Generalized Autoregressive Moving Average Research Articles

Identifying the time of step change in the mean of autocorrelated processes

Control charts are used to detect changes in a process. Once a change is detected, knowledge of the change point would simplify the search for and identification of the special cause. Consequently, having an estimate of the process change point following a control chart signal would be useful to process analysts. Change-point methods for the uncorrelated process have been studied extensively in the literature; however, less attention has been given to change-point methods for autocorrelated processes. Autocorrelation is common in practice and is often modeled via the class of autoregressive moving average (ARMA) models. In this article, a maximum likelihood estimator for the time of step change in the mean of covariance-stationary processes that fall within the general ARMA framework is developed. The estimator is intended to be used as an “add-on” following a signal from a phase II control chart. Considering first-order pure and mixed ARMA processes, Monte Carlo simulation is used to evaluate the performance of the proposed change-point estimator across a range of step change magnitudes following a genuine signal from a control chart. Results indicate that the estimator provides process analysts with an accurate and useful estimate of the last sample obtained from the unchanged process. Additionally, results indicate that if a change-point estimator designed for the uncorrelated process is applied to an autocorrelated process, the performance of the estimator can suffer dramatically.

Statistical modeling of valley fever data in Kern County, California

Coccidioidomycosis (valley fever) is a fungal infection found in the southwestern US, northern Mexico, and some places in Central and South America. The fungus that causes it (Coccidioides immitis) is normally soil-dwelling but, if disturbed, becomes air-borne and infects the host when its spores are inhaled. It is thus natural to surmise that weather conditions that foster the growth and dispersal of the fungus must have an effect on the number of cases in the endemic areas. We present here an attempt at the modeling of valley fever incidence in Kern County, California, by the implementation of a generalized auto regressive moving average (GARMA) model. We show that the number of valley fever cases can be predicted mainly by considering only the previous history of incidence rates in the county. The inclusion of weather-related time sequences improves the model only to a relatively minor extent. This suggests that fluctuations of incidence rates (about a seasonally varying background value) are related to biological and/or anthropogenic reasons, and not so much to weather anomalies.

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.

EMPIRICAL CHARACTERISTIC FUNCTION IN TIME SERIES ESTIMATION

Because the empirical characteristic function (ECF) is the Fourier transform of the empirical distribution function, it retains all the information in the sample but can overcome difficulties arising from the likelihood. This paper discusses an estimation method via the ECF for strictly stationary processes. Under some regularity conditions, the resulting estimators are shown to be consistent and asymptotically normal. The method is applied to estimate the stable autoregressive moving average (ARMA) models. For the general stable ARMA model for which the maximum likelihood approach is not feasible, Monte Carlo evidence shows that the ECF method is a viable estimation method for all the parameters of interest. For the Gaussian ARMA model, a particular stable ARMA model, the optimal weight functions and estimating equations are given. Monte Carlo studies highlight the finite sample performances of the ECF method relative to the exact and conditional maximum likelihood methods.

Seasonal Moving‐average Unit Root Tests in the Presence of a Linear Trend

To distinguish stochastic from deterministic seasonality, test procedures are developed for a unit root in the integrated seasonal moving‐average (SMA) model when an underlying deterministic trend is present. Locally best invariant unbiase (LBIU) and point optimal invariant tests are considered. Their asymptotic distributions are developed and are found to differ from those for the no‐linear‐trend case. The limiting distribution of the LBIU statistic is expressed as a functional of Brownian motions. The procedures are extended to more general seasonal autoregressive moving‐average (ARMA) models, and to the inclusion of exogenous regressors. Finite‐sample distributions are also derived for the SMA(1) model. Simulations suggest that these distributions provide accurate approximations for more general ARMA models. A numerical example is included to illustrate the tests.

On the expectation of estimators for general ARMA processes

A simple technique is presented for obtaining explicit expressions for the approximate expectation of estimators for general stationary autoregressive moving average (ARMA) processes. The technique is based on a Taylor series expansion of the log-likelihood function in terms of the expected values of the sample autocovariances or in terms of the expected value of the periodogram ordinates, depending upon whether the estimation makes use of time-domain methods. For illustrative purpose the expectation of estimators arising from two particular approximations of the likelihood function is obtained , although the proposed technique can be easily extended to other types of log-likelihood functions as well. The results may be used to provide information about the bias of estimators in non-seasonal and seasonal ARMA models.

Accounts Receivable Policy Under Stochastic Inflation

This paper investigates the problem of managing accounts receivable under uncertain inflation. In particular, we derive expressions for the cash discount and the length of the credit period under the assumption that the inflation rate can be modeled by a general autoregressive moving-average (ARMA) process. A number of examples illustrate the size of the change required to keep the real rate of interest, implicit in the terms of sale, constant.

Asymptotic Properties of Multivariate Nonstationary Processes with Applications to Autoregressions

Asymptotic properties of multivariate time series with characteristic roots on the unit circle are considered. For a vector autoregressive moving average (ARMA) process, we derive the limiting distributions of certain statistics which are useful in understanding nonstationary processes. These distributions are derived in a unified manner for all types of characteristic roots and are expressed in terms of stochastic integrals of Brownian motions. For applications, we use the limiting distributions to establish the consistency properties of the ordinary least squares (LS) estimates of various autoregressions of a vector process, e.g., the ordinary, forward and shifted autoregressions. For a purely nonstationary vector ARMA($p, q$) process, the LS estimates are consistent if the order of the fitted autoregression is $p$; for a general ARMA model, the limits of the LS estimates exist, but these estimates can only provide consistent estimates of the nonstationary characteristic roots.

The iterative NCDE algorithm for ARMA system identification and spectral estimation

The problem of identifying autoregressive moving average (ARMA) models with observational output data is addressed within this report. In the absence of actual input data, the ARMA identification problem is nonlinear in the parameters. The new general ARMA algorithm derived within, entitled NCDE, makes use of the Yule-Walker equations for input estimation and a least squares input-output ARMA algorithm for initial parameter estimation. The NCDE algorithm has been tested and results show that it is both effective and efficient for autoregressive (AR), moving average (MA) and ARMA system identification via the application of an ARMA model.