Vector Autoregressive Moving Average Models Research Articles

An Interpretable and Efficient Infinite-Order Vector Autoregressive Model for High-Dimensional Time Series

As a special infinite-order vector autoregressive (VAR) model, the vector autoregressive moving average (VARMA) model can capture much richer temporal patterns than the widely used finite-order VAR model. However, its practicality has long been hindered by its non-identifiability, computational intractability, and difficulty of interpretation, especially for high-dimensional time series. This article proposes a novel sparse infinite-order VAR model for high-dimensional time series, which avoids all above drawbacks while inheriting essential temporal patterns of the VARMA model. As another attractive feature, the temporal and cross-sectional structures of the VARMA-type dynamics captured by this model can be interpreted separately, since they are characterized by different sets of parameters. This separation naturally motivates the sparsity assumption on the parameters determining the cross-sectional dependence. As a result, greater statistical efficiency and interpretability can be achieved with little loss of temporal information. We introduce two l 1 -regularized estimation methods for the proposed model, which can be efficiently implemented via block coordinate descent algorithms, and derive the corresponding nonasymptotic error bounds. A consistent model order selection method based on the Bayesian information criteria is also developed. The merit of the proposed approach is supported by simulation studies and a real-world macroeconomic data analysis. Supplementary materials for this article are available online including a standardized description of the materials available for reproducing the work.

EPCMSDB: Design of an ensemble predictive control model for solar PV MPPT deployments via dual bioinspired optimizations

With the increasing demand for renewable energy, solar power has emerged as a promising option for sustainable power generation. However, the effectiveness and efficiency of solar power systems rely on the ability to effectively manage their performance, making it essential to develop efficient control models. This paper proposes a novel ensemble predictive control model for solar deployments using bio-inspired optimizations to improve load-connected solar deployments’ performance. The proposed model integrates multiple control devices, including Maximum Power Point Tracker, Proportional-Integral-Derivative, Proportional-Integral, and Fuzzy Logic Controllers, to selectively control the solar Photovoltaic systems. The proposed model incorporates a predictive control operation utilizing an LSTM-GRU (Long Short-Term Memory-Gated Recurrent Unit) with the VARMA (Vector Auto-Regressive Moving Average) model, which can accurately predict the future power generation of the solar system. This feature can facilitate efficient energy management and increase the system’s performance for different use cases. Implement a SEPIC (Single Ended Primary Inductor Capacitor) converter design to improve the system’s overall efficiency levels. To validate the effectiveness of the proposed approach, the author conducted experiments using real-world data and compared the proposed results with other control strategies. The results demonstrate that the ensemble predictive control model based on bio-inspired optimizations outperforms the existing control models regarding accuracy, efficiency, and stability levels. The proposed model has the potential to significantly improve the performance of load-connected solar deployments, offering a more practical approach to solar power generation. The combination of predictive control operations with bio-inspired optimizations can facilitate the design of sustainable energy systems with higher efficiency and accuracy.

Forecasting Parameters in the SABR Model

<p>We present two approaches to forecasting parameters in the SABR model. The first approach is the vector autoregressive moving-average model (VARMA) for the time series of the in-sample calibrated parameters, and the second is based on machine learning techniques called epsilon-support vector regression (ε-SVR). Using daily data of S&P 500 ETF option prices from January 1, 2014, to December 31, 2018, we first calibrate the daily values of the model parameters from the training sample, then conduct out-of-sample forecasting of parameters and pricing of options. Both approaches produce good fits between the forecasted and calibrated parameters for out-of-sample dates. A comparison study shows that using forecasted parameters as inputs, the SABR model generates better pricing results than assuming constant parameters or using lag parameters. We also discuss the market conditions under which one approach outperforms the other.</p>

Multivariate time series modelling for urban air quality

We introduce a spatio-temporal model to represent development of atmospheric pollution in an urban area. An important element of this is that recorded measurements are often incomplete which undermines time-series approaches. We identify the multiple imputation by chained equation (MICE) method as effective to complete data sequences synthetically. Following on from this, we develop a vector autoregressive moving average (VARMA) model for the spatio-temporal development of atmospheric pollution in urban areas. This model was fitted to hourly measurements of four pollutants (NO, NO2, NOx and PM10) for the whole of the calendar year 2017 at 30 stations across London, completed by MICE as required. We show by cross-validation that the VARMA model is more effective than other formulations, including the Kriging method of spatial interpolation, and seasonal ARMA models for individual stations with either daily or weekly trends. The resulting model can be used for prediction of air quality in different periods and as the basis for assessment of policy interventions such as increasing vehicle emission standards, and traffic management and control policies such as low and ultra-low emission zones.

COVID-19 incidence and mortality in Nigeria: gender based analysis.

BackgroundCoronavirus Disease 2019 (COVID-19) has been surging globally. Risk strata in medical attention are of dynamic significance for apposite assessment and supply distribution. Presently, no known cultured contrivance is available to fill this gap of this pandemic. The aim of this study is to develop a predictive model based on vector autoregressive moving average (VARMA) model of various orders for gender based daily COVID-19 incidence in Nigeria. This study also aims to proffer empirical evidence that compares incidence between male and female for COVID-19 risk factors.MethodsWilcoxon signed-rank test is employed to investigate the significance of the difference in the gender distributions of the daily incidence. A VARMA model of various orders is formulated for the gender based daily COVID-19 incidence in Nigeria. The optimal VARMA model is identified using Bayesian information criterion. Also, a predictive model based on univariate autoregressive moving average model is formulated for the daily death cases in Nigeria. Fold change is estimated based on crude case-fatality risk to investigate whether there is massive underreporting and under-testing of COVID-19 cases in Nigeria.ResultsDaily incidence is higher in males on most days from 11 April 2020 to 12 September 2020. Result of Wilcoxon signed-rank test shows that incidence among male is significantly higher than female (p-value < 2.22 × 10−16). White neural network test shows that daily female incidence is not linear in mean (p-value = 0.00058746) while daily male incidence is linear in mean (p-value = 0.4257). McLeod-Li test shows that there is autoregressive conditional heteroscedasticity in the female incidence (Maximum p-value = 1.4277 × 10−5) and male incidence (Maximum p-value = 9.0816 × 10−14) at 5% level of significance. Ljung-Box test (Tsay, 2014) shows that the daily incidence cases are not random (p-value=0.0000). The optimal VARMA model for male and female daily incidence is VARMA (0,1). The optimal model for the Nigeria’s daily COVID-19 death cases is identified to be ARIMA (0,1,1). There is no evidence of massive underreporting and under-testing of COVID-19 cases in Nigeria.ConclusionsComparison of the observed incidence with fitted data by gender shows that the optimal VARMA and ARIMA models fit the data well. Findings highlight the significant roles of gender on daily COVID-19 incidence in Nigeria.

Spatial transmission network construction of influenza-like illness using dynamic Bayesian network and vector-autoregressive moving average model

BackgroundAlthough vaccination is one of the main countermeasures against influenza epidemic, it is highly essential to make informed prevention decisions to guarantee that limited vaccination resources are allocated to the places where they are most needed. Hence, one of the fundamental steps for decision making in influenza prevention is to characterize its spatio-temporal trend, especially on the key problem about how influenza transmits among adjacent places and how much impact the influenza of one place could have on its neighbors. To solve this problem while avoiding too much additional time-consuming work on data collection, this study proposed a new concept of spatio-temporal route as well as its estimation methods to construct the influenza transmission network.MethodsThe influenza-like illness (ILI) data of Sichuan province in 21 cities was collected from 2010 to 2016. A joint pattern based on the dynamic Bayesian network (DBN) model and the vector autoregressive moving average (VARMA) model was utilized to estimate the spatio-temporal routes, which were applied to the two stages of learning process respectively, namely structure learning and parameter learning. In structure learning, the first-order conditional dependencies approximation algorithm was used to generate the DBN, which could visualize the spatio-temporal routes of influenza among adjacent cities and infer which cities have impacts on others in influenza transmission. In parameter learning, the VARMA model was adopted to estimate the strength of these impacts. Finally, all the estimated spatio-temporal routes were put together to form the final influenza transmission network.ResultsThe results showed that the period of influenza transmission cycle was longer in Western Sichuan and Chengdu Plain than that in Northeastern Sichuan, and there would be potential spatio-temporal routes of influenza from bordering provinces or municipalities into Sichuan province. Furthermore, this study also pointed out several estimated spatio-temporal routes with relatively high strength of associations, which could serve as clues of hot spot areas detection for influenza surveillance.ConclusionsThis study proposed a new framework for exploring the potentially stable spatio-temporal routes between different places and measuring specific the sizes of transmission effects. It could help making timely and reliable prediction of the spatio-temporal trend of infectious diseases, and further determining the possible key areas of the next epidemic by considering their neighbors’ incidence and the transmission relationships.

An indirect proof for the asymptotic properties of VARMA model estimators

Strong consistency and asymptotic normality of a Gaussian quasi-maximum likelihood estimator for the parameters of a causal, invertible, and identifiable vector autoregressive-moving average (VARMA) model are established in an indirect way. The proof is based on similar results for a much wider class of VARMA models with time-dependent coefficients, hence in the context of non-stationary and heteroscedastic time series. For that reason, the proof avoids spectral analysis arguments and does not make use of ergodicity. The results presented are also applicable to ARMA models.

IoT-Enabled Particulate Matter Monitoring and Forecasting Method Based on Cluster Analysis

In recent years, particulate matter (PM) having a diameter smaller than 2.5 μm has become a significant issue due to its severe impact on human health. With the advent of IoT-enabling technologies, a ubiquitous IoT sensing infrastructure is now used to constantly monitor aspects of our surrounding environment, such as ambient air pollution. In this article, we introduce a PM-sensing system composed of off-the-shelf LoRa-based wireless hardware boards and low-cost PM sensors. By leveraging software platforms that are compliant with an IoT standard called oneM2M, PM data sets can be collected and accessed in a standardized manner, i.e., via oneM2M-defined representational state transfer application programmable interfaces. Also, for reliable PM monitoring, a short-term (i.e., within 2 h) PM forecasting method based on autoregressive integrated moving average and vector autoregressive moving average (VARMA) models is proposed and evaluated with a 30-day PM data set collected from 15 LoRa-based PM sensor nodes installed at a university campus. The experimental results show that the overall root-mean square error and correlation coefficient of the VARMA models integrated with hierarchical clustering are improved by 7.77% and 3.7%, respectively, compared with the single node-based forecast model.

Evolution and forecasting of PM10 concentration at the Port of Gijon (Spain)

The name PM10 refers to small particles with a diameter of less than 10 microns. The present research analyses different models capable of predicting PM10 concentration using the previous values of PM10, SO2, NO, NO2, CO and O3 as input variables. The information for model training uses data from January 2010 to December 2017. The models trained were autoregressive integrated moving average (ARIMA), vector autoregressive moving average (VARMA), multilayer perceptron neural networks (MLP), support vector machines as regressor (SVMR) and multivariate adaptive regression splines. Predictions were performed from 1 to 6 months in advance. The performance of the different models was measured in terms of root mean squared errors (RMSE). For forecasting 1 month ahead, the best results were obtained with the help of a SVMR model of six variables that gave a RMSE of 4.2649, but MLP results were very close, with a RMSE value of 4.3402. In the case of forecasts 6 months in advance, the best results correspond to an MLP model of six variables with a RMSE of 6.0873 followed by a SVMR also with six variables that gave an RMSE result of 6.1010. For forecasts both 1 and 6 months ahead, ARIMA outperformed VARMA models.

Nonparametric estimate of spectral density functions of sample covariance matrices generated by VARMA models

The density function of the limiting spectral distribution(LSD) of sample covariance matrices is widely used in large scale statistical inference when the sample size and dimension both tend to infinity. However, there are no explicit expressions for the density function generated by vector autoregressive moving average(VARMA) models. For such models whose sample covariance matrices do not have independence structure in columns, we propose to use modified kernel estimators which are proved to be consistent. A simulation study is also conducted to show the performance of the estimators.

ECM Algorithm for Estimating Vector ARMA Model with Variance Gamma Distribution and Possible Unbounded Density

The simultaneous analysis of several financial time series is important in portfolio setting and risk management. This paper proposes a novel alternating Expectation conditional Maximisation (AECM) algorithm to estimate the vector autoregressive moving average (VARMA) model with variance gamma (VG) error distribution in the multivariate skewed setting. We explain why VARMA-VG model is suitable for high frequency returns (HFRs) because VG distribution provides thick tails to capture the high kurtosis in the data and the unbounded central density further captures the majority of near zero HFRs. The distribution can also be expressed in normal mean-variance mixtures which facilitate model implementation using Bayesian and expectation maximisation (EM) approaches. We adopt the EM approach which avoids the time consuming Markov chain Monto Carlo sampling and solve the unbounded density problem in the classical maximum likelihood estimation. We discuss some properties of VARMA model, conduct extensive simulation studies to evaluate the accuracy of the proposed AECM estimator and apply the models to analyse the dependency between several HFR series.

Projected Climatic and Hydrologic Changes to Lake Victoria Basin Rivers under Three RCP Emission Scenarios for 2015–2100 and Impacts on the Water Sector

Rivers in the Lake Victoria Basin support a multitude of ecosystem services, and the economies of the riparian countries (Kenya, Tanzania, Uganda, Rwanda, and Burundi) rely on their discharge, but projections of their future discharges under various climate change scenarios are not available. Here, we apply Vector Autoregressive Moving Average models with eXogenous variables (VARMAX) statistical models to project hydrological discharge for 23 river catchments for the 2015–2100 period, under three representative concentration pathways (RCPs), namely RCPs 2.6, 4.5, and 8.5. We show an intensification of future annual rainfall by 25% in the eastern and 5–10% in the western part of the basin. At higher emission scenarios, the October to December season receives more rainfall than the March to May season. Temperature projections show a substantial increase in the mean annual minimum temperature by 1.3–4.5 °C and warming in the colder season (June to September) by 1.7–2.9 °C under RCP 4.5 and 4.9 °C under RCP 8.5 by 2085. Variability in future river discharge ranges from 5–267%, increases with emission intensity, and is the highest in rivers in the southern and south eastern parts of the basin. The flow trajectories reveal no systematic trends but suggest marked inter-annual variation, primarily in the timing and magnitude of discharge peaks and lows. The projections imply the need for coordinated transboundary river management in the future.

MODELING AND FORECASTING BY THE VECTOR AUTOREGRESSIVE MOVING AVERAGE MODEL FOR EXPORT OF COAL AND OIL DATA (CASE STUDY FROM INDONESIA OVER THE YEARS 2002-2017)

The vector autoregressive moving average (VARMA) model is one of the statistical analyses frequently used in several studies of multivariate time series data in economy, finance, and business. It is used in numerous studies because of its simplicity. Moreover, the VARMA model can explain the dynamic behavior of the relationship among endogenous and exogenous variables or among endogenous variables. It can also explain the impact of a variable or a set of variables by means of the impulse response function and Granger causality. Furthermore, it can be used to predict and forecast time series data. In this study, we will discuss and develop the best model that describes the relationship between two vectors of time series data export of Coal and data export of Oil in Indonesia over the period 2002-2017. Some models will be applied to the data: VARMA (1,1), VARMA (2,1), VARMA (3,1), and VARMA (4,1). On the basis of the comparison of these models using information criteria AICC, HQC, AIC, and SBC, it was found that the best model is VARMA (2,1) with restriction on some parameters: AR2_1_2 = 0, AR2_2_1 = 0, and MA1_2_1 = 0. The dynamic behavior of the data is studied through Granger causality analysis. The forecasting of the series data is also presented for the next 12 months.

Non-Gaussian VARMA model with stochastic volatility and applications in stock market bubbles

Abstract In order to analyze the stock market bubble phenomenon, the vector autoregressive moving average (VARMA) model with non-Gaussian innovations and stochastic volatility components (VARMA-t-SV) is constructed for financial modeling. Considering the estimation complexity of VARMA-t-SV model, the Kronecker structure of likelihood function is employed to speed up computation. Then we develop the corresponding Markov chain Monte Carlo (MCMC) sampling method to test the covariance structure specifications. Model comparisons illustrate that the VARMA model with flexible covariance structures perform better performances. The model parameter estimation results show that the fat tail and the heteroscedasticity features are useful in raising the performances compared to the standard form. Finally, using Chinese financial markets data, the effects of monetary policy on stock market bubbles are analyzed based on the VARMA-t-SV model. The empirical studies provide evidence to support the rational asset price bubble theory, namely, the tightening monetary policy may not succeed in shrinking the asset price bubble, which provides suggestions for regulators and investors.

A hybrid ARIMA–SVM model for the study of the remaining useful life of aircraft engines

In this research, an algorithm is presented for predicting the remaining useful life (RUL) of aircraft engines from a set of predictor variables measured by several sensors located in the engine. RUL prediction is essential for the safety of those aboard, but also to reduce engine maintenance and repair costs. The algorithm combines time series analysis methods to forecast the values of the predictor variables with machine learning techniques to predict RUL from those variables. First, an auto-regressive integrated moving average (ARIMA) model is used to estimate the values of the predictor variables in advance. Then, we use the result of the previous step as the input of a support vector regression model (SVM), where RUL is the response variable. The validity of the method was checked on an extensive public database, and the results compared with those obtained using a vector auto-regressive moving average (VARMA) model. Our algorithm showed a high prediction capability, far greater than that provided by the VARMA model. • A method to forecast the remaining useful life of aircraft engines is proposed. • The predictor variables were obtained from sensors located in the engine. • The proposed method combines ARIMA and SVM models. • Results of our method unsurpassed those obtained using a VARMA model.

Online learning for vector autoregressive moving-average time series prediction

Abstract Multivariate time series analysis considers simultaneously multiple time series, which is much more complicated than the univariate time series analysis in general. VARMA (vector autoregressive moving-average) is one of the most mainstream multivariate time series model for time series prediction. However, the parameters of VARMA are often estimated in a batch manner in traditional multivariate statistical analysis and the noise terms are assumed Gaussian. In addition, the batch methods cannot perform satisfactorily in real-time prediction and the noise terms are unknown to us in real world. In this paper, we propose a novel online time series prediction framework for VARMA. We prove that VAR (vector autoregressive) can be used to mimic the underlying VARMA model under the online settings. Under this framework, we develop two effective algorithms VARMA-OGD and VARMA-ONS for this prediction problem assuming that the noise terms are generated stochastically and independently. The VARMA-OGD algorithm is based on the OGD (online gradient descent) algorithm, which is valid for general convex loss function. While the VARMA-ONS algorithm adopting the ONS (online newton step) algorithm is only valid for exp-concave loss function. Theoretical analysis shows that the regret bounds against the best VARMA prediction in hindsight of the proposed algorithms are sublinear of the number of samples. Furthermore, our experimental results further validate the effectiveness and robustness of our algorithms.

Estimation and forecasting in vector autoregressive moving average models for rich datasets

We address the issue of modelling and forecasting macroeconomic variables using rich datasets by adopting the class of Vector Autoregressive Moving Average (VARMA) models. We overcome the estimation issue that arises with this class of models by implementing an iterative ordinary least squares (IOLS) estimator. We establish the consistency and asymptotic distribution of the estimator for weak and strong VARMA(p,q) models. Monte Carlo results show that IOLS is consistent and feasible for large systems, outperforming the MLE and other linear regression based efficient estimators under alternative scenarios. Our empirical application shows that VARMA models are feasible alternatives when forecasting with many predictors. We show that VARMA models outperform the AR(1), ARMA(1,1), Bayesian VAR, and factor models, considering different model dimensions.

Dirichlet ARMA models for compositional time series

A compositional time series is a multivariate time series in which the observation vector at each time point is a set of proportions that sum to 1. Traditionally, such time series are modeled by taking a log-ratio transformation of the observations and then modeling them with a Gaussian vector autoregressive moving average (ARMA) model. In this paper, a new class of models is proposed by assuming that the proportions follow a time-varying Dirichlet distribution, and that the corresponding time-varying parameters, after a proper transformation, assume an ARMA-type of dynamic structure. The new model is referred to as the Dirichlet autoregressive moving average (DARMA) model. Under this model, after a proper transformation, the original data follow a vector ARMA model with a martingale difference sequence as its noise series. Two specific transformations are studied under the DARMA framework. Estimation procedures are developed and their numerical properties are investigated. Simulation studies and real examples are presented to demonstrate the properties of the proposed models, and comparisons are made with the existing modeling approaches.

Estimation and Forecasting in Vector Autoregressive Moving Average Models for Rich Datasets

We address the issue of modelling and forecasting macroeconomic variables using rich datasets, by adopting the class of Vector Autoregressive Moving Average (VARMA) models. We overcome the estimation issue that arises with this class of models by implementing an iterative ordinary least squares (IOLS) estimator. We establish the consistency and asymptotic distribution of the estimator for strong and weak VARMA(p,q) models. Monte Carlo results show that IOLS is consistent and feasible for large systems, outperforming the MLE and other linear regression based efficient estimators under alternative scenarios. Our empirical application shows that VARMA models are feasible alternatives when forecasting with many predictors. We show that VARMA models outperform the AR(1), BVAR and factor models, considering different model dimensions.

DYNAMIC ECONOMIC RELATIONS AMONG STEEL PRICE INDICES

Bulk shipping is now a globalized industry, with market prices entirely dependent on the world's economic climate. For the bulk shipping market, the steel price index is a crucial standard of reference. In a global industry, the ability to understand the international relationship between changes in information and their effects on economics can help to reduce losses caused by the uncertainty of risk. Because the types of ships and contracts used in bulk shipping are wide-ranging, market participants tend to have difficulty in determining market price changes and trends, and are therefore forced to face greater market uncertainty and volatility. Because steel price is one of the leading indicators used in the bulk shipping market, a clear understanding of steel price index fluctuations can help to reduce the risks associated with bulk shipping. This study uses the vector autoregressive moving-average model as the principle method of analysis to determine how three major economic variables representative of Asian Steel Price Index (ASI), European Steel Price Index (ESI), and North American Steel Price Index (NASI) might influence one another. The results are as follows: (1) ASI is affected by its own positive changes as well as those of NASI, whereas the ESI is only affected by its own moving average. (2) The ASI and NASI are considered as leading indicators. This study aims to provide bulk shipping participants with additional ideas to determine the ideal time to enter the market.