ARMA model

ARIMA modeling in contrast with the ARMA model, includes trend or seasonality of time series. For such series, the trend can be removed by one-step differencing. Seasonality may be removed by 12-step differencing. After differencing the time series one gets a stationary time series which can be described as an ARMA process. 

It is easy to interpret Eq. 6-49, difficult as it may look. The series for the moving average term equals a combination of the current disturbance and of the disturbance of one previous seasonal period. The original series, therefore, follows a model integrated from the ARMA model of the stationary series, see Eq. 6-47. 

Note that in a full ARMA model, < , itself is dependent on past values of noise. 

The d3Tiamic response of e k) can be expressed as an autoregressive moving average (ARMA) model or a moving average (MA) time series model ... 

I MPC is a stochastic variable and statistically significant changes in the controller performance can be detected by statistical analysis. Impc is assumed to be generated by an ARMA model... 

Cigizoglu, H. K. Incorporation of ARMA models into flow forecasting by artificial neural networks,... 

This section describes the class of the most common ARMA models and some of their extensions. The term ARMA combines both basic types of time-dependencies, the autoregressive (AR) model and the moving average (MA) model. Suppose a time series y = collected over T periods with zero mean. Autoregressive dependency means that any observation yt depends on previous observations yt-i of this time series with i = 1,. ..,p such that... 

However, setting 6 = yields exactly the same auto-correlation p. Hence, there are at least two possible values for 9 producing exactly the same time series w.r.t. the auto-correlation struotureJ This problem is related to the stationarity condition of AR processes. To solve this problem the invertibiUty condition is introduced. An MA process must be invertible into an infinite AR process. This holds if and only if the characteristic equation for the characteristic polynomial 9 x) = l + 6i-x + 62-x +. .. + 6q-x has roots with absolute value larger than IJ Given a special MA process with known order and unknown parameter (set), there exists only one parameter (set) such that this MA process is invertible. Both types of auto-correlation models are rarely found in real world problems in genuine form, but in combination they build a huge class of time series patterns summarized as so-called ARMA models. An ARMA(p,g) model can be formalized as ... 

It can be shown that ARCH and GARCH models are able to approximate stochastic differential processes if the latter fulfil certain properties. Albeit the goodness of fit is limited, both types of methods are related and can be converted into each other. Moreover, simple stochastic processes show quite simple auto-correlation structures similar to basic ARMA models. For instance, the Ornstein-Uhlenbeck process can be seen as the continuous equivalent of the AR(1) process. In other words, an Ornstein-Uhlenbeck process measured in discrete intervals can be interpreted/modeUed as an AR(1) process (see also (2.23), (2.60), and (2.61)). ... 

This subsection briefly introduces GARCH models as discrete counterpart of continnous stochastic processes. In contrast to ARMA models, the basic idea is that the variance/ volatility in time is no longer deterministic and constant bnt depends on previous errors and volatility, i.e. 

It has to be noted that the estimation of ARMA-GARCH models requires some more sophisticated methods compared to simple ARMA models, see Francq and Zakoian (2004). 

Jarque-Bera s test for normality confirms that the hypothesis of normally distributed residuals cannot be rejected (at a 5% level), i.e. it -is assumed that e A (0, 0.002 ). Due to the already convincing results of the ARX(l) model (which has minimal order) no other ARMA models of higher order are estimated and (2.66) is taken as the best m,odel for the Naphtha time series. 

After calculation, each model meets the conditions of stationary and invertible in the ARMA modeling process. At the same time, the models are reasonably defined and desirably fitting the data. Among these models, the AIC value of the 3rd model is the smallest. Therefore, it is appropriate to choose the 3rd model ARIMA(4,1,1)(1, 1, as the final model to forecast. 

Stationary test of the residual sequence Before modeling on the residual sequence, use Eviews6.0 software to check whether the residual sequence is smooth or not, if so, directly establish ARMA model, if not, it has to be taken some steps of difference. After it s smooth, use the ARMA model and establish ARIMA model. Residual sequence is tested on the unit root test, the results are shown in Figure 1 ... 

The Figure 1 shows that the values of t-statistic are less than the critical values of significance level, it means sequence is smooth. So do not use the difference, but ARMA sequence fitting. Consider to use ARMA model. Select the ARMA (2, 2) model as a predictive model. Parameter estimation results are shown in Table 4. 

Much attention has also been devoted to modal identification without measuring the input time history. In particular, a lot of effort has been dedicated to the case of free vibration (or impulse response) and to the case of ambient vibration. In the former case, often time-domain methods based on auto-regressive moving average (ARMA) models are employed, using least squares as the core ingredient in their formulations. However, it was found that the least-squares method yields biased estimates [76], A number of methods have been developed to eliminate this bias, including the instrumental matrix with delayed observations method [76], the correlation fit method [275], the double least-squares method [114,202] and the total least-squares method [92]. A detailed comparison of these methods can be found in Cooper [61],... 

Conte, J. P. and Kumar, S. Statistical system identification of structures using ARMA models. In Proceedings of 7th ASCE Specialty Conference on Probabilistic Mechanics and Structural Reliability (Worcester, MA, 1996), pp. 142-145. 

Pi, Y. L. and Mickleborough, N. C. Modal identification of vibrating structures using ARMA models. Journal of Engineering Mechanics (ASCE) 115(10) (1989), 2232-2250. 

Maximum-Likelihood Parameter Estimates for ARMA Models... 

N and A/are integers representing the number of inputs to the neural network. If the function f is linear then the model is an ARMAX model, as discussed in chapter 24. It is good practice to start with a linear ARMA model first, if poor modeling results are obtained, the non-linear version should be investigated. 

Fault diagnosis Nonstationary random vibration Signal-based modeling (identification) Structural Health Monitoring Time-dependent ARMA modeling Time-frequency analysis... 

Time series methods operate by identifying ARMA models of the form... 

---