Autoregressive process

Terms of this equation are defined similarly to those for the LSAR interpolator of section 4.3.3, with subscript V referring to the autoregressive process for the noise pulses. Av is a diagonal matrix whose mth diagonal element Xy m is the variance of the with noise excitation component, i.e. 

Vol. 536 R. Brtiggemann, Model Reduction Methods for Vector Autoregressive Processes. X, 218 pages. 2004. 

A more general definition of such an autoregressive process is ... 

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. 

Autoregressive processes are stochastic processes with a memory effect. The basic equation is known from regression analysis ... 

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

One can imagine this operator as a lag which is observed in the time series. The new notation for autoregressive processes as shown in Eq. 6-40, using the back shift operator is ... 

These autoregressive processes were explained in more detail in Section 6.6.2. 

The ARMA method offers the possibility of combining both autoregressive and moving average processes. The disturbance term from autoregressive processes may be described by a moving average. 

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 result is a process which determines the values from the previous values (autoregressive process), and smoothes the errors using a 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. 

In the equation, p represents the order of Seasonal Autoregressive Processes, and q represents the order of Seasonal Moving Average Processes. Moreover, p(B ) and Qp(B ) respectively represent the operators of the Seasonal AR(p) and MA(q) models. 

This EM procedure is summarized as follows Consider a scalar autoregressive process Yt, t = 1,2,... of order r and coefficients a = (ai,..., ) Let V denote an independent identically distributed (iid)... 

This section will examine the properties of an autoregressive process, AR(p), defined as... 

The mean value of the autoregressive process can be computed using Eq. (5.50),... 

Theorem 5.2 Autocovariance of an Autoregressive Process. The autocovariance of an autoregressive process can be written as ... 

Proof Consider a causal, autoregressive process that can be written as follows ... 

Noting that, for any causal autoregressive process where 0 < 1,... 

This result clearly shows the behaviour of the autocorrelation function for an autoregressive process and its difference from the moving-average process. 

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