Autoregressive Moving-average Processes Research Articles

An alternative method for estimating Hurst exponent of control signals based on system dynamics

One of the indices for evaluating the performance of control systems based on the routine operation data is the Hurst index. It is shown that the Hurst index value is very close to that of the minimum variance index which measures the performance of a control loop in terms of minimum variance benchmark. The advantage of Hurst index over minimum variance index is that it does not require the system model and its parameters. The Hurst index is based on the Hurst exponent, which measures the time series correlations. This paper analytically scrutinizes the effect of parameters and dynamics of a stochastic control system on the output Hurst exponent, which helps better comprehend the Hurst index and its relationship with the minimum variance index. For this purpose, the control system output is modeled as an Autoregressive Moving Average (ARMA) process, and the relationship between the parameters of the ARMA process and the output Hurst exponent is examined. Furthermore, the characteristics of the autocorrelation function of a signal under minimum variance conditions are extracted. This paper also proposes a new technique for estimating the Hurst exponent of control signals based on the autocorrelation function. Simulation examples illustrate the efficiency of the proposed approaches.

The impact of long memory in mortality differentials on index-based longevity hedges

AbstractIn multi-population mortality modeling, autoregressive moving average (ARMA) processes are typically used to model the evolution of mortality differentials between different populations over time. While such processes capture only short-term serial dependence, it is found in our empirical work that mortality differentials often exhibit statistically significant long-term serial dependence, suggesting the necessity for using long memory processes instead. In this paper, we model mortality differentials between different populations with long memory processes, while preserving coherence in the resulting mortality forecasts. Our results indicate that if the dynamics of mortality differentials are modeled by long memory processes, mean reversion would be much slower, and forecast uncertainty over the long run would be higher. These results imply that the true level of population basis risk in index-based longevity hedges may be larger than what we would expect when ARMA processes are assumed. We also study how index-based longevity hedges should be calibrated if mortality differentials follow long memory processes. It is found that delta hedges are more robust than variance-minimizing hedges, in the sense that the former remains effective even if the true processes for mortality differentials are long memory ones.

Generalized gamma ARMA process for synthetic aperture radar amplitude and intensity data

AbstractWe propose a new autoregressive moving average (ARMA) process with generalized gamma (G) marginal law, called G‐ARMA. We derive some of its mathematical properties: moment‐based closed‐form expressions, score function, and Fisher information matrix. We provide a procedure for obtaining maximum likelihood estimates for the G‐ARMA parameters. Its performance is quantified and discussed using Monte Carlo experiments, considering (among others) various link functions. Finally, our proposal is applied to solve remote sensing problems using synthetic aperture radar (SAR) imagery. In particular, the G‐ARMA process is applied to real data from images taken in the Munich and San Francisco regions. The results show that G‐ARMA describes the neighborhoods of SAR features better than the gamma‐ARMA process (a reference for asymmetric positive data). For pixel ray modeling, our proposal outperforms and gamma‐ARMA.

Multivariate continuous-time autoregressive moving-average processes on cones

In this article we study multivariate continuous-time autoregressive moving-average (MCARMA) processes with values in convex cones. More specifically, we introduce matrix-valued MCARMA processes with Lévy noise and present necessary and sufficient conditions for processes from this class to be cone valued. We derive specific hands-on conditions in the following two cases: First, for classical MCARMA on Rd with values in the positive orthant Rd+. Second, for MCARMA processes on real square matrices assuming values in the cone of symmetric and positive semi-definite matrices. Both cases are relevant for applications and we give several examples of positivity ensuring parameter specifications. In addition to the above, we discuss the capability of positive semi-definite MCARMA processes to model the spot covariance process in multivariate stochastic volatility models. We justify the relevance of MCARMA based stochastic volatility models by an exemplary analysis of the second order moment structure of positive semi-definite well-balanced Ornstein–Uhlenbeck based models.

Model of rough surfaces with Gaussian processes

Surface roughness plays a critical role and has effects in, e.g. fluid dynamics or contact mechanics. For example, to evaluate fluid behavior at different roughness properties, real-world or numerical experiments are performed. Numerical simulations of rough surfaces can speed up these studies because they can help collect more relevant information. However, it is hard to simulate rough surfaces with deterministic or structured components in current methods. In this work, we present a novel approach to simulate rough surfaces with a Gaussian process (GP) and a noise model because GPs can model structured and periodic elements. GPs generalize traditional methods and are not restricted to stationarity so they can simulate a wider range of rough surfaces. In this paper, we summarize the theoretical similarities of GPs with auto-regressive moving-average processes and introduce a linear process view of GPs. We also show examples of ground and honed surfaces simulated by a predefined model. The proposed method can also be used to fit a model to measurement data of a rough surface. In particular, we demonstrate this to model turned profiles and surfaces that are inherently periodic.

Performance bound for myopic order-up-to inventory policies under stationary demand processes

We consider a single-item infinite-horizon inventory system operating in discrete time and study the performance of the popular myopic order-up-to policy when demand is driven by a general autoregressive moving average (ARMA) stationary process. We derive a suboptimality bound for a system that operates under full demand backlogging, linear holding and backordering costs, and a constant replenishment lead time. We illustrate our results for the case in which demand follows an ARMA(1,1) process, which includes two commonly used demand models, MA(1) and AR(1), as special cases.

Examining AGN UV/Optical Variability beyond the Simple Damped Random Walk

We present damped harmonic oscillator (DHO) light-curve modeling for a sample of 12,714 spectroscopically confirmed quasars in the Sloan Digital Sky Survey Stripe 82 region. DHO is a second-order continuous-time autoregressive moving-average process, which can be fully described using four independent parameters: a natural oscillation frequency (ω 0), a damping ratio (ξ), a characteristic perturbation timescale (τ perturb), and an amplitude for the perturbing white noise (σ ϵ ). The asymptotic variability amplitude of a DHO process is quantified by σ DHO—a function of ω 0, ξ, τ perturb, and σ ϵ . We find that both τ perturb and σ ϵ follow different dependencies with rest-frame wavelength (λ RF) on either side of 2500 Å, whereas σ DHO follows a single power-law relation with λ RF. After correcting for wavelength dependence, σ DHO exhibits anticorrelations with both the Eddington ratio and the black hole mass, while τ perturb—with a typical value of days in the rest frame—shows an anticorrelation with the bolometric luminosity. Modeling active galactic nuclei (AGN) variability as a DHO offers more insight into the workings of accretion disks close to the supermassive black holes at the center of AGN. The newly discovered short-term variability (characterized by τ perturb and σ ϵ ) and its correlation with bolometric luminosity pave the way for new algorithms that will derive fundamental properties (e.g., Eddington ratio) of AGN using photometric data alone.

Putting the Significance of Spectral Peaks on the Level: Implications for the 1470-Yr Peak in Greenland δ18O

Abstract Spectral analysis of the Greenland Ice Sheet Project 2 (GISP2) δ18O record has been interpreted to show a 1/(1470 yr) spectral peak that is highly statistically significant (p < 0.01). The presence of such a peak, if accurate, provides an important clue about the mechanisms controlling glacial climate. As is standard, however, statistical significance was judged relative to a null model, H0, consisting of an autoregressive order one process, AR(1). In this study, H0 is generalized using an autoregressive moving-average process, ARMA(p, q). A rule of thumb is proposed for evaluating the adequacy of H0 involving comparing the expected and observed variances of the logarithm of a spectral estimate, which are generally consistent insomuch as removal of the ARMA structure from a time series results in an approximately level spectral estimate. An AR(1), or ARMA(1, 0), process is shown to be an inadequate representation of the GISP2 δ18O structure, whereas higher-order ARMA processes result in approximately level spectral estimates. After suitably leveling GISP2 δ18O and accounting for multiple hypothesis testing, multitaper spectral estimation indicates that the 1/(1470 yr) peak is insignificant. The seeming prominence of the 1/(1470 yr) peak is explained as the result of evaluating a spectrum involving higher-order ARMA structure and the peak having been selected on the basis of its seeming anomalous. The proposed technique for evaluating the significance of spectral peaks is also applicable to other geophysical records. Significance Statement A suitable null hypothesis is necessary for obtaining accurate test results, but a means for evaluating the adequacy of a null hypothesis for a spectral peak has been lacking. A generalized null model is presented in the form of an autoregressive, moving-average process whose adequacy can be gauged by comparing the observed and expected variance of log spectral density. Application of the method to the GISP2 δ18O record indicates that spectral structure found at 1/(1470 yr) is statistically insignificant.

Generalized maximum entropy based identification of graphical ARMA models

This paper focuses on the joint estimation of parameters and topologies of multivariate graphical autoregressive moving-average (ARMA) processes. Since the graphical structure of a model can be characterized by the sparsity pattern of its inverse spectral density matrix, a generalized maximum entropy optimization model is derived by imposing a sparse regularization on the inverse spectrum from which the parameters of the AR part and the graph topology can be simultaneously estimated. Then, the whole graphical ARMA model is identified by alternatingly estimating the graphical AR part by solving the regularized maximum entropy problem and the MA part by applying the moment estimation method. The weight to the sparsity regularization term is chosen according to the information theoretic model selection criterion, and simulation examples are provided to illustrate the effectiveness of the proposed identification approach.

On the Paradox of the Duality of Autoregressive and Moving Average Processes

A widely held view in time series analysis is the concept of duality that a finite order stationary autoregressive process of order p (AR(p)) is equivalent to an infinite order moving average (MA) process and a finite order invertible moving average of order q (MA(q)) is equivalent to an infinite order autoregressive (AR) process. The purpose of this paper is to demonstrate that the concept is not universally true. Thus, a finite order stationary autoregressive process of order p (AR(p)) can be written as an finite order moving average process and a finite order moving average process of order q (MA(q)) can be written as a finite order stationary autoregressive process. The regions of breakdown of concept of duality were determined for p = q = 1,2 using method of moments. The method involves equating non-zero autocovariances of the stationary AR(p) to the equivalent non-zero autocovariances of the invertible MA(p) to determine the region of non-duality. In such region of breakdown in duality, 1) both the Autocorrelation function and the Partial Autocorrelation function of the AR process and MA process cuts off after equal lags 2) a finite AR model can be adequately represented by a finite MA model of equal order and conversely with the same error variance and 3) negative values of the parameters of the AR process are equal in magnitude but opposite in direction to the parameters of the equivalent MA process and conversely. Empirical examples (simulation and real life examples) were used to illustrate these. Therefore, it has been recommended that caution should be exercised in using the concept of duality in time series analysis until future research proves otherwise.

The bullwhip effect in a manufacturing/remanufacturing supply chain under a price-induced non-standard ARMA(1,1) demand process

This study examines the bullwhip effect in a manufacturing/remanufacturing supply chain consisting of one retailer, one manufacturer, one supplier, and one third-party collector. Products returned from the end users are converted to ‘as-good-as-new’ products through remanufacturing. Both manufactured and remanufactured products are used to meet the stochastic demand of the retailer. We incorporate the price of the product in the demand through an autoregressive moving average (ARMA) pricing process, realizing a market demand that evolves according to a non-standard ARMA(1,1) process. For this demand model, we employ the order-up-to inventory policy with a minimum mean square error forecasting scheme and measure the bullwhip effect at each echelon of the supply chain. We investigate the behaviour of the bullwhip effect analytically, and a numerical evaluation shows that the bullwhip effect or the anti-bullwhip effect may occur at each echelon depending on the values of the autoregressive and moving average parameters of the underlying demand process. We also investigate the effects of the remanufacturing yield, the variance of the market shock on price, and the lead times on the bullwhip effect.

Post-Processing of High Formwork Monitoring Data Based on the Back Propagation Neural Networks Model and the Autoregressive—Moving-Average Model

Many high formwork systems are currently equipped with health monitoring systems, and the analysis of the data obtained can determine whether high formwork is a hazard. Therefore, the post-processing of monitoring data has become an issue of widespread concern. In this paper, we discussed the fitting effect of the symmetrical high formwork monitoring data using the autoregressive–moving-average (ARMA) model and the back propagation neural networks (BPNN) combined model to process. In the actual project, the symmetry of the high formwork system allows the analysis of local monitoring results to be well extended to the whole. For the establishment of the ARMA model, the accurate judgment of the model order has a significant impact. In this paper, back propagation neural networks (BPNN) are used to simulate the ARMA process. The order of the ARMA model is estimated by determining the optimal neural network structure, which is suitable for linear or nonlinear sequences. We validated this approach from the ARMA model data simulated in Monte Carlo and compared it with the Akaike information criterion (AIC) and Bayesian information criterion (BIC). The length of the sequence, the coefficients and the order of the ARMA model are considered as factors that influence the judgment effect. Under different conditions, the BPNN always shows an accuracy rate of more than 90%, while the BIC only has a higher accuracy rate when the model order is low and the judgment efficiency of the AIC is below 50%. Finally, the proposed method successfully modeled the stress sequence and obtained the stress change trend. Compared with AIC and BIC, the efficiency of the processing time series is increased by about 50% when an order is obtained by BPNN.

SPHARMA approximations for stationary functional time series on the sphere

In this paper, we focus on isotropic and stationary sphere-cross-time random fields. We first introduce the class of spherical functional autoregressive-moving average processes (SPHARMA), which extend in a natural way the spherical functional autoregressions (SPHAR) recently studied in Caponera and Marinucci (Ann Stat 49(1):346–369, 2021) and Caponera et al. (Stoch Process Appl 137:167–199, 2021); more importantly, we then show that SPHAR and SPHARMA processes of sufficiently large order can be exploited to approximate every isotropic and stationary sphere-cross-time random field, thus generalizing to this infinite-dimensional framework some classical results on real-valued stationary processes. Further characterizations in terms of functional spectral representation theorems and Wold-like decompositions are also established.

ARMA process for speckled data

Synthetic aperture radar (SAR) systems are efficient to deal with remote sensing issues. In contrast, SAR images are affected by speckle noise, due to the use of coherent illumination in their capturing. This noise imposes both a granular interference on such images (precluding their interpretability) and a multiplicative and non-Gaussian nature on their data. The multiplicative modelling is often used to surpass previous difficulties, mainly its particular case the distribution. In this paper, we introduce a new time series modelling for SAR imagery, called autoregressive-moving-average (ARMA) process. We derive some of its mathematical properties: score vector, Fisher information matrix, residual analysis and prediction equations. The maximum likelihood estimation procedure for -ARMA parameters is discussed and some asymptotic behaviours for its estimates are quantified by Monte Carlo experiments. Applications to Munich and Foulum SAR actual images are made. Results show the proposed model can outperform the Γ-ARMA model.

AR and ARMA model order selection for time-series modeling with ImageNet classification

We propose model order selection methods for autoregressive (AR) and autoregressive moving average (ARMA) time-series modeling based on ImageNet classifications with a 2-dimensional convolutional neural network (2-D CNN). We designed two models for two realistic scenarios: (1) a general model which emulates the scenario that validation and test datasets do not necessarily have the same dynamics as the training data, (2) a specific model which emulates the opposite scenario—the validation and test datasets share the dynamics of the training data. The results were compared to those of both Akaike Information criterion (AIC) and Bayesian Information criterion (BIC). Using simulation examples, we trained 2-D CNN-based Inception-v3 and ResNet50-v2 models for either AR or ARMA order selection for each of the two scenarios. The proposed ResNet50-v2 to use both time-frequency and the original time series data outperformed AIC and BIC for all scenarios. For the general model, the average of relative error reduction (ARER) when compared to the BIC method in the clean and three noisy environments was 19.07% (±14.22%) for the AR order for an AR process, and 5.67% (±2.83%) for the ARMA order for an ARMA process. The ARERs significantly improved to 73.92% (±30.95%) and 65.58% (±38.61%) for the AR and ARMA models, respectively, for the specific model scenario.

Continuous Time Autoregressive Moving Average Processes Driven by Semi-Levy Process

Continuous Time Autoregressive Moving Average Processes Driven by Semi-Levy Process

Identification of the components of the drinking water model: The use of ARMA models

In this paper, we consider the modeling of time series corresponding to the methodology of Box and Jenkins for building forecasting modelling from ARMA (Auto Regressive Moving Average)processes. This involves the analysis and forecasting of drinking water production data. The purpose of this study is to modelling a time series of drinking water production data according to its past and present values in order to determine the adequate ARMA process by the principle of parsimony. Using the Eviews for Windows software, the results show that the exploration of monthly drinking water consumption data provided by the Biskra Management Authority (A.D.E) during the period from January2009 to January2016, Revealed characteristics such as the non-stochastic non-stationarity of the drinking water production series.

Time-series modeling of fishery landings in the Colombian Pacific Ocean using an ARIMA model

Seer fish (Scomberomorus sierra) and mullet (Mugil cephalus) are some of the most important marine fishery resources along the Colombian Pacific Ocean. The objective of this study was to forecast the landings of seer fish and mullet based on data from time-series annual landings reported by the Food and Agriculture Organization of the United Nations (FAO) from 1971 to 2014. The study considered autoregressive integrated moving-average (ARIMA) processes to forecast the landings of the species. The ARIMA model (5,1,5) for seer fish and ARIMA model (2,2,1) for mullet showed good agreement concerning the observed data on landings based on the Akaike information criterion. The results show the ARIMA model to be a suitable method for analyzing statistics. In data-poor fisheries situations, this method can support potential evaluations of fishery production for decision making and management.

An adaptive variable-parameters scheme for the simultaneous monitoring of the mean and variability of an autocorrelated multivariate normal process

ABSTRACTDue to advances in technology, sampling procedures and short lag times between successive sampling, autocorrelation among the measured data has become common in most applications. Neglecting autocorrelation leads to a poor false alarm performance. In the current paper, the effect of the autocorrelation on the performance of a variable-parameters multivariate single control chart is investigated in the case of the simultaneous monitoring of the mean and variability. At first, formulas for the sample mean and variability of a multivariate autoregressive-moving average process are derived. Then, a variable-parameters single control chart is developed for the simultaneous monitoring of the mean vector and the covariance matrix of an autocorrelated multivariate normal process. Next, the performance of the proposed control chart is evaluated by using eight performance measures based on a dedicated Markov chain model. Finally, by presenting an illustrative example, the application of the proposed scheme is demonstrated in practice.

An empirical study on the parsimony and descriptive power of TARMA models

In linear time series analysis, the incorporation of the moving-average term in autoregressive models yields parsimony while retaining flexibility; in particular, the first order autoregressive moving-average model, ARMA(1,1) is notable since it retains a good approximating capability with just two parameters. In the same spirit, we assess empirically whether a similar result holds for threshold processes. First, we show that the first order threshold autoregressive moving-average process, TARMA(1,1) exhibits complex, high-dimensional, behaviour with parsimony, by comparing it with threshold autoregressive processes, TAR(p), with possibly large autoregressive order p. Second, we study the descriptive power of the TARMA(1,1) model with respect to the class of autoregressive models, seen as universal approximators: in several situations, the TARMA(1,1) model outperforms AR(p) models even when p is large. Lastly, we analyze two real world data sets: the sunspot number and the male US unemployment rate time series. In both cases, we show that TARMA models provide a better fit with respect to the best TAR models proposed in literature.