Time-varying Covariance Matrix Research Articles

Understanding the state-dependent impact of species correlated responses on community sensitivity to perturbations.

Understanding how communities respond to perturbations requires us to consider not only changes in the abundance of individual species but also correlated changes that can emerge through interspecific effects. However, our knowledge of this phenomenon is mostly constrained to situations where interspecific effects are fixed. Here, we introduce a framework to disentangle the impact of species correlated responses on community sensitivity to perturbations when interspecific effects change over time due to cyclic or chaotic population dynamics. We partition the volume expansion rate of perturbed abundances (community sensitivity) into contributions of individual species and of species correlated responses by converting the time-varying Jacobian matrix containing interspecific effects into a time-varying covariance matrix. Using population dynamics models, we demonstrate that species correlated responses change considerably across time and continuously alternate between reducing and having no impact on community sensitivity. Importantly, these alternating impacts depend on the abundance of particular species and can be detected even from noisy time series. We showcase our framework using two experimental predator-prey time series and find that the impact of species correlated responses is modulated by prey abundance-as theoretically expected. Our results provide new insights into how and when species interactions can dampen community sensitivity when abundances fluctuate over time.

Sparse DDK: A Data-Driven Decorrelation Filter for GRACE Level-2 Products

High-frequency and correlated noise filtering is one of the important preprocessing steps for GRACE level-2 products before calculating mass anomaly. Decorrelation and denoising kernel (DDK) filters are usually considered as such optimal filters to solve this problem. In this work, a sparse DDK filter is proposed. This is achieved by replacing Tikhonov regularization in traditional DDK filters with weighted L1 norm regularization. The proposed sparse DDK filter adopts a time-varying error covariance matrix, while the equivalent signal covariance matrix is adaptively determined by the Gravity Recovery and Climate Experiment (GRACE) monthly solution. The covariance matrix of the sparse DDK filtered solution is also developed from the Bayesian and error-propagation perspectives, respectively. Furthermore, we also compare and discuss the properties of different filters. The proposed sparse DDK has all the advantages of traditional filters, such as time-varying, location inhomogeneity, and anisotropy, etc. In addition, the filtered solution is sparse; that is, some high-degree and high-order terms are strictly zeros. This sparsity is beneficial in the following sense: high-degree and high-order sparsity mean that the dominating noise in high-degree and high-order terms is completely suppressed, at a slight cost that the tiny signals of these terms are also discarded. The Center for Space Research (CSR) GRACE monthly solutions and their error covariance matrices, from January 2004 to December 2010, are used to test the performance of the proposed sparse DDK filter. The results show that the sparse DDK can effectively decorrelate and denoise these data.

Forecasting performance of Bayesian VEC-MSF models for financial data in the presence of long-run relationships

The paper is focused on comparing the forecasting performance of two relatively new types of Vector Error Correction - Multiplicative Stochastic Factor (VEC-MSF) specifications: VEC-MSF with constant conditional correlations, and VEC-MSF-SBEKK with time-varying conditional correlations. For the sake of comparison, random walks, vector autoregressions (VAR) with constant conditional covariance matrix, and VAR-SBEKK models are also considered. Based on daily quotations on three exchange rates: PLN/EUR, PLN/USD, and EUR/USD, where the cointegrating vector may be assumed to be known a priori, we show that in econometric models it can be more important to allow for cointegration relationships than for time-varying conditional covariance matrix.

A Compact Noise Covariance Matrix Model for MVDR Beamforming

Acoustic beamforming is routinely used to improve the SNR of the received signal in applications such as hearing aids, robot audition, augmented reality, teleconferencing, source localisation and source tracking. The beamformer can be made adaptive by using an estimate of the time-varying noise covariance matrix in the spectral domain to determine an optimised beam pattern in each frequency bin that is specific to the acoustic environment and that can respond to temporal changes in it. However, robust estimation of the noise covariance matrix remains a challenging task especially in non-stationary acoustic environments. This paper presents a compact model of the signal covariance matrix that is defined by a small number of parameters whose values can be reliably estimated. The model leads to a robust estimate of the noise covariance matrix which can, in turn, be used to construct a beamformer. The performance of beamformers designed using this approach is evaluated for a spherical microphone array under a range of conditions using both simulated and measured room impulse responses. The proposed approach demonstrates consistent gains in intelligibility and perceptual quality metrics compared to the static and adaptive beamformers used as baselines.

ZUPT-Aided INS Bypassing Stance Phase Detection by Using Foot-Instability-Based Adaptive Covariance

In this paper, we propose a Foot-Instability-Based Adaptive (FIBA) covariance to dynamically adjust the covariance matrix for the pseudo-zero-velocity measurements in the Zero velocity UPdaTe (ZUPT)-aided Inertial Navigation Systems (INS). The proposed ZUPT-aided INS using the FIBA covariance is implemented in an Extended Kalman Filter (EKF) framework, which utilizes a time-varying measurement covariance matrix that is updated in each iteration according to the FIBA covariance. The FIBA covariance is designed to have a very high value during the swing phases in a gait cycle, and the value significantly decreases during the stance phases. As a result, the proposed method eliminates a need to use a binary stance phase detector in implementation of the ZUPT-aided INS. Properties of EKF innovation sequences in the algorithm were studied, and two series of indoor pedestrian navigation experiments were conducted to demonstrate the navigation performance of the proposed system. In the first series of experiments, which included cases of walking and running, localization solutions produced by the system using the FIBA covariance demonstrated 36% and 64% improvements in navigation accuracy along the horizontal and vertical directions, respectively. In the second series of experiments, which included a pedestrian walking on different indoor terrains, such as flat planes, stairs, and ramps, the navigation accuracy of the system using the FIBA covariance reduced horizontal and vertical position errors by 12% and 45%, respectively, as compared to the conventional ZUPT-aided INS.

Pricing Implications of Covariances and Spreads in Currency Markets

Abstract We introduce a covariance and spread (i.e., exchange rate forward discount) adjusted carry factor that prices the cross-section of FX market returns, where many other single- and multifactor models fail. Both the covariance matrix of exchange rate growths and forward discounts contain important information for pricing that is not captured by well-known factors. The time-varying conditional covariance matrix and forward discounts forecast future realized currency returns. (JEL F31, F37, G12, G15, G17)

Scenario Forecasting for Global Tourism

This study provides innovative forecasts of the probabilities of certain scenarios of tourism demand. The scenarios of interest are constructed in relation to tourism growth and economic growth. The probability forecasts based on these scenarios provide valuable information for destination policy makers. The time-varying parameter panel vector autoregressive (TVP-PVAR) model is adopted for scenario forecasting. Both the accuracy rate and the Brier score are used to evaluate the forecasting performance. A global set of 25 tourism destinations is empirically examined, and the results confirm that the TVP-PVAR model with a time-varying error covariance matrix is generally a promising tool for forecasting. Our study contributes to tourism forecasting literature in advocating the use of scenario forecasting to facilitate industry decision making in situations wherein forecasts are defined by two or more dimensions simultaneously. In addition, it is the first study to introduce the TVP-PVAR model to tourism demand forecasting.

Estimation of Autoregressive Parameters from Noisy Observations Using Iterated Covariance Updates.

Estimating the parameters of the autoregressive (AR) random process is a problem that has been well-studied. In many applications, only noisy measurements of AR process are available. The effect of the additive noise is that the system can be modeled as an AR model with colored noise, even when the measurement noise is white, where the correlation matrix depends on the AR parameters. Because of the correlation, it is expedient to compute using multiple stacked observations. Performing a weighted least-squares estimation of the AR parameters using an inverse covariance weighting can provide significantly better parameter estimates, with improvement increasing with the stack depth. The estimation algorithm is essentially a vector RLS adaptive filter, with time-varying covariance matrix. Different ways of estimating the unknown covariance are presented, as well as a method to estimate the variances of the AR and observation noise. The notation is extended to vector autoregressive (VAR) processes. Simulation results demonstrate performance improvements in coefficient error and in spectrum estimation.

Modified models and simulations for estimating dynamic functional connectivity in resting state functional magnetic resonance imaging.

As understanding the nature of brain networks through dynamic functional connectivity (dFC) estimation is of paramount significant, the introduction and revision of blood-oxygen-level dependent (BOLD) signal simulation methods in brain regions and dFC estimation methods have gained significant ground in recent years. Based on the observation of BOLD signals with multivariate nonnormal distribution in functional magnetic resonance imaging (fMRI) images, we first propose a copula-based method for the production of these signals, in which nonnormal data are generated with a selected time-varying covariance matrix. Therefore, we can compare the performance of models in the cases where brain signals have a multivariate nonnormal distribution. Then, two kendallized exponentially weighted moving average (KEWMA) and kendallized dynamic conditional correlation (KDCC) multivariate volatility models are introduced which are based on two well-known and commonly used exponentially weighted moving average (EMWA) and dynamic conditional correlation (DCC) models. The results show that KDCC model can estimate conditional correlation significantly far better than the former ones (ie, DCC, standardized dynamic conditional correlation, EWMA, and standardized exponentially weighted moving average) on both types of data (ie, multivariate normal and nonnormal). In the next step, the bivariate normal distribution in Iranian resting state fMRI data is confirmed by using statistical tests, and it is shown that the dynamic nature of FC is not optimally detected using prevalent methods. Two alternative Portmanteau and rank-based tests are proposed for the examination of conditional heteroscedasticity in data. Finally, dFC in these data is estimated by employing the KDCC model.

The Five Trolls Under the Bridge: Principal Component Analysis With Asynchronous and Noisy High Frequency Data

We develop a principal component analysis (PCA) for high frequency data. As in Northern fairy tales, there are trolls waiting for the explorer. The first three trolls are market microstructure noise, asynchronous sampling times, and edge effects in estimators. To get around these, a robust estimator of the spot covariance matrix is developed based on the smoothed two-scale realized variance (S-TSRV). The fourth troll is how to pass from estimated time-varying covariance matrix to PCA. Under finite dimensionality, we develop this methodology through the estimation of realized spectral functions. Rates of convergence and central limit theory, as well as an estimator of standard error, are established. The fifth troll is high dimension on top of high frequency, where we also develop PCA. With the help of a new identity concerning the spot principal orthogonal complement, the high-dimensional rates of convergence have been studied after eliminating several strong assumptions in classical PCA. As an application, we show that our first principal component (PC) closely matches but potentially outperforms the S&P 100 market index. From a statistical standpoint, the close match between the first PC and the market index also corroborates this PCA procedure and the underlying S-TSRV matrix, in the sense of Karl Popper. Supplementary materials for this article are available online.

Robust Inference In Time-Varying Structural VAR Models: The DC-Cholesky Multivariate Stochastic Volatility Model

This paper investigates how the ordering of variables affects properties of the time-varying covariance matrix in the Cholesky multivariate stochastic volatility model. It establishes that systematically different dynamic restrictions are imposed when the ratio of volatilities is time-varying. Simulations demonstrate that estimated covariance matrices become more divergent when volatility clusters idiosyncratically. It is illustrated that this property is important for empirical applications. Specifically, alternative estimates on the evolution of U.S. systematic monetary policy and in ation-gap persistence indicate that conclusions may critically hinge on a selected ordering of variables. The dynamic correlation Cholesky multivariate stochastic volatility model is proposed as a robust alternative.

Privacy Preservation in Industrial IoT via Fast Adaptive Correlation Matrix Completion

The Industrial Internet of Things (IIoT) is a key element of industry 4.0, bringing together modern sensor technology, fog and cloud computing platforms, and artificial intelligence to create smart, self-optimizing industrial equipment and facilities. Though, the scale and sensitivity degree of information continuously increases, giving rise to serious privacy concerns. The scope of this article is to provide efficient privacy preservation techniques, by tracking the correlation of multivariate streams recorded in a network of IIoT devices. The time-varying data covariance matrix is used to add noise that cannot be easily removed by filtering, generating obfuscated measurements and, thus, preventing unauthorized access to the original data. To improve communication efficiency between connected IoT devices, we exploit inherent properties of the correlation matrices, and track the essential correlations from a small subset of correlation values. Extensive simulation studies using constrained IIoT devices validate the robustness, efficiency, and effectiveness of our approach.

Improved Covariance Matrix Estimation for Portfolio Risk Measurement: A Review

The literature on portfolio selection and risk measurement has considerably advanced in recent years. The aim of the present paper is to trace the development of the literature and identify areas that require further research. This paper provides a literature review of the characteristics of financial data, commonly used models of portfolio selection, and portfolio risk measurement. In the summary of the characteristics of financial data, we summarize the literature on fat tail and dependence characteristic of financial data. In the portfolio selection model part, we cover three models: mean-variance model, global minimum variance (GMV) model and factor model. In the portfolio risk measurement part, we first classify risk measurement methods into two categories: moment-based risk measurement and moment-based and quantile-based risk measurement. Moment-based risk measurement includes time-varying covariance matrix and shrinkage estimation, while moment-based and quantile-based risk measurement includes semi-variance, VaR and CVaR.

The DC-Cholesky Multivariate Stochastic Volatility Model

This paper investigates how the ordering of variables affects properties of the time-varying covariance matrix in the Cholesky multivariate stochastic volatility model. It establishes that estimated covariance matrices, obtained under alternative orderings of variables, are systemically different when the data exhibits independent volatility dynamics. Specifically, simulations show that estimated covariances and correlations become more different, the larger the ratio of individual volatility paths becomes. This paper shows that this property is important for empirical applications as alternative estimates on the evolution of U.S. systematic monetary policy and on inflation-gap persistence indicate that conclusions may hinge on a selected ordering of variables. The dynamic correlation Cholesky multivariate stochastic volatility model is proposed as a robust alternative.

Recursive Spectral Meta-Learner for Online Combining Different Fault Classifiers

This paper considers the problem of fault classification when different fault classifiers are performed simultaneously. Based on spectral meta-learner (SML) proposed by Parisi et al. , its recursive version, i.e., recursive SML (RSML) is developed for online combining the potentially conflicting classification information. Considering different statistical properties of faults occurring at different time intervals, the binary classification information is recursively utilized. By introducing a forgetting factor, the leading eigenvector of the estimate of the time-varying covariance matrix is used as the weight vector for each classifier. Rank-one modification is then used for calculating the eigenvector in order to reduce the online computational complexity. The performance of RSML is strictly analyzed in a statistical sense, including the effect of recursive calculation and conditional dependence of different classifiers on the weight vector. Compared with majority voting and SML, a higher balanced accuracy of RSML can be verified by the benchmark Tennessee Eastman process.

The Five Trolls Under the Bridge: Principal Component Analysis with Asynchronous and Noisy High Frequency Data

We develop a principal component analysis (PCA) for high frequency data. As in Northern fairly tales, there are trolls waiting for the explorer. The first three trolls are market microstructure noise, asynchronous sampling times, and edge effects in estimators. To get around these, a robust estimator of spot covariance matrix is developed based on the Smoothed TSRV (Mykland et al. (2017)). The fourth troll is how to pass from estimated time-varying covariance matrix to PCA. Under finite dimensionality, we develop this methodology through the estimation of realized spectral functions. Rates of convergence and central limit theory, as well as an estimator of standard error, are established. The fifth troll is high dimension on top of high frequency, where we also develop PCA. With the help of a new identity concerning the spot principal orthogonal complement, the high-dimensional rates of convergence have been studied by freeing several strong assumptions in classical PCA. As an application, we show that our first principal component (PC) potentially outperforms the S&P 100 market index, while three of the next four PCs are cointegrated with two of the Fama-French non-market factors.

Network Inference via the Time-Varying Graphical Lasso.

Many important problems can be modeled as a system of interconnected entities, where each entity is recording time-dependent observations or measurements. In order to spot trends, detect anomalies, and interpret the temporal dynamics of such data, it is essential to understand the relationships between the different entities and how these relationships evolve over time. In this paper, we introduce the time-varying graphical lasso (TVGL), a method of inferring time-varying networks from raw time series data. We cast the problem in terms of estimating a sparse time-varying inverse covariance matrix, which reveals a dynamic network of interdependencies between the entities. Since dynamic network inference is a computationally expensive task, we derive a scalable message-passing algorithm based on the Alternating Direction Method of Multipliers (ADMM) to solve this problem in an efficient way. We also discuss several extensions, including a streaming algorithm to update the model and incorporate new observations in real time. Finally, we evaluate our TVGL algorithm on both real and synthetic datasets, obtaining interpretable results and outperforming state-of-the-art baselines in terms of both accuracy and scalability.

A suggestion for constructing a large time-varying conditional covariance matrix

The construction of large conditional covariance matrices has posed a problem in the empirical literature because the direct extension of the univariate GARCH model to a multivariate setting produces large numbers of parameters to be estimated as the number of equations rises. A number of procedures have previously aimed to simplify the model and restrict the number of parameters, but these procedures typically involve either invalid or undesirable restrictions. This paper suggests an alternative way forward, based on the GARCH approach, which allows conditional covariance matrices of unlimited size to be constructed. The procedure is computationally straightforward to implement. At no point in the procedure is it necessary to estimate anything other than a univariate GARCH model.

Bayesian nonparametric covariance regression

Capturing predictor-dependent correlations amongst the elements of a multivariate response vector is fundamental to numerous applied domains, including neuroscience, epidemiology, and finance. Although there is a rich literature on methods for allowing the variance in a univariate regression model to vary with predictors, relatively little has been done in the multivariate case. As a motivating example, we consider the Google Flu Trends data set, which provides indirect measurements of influenza incidence at a large set of locations over time (our predictor). To accurately characterize temporally evolving influenza incidence across regions, it is important to develop statistical methods for a time-varying covariance matrix. Importantly, the locations provide a redundant set of measurements and do not yield a sparse nor static spatial dependence structure. We propose to reduce dimensionality and induce a flexible Bayesian nonparametric covariance regression model by relating these location-specific trajectories to a lower-dimensional subspace through a latent factor model with predictor-dependent factor loadings. These loadings are in terms of a collection of basis functions that vary nonparametrically over the predictor space. Such low-rank approximations are in contrast to sparse precision assumptions, and are appropriate in a wide range of applications. Our formulation aims to address three challenges: scaling to large p domains, coping with missing values, and allowing an irregular grid of observations. The model is shown to be highly exible, while leading to a computationally feasible implementation via Gibbs sampling. The ability to scale to large p domains and cope with missing values is fundamental in analyzing the Google Flu Trends data.

Heavy-Traffic Limits for a Many-Server Queueing Network with Switchover

We study a multiclass Markovian queueing network with switchover across a set of many-server stations. New arrivals to each station follow a nonstationary Poisson process. Each job waiting in queue may, after some exponentially distributed patience time, switch over to another station or leave the network following a probabilistic and state-dependent mechanism. We analyze the performance of such networks under the many-server heavy-traffic limiting regimes, including the critically loaded quality-and-efficiency-driven (QED) regime, and the overloaded efficiency-driven (ED) regime. We also study the limits corresponding to mixing the underloaded quality-driven (QD) regime with the QED and ED regimes. We establish fluid and diffusion limits of the queue-length processes in all regimes. The fluid limits are characterized by ordinary differential equations. The diffusion limits are characterized by stochastic differential equations, with a piecewise-linear drift term and a constant (QED) or time-varying (ED) covariance matrix. We investigate the load balancing effect of switchover in the mixed regimes, demonstrating the migration of workload from overloaded stations to underloaded stations and quantifying the load balancing impact of switchover probabilities.