvalues of the channel correlation matrix. We derive an approximated formula for the probability density function of all eigenvalues linked to the space diversity. It is shown that the calculated values based on the proposed method agrees very well with the simulated values.

Pre-processing technique for improving estimation of number of paths embedded in noisy observations

Numerous algorithms based on correlation matrix eigenvalues have been developed to estimate the number of paths in a multipath environment. To improve their efficiency, the use of the interlacing properties of the eigenvalues both of a Hermitian matrix and of its main submatrices is proposed.

Scaling and correlations in foreign exchange market

We observe that the distribution of the relative return, describing the variation of a certain currency, of 74 global currencies obeys a power-law. By using the random matrix theory we find that the distribution of eigenvalues of correlation matrix of relative return also follows a power-law. Using a scaled factorial moment we investigate the distribution of correlation coefficients of the relative return and observe intermittence phenomenon. Furthermore, we define the influence strength for a certain currency, which reflects the influence of its price change to the community interested. By doing that, we find that the distribution of influence strength is again a power-law. Beyond that, we compare the influence strength of Chinese Yuan (RMB) to those of other seven important currencies, which may have some interesting indications.

Eigenvalue‐based model selection during latent semantic indexing

AbstractIn this study amended parallel analysis (APA), a novel method for model selection in unsupervised learning problems such as information retrieval (IR), is described. At issue is the selection of k, the number of dimensions retained under latent semantic indexing (LSI). Amended parallel analysis is an elaboration of Horn's parallel analysis, which advocates retaining eigenvalues larger than those that we would expect under term independence. Amended parallel analysis operates by deriving confidence intervals on these “null” eigenvalues. The technique amounts to a series of nonparametric hypothesis tests on the correlation matrix eigenvalues. In the study, APA is tested along with four established dimensionality estimators on six standard IR test collections. These estimates are evaluated with regard to two IR performance metrics. Additionally, results from simulated data are reported. In both rounds of experimentation APA performs well, predicting the best values of k on 3 of 12 observations, with good predictions on several others, and never offering the worst estimate of optimal dimensionality.

Convergence properties of optimal adaptive carrier phase jitter predictors for sinusoidal jitter

Carrier recovery systems in high performance digital modem receivers must remove carrier phase jitter as well as track carrier phase and frequency offsets. A popular approach to removing periodic phase jitter is to use an adaptive LMS FIR linear predictor with baud spaced taps to predict the jitter angle one baud in the future. Experimentally, the convergence rates for such predictors were found to depend heavily on the jitter frequency. It is well known in adaptive filter theory that the predictor convergence rate depends on the eigenvalues of the correlation matrix of the input signal. By allowing jitter predictors with a given number of taps to have tap spacings greater than one baud and by deriving a general formula for the eigenvalues of the appropriate correlation matrix, it is shown how to design predictors exhibiting optimal convergence rates over a range of commonly observed jitter frequencies.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Systematic QSAR Procedures with Quantum Chemical Descriptors

AbstractSystematic QSAR procedures with quantum chemical descriptors were described by showing practical calculations on the inhibition of thiopurine methyltransferase by benzoic acid derivatives. In the QSAR analysis, a large number of descriptors relating to reactivity and interaction efficiency are handled, and the collinearities inherently involved among them should be removed to get the stable regression equations. Collinearity was detected by the A value which is the sum of the reciprocals of the eigenvalues of correlation matrix of descriptors divided by the number of descriptors in the equation. In the method of evaluation of all possible equations, the candidate equations selected on the basis of the correlation coefficient with activity and/or predicted sum of square should be examined one by one by the A testing. In the stepwise methods, noncollinear descriptor set is required. The collinearity between two descriptors was removed by a simple method, deleting one of the two descriptors whose correlation is large. It was recommended that the A testing routine was introduced into each step to find out multicollinearity.