Autoregression

Autoregressive Integrated Moving Average Model (ARIMA) ... 

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. 

A general approach was developed by G.E.P. Box and G.M. Jenkins (S) which combines these various methods into an analysis which permits choice of the most appropriate model, checks the forecast precision, and allows for interpretation. The Box-Jenkins analysis is an autoregressive integrated moving average model (ARIMA). This approach, as implemented in the MINITAB computer program is one used for the analyses reported here. 

There are three adjustable parameters in a Box-Jenkins analysis, one each for autoregression, differencing, and moving average terms. Corrections for cyclical behavior may be added as three optional terms. The approach is flexible, and provides much information. 

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. 

Calculation of Model. Examination of Figs. 1 and 2 suggest the initial choice of p=l for the autoregression part, and the use of 1st differences, i.e. an ARIMA (1 1 0) model. The potential vs. time data was fit using this model. 

When experimental data are collected over time or distance there is always a chance of having autocorrelated residuals. Box et al. (1994) provide an extensive treatment of correlated disturbances in discrete time models. The structure of the disturbance term is often moving average or autoregressive models. Detection of autocorrelation in the residuals can be established either from a time series plot of the residuals versus time (or experiment number) or from a lag plot. If we can see a pattern in the residuals over time, it probably means that there is correlation between the disturbances. 

Hurvich, C. and C. L. Tsai. A Corrected Akaike Information Criterion for Vector Autoregressive Model Selection. J Time Series Anal 14, 271-279 (1993). 

An autoregressive time series model (16) seems to be less suitable for cumulative distribution data. This technique is primarily designed for finding trends and/or cycles for data recorded in a time sequence, under the null-hypothesis that the sequence has no effect. 

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. 

By multiplying through by the denominator of the lag function, we obtain an autoregressive form... 

The model can be estimated as an autoregressive or distributed lag equation. Consider, first, the autoregressive fonn. Multiply through by (1 - yL)(l - < )L) to obtain... 

Autoregressive (AR) model-based Click Detection. In this method ([Vaseghi and Rayner, 1988, Vaseghi, 1988, Vaseghi and Rayner, 1990]) the underlying audio data. v n is assumed to be drawn from a short-term stationary autoregressive (AR) process (see equation (4.1)). The AR model parameters a and the excitation variance <52e are estimated from the corrupted data x[n using some procedure robust to impulsive noise, such as the M-estimator (see section 4.2). 

The general Volterra and NARMA models suffer from two problems from the point of view of distortion correction. They are unnecessarily complex and even after identifying the parameters of the model it is still necessary to recover the undistorted signal by some means. In section 4.2 it was noted that audio signals are well-represented by the autoregressive (AR) model defined by equation 4.1 ... 

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. 

Auditory scene analysis, 24, 26 Auditory system, 8, 10, 16 Auralization, 87 Automatic gain control, 255 Autoregressive (AR) model, 135, 142, 159, 164 Autoregressive (AR) model, Interpolation (See Restoration,Interpolation)... 

A model which has found application in many areas of time series processing, including audio restoration (see sections 4.3 and 4.7), is the autoregressive (AR) or allpole model (see Box and Jenkins [Box and Jenkins, 1970], Priestley [Priestley, 1981] and also Makhoul [Makhoul, 1975] for an introduction to linear predictive analysis) in which the current value of a signal is represented as a weighted sum of P previous signal values and a white noise term ... 

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