ARMA process model

Godsill, 1997a] Godsill, S. J. (1997a). Bayesian enhancement of speech and audio signals which can be modelled as ARMA processes. International Statistical Review, 65(1) 1—21. 

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. 

The remainder of this section introduces the relevant notation with an additional focus on the extension to heteroscedastic models (so-called (G)ARCH and ARMA-GARCH models) as these can be seen as the discrete-time counterpart of continuous stochastic processes formulated in terms of SDEs. 

Assume the SISO model (2.25) where is coloured noise and follows an ARMA process with lag polynomials < (B) and 0 B). Without loss of generality fi is assumed to be zero and (2.25) changes to... 

Unfortunately, neither PACF nor ACF lead to directly interpretable results for ARMA processes. The extended ACF tries to overcome this drawback by jointly providing information about the order of both components. For each AR order tested, the EACF first determines estimates of the AR coefficients by a sequence of regression models. Afterwards, the residuals ACF is calculated. The results are presented in a table indicating significant or non-significant auto-correlations (typically denoted by an x and o, respectively). In such a table, the rows represent the AR order p whereas columns represent... 

Fig. 2 The autocorrelation function (correlation ver.vtM lag) and power. spectrum (log2(power) versus log2(frequency)) of the wavelet coefficients for an ARMA(Iff) process with the model y, = 0.8y, i + a, — 0.3a,-1. where a is white noi.se of variance one.

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. 

An increase in volatiUty leads to an increase in the mean. Similarly, for an ARMA-GARCH(P,Q)(p,g) the mean process follows an ARMA(P,Q) process whereas the volatility is modelled by a GARCH(p, ) process ... 

The SIC is deduced from Bayesian arguments. It consistently estimates the true order of ARMA(p, q) processes and is probably the most widely used information criterion in univariate time series analysis. The HQIC is the most recent IC and especially designed for multivariate time series models. In practice, multiple ICs are simultaneously calculated which allows the analyst to cross-check the recommendations of the various ICs. Strongly deviating recommendations may indicate an inappropriate model structure. 

Also, a time series changing seasonally couldn t be modeled by ARMA immediately. Firstly, it should be differed to remove the seasonality of the series, and the length of one difference step is one seasonal cycle. Generally, if the series is processed by a D-th order difference the cycle length of which is s, the seasonality will almost be removed. Furthermore, if the series needs to be turned stationary by a d-th order difference before it s impacted by the seasonal difference, a model called the Multiplicative Seasonal Mod-el ARIMA(p, d, q) (P, D, Q) could analyze the original series, and the model is described as below ... 

As the trend in the original series log(x) is removed by a first-order difference, the value of d should be equal to 1 And similarly, the seasonality in the series ilx disappears after it s processed by a seasonal difference, so D is also equal to 1 and the ARMIA(p, d, q) (P, D, Q) is thereby selected. After observing the autocorrelogram and partial-autocorrelogram of series silx, we prefer to pick q=l,p = 2orp = 3. Compared to the non-linear estimation possessed by MA(q) and ARMA(p, q) processes, the AR(p) model as a linear equation is easier to estimate, explain and forecast. Consequently, in the practi-... 

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. 

Even under the relatively simple type of model presented above, the process At does not follow an ARMA(1,1) evolution as under the case in which the demand evolves according to the elementary auto-regressive process of Example 1. In fact, under (10.19), the orders At maintain the following recursive scheme ... 

The autoregressive, moving-average process denoted as ARMA(p, q) is one of the most common times series models that can be used. It has the general form given as... 

From an initial attempt to model the mean summer temperature in Edmonton as an ARMA(8,8) process, the parameter estimates and their standard deviatimi were determined as... 

Black box models or empirical models do not describe the physical phenomena of the process, they are based on input/output data and only describe the relationship between the measured input and output data of the process. These models are useful when limited time is available for model development and/or when there is insufficient physical understanding of the process. Mathematical representations include time series models (such as ARMA, ARX, Box and Jenkins models, recurrent neural network models, recurrent fuzzy models). 

Shinozuka M, Sato Y (1967) Simulation of nonstationary random process. J Eng Mech ASCE 93 11-40 Solomos GP, Spanos PD (1984) Oscillator response to non-stationary excitation. Am Soc Mech Eng Appl Mech Div AMD 65 159-170 Spanos PD (1983) ARMA algtmthms for ocean wave modeling. Trans ASME J Energy Resour Technol 105 300-309 Spanos PD (1986) Filter approach to wave kinematics approximations. Appl Ocean Res 8 2-7 Spanos PD, Miller SM (1994) Flilbert transform generalization of a classical random vibration integral. J Appl Mech 61 575-581... 

---