Spectral analysis of high-dimensional spot volatility matrix with applications

In the research area of the high-frequency financial data analysis, estimation and prediction of the high-dimensional volatility matrix are challenging problems, especially when the assets of interest have a naturally given group structure. In order to address this issue, we propose a novel GARCH-Itôgrouped factor model, in which we present the log price series of grouped assets with common factors, group-specific factors, and an asset-specific error term. We then embed the discrete GARCH structure into the volatility of the eigenvalue processes to capture the volatility dynamics of the observed price data. We propose a quasi maximum likelihood method for parameter estimation, establish its asymptotic properties and illustrate its good finite-sample performance with simulation. In real data analysis, we compare our method with the ungrouped factor model and the nonparametric high-frequency volatility estimator MSRV using the data from the SSE Main Board and the SZSE ChiNext market, where the proposed method outperforms its competitors in terms of prediction of the volatility matrix.

Numerical computing of the rank of a matrix is a fundamental problem in scientific computation. The datasets generated by the Internet often correspond to the analysis of high-dimensional sparse matrices. Notwithstanding recent advances in the promotion of traditional singular value decomposition (SVD), an efficient estimation algorithm for the rank of a high-dimensional sparse matrix is still lacking. Inspired by the controllability theory of complex networks, we converted the rank of a matrix into maximum matching computing. Then, we established a fast rank estimation algorithm by using the cavity method, a powerful approximate technique for computing the maximum matching, to estimate the rank of a sparse matrix. In the merit of the natural low complexity of the cavity method, we showed that the rank of a high-dimensional sparse matrix can be estimated in a much faster way than SVD with high accuracy. Our method offers an efficient pathway to quickly estimate the rank of the high-dimensional sparse matrix when the time cost of computing the rank by SVD is unacceptable.

In order to ensure that a perimeter security system can work effectively, a convenient and effective event detection algorithm has an important engineering significance. Given the above background, in this paper, we propose a high reliability intrusion event recognition method and vibration sensing system, based on ultra-weak fiber Bragg grating array, by using high dimensional random matrix. We obtain a high sensitivity optical interference signal by constructing a patch-matched optical interference system, then compose the demodulated interference signal into a high-dimensional random matrix. The statistical characteristics of the matrix for the Marcenko-Pastur (M-P) law and ring law are used to confirm the presence of intrusion events efficiently, which can reflect the limit spectrum distribution of the high-dimensional random matrix; meanwhile, the abnormal state quantity and moment are obtained. Further, the average spectral radius value is used to judge the fault cause. Field experimental results show that the proposed method can effectively obtain the correct monitoring data for the sensor array. By comparing the monitoring results of normal operation and crusher operation, we can detect the intrusion event in 4.5 s, and the accuracy rate can reach more than 90%, which verifies that the proposed high-dimensional random matrix analysis method can work properly, proving a practical engineering application prospect.

A fast non-negative latent factor (FNLF) model for a high-dimensional and sparse (HiDS) matrix adopts a Single Latent Factor-dependent, Non-negative, Multiplicative and Momentum-incorporated Update (SLF-NM <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> U) algorithm, which enables its fast convergence. It is crucial to achieve a rigorously theoretical proof regarding its fast convergence, which has not been provided in prior research. Aiming at addressing this critical issue, this work theoretically proves that with an appropriately chosen momentum coefficient, SLF-NM <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> U enables the fast convergence of an FNLF model in both continuous and discrete time cases. Empirical analysis of HiDS matrices generated by representative industrial applications provides empirical evidences for the theoretical proof. Hence, this study represents an important milestone in the field of HiDS matrix analysis.

We propose a new procedure to monitor and forecast the onset of transitions in high-dimensional complex systems. We describe our procedure by an application to the tangled nature model of evolutionary ecology. The quasistable configurations of the full stochastic dynamics are taken as input for a stability analysis by means of the deterministic mean-field equations. Numerical analysis of the high-dimensional stability matrix allows us to identify unstable directions associated with eigenvalues with a positive real part. The overlap of the instantaneous configuration vector of the full stochastic system with the eigenvectors of the unstable directions of the deterministic mean-field approximation is found to be a good early warning of the transitions occurring intermittently.

The joint volatility matrix of assets is an important statistic for resource allocation and risk management. Accurate estimation of the joint volatility matrix is one of the hot issues in financial statistics and risk measurement. In this paper, we study theintegral volatility matrix estimation of logarithmic price data with jumps under microstructure noise including market information. When the prices are not synchronized, and the number of assets and sample size tend to infinity, four estimation methods of high-dimensional integral volatility matrices are proposed by using the non-overlapping interval method and sparse characteristics. The convergence rate can reach the optimal convergence rate of the existing high-dimensional integral volatility matrix estimation. At the same time, the proposed adjusted estimators are consistent and semi-positive definite. The advantages and disadvantages of these estimators are compared in the simulation study. Finally the proposed methods are applied to the empirical study of Shanghai Securities Index data.

We propose a locally stationary linear model for the evolution of high-dimensional financial returns, where the time-varying volatility matrix is modelled as a piecewise constant function of time. We introduce a new wavelet-based technique for estimating the volatility matrix, which 10 combines four ingredients: a Haar wavelet decomposition, variance stabilization of the Haar coefficients via the Fisz transform prior to thresholding, a bias correction, and extra time-domain thresholding, soft or hard. Under the assumption of sparsity, we demonstrate the interval-wise consistency of the proposed estimators of the volatility matrix and its inverse in the operator norm, with rates which adapt to the features of the target matrix. We also propose a version of 15 the estimators based on the polarization identity, which permits a more precise derivation of the thresholds. We discuss the practicalities of the algorithm, including parameter selection and how to perform it online. A simulation study shows the benefits of the method, which is illustrated using a stock index portfolio.

Vast Volatility Matrix Estimation using High Frequency Data for Portfolio Selection

Portfolio allocation with gross-exposure constraint is an effective method to increase the efficiency and stability of portfolios selection among a vast pool of assets, as demonstrated by Fan, Zhang, and Yu. The required high-dimensional volatility matrix can be estimated by using high-frequency financial data. This enables us to better adapt to the local volatilities and local correlations among a vast number of assets and to increase significantly the sample size for estimating the volatility matrix. This article studies the volatility matrix estimation using high-dimensional, high-frequency data from the perspective of portfolio selection. Specifically, we propose the use of “pairwise-refresh time” and “all-refresh time” methods based on the concept of “refresh time” proposed by Barndorff-Nielsen, Hansen, Lunde, and Shephard for the estimation of vast covariance matrix and compare their merits in the portfolio selection. We establish the concentration inequalities of the estimates, which guarantee desirable properties of the estimated volatility matrix in vast asset allocation with gross-exposure constraints. Extensive numerical studies are made via carefully designed simulations. Comparing with the methods based on low-frequency daily data, our methods can capture the most recent trend of the time varying volatility and correlation, hence provide more accurate guidance for the portfolio allocation in the next time period. The advantage of using high-frequency data is significant in our simulation and empirical studies, which consist of 50 simulated assets and 30 constituent stocks of Dow Jones Industrial Average index.

It is significant to perform structural analysis of large sparse matrices in order to obtain scalable direct solvers. In this paper, we focus on spectral analysis of large sparse matrices. We believe that the approach for exception handling of challenging matrices via Gerschgorin circles and using tuned parameters is beneficial and practical to stabilize the performance of sparse direct solvers. Nearly defective matrices are among challenging matrices for the performance of solver. Such matrices should be handled separately in order to get rid of potential performance bottleneck. Clustered eigenvalues observed via Gerschgorin circles may be used to detect nearly defective matrix. We observe that the usage of super-nodal storage parameters affects the number of fill-ins and memory usage accordingly.KeywordsSpectral analysisSparse solverDefective matrices

A linear full elliptic second-order Partial Differential Equation (PDE), defined on a d d -dimensional domain Ω \Omega , is approximated by the isogeometric Galerkin method based on uniform tensor-product B-splines of degrees ( p 1 , … , p d ) (p_1,\ldots ,p_d) . The considered approximation process leads to a d d -level stiffness matrix, banded in a multilevel sense. This matrix is close to a d d -level Toeplitz structure if the PDE coefficients are constant and the physical domain Ω \Omega is the hypercube ( 0 , 1 ) d (0,1)^d without using any geometry map. In such a simplified case, a detailed spectral analysis of the stiffness matrices has already been carried out in a previous work. In this paper, we complete the picture by considering non-constant PDE coefficients and an arbitrary domain Ω \Omega , parameterized with a non-trivial geometry map. We compute and study the spectral symbol of the related stiffness matrices. This symbol describes the asymptotic eigenvalue distribution when the fineness parameters tend to zero (so that the matrix-size tends to infinity). The mathematical tool used for computing the symbol is the theory of Generalized Locally Toeplitz (GLT) sequences.

This work presents a deuterium magic angle spinning (MAS) NMR study of the adsorption–desorption dynamics of glycine-(2,2)-d2-alanine dipeptide adsorbed at the inner surfaces of mesoporous SBA-15 silica under different hydration levels and temperatures. The experimental and theoretical challenges posed by the strong quadrupolar interaction of the rigid CD2 group, 3-fold bigger than that of the rotating methyl CD3, were addressed. Deuterium MAS NMR spectra modulated by exchange were analyzed using theoretically calculated exchange spectra based on the two-site Bloch–McConnel exchange equation represented in Floquet space. To solve this equation, which is composed of a high dimensional Floquet exchange matrix, our former computational approach was modified to reduce the overall computation time by orders of magnitude so as to yield more accurate exchange parameters from the spectral analysis. The adsorption–desorption kinetics of minutely hydrated silica surfaces is understood to originate from the diffusion of water molecules into and out of adsorbate binding sites, thereby gating the dynamic behavior of the adsorbate via increase or reduction of the size of the surrounding water cluster. Molecular dynamic (MD) simulations were employed to model the dynamic behavior of the adsorbate at the two states. Deviations between the MD and experimental observations are attributed to the simplified surface modeling, thereby highlighting the importance of experimental MAS NMR data to improve future modeling of realistic functional surfaces.

The susceptible-infected-susceptible (SIS) model is one of the simplest memoryless systems for describing information or epidemic spreading phenomena with competing creation and spontaneous annihilation reactions. The effect of quenched disorder on the dynamical behavior has recently been compared to quenched mean-field (QMF) approximations in scale-free networks. QMF can take into account topological heterogeneity and clustering effects of the activity in the steady state by spectral decomposition analysis of the adjacency matrix. Therefore, it can provide predictions on possible rare-region effects, thus on the occurrence of slow dynamics. I compare QMF results of SIS with simulations on various large dimensional graphs. In particular, I show that for Erdős-Rényi graphs this method predicts correctly the occurrence of rare-region effects. It also provides a good estimate for the epidemic threshold in case of percolating graphs. Griffiths Phases emerge if the graph is fragmented or if we apply a strong, exponentially suppressing weighting scheme on the edges. The latter model describes the connection time distributions in the face-to-face experiments. In case of a generalized Barabási-Albert type of network with aging connections, strong rare-region effects and numerical evidence for Griffiths Phase dynamics are shown. The dynamical simulation results agree well with the predictions of the spectral analysis applied for the weighted adjacency matrices.

The detection and analysis of low signal-to-noise-ratio (SNR) events are valuable for source positions mapping of microseismicity and understanding of underground reservoirs. In the previous study, we applied the spectral matrix (SPM) analysis to characterize the motion of 3D particles in the time-frequency domain by eigenvector of SPM and detected the P-wave arrivals of low SNR microseismic events. In the present study, we further extended low-SNR event detection based on the spectral matrix analysis with the wavelet analysis and time delay coordinate. By using wavelet analysis, the optimum length of the time window for frequency analysis is automatically applied for each frequency band. In addition, the introduction of time-delay coordinates increases the dimension of the SPM matrix and improves the SNR in SPM analysis. Particularly, the time delay is beneficial when event signals are coherent. The characteristics of the proposed method were evaluated by applying the method to synthetic signals. Synthetic signals include sinusoidal event signals and colored noise, and the SNR sensitivity test was conducted by changing the intensity of the noise component. The proposed method is robust to noise and shows the potential to detect low SNR coherent signals efficiently.

On the exact constant in the [formula omitted] Markov inequality