Multivariate Autoregressive Moving Average Research Articles

Simultaneous monitoring of mean vector and covariance matrix of auto‐correlated multivariate multiple linear profiles

AbstractThe aim of this paper is to develop control charts for a simultaneous monitoring of the mean vector and covariance matrix of multivariate multiple linear regression profiles in phase II, when the independence assumption of the observations within each profile is violated, and there is multivariate autoregressive moving average (MARMA)(1,1) autocorrelation structure within each profile. A transformation method is applied to eliminate the autocorrelation, and six control charts for a simultaneous monitoring of the mean vector and covariance matrix of auto‐correlated multivariate multiple linear regression profiles are proposed. Then, through simulation runs, in terms of the average run length (ARL) criterion, it is shown that the proposed methods have better performance in comparison with the competing methods in detecting shifts in the parameters of regression profiles. Finally, the application of the proposed control charts is shown using real data in a machining process.

Improved condition monitoring for an FPSO system with multiple correlated components

It is expected that a resilient condition monitoring (CM) system can identify the faults of sensors and key components, assess the system risk level based upon multichannel monitoring data, and recover missing real data when complete sensor failure occurs. As for CM of an offshore system, sensors are usually less reliable than the components being monitored. Without CM resilience, the whole system will be vulnerable to accidents and different types of disturbance. This paper provides a statistical and artificial intelligence approach to realizing resilient CM for offshore systems with multiple correlated components. This approach enables identifying sensor faults, damage and degradation of key components through frequency-centered wavelet analysis along with an artificial neural network-based dynamic threshold under different operating environments. More importantly, the proposed method enables the recovery of missing real data based on fusion data from other normal sensors and multivariate Autoregressive Moving Average (ARMA) model. The method is applied to a hawsers’ CM system in a Floating Production, Storage and Offloading (FPSO) oil offloading system. The numerical results illustrate the feasibility and effectiveness of the proposed method in realizing CM resilience for such complex and critical systems.

Closed-form expression for finite predictor coefficients of multivariate ARMA processes

We derive a closed-form expression for the finite predictor coefficients of multivariate ARMA (autoregressive moving-average) processes. The expression is given in terms of several explicit matrices that are of fixed sizes independent of the number of observations. The significance of the expression is that it provides us with a linear-time algorithm to compute the finite predictor coefficients. In the proof of the expression, a correspondence result between two relevant matrix-valued outer functions plays a key role. We apply the expression to determine the asymptotic behavior of a sum that appears in the autoregressive model fitting and the autoregressive sieve bootstrap. The results are new even for univariate ARMA processes.

Improving the sensitivity of neurovascular coupling assessment using motor‐cognitive paradigms post‐stroke (1068.4)

Neurovascular coupling (NVC) has been used as an index of synaptic activity in health and disease. Cerebral blood flow velocity (CBFv) responses to neural stimulation can be misleading due to influences from peripheral co‐variates, leading to inaccurate NVC assessment. A dynamic multivariate autoregressive‐moving average (ARMA) model was used to calculate the contributions of blood pressure (BP), PaCO2 and the stimulus itself to changes in CBFv during active (AP), passive (PP) and motor imagery (MIP) paradigms of elbow flexion/ extension. Continuous recordings of CBFv, BP, heart rate and end‐tidal CO2 were performed in 20 stroke patients and 17 control subjects during the three paradigms. Though the paradigms led to a significant bilateral CBFv increase in both groups, differences between stroke and controls were found only during the PP (p=0.02). ARMA analysis revealed a marked BP contribution to the initial peaked CBFv response in both groups during all paradigms, as well as a decrease in the amplitude of the stimulation component after stroke (p<0.001). PaCO2 did not interfere with the CBFv responses during either the MIP in controls or any of the paradigms in the stroke group. However, the responses to AP and MIP in controls were influenced by a significant reduction in PaCO2 (AP reduction of 2%; MIP reduction of 2.5%). By removing the influences of BP and PaCO2, it was possible to detect differences in the effects of stimulation between controls and strokes. We concluded that, independent of the paradigm adopted for NVC assessment, factoring out the influences of BP and PaCO2 can lead to improvements in sensitivity.Grant Funding Source: CAPES ‐ Ministry of Education of Brazil

MARKET VOLATILITY: A STUDY OF EQUITY MARKETS OF USA, CANADA, GERMANY AND CHINA

ABSTRACT If equity markets are integrated, an unexpected event in one market may influence not only returns but also volatility in the other markets. The analysis of volatility is particularly important because of the information it provides for the riskiness of assets. This paper studies relationships among the volatilities of the four different equity markets: USA and Canada from North America; Germany from Europe and China from Asia. This study uses data for the period of January 2008-December 2010. Multivariate Autoregressive Moving Average (MARMA) method was applied to three month rolling daily volatilities, which were calculated using extended logarithmic Garman-Klass formula. The results indicate that the US market volatility had the most marked effect on the volatilities of the other markets. While both Canadian and German market volatilities seem important to explain the volatilities in other markets (US and to a lesser extend China), spillovers seem to become more pronounced as we moved from 2008 to 2010. Keywords Equity market volatility, Global market integration and correlations, Volatility spillovers, Portfolio diversification, Hedging, Exchange-Traded Funds

Asymptotically Equivalent Sequences of Matrices and Multivariate ARMA Processes

The present paper considers a special class of vector random processes that we call multivariate asymptotically wide sense stationary (WSS) processes. A multivariate random process is said to be asymptotically WSS if it has constant mean and the sequence of its autocorrelation matrices is asymptotically equivalent (a.e.) to the sequence of autocorrelation matrices of some multivariate WSS process. It is shown that this class of processes contains meaningful processes other than multivariate WSS processes. In particular, we give sufficient conditions for multivariate moving average (MA) processes, multivariate autoregressive (AR) processes and multivariate autoregressive moving average (ARMA) processes to be asymptotically WSS. Furthermore, in order to solve multiple-input-multiple-output (MIMO) problems in communications and signal processing involving this kind of processes, we extend the Gray definition of a.e. sequences of matrices and his main results on these sequences to non-square matrices. As an example, the derived results on a.e. sequences of non-square matrices are applied to compute the differential entropy rate and the minimum mean square error (MMSE) for a linear predictor of a multivariate asymptotically WSS process.

Comparative Forecasting Models Accuracy of Short-term Natural Rubber Prices

The study describes a number of short-term expost forecasts of econometric models and univariate model of autoregressive-integrated-moving average (ARIMA) model of Natural Rubber (NR) prices SMR20 (Standard Malaysia Rubber Grade 20) in the Malaysia market. The econometric models include single equation model using Vector Error Correction Method (VECM) and multivariate autoregressive-moving average (MARMA) model. The models were developed to determine the inter-relationships between NR production, consumption and prices, to forecast the price of NR (SMR20) and to determine which model is more efficient in their price forecasting accuracy. The models based on data from period January 1990-December 2008. Comparative forecasting models accuracy between single equation model, MARMA model and univariate model of ARIMA, were made in terms of their estimation accuracy based on RMSE, MAE, RMPE and (U-Theil) criteria. The results revealed that the values of the RMSE, MAE, RMPE and U of MARMA model were comparatively smaller than the values generated by single equation model and ARIMA model. These statistics suggested that MARMA model is more accurate and efficient measured in terms of its statistical criteria than single equation model and ARIMA model in forecasting the NR price of SMR20 in the next 6 months or so.

Application of the multi-model partitioning theory for simultaneous order and parameter estimation of multivariate ARMA models

In this paper, a study on how to perform simultaneous order and parameter estimation of multivariate (MV) ARMA (autoregressive moving average) models under the presence of noise is addressed. The proposed method, which is computationally efficient, is an extension of a previously presented method for MV AR models and is based on the well established and widely applied multi-model partitioning theory. A series of computer simulations indicate that the method is infallible in selecting the correct model order in very few steps. The simultaneous estimation of the multivariate ARMA parameters is also another benefit of the proposed method. The results are compared with two other established order selection criteria namely Akaike's Information Criterion (AIC) and Schwarz's Bayesian Information Criterion (BIC). Finally, it is shown that the method is also successful in tracking model order changes, in real time.

Multivariate stable ARMA processes with time dependent coefficients

This paper considers the class of m-variate autoregressive moving average (ARMA) processes with stable innovations and time dependent coefficients. A set of suitable AR and MA regularity conditions is given to ensure existence and uniqueness of valid solutions. A simple form of the above solution is expressed in terms of one sided Green's matrix functions associated with the AR operator. We solve the prediction problem arising in this class of models. A few examples are added to support the general theory.

Recursive Prediction and Likelihood Evaluation for Periodic ARMA Models

This paper explores recursive prediction and likelihood evaluation techniques for periodic autoregressive moving‐average (PARMA) time series models. The innovations algorithm is used to develop a simple recursive scheme for computing one‐step‐ahead predictors and their mean squared errors. The asymptotic form of this recursion is explored. The prediction results are then used to develop an efficient (and exact) PARMA likelihood evaluation algorithm for Gaussian series. We then show how a multivariate autoregressive moving average (ARMA) likelihood can be evaluated by writing the multivariate ARMA model in PARMA form. Explicit calculations for PARMA(1, 1) models and periodic autoregressions are included.

RECURSIVE COMPUTATION OF THE PARAMETERS OF PERIODIC AUTOREGRESSIVE MOVING‐AVERAGE PROCESSES

Abstract. An algorithm for recursive computation of the parameters of periodic autoregressive moving‐average (ARMA) processes is given. It also provides recursions for stationary multivariate ARMA processes. A procedure for simultaneous estimation of the order and the parameters of a periodic ARMA process is outlined.

Multivariate ARMA modeling by scalar algorithms

An algorithm for multichannel autoregressive moving average (ARMA) modeling which uses scalar computations only and is well suited for parallel implementation is proposed. The given ARMA process is converted to an equivalent scalar, periodic ARMA process. The scalar autoregressive (AR) parameters are estimated by first deriving a set of modified Yule-Walker-type equations and then solving them by a parallel, order recursive algorithm. The moving average (MA) parameters are estimated by a least squares method from the estimates of the input samples obtained via a high-order, periodic AR approximation of the scalar process.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Recursive Parameter Estimation for ARMA Simulations

The parameter estimation algorithms for the autoregressive moving average (ARMA) simulation of multivariate random processes is considered. The recursive procedure for estimating the parameters of multivariate AR models is extended to the parameter estimation of multivariate ARMA models of same order for both the AR and MA components, which are usually used in the ARMA simulation of stationary multivariate random processes. The recursive parameter estimation procedure for ARMA models relies on the knowledge of the input-output cross correlation of the model, which is obtained with a procedure from the two-stage least-squares method, in that an AR model of high order is used to estimate the cross correlation. A numerical example shows that the recursive procedure for multivariate ARMA models leads to good ARMA representation of random processes, which is characterized by a specified target spectrum.

MAXIMUM LIKELIHOOD ESTIMATORS IN THE MULTIVARIATE AUTOREGRESSIVE MOVING‐AVERAGE MODEL FROM A GENERALIZED LEAST SQUARES VIEWPOINT

Abstract. Explicit expressions are derived for the gradient vector and (approximate) Hessian matrix of the log likelihood function for the multivariate autoregressive moving‐average (ARMA) model. Based on these expressions an explicit description of the Gauss‐Newton iterative procedure to obtain maximum likelihood (ML) estimates of the parameters in the multivariate ARMA model is presented. The resulting computational procedure has the form of a generalized least squares (GLS) estimation involving lagged values of the observed vector series and of the residual series as independent variables. This direct form of the estimator is found to be appealing and useful in understanding and interpreting the ML estimation procedure from a regression point of view, and in comparing the ML procedure with other ‘linear’ estimation procedures that have recently been presented. Simulation results are also presented for a univariate and a multivariate ARMA model to illustrate the ML‐GLS estimation procedure and to compare it with other linear estimation procedures.

Computation of Theoretical Autocovariance Matrices of Multivariate Autoregressive Moving Average Time Series

SUMMARY Matrix expressions relating the theoretical autocovariances of autoregressive moving average (ARMA) processes to their parameters are derived and used to design an efficient procedure for computing autocovariance sequences of multivariate ARMA processes. The method proposed is more efficient than others suggested in the literature and, in particular, reduces the computational burden associated with exact maximum likelihood estimation of ARMA models. The closed form expressions facilitate the implementation of algorithms for computing multivariate autocovariances.

Multivariate Periodic ARMA(1,1) Processes

The main objective of this paper is to investigate the properties of multivariate periodic autoregressive moving average (ARMA)(l, 1) processes. Such properties include the covariance structure, the parameter space, and estimation. The parameter space of such periodic processes is derived by aggregation (for instance, monthly models will aggregate to annual models). In particular, the parameter space and estimation are analyzed for contemporaneous ARMA (CARMA) models. It is shown that in general, the aggregation of a multivariate periodic ARMA(1, 1) model leads to a stationary multivariate ARMA(1, 1) model. However, not always a periodic CARMA(1, 1) model aggregates into a stationary CARMA(1, 1) model. Furthermore, it is shown that in estimating the aggregated model parameters, the method based on the periodic model and corresponding parameters, is superior to that based on the aggregated series.

Multisite ARMA(1,1) and Disaggregation Models for Annual Streamflow Generation

Disaggregation and multisite autoregressive moving average (ARMA)(1,1) time‐series models provide simple and efficient frameworks for generation of multisite synthetic streamflow sequences that exhibit long‐term persistence. This paper considers multisite ARMA(1,1) models whose Φ and Θ matrices are diagonal; a Monte Carlo study examined the efficiency of three procedures for estimating individual Φ‐θ values for each site and two estimators of the covariance matrix of the innovations. Also included in the study was a univariate ARMA(1,1) model of the aggregate flows with a simple disaggregation algorithm to generate flows at the individual sites. In the realm of most hydrologic interest, simple diagonal multisite ARMAl(1,1) models performed adequately and it is not necessary to fit the more cumbersome nondiagonal models. The disaggregation procedure coupled with an ARMA(1,1) aggregate flow model did as well as the multivariate diagonal ARMA(1,1) models.

The Multivariate Portmanteau Statistic

Abstract Box and Pierce have derived a goodness-of-fit test, the portmanteau test, for univariate autoregressive moving-average (ARMA) time series models. This test is here extended to multivariate ARMA models; the test statistic may be conveniently expressed as a function of the covariances between the residuals of the fitted model. A modified form of the statistic designed to have superior properties in small samples is derived, and the two forms of the statistic are compared via computer simulation.

Parametric time series models for multivariate EEG analysis

The autoregressive (AR) and mixed autoregressive-moving average (AR-MA) parametric models of stationary time series are of current interest for the purposes of spectral analysis and for the extraction of features for automatic EEG classification. Procedures for computing AR models and a new two-stage least-squares procedure for computing AR-MA models of multivariate time series are shown. The results of spectral estimation of EEGs using the multivariate AR, AR-MA, and conventional windowed periodogram analysis are compared.