Autoregressive order

P. M. T. Broersen, Rnite sample criteria for autoregressive order selection, IEEE Trans Signal Proc. 48 3550-3558 (2000). 

In the easiest case, a first order autoregressive model, the effects of variations in the past are contained and accounted for in the most immediate value. This value becomes an independent variable in generalized regression analysis. 

Examine the autocorrelation function. The high autocorrelations will indicate the order of the autoregressive part if any. The rate of decay of the autocorrelations will indicate a need for differencing. 

For the first order autoregressive model, the autocorrelation is p. Consider the first difference, v, =... 

It is commonly asserted that the Durbin-Watson statistic is only appropriate for testing for first order autoregressive disturbances. What combination of the coefficients of the model is estimated by the Durbin-Watson statistic in each of the following cases AR(1), AR(2), MA(1) In each case, assume that the regression model does not contain a lagged dependent variable. Comment on the impact on your results of relaxing this assumption. 

The linear terms in x[n - i i] have not been included since they are represented by the linear terms in the AR model. This model will be referred to as Autoregressive Nonlinear Autoregressive (AR-NAR) model in general and as AR(P)-NAR(Q) model in which the AR section has order P and only Q of the non-linear terms from equation 4.29 are included. Note that the undistorted signal s[ ] can be recovered from the distorted signal x n by use of equation 4.29 provided that the parameter values can be identified. 

A reasonable assumption is that there is negligible distortion for low-level signals, ie x[n = s[n, for s[n 0 so that k0 = 1. (Note that this assumption would not be valid for crossover distortion). This model will be referred to in general as the Autoregressive-Memoryless Non-linearity (AR-MNL) model and as the AR(P)-MNL(Q) to denote a AR model of order P and a memoryless non-linearity of order... 

The current value of a time series is a linear combination of a number of previous observations of the time series. The number of significant coefficients, a, is called the order of the autoregressive process. 

The partial correlation function overcomes the correlation transfer effect as described above and shows, in contrast to the autocorrelation function, only one spike at t — 1 for first order autoregressive processes and spikes at t = 1 and x = 2 for second order autoregressive processes, and so on. 

For practical computations one has to determine the order of the autoregressive process. 

The storage reservoir and the feeder stream both show the order one for autoregression, but a time lag of two months between the two series is detected by the cross-correlation function (Fig. 6-15). For this reason, it is necessary to modify the general model to ... 

First an ARMA (autoregressive moving average) model will be explained without taking into account trends and seasonal effects in order to get a better understanding of the method. 

Stationary time series can be described by an ARMA process. The ARMA formula of a first-order autoregressive process and a first-order moving average is the following ... 

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. 

Autoregressive processes have an exponentially decreasing autocorrelation function and one or more spikes in the partial autocorrelation function. The number of spikes in the partial autocorrelation function indicates the order of autoregression. 

The PACF illustrates the order 1 for the AR component, but at this stage of estimation of the model it is unknown if the trend or the seasonal model follow the autoregression with the order of one. No moving average component can be found from the PACF. Deduced possible models are ARIMA (1,1,0)( 1,0,0), ARIMA (0,1,0)( 1,0,0), or ARIMA (1,1,0)(0,0,0). 

PACF of the errors indicates a first-order autoregression component, therefore the parameter p is set to unity. The spikes at lags 13 and 25 are a consequence of the multiplicative seasonal model. 

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 ... 

Comparing this with equation (3) shows that this can be considered as the output of a first order transfer function in response to a random input sequence. More generally, most stochastic disturbances can be modelled by a general autoregressive-integrated moving-average (ARIMA) time series model of order (p,d,q), that is,... 

The autoregressive component relates the noise to the observed value of the response at one or more previous times. A model of order p is given by... 

The effects of autocorrelation on monitoring charts have also been reported by other researchers for Shewhart [186] and CUSUM [343, 6] charts. Modification of the control limits of monitoring charts by assuming that the process can be represented by an autoregressive time series model (see Section 4.4 for terminology) of order 1 or 2, and use of recursive Kalman filter techniques for eliminating autocorrelation from process data have also been proposed... 

Past values of the observed variable Autoregressive terms (up to order p) AR... 

A variety of within-subject covariance matrices have been proposed (Table 6.2). The most common are the simple, unstructured, compound symmetry, first-order autoregressive [referred to as AR(1)], and Toeplitz covariance. The simple covariance assumes that observations within a subject are uncorrelated and have constant variance, like in Eq. (6.28). Unstructured assumes... 

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