Generalized Autoregressive Moving Average Models Research Articles

Estimation Approach for a Linear Quantile-Regression Model with Long-Memory Stationary GARMA Errors

The aim of this paper is to assess the significant impact of using quantile analysis in multiple fields of scientific research . Here, we focus on estimating conditional quantile functions when the errors follow a GARMA (Generalized Auto-Regressive Moving Average) model. Our key theoretical contribution involves identifying the Quantile-Regression (QR) coefficients within the context of GARMA errors. We propose a modified maximum-likelihood estimation method using an EM algorithm to estimate the target coefficients and derive their statistical properties. The proposed procedure yields estimators that are strongly consistent and asymptotically normal under mild conditions. In order to evaluate the performance of the proposed estimators, a simulation study is conducted employing the minimum bias and Root Mean Square Error (RMSE) criterion. Furthermore, an empirical application is given to demonstrate the effectiveness of the proposed methodology in practice.

Generalized autoregressive and moving average models: multicollinearity, interpretation and a new modified model

ABSTRACTIn this paper, we call attention of two observed features in practical applications of the Generalized Autoregressive Moving Average (GARMA) model due to the structure of its linear predictor. One is the multicollinearity which may lead to a non-convergence of the maximum likelihood, using iteratively reweighted least squares, and the inflation of the estimator's variance. The second is that the inclusion of the same lagged observations into the autoregressive and moving average components confounds the interpretation of the parameters. A modified model, GAR-M, is presented to reduce the multicollinearity and to improve the interpretation of the parameters. The expectation and variance under stationarity conditions are presented for the identity and logarithm link function. In a general sense, simulation studies show that the maximum likelihood estimators based on the GARMA and GAR-M models are equivalent but the GAR-M estimators presented a little lower standard errors and some restrictions in the parametric space are imposed to guarantee the stationarity of the process. Also, a real data analysis illustrates the GAR-M fit for daily hospitalization rates of elderly people due to respiratory diseases from October 2012 to April 2015 in São Paulo city, Brazil.

Forecasting demand for health services: Development of a publicly available toolbox

Efficient health care delivery systems aim to match resources to demand for services over time. Resource allocation decisions must be made under stochastic uncertainty. This includes uncertainty in the number of individuals (i.e., counts) in need of services over discrete time intervals. Examples include counts of patients arriving to emergency departments and counts of prescription medications distributed by pharmacies. Accurately forecasting count data in health care systems allows decision-makers to anticipate the need for services and make informed decisions about how to manage resources and purchase supplies over time.A publicly available toolbox to forecast count data is developed in this work. The toolbox is implemented in MATLAB environment with the newly developed generalized autoregressive moving average (GARMA) models with discrete-valued distributions. GARMA models treat count data in a mathematically coherent manner compared to Gaussian models, often inappropriately applied in health care applications. GARMA models can incorporate none to many exogenous variables hypothesized to influence the predicted responses (i.e., counts forecasted). The toolbox’s primary purpose is to deliver one to multiple-steps ahead forecasts, but also gives information for model inference and validation. The toolbox uses the maximum likelihood method to estimate model parameters from the data. We demonstrate toolbox application and validity on two example health care count data sets and show how using integer-valued conditional distributions as offered by GARMA models can produce forecast models that outperform the traditional Gaussian models.

Generalized ARMA models with martingale difference errors

The analysis of non-Gaussian time series has been studied extensively and has many applications. Many successful models can be viewed as special cases or variations of the generalized autoregressive moving average (GARMA) models of Benjamin et al. (2003), where a link function similar to that used in generalized linear models is introduced and the conditional mean, under the link function, assumes an ARMA structure. Under such a model, the ‘transformed’ time series, under the same link function, assumes an ARMA form as well. Unfortunately, unless the link function is an identity function, the error sequence defined in the transformed ARMA model is usually not a martingale difference sequence. In this paper we extend the GARMA model in such a way that the resulting ARMA model in the transformed space has a martingale difference sequence as its error sequence. The benefit of such an extension are four-folds. It has easily verifiable conditions for stationarity and ergodicity; its Gaussian pseudo-likelihood estimator is consistent; standard time series model building tools are ready to use; and its MLE’s asymptotic distribution can be established. We also proposes two new classes of non-Gaussian time series models under the new framework. The performance of the proposed models is demonstrated with simulated and real examples.

Intrusion Detection Forecasting Using Time Series for Improving Cyber Defence

The strength of time series modeling is generally not used in almost all current intrusion detection and prevention systems. By having time series models, system administrators will be able to better plan resource allocation and system readiness to defend against malicious activities. In this paper, we address the knowledge gap by investigating the possible inclusion of a statistical based time series modeling that can be seamlessly integrated into existing cyber defense system. Cyber-attack processes exhibit long range dependence and in order to investigate such properties a new class of Generalized Autoregressive Moving Average (GARMA) can be used. In this paper, GARMA (1, 1; 1, ±) model is fitted to cyber-attack data sets. Two different estimation methods are used. Point forecasts to predict the attack rate possibly hours ahead of time also has been done and the performance of the models and estimation methods are discussed. The investigation of the case-study will confirm that by exploiting the statistical properties, it is possible to predict cyber-attacks (at least in terms of attack rate) with good accuracy. This kind of forecasting capability would provide sufficient early-warning time for defenders to adjust their defense configurations or resource allocations.

Stationarity of generalized autoregressive moving average models

Time series models are often constructed by combining nonstationary effects such as trends with stochastic processes that are believed to be stationary. Although stationarity of the underlying process is typically crucial to ensure desirable properties or even validity of statistical estimators, there are numerous time series models for which this stationarity is not yet proven. A major barrier is that the most commonly-used methods assume φ-irreducibility, a condition that can be violated for the important class of discrete-valued observation-driven models. We show (strict) stationarity for the class of Generalized Autoregressive Moving Average (GARMA) models, which provides a flexible analogue of ARMA models for count, binary, or other discrete-valued data. We do this from two perspectives. First, we show conditions under which GARMA models have a unique stationary distribution (so are strictly stationary when initialized in that distribution). This result potentially forms the foundation for broadly showing consistency and asymptotic normality of maximum likelihood estimators for GARMA models. Since these conclusions are not immediate, however, we also take a second approach. We show stationarity and ergodicity of a perturbed version of the GARMA model, which utilizes the fact that the perturbed model is φ-irreducible and immediately implies consistent estimation of the mean, lagged covariances, and other functionals of the perturbed process. We relate the perturbed and original processes by showing that the perturbed model yields parameter estimates that are arbitrarily close to those of the original model.

Generalized Autoregressive Moving Average Models

A class of generalized autoregressive moving average (GARMA) models is developed that extends the univariate Gaussian ARMA time series model to a flexible observation-driven model for non-Gaussian time series data. The dependent variable is assumed to have a conditional exponential family distribution given the past history of the process. The model estimation is carried out using an iteratively reweighted least squares algorithm. Properties of the model, including stationarity and marginal moments, are either derived explicitly or investigated using Monte Carlo simulation. The relationship of the GARMA model to other models is shown, including the autoregressive models of Zeger and Qaqish, the moving average models of Li, and the reparameterized generalized autoregressive conditional heteroscedastic GARCH model (providing the formula for its fourth marginal moment not previously derived). The model is demonstrated by the application of the GARMA model with a negative binomial conditional distribution to a well-known time series dataset of poliomyelitis counts.