On the asymptotic behavior of the eigenvalue distribution of block correlation matrices of high-dimensional time series

2021 ◽  
Author(s):  
Philippe Loubaton ◽  
Xavier Mestre
Keyword(s):  
Time Series ◽  
Correlation Matrix ◽  
Cross Correlation ◽  
Eigenvalue Distribution ◽  
Time Lags ◽  
Correlation Matrices ◽  
Consecutive Time ◽  
Spectral Statistics ◽  
Mutually Independent ◽  
Scalar Time Series

We consider linear spectral statistics built from the block-normalized correlation matrix of a set of [Formula: see text] mutually independent scalar time series. This matrix is composed of [Formula: see text] blocks. Each block has size [Formula: see text] and contains the sample cross-correlation measured at [Formula: see text] consecutive time lags between each pair of time series. Let [Formula: see text] denote the total number of consecutively observed windows that are used to estimate these correlation matrices. We analyze the asymptotic regime where [Formula: see text] while [Formula: see text], [Formula: see text]. We study the behavior of linear statistics of the eigenvalues of this block correlation matrix under these asymptotic conditions and show that the empirical eigenvalue distribution converges to a Marcenko–Pastur distribution. Our results are potentially useful in order to address the problem of testing whether a large number of time series are uncorrelated or not.

RANDOM MATRIX THEORY AND FINANCIAL CORRELATIONS

2000 ◽  
Vol 03 (03) ◽  
pp. 391-397 ◽  
Author(s):  
LAURENT LALOUX ◽  
PIERRE CIZEAU ◽  
MARC POTTERS ◽  
JEAN-PHILIPPE BOUCHAUD
Keyword(s):  
Time Series ◽  
Risk Management ◽  
Theoretical Prediction ◽  
Correlation Matrix ◽  
Matrix Theory ◽  
Financial Time Series ◽  
Statistical Structure ◽  
Correlation Matrices ◽  
Financial Time ◽  
Remarkable Agreement

We show that results from the theory of random matrices are potentially of great interest when trying to understand the statistical structure of the empirical correlation matrices appearing in the study of multivariate financial time series. We find a remarkable agreement between the theoretical prediction (based on the assumption that the correlation matrix is random) and empirical data concerning the density of eigenvalues associated to the time series of the different stocks of the S&P500 (or other major markets). Finally, we give a specific example to show how this idea can be sucessfully implemented for improving risk management.

scholarly journals ECONOPHYSICS: WHAT CAN PHYSICISTS CONTRIBUTE TO ECONOMICS?

2000 ◽  
Vol 03 (03) ◽  
pp. 335-346 ◽  
Author(s):  
H. EUGENE STANLEY ◽  
LUÍS A. NUNES AMARAL ◽  
PARAMESWARAN GOPIKRISHNAN ◽  
YANHUI LIU ◽  
VASILIKI PLEROU ◽  
...  
Keyword(s):  
Time Series ◽  
Power Law ◽  
Random Matrix Theory ◽  
Random Matrix ◽  
Correlation Matrix ◽  
Cross Correlation ◽  
Matrix Theory ◽  
Financial Time Series ◽  
Price Fluctuations ◽  
Financial Time

In recent years, a considerable number of physicists have started applying physics concepts and methods to understand economic phenomena. The term "Econophysics" is sometimes used to describe this work. Economic fluctuations can have many repercussions, and understanding fluctuations is a topic that many physicists have contributed to in recent years. Further, economic systems are examples of complex interacting systems for which a huge amount of data exist and it is possible that the experience gained by physicists in studying fluctuations in physical systems might yield new results in economics. Much recent work in econophysics is focused on understanding the peculiar statistical properties of price fluctuations in financial time series. In this talk, we discuss three recent results. The first result concerns the probability distribution of stock price fluctuations. This distribution decreases with increasing fluctuations with a power-law tail well outside the Lévy stable regime and describes fluctuations that differ by as much as 8 orders of magnitude. Further, this nonstable distribution preserves its functional form for fluctuations on time scales that differ by 3 orders of magnitude, from 1 min up to approximately 10 days. The second result concerns the accurate quantification of volatility correlations in financial time series. While price fluctuations themselves have rapidly decaying correlations, the volatility estimated by using either the absolute value or the square of the price fluctuations has correlations that decay as a power-law and persist for several months. The third result bears on the application of random matrix theory to understand the correlations among price fluctuations of any two different stocks. We compare the statistics of the cross-correlation matrix constructed from price fluctuations of the leading 1000 stocks and a matrix with independent random elements, i.e., a random matrix. Contrary to first expectations, we find little or no deviation from the universal predictions of random matrix theory for all but a few of the largest eigenvalues of the cross-correlation matrix.

Detrended Correlogram Method for Non-Stationary Time-Series Analysis

2021 ◽  
pp. 2250012
Author(s):  
G. F. Zebende ◽  
E. F. Guedes
Keyword(s):  
Time Series ◽  
Time Series Analysis ◽  
Correlation Coefficient ◽  
Memory Effect ◽  
Cross Correlation ◽  
Time Lag ◽  
Stationary Time Series ◽  
Time Lags ◽  
Statistical Tool ◽  
Auto Correlation

A correlogram is a statistical tool that is used to check time-series memory by computing the auto-correlation coefficient as a function of the time lag. If the time-series has no memory, then the auto-correlation must be close to zero for any time lag, otherwise if there is a memory, then the auto-correlations must be significantly different from zero. Therefore, based on the robust detrended cross-correlation coefficient, [Formula: see text], we propose the detrended correlogram method in this paper, which will be tested for some time-series (simulated and empirical). This new statistical tool is able to visualize a complete map of the auto-correlation for many time lags and time-scales, and can therefore analyze the memory effect for any time-series.

Generation of synthetic cross-correlated water demand time series

2013 ◽  
Vol 13 (4) ◽  
pp. 977-986 ◽  
Author(s):  
N. Ansaloni ◽  
S. Alvisi ◽  
M. Franchini
Keyword(s):  
Time Series ◽  
Water Demand ◽  
Distribution System ◽  
Water Distribution ◽  
Cross Correlation ◽  
Time Lags ◽  
Cross Correlations ◽  
The Cross ◽  
Water Districts

This paper presents a procedure for generating synthetic district-level series of hourly water demand coefficients cross-correlated in space (between districts) and time. The procedure consists of two steps: (1) generation of hourly water demand coefficients which respect, for each hour of the day, pre-assigned means and variances; and (2) introduction of the cross-correlation at different time lags through the application of a method which implies the reordering of the data generated at step 1. The procedure was applied to a case study of the Ferrara water distribution system with the aim of generating cross-correlated synthetic series of hourly water demand coefficients for the 19 water districts making it up. It was observed that the application of the method for introducing the cross-correlation (step 2) causes numerical problems when a large number of water districts are involved and the cross-correlations are considered at many time lags; this problem is solved by carrying out an appropriate regularization of the observed cross-correlation matrix. The results obtained show that overall the proposed procedure constitutes a valid tool for generating synthetic water demand time series with pre-assigned characteristics in terms of means, variances and cross-correlation at different time lags.

scholarly journals GRACE-Derived Time Lag of Mekong Estuarine Freshwater Transport in the Western South China Sea Validated by Isotopic Tracer Age

2021 ◽  
Vol 13 (6) ◽  
pp. 1193
Author(s):  
Zhongtian Ma ◽  
Hok Sum Fok ◽  
Linghao Zhou
Keyword(s):  
Time Series ◽  
South China Sea ◽  
South China ◽  
Cross Correlation ◽  
Time Lag ◽  
Time Lags ◽  
Substantial Impact ◽  
China Sea ◽  
Freshwater Transport ◽  
Western South China Sea

Estuarine freshwater transport has a substantial impact on the near-shore ecosystem and coastal ocean environment away from the estuary. This paper introduces two independent methods to track the Mekong freshwater-induced mass transport by calculating the time lag (or equivalently, the phase) between in situ Mekong basin runoff and the equivalent water height (EWH) time series over the western South China Sea from a gravity recovery and climate experiment (GRACE). The first method is the harmonic analysis that determines the phase difference between annual components of the two time series (called the P-method), and the other is the cross-correlation analysis that directly obtains the time lag by shifting the lagged time series forward to attain the highest cross-correlation between the two time series (called the C-method). Using a three-year rolling window, the time lag variations in three versions of GRACE between 2005 and 2012 are computed for demonstrating the consistency of the results. We found that the time lag derived from the P-method is, on average, slightly larger and more variable than that from the C-method. A comparison of our gridded time lag against the age determined via radium isotopes in September, 2007 by Chen et al. (2010) revealed that our gridded time lag results were in good agreement with most isotope-derived ages, with the largest difference less than 6 days. Among the three versions of the GRACE time series, CSR Release 05 performed the best. The lowest standard deviation of time lag was ~1.6 days, calculated by the C-method, whereas the mean difference for all the time lags from the isotope-derived ages is ~1 day by P-method. This study demonstrates the potential of monitoring Mekong estuarine freshwater transport over the western South China Sea by GRACE.

scholarly journals An Efficient Framework for Estimating the Direction of Multiple Sound Sources Using Higher-Order Generalized Singular Value Decomposition

Sensors ◽  
2019 ◽  
Vol 19 (13) ◽  
pp. 2977 ◽  
Author(s):  
Bandhit Suksiri ◽  
Masahiro Fukumoto
Keyword(s):  
Singular Value Decomposition ◽  
Correlation Matrix ◽  
Cross Correlation ◽  
Estimation Algorithm ◽  
Higher Order ◽  
Singular Value ◽  
Generalized Singular Value Decomposition ◽  
Sound Sources ◽  
Correlation Matrices ◽  
Value Decomposition

This paper presents an efficient framework for estimating the direction-of-arrival (DOA) of wideband sound sources. The proposed framework provides an efficient way to construct a wideband cross-correlation matrix from multiple narrowband cross-correlation matrices for all frequency bins. In addition, the proposed framework is inspired by the coherent signal subspace technique with further improvement of linear transformation procedure, and the new procedure no longer requires any process of DOA preliminary estimation by exploiting unique cross-correlation matrices between the received signal and itself on distinct frequencies, along with the higher-order generalized singular value decomposition of the array of this unique matrix. Wideband DOAs are estimated by employing any subspace-based technique for estimating narrowband DOAs, but using the proposed wideband correlation instead of the narrowband correlation matrix. It implies that the proposed framework enables cutting-edge studies in the recent narrowband subspace methods to estimate DOAs of the wideband sources directly, which result in reducing computational complexity and facilitating the estimation algorithm. Practical examples are presented to showcase its applicability and effectiveness, and the results show that the performance of fusion methods perform better than others over a range of signal-to-noise ratios with just a few sensors, which make it suitable for practical use.

scholarly journals Mutual antenna coupling influence on the channel correlation matrix for linear antenna arrays

2021 ◽  
Vol 10 (2) ◽  
pp. 820-827
Author(s):  
Tatiana K. Artemova ◽  
Aleksey S. Gvozdarev ◽  
Konstantin S. Artemov
Keyword(s):  
Antenna Arrays ◽  
Correlation Matrix ◽  
Communication Systems ◽  
Cross Correlation ◽  
Spatial Separation ◽  
Correlation Matrices ◽  
Antenna Coupling ◽  
Quantitative Results ◽  
Multiantenna Communication ◽  
The Impact

The paper presents the results of the research of electromagnetic mutual coupling impact on the structure of the correlation matrices in multiantenna communication systems. Classical correlation structures employed in most of the up-to-date communication systems descriptions and designs usually assume unit autocorrelation and exponentially decreasing cross-correlation of antenna elements in the receiving/transmitting array. At the same time numerous studies had shown that these assumptions may not hold under certain conditions. The performed research relates the correlation effects with the imbalances of the array impedance matrix terms and studies the impact of antenna elements mutual electromagnetic interaction upon the diagonal (autocorrelation) and off-diagonal (cross-correlation) terms of correlation matrix, depending of the geometry of the array: number of elements and their spatial separation. To exemplify quantitative results the analysis was carried out for the 5G NR #78 band, being one of the most wideband subchannels in Under-6 GHz regime for 5G systems. The obtained results also justified the applicability of the banded correlation matrix model for wireless communications.

scholarly journals MULTISCALED CROSS-CORRELATION DYNAMICS IN FINANCIAL TIME-SERIES

2009 ◽  
Vol 12 (04n05) ◽  
pp. 439-454 ◽  
Author(s):  
T. CONLON ◽  
H. J. RUSKIN ◽  
M. CRANE
Keyword(s):  
Time Series ◽  
Cross Correlation ◽  
Financial Time Series ◽  
Time Windows ◽  
Equity Returns ◽  
Discrete Wavelet ◽  
Eigenvalue Spectrum ◽  
Correlation Matrices ◽  
Multiple Interactions ◽  
Insight Into

The cross-correlation matrix between equities comprises multiple interactions between traders with varying strategies and time horizons. In this paper, we use the Maximum Overlap Discrete Wavelet Transform to calculate correlation matrices over different time–scales and then explore the eigenvalue spectrum over sliding time-windows. The dynamics of the eigenvalue spectrum at different times and scales provides insight into the interactions between the numerous constituents involved. Eigenvalue dynamics are examined for both medium, and high-frequency equity returns, with the associated correlation structure shown to be dependent on both time and scale. Additionally, the Epps effect is established using this multivariate method and analyzed at longer scales than previously studied. A partition of the eigenvalue time-series demonstrates, at very short scales, the emergence of negative returns when the largest eigenvalue is greatest. Finally, a portfolio optimization shows the importance of time–scale information in the context of risk management.

Sign in / Sign up

Export Citation Format

Share Document