ARIMA models

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 

Alternatively, the noise type can be ignored applying so-called non-parsimoniovs finite impulse-response (FIR) models by increasing N, the number of lagged observations of the control variable(s). This method has some serious drawbacks e.g. biased and instable estimates/forecasts in case of finite samples. See e.g. Dayal and MacGregor (1996) and references therein. 

this process is not stationary due to the time-varying, unbounded variance and covariance. Only AR(1) processes with (() 1 are stationary. This result can be generalized for any AR(p) model for which the following conditions must hold  

In the moving average (MA) model yt depends on the previous errors instead of previous observations. An MA model of order q (MA(g)) is represented by 

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  

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Summary of Model Process. The basic steps in building an ARIMA model are the following ... 

The values of the three electrochemical measurements, potential, resistance, and current were measured for the four coatings over time. The resultant time series for each measurement and coating combination were analyzed by the Box-Jenkins ARIMA procedure. Application of the ARIMA model will be demonstrated for the poly(urethane) coating. Similar prediction results were obtained for all coatings and measurements, however, not all systems were modeled by the same order of ARIMA process. 

ARIMA modeling in the analysis of water quality data is discussed in detail by ZET-TERQVIST [1989]. Similar models were applied to the analysis of acid rain data [TERRY et al., 1985] and to the analysis of air quality data [JOHNSON and WIJN-BERG, 1981]. In general, these models are very rarely used in environmental analysis. We want to demonstrate the power of this kind of time series model and will also supply candidates suitable for use of these models for evaluation of time series. 

In order to calculate the fitted values of the drinking water in the storage reservoir by ARIMA modeling, the data set was shortened for the explanatory variable, the feeder stream. All following time series analytical procedures only use the values from the nitrate concentration series in the drinking water reservoir. 

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. 

Application of ARIMA Models 6.7.2.1 Specification of ARIMA Models... 

The specification of ARIMA models is very expensive for the operator who analyzes time series. The first phase is the estimation of the order of three inherent processes, autoregression, integration, and moving average. 

This differencing is continued until the time series is stationary (mean and variance are not dependent on time). Frequently, single time differencing, i.e. first order for the seasonal ARIMA model, is sufficient. Second order differencing is necessary for quadratic trends. Please note the loss of values after differencing (e.g. after first-order seasonal differencing twelve values will be lost). 

In reality, these functions are more complex and the operator has to use the trial and error mode. Practical criteria which improve the likelihood of correct selection of the parameters of the ARIMA model are the autocorrelation and the partial autocorrelation function of the errors of the resulting ARIMA fit. If they do not have significant spikes the model is satisfactory. 

Improving the Noncorrelation between the ARIMA Model Components... 

Application of the ARIMA Modeling to the Example Time Series... 

First, the series of the nitrate concentrations within the storage reservoir is made stationary in order to obtain the parameters d and sd for the trend and the seasonal ARIMA model. With one-time differencing at the differences 1, the series becomes stationary and the parameter d is set to unity (Fig. 6-24), but seasonal fluctuations are present. With one-time differencing of the original nitrate series at the difference 12, the seasonal fluctuations disappear, but the trend is present (Fig. 6-25). It is, therefore, necessary to include the seasonal ARIMA component in the model, the parameter sd is set to zero. The deduced possible model is ARIMA ( ,1, )( ,0, ). 

These were the general conclusions from ACF and PACF of the time series for modeling ARIMA (p,d,q) sp,sd,sq). Now the second term of the multiplicative ARIMA model - the seasonal ARIMA component, ARIMA (()S)S))(sp,sd,sq) - must be estimated. 

The patterns of both ACF (Fig. 6-27) and PACF (Fig. 6-28) of the errors from the previous model are used to find the complete multiplicative trend and the seasonal ARIMA model. The following conclusions can be drawn ... 

In the example with the ARIMA model (1,1,0)(1,0,0), the correlation coefficient between the autoregression and the seasonal autoregression component is -0.19. This means they are not correlated with an error probability of 0.002. Therefore, this model is valid. 

In ARIMA modeling, the order of the autoregressive component is frequently zero, one or sometimes two. Therefore, only short forecasting intervals are of any use. Disturbances in future values, normally smoothed by the moving average, were set to zero. The following example demonstrates this fact ... 

A useful extension to classic ARIMA models are models incorporating exogenous explanatory variables, so called ARIMAX models. As the name suggests, additional parameters are added to the ARIMA model. The use of explanatory variables usually results from knowledge about the process external dependencies, but sometimes it is also reasonable to model outliers or structural changes of the time series using auxiliary exogenous variables. ... 

The first approach to solve such a problem is to perform a regression analysis between the time series y and the regression variable x ignoring the fact of auto-correlated, stochastic variables. Afterwards, the residuals from this hrst step could be obtained and a time series could be fitted for these residuals, e.g. using ARIMA models. Unfortunately, this approach can lead to biased and inefficient estimates even if the sample is large. Due to the time series characteristics of the dependent and independent variables, the cross-correlations between x and y might be overlaid by the individual temporal dependencies of X and y. To obtain (reasonable) estimates for the impulse response parameters V = (vo,, the time series y has to be cleaned from the time series effects of the re-... 

Table 8 is an extension of Table 3 but adding a column with the results when considering the ARIMA models to forecast demand in the first level of the supply chain (BW5). Furthermore, we show the reduction achieved in each case. 

Database Name Number of data ARIMA model... 

Table 8. Results of the tests using ARIMA models.

The results presented in this section show that the use of advanced forecasting methods leads to the reduction of Bullwhip Effect. Thus, the inclusion of ARIMA models at the lowest level of the supply chain provides very interesting results, and it can significantly reduce, in many cases, the Bullwhip Effect. In these circumstances, we... 

Figure 17 depicts, by way of example, the results obtained for the serie BJ06. It is compared with Fig. 16. It is possible to see how the use of ARIMA models significantly reduces, above 35 %, the variability of orders along the supply chain. Table 9 shows, in each case, the optimal policy for each level of the supply chain.

To develop the tool, we have considered only simple forecasting methods, such as moving averages and exponential smoothing, so that each level of the chain uses the best one that suits the demand it should deal with. With them, it is possible to achieve great results in reducing Bullwhip Effect. Even so, we have also shown that the inclusion of more advanced forecasting methods (ARIMA models) allows an even better system performance. 

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