A Portmanteau Test for ARMA Processes with Infinite Variance

The methods behind the predefined impulse response function in continuous time (PIRFICT) time series model are extended to cover more complex situations where multiple stresses influence ground water head fluctuations simultaneously. In comparison to autoregressive moving average (ARMA) time series models, the PIRFICT model is optimized for use on hydrologic problems. The objective of the paper is twofold. First, an approach is presented for handling multiple stresses in the model. Each stress has a specific parametric impulse response function. Appropriate impulse response functions for other stresses than precipitation are derived from analytical solutions of elementary hydrogeological problems. Furthermore, different stresses do not need to be connected in parallel in the model, as is the standard procedure in ARMA models. Second, general procedures are presented for modeling and interpretation of the results. The multiple-input PIRFICT model is applied to two real cases. In the first one, it is shown that this model can effectively decompose series of ground water head fluctuations into partial series, each representing the influence of an individual stress. The second application handles multiple observation wells. It is shown that elementary physical knowledge and the spatial coherence in the results of multiple wells in an area may be used to interpret and check the plausibility of the results. The methods presented can be used regardless of the hydrogeological setting. They are implemented in a computer package named Menyanthes (www.menyanthes.nl).

This study presents a stochastic approach for simulating earthquake ground motions using time-varying ARMA models with parameters fitted to real accelerograms. The ARMA model identification relies on the autocorrelation function (ACF), partial autocorrelation function (PACF), and the Akaike Information Criterion (AIC) to ensure optimal model order selection. The maximum likelihood technique is applied for parameter estimation. Near-field earthquakes recorded on dense soils in Algeria Boumerdes (2003) are synthesized using a moving time-window technique. Random sets of synthetic earthquakes are generated for each event to establish statistically valid structural response spectra. Key damage predictors, including peak linear displacement, ductility demand, and hysteretic energy, are computed from the mean and variance of response spectral ordinates and compared with spectra based on single earthquake records. Results show that low-order ARMA (2,1) models, excited by Gaussian white noise and amplitude-modulated with a simple envelope function, reliably reproduce both the nonstationary amplitude and frequency content of real earthquake records. This highlights the effectiveness of the proposed methodology for seismic motion simulation and structural damage assessment.

Impact of solar activity on precipitation in the United States

Three important parametric models for describing the correlation functions and spectra of stationary stochastic processes are the autoregressive (AR), moving average (MA), and autoregressive-moving average (ARMA) models. Quite recently, the MATLAB toolbox “ARMASA” has been made publicly available. This toolbox provides state-of-the-art algorithms to perform automatic identification and selection between the models based on the estimated prediction error. ARMASA works on a single segment of data, whereas in some applications, the data are available as multiple segments. We could process each segment independently and average the estimated autocorrelation functions or spectra afterward. Better performance, however, can be expected when all segments are processed simultaneously, for two reasons. Initially, the bias in the estimated model parameters depends on the number of observations in a segment. Averaging models from segments does not reduce this bias, in contrast to identification from all the data at once. In either case, the variance of the estimated model parameters will decrease. By considering all the data at once this variance reduction can be considered in the order selection stage, decreasing the selection bias. In this paper, we extend the parameter estimation algorithms of the ARMASA toolbox to the case of multiple segments. AR parameter estimation from segments has been considered before, but not for the MA and ARMA models. In addition, in the AR case, model order selection can be done easily when the segments are independent. But, in the application we are primarily interested in, the segments are formed by lines of scanning transmission electron microscopy (STEM) images, which can be correlated with each other. Because the correlation among data segments influences the effective number of independent observations available it becomes necessary to estimate that number. This is not an easy problem in general. For observations from a stochastic process with an isotropic 2-D autocorrelation function we can, however, formulate an accurate estimator in terms of the line autocorrelation function. This situation arises, for example, in STEM images of amorphous specimens. From the AR model identification for segments, the extension to the MA and ARMA models is rather straightforward because MA and ARMA algorithms are based on Durbin's methods, whose first step is the estimation of a long AR model. We compare the segment-based algorithms to conventional ARMASA with simulations of stationary stochastic processes. An illustrative example involving simulated STEM images shows large potential benefits.

An autoregressive moving average (ARMA) is a time series model that is applied in everyday life for pattern recognition and forecasting. The ARMA model contains a noise which is assumed to have a specific distribution. The noise is often considered to have a Gaussian distribution. However in applications, the noise is sometimes found that does not have a Gaussian distribution. The first objective is to develop the ARMA model in which noise has a Laplacian distribution. The second objective is to estimate the parameters of the ARMA model. The ARMA model parameters include ARMA model orders, ARMA model coefficients, and noise variance. The parameter estimation of the ARMA model is carried out in the Bayesian framework. In the Bayesian framework, the ARMA model parameters are treated as a variable that has a prior distribution. The prior distribution for the ARMA model parameters is combined with the likelihood function for the data to get the posterior distribution for the parameter. The posterior distribution for parameters has a complex form so that the Bayes estimator cannot be determined analytically. The reversible jump Markov chain Monte Carlo (MCMC) algorithm was adopted to determine the Bayes estimator. The first result, the ARMA model can be developed by assuming Laplacian distribution noise. The second result, the performance of the algorithm was tested using simulation studies. The simulation shows that the reversible jump MCMC algorithm can estimate the parameters of the ARMA model correctly.

Background/Objective: To provide an efficient predictive technique to foresee future workload as well as to handle the resources efficiently by performing hybrid auto scaling for Cloud applications. Cloud applications might expertise completely different workload at different times, automatic provisioning has to work with efficiency at any point of time. Auto scaling is a feature of cloud computing that potentially scale the resources in line on demand. Considering this expectation, they are generally categorized into Reactive scaling which adds or reduces resources based on a fixed threshold value. The predictive scaling is used provide necessary scaling actions beforehand. Methods/Statistical Analysis: To perform the hybrid auto scaling (reactive plus predictive auto scaling), a time series technique should be used. Auto-regressive Moving Average (ARMA) model, the Exponential Smoothing (ES) model, the Autoregressive model (AR), the Moving Average model (MA) and the Trend- Adjusted Exponential Smoothing (TAES), Auto Regressive Integrated Moving Average (ARIMA) Time-series model, Naïve bayes algorithm, Recurrent Neural Network- Long Short Term Memory (RNN-LSTM), Independent Recurrent Neural Network (IndRNN) are time series techniques used to foresee the future workload. To find the effectiveness of predictive techniques, Mean Absolute Error (MAE), Mean Squared Error (MSE) and Mean Absolute Percentage Error (MAPE) performance metrics are evaluated. Findings: Based on the evaluation, IndRNN gives the minimum error rate. IndRNN is used to predict the future resource requisites in order to ascertain adequate resource are available ahead of time. Application: The predicted result from IndRNN method is integrated on private cloud to autoscale the resources for cloud applications. Keywords: Cloud Applications, Hybrid Autoscaling, Independent Recurrent Neural Network (IndRNN), Private Cloud, Workload

Comparison of the ARMA, ARIMA, and the autoregressive artificial neural network models in forecasting the monthly inflow of Dez dam reservoir

Asymmetry in economic time series and the simultaneous switching autoregressive model

Autoregressive moving average models of crown profiles for two California hardwood species

The Internet environmenthas provided massive data to the actualindustrial production process. It not only has large amounts of databut also has a high data dimension, which brings challenges to thetraditional statistical process monitoring. Aiming at the nonlinearityand dynamics of industrial large-scale high-dimensional data, an efficientiterative multiple dynamic kernel principal component analysis (IMDKPCA)method is proposed to monitor the complex industrial process withsuper-large-scale high-dimensional data. In KPCA, a new KKT matrix is first created by using kernel matrix K. According to theproperties of the symmetric matrix, the newly constructed matrix hasthe same eigenvector as the original matrix K; hence, each columnof the matrix K can be used as the input sample of the iteration algorithm.After iterative operation, the kernel principal component can be deducedfleetly without the eigen decomposition. Because the kernel matrixis not stored in the algorithm beforehand, it can effectively reducethe computation complexity of the kernel. Especially for a tremendousdata scale, the traditional eigen decomposition technology is no longerappropriate, yet the presented method can be solved quickly. The autoregressivemoving average (ARMA) time series model and kernel principal componentanalysis (KPCA) are combined to build the IDKPCA model for dealingwith the dynamics and nonlinearity in the industrial process. Eventually,it is applied to monitor faults in the penicillin fermentation processand compared with MKPCA to certify the accuracy and applicabilityof the proposed method.

In the recent years, artificial intelligence techniques have attracted much attention in hydrological studies, while time series models are rarely used in this field. The present study evaluates the performance of artificial intelligence techniques including gene expression programming (GEP), Bayesian networks (BN), as well as time series models, namely autoregressive (AR) and autoregressive moving average (ARMA) for estimation of monthly streamflow. In addition, simple multiple linear regression (MLR) was also used. To fulfill this objective, the monthly streamflow data of Ponel and Toolelat stations located on Shafarood and Polrood Rivers, respectively in Northern Iran were used for the period of October 1964 to September 2014. In order to investigate the models’ accuracy, root mean square error (RMSE), mean absolute error (MAE) and coefficient of determination (R2) were employed as the error statistics. The obtained results demonstrated that the single AR and ARMA time series models had better performance in comparison with the single GEP, BN and MLR methods. Furthermore, in this study, six hybrid models known as GEP-AR, GEP-ARMA, BN-AR, BN-ARMA, MLR-AR and MLR-ARMA were developed to enhance the estimation accuracy of the monthly streamflow. It was concluded that the developed hybrid models were more accurate than the corresponding single artificial intelligence and time series models. The obtained results confirmed that the integration of time series models and artificial intelligence techniques could be of use to improve the accuracy of single models in modeling purposes related to the hydrological studies.

In this chapter, we deal with linear dynamic factor models and related topics, such as dynamic principal component analysis (dynamic PCA). The main motivation for the use of such models is the so-called “curse of dimensionality” plaguing modeling of high-dimensional time series by “ordinary” multivariate AR or ARMA models. For instance, consider an AR system for, a say, 20-dimensional time series. Then each of the coefficient matrices contains 400 “free” parameters, if no additional restrictions on the parameter space have been imposed, i.e. in such a case the parameter spaces grow with the square of the output dimension n, whereas the data, for given sample size, grow linearly with n. Thus for moderate sample size and large n (as is the case, e.g. in many situations faced in macroeconomics), reliable parameter estimation in “fully parametrized” AR(X) or ARMA(X) models is hardly possible. On the other hand, e.g. in macroeconomics, for analysis and in particular for short-term forecasting, modeling of “comovement” and of “cross-sectional dependencies” between a large number of univariate time series recently has received increasing attention and appropriate tools for modeling of high-dimensional time series have been developed. Correspondingly, during the last 25 years, a substantial literature has emerged, dealing with such models, methods and applications, in particular for factor models in this context. The first section introduces a general framework for linear dynamic factor models. In the second section, we describe dynamic principal component analysis, which is a generalization of the well-known static principal component analysis to the dynamic case. In practical applications often the generalized dynamic factor model is used, which allows for cross-sectionally weakly dependent noise and assumes strong dependence in latent variables. This model class is suited for very high-dimensional time series.

OF THE DISSERTATION ON ORDER IDENTIFICATION OF TIME SERIES MODELS AND ITS APPLICATIONS by Shuhao Chen Dissertation Director: Rong Chen My thesis focuses on the order identification schemes of the widely-used time series model Autoregressive Integrated Moving-Average (ARIMA) model and the applications of the order determination methods. The first part investigates the impact of dependent but uncorrelated innovations (errors) on the traditional autoregressive integrated moving average (ARIMA) model order determination schemes such as autocorrelation function (ACF), partial autocorrelation function (PACF), extended autocorrelation function (EACF) and unit-root test. We also propose a new order determination scheme to address those impacts and can be used to time series sequences with uncorrelated innovations. In the second part, a unified approach for the tentative specification of both the seasonal and nonseasonal orders of general multiplicative seasonal model is proposed. This new approach has the advantages of determining the seasonal and nonseasonal orders simultaneously and automatically.

Model order estimation is the most important but challenging step for system identification using an autoregressive moving average (ARMA) model. In this paper, we propose an artificial neural network (ANN) structure to estimate the best model order for ARMA modeling of linear, time-invariant systems using the system’s input and output data. The proposed algorithm creates an equivalent ANN structure corresponding to an ARMA model and chooses the best model order using the neural network’s mean squared error (MSE) loss function. The proposed method is validated on simulated ARMA model data and the performance is compared with the Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC). We considered three hypothetical linear systems and performed 100 Monte Carlo simulations for each model, with different data lengths, and with additive noise. For each of the three simulation models, the proposed method significantly outperformed the AIC and BIC in terms of the correct model order selection. Finally, the proposed ANN-based model order estimator was successfully applied to determine the dynamic relationship between heart rate (HR) and instantaneous lung volume (ILV) using an ARMA model. The results indicate that physiological and biological systems can be modeled with appropriate ARMA models obtained by the proposed algorithm to better understand the system dynamics.

The seemingly complex nature of river flow and the significant variability it exhibits in both time and space, have largely led to the development and application of the stochastic process concept for its modelling, forecasting, and other ancillary purposes. Towards this end, in this study, attempt was made at stochastic modelling of the daily streamflow process of the Benue River. In this regard, Autoregressive Moving Average (ARMA) models and its derivative, the Periodic Autoregressive (PAR) model were developed and used for forecasting. Comparative forecast performances of the different models indicate that despite the shortcomings associated with univariate time series, reliable forecasts can be obtained for lead times, 1 to 5 day-ahead. The forecast results also showed that the traditional ARMA model could not robustly simulate high flow regimes unlike the periodic AR (PAR). Thus, for proper understanding of the dynamics of the river flow and its management, especially, flood defense, in the light of this study, the traditional ARMA models may not be suitable since they do not allow for real-time appraisal. To account for seasonal variations, PAR models should be used in forecasting the streamflow processes of the Benue River. However, since almost all mechanisms involved in the river flow processes present some degree of nonlinearity thus, how appropriate the stochastic process might be for every flow series may be called to question.