Time autoregression

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. 

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 ... 

Another way to characterize the LPC filter is as an autoregressive (AR) spectral envelope model [Kay, 1988], The error minimized by LPC (time-waveform prediction error) forces the filter to model parametrically the upper spectral envelope of the speech waveform [Makhoul, 1975], Since the physical excitation of the vocal tract is not spectrally flat, the filter obtained by whitening the prediction error is not a physical model of the vocal tract. (It would be only if the glottal excitation were an impulse... 

McCulloch and Tsay, 1994] McCulloch, R. E. and Tsay, R. S. (1994). Bayesian analysis of autoregressive time series via the Gibbs sampler. Journal of Time Series Analysis, 15(2) 235-250. 

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 autoregression technique as the alternative was developed for such autocorrelated variables and errors, which are frequently available in time series analysis. 

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 ... 

A special model from this type (autoregression with an explanatory variable), an autoregression model combined with a moving average model was applied by VAN STRA-TEN and KOUWENHOVEN [1991] to the time dependence of dissolved oxygen in lakes. 

Weida storage reservoir —Autoregression prediction from feeder stream Fig. 6-18. Autoregression fit of the example time series... 

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 ... 

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. 

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,... 

Burg135 derived a means of estimating a power spectrum from the limited number of autocorrelation coefficients of a short time series, without adding extra information. It can be shown that this approach leads to equations identical to an autoregressive LP method, with the power spectrum P(f) given by... 

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... 

The d3Tiamic response of e k) can be expressed as an autoregressive moving average (ARMA) model or a moving average (MA) time series model ... 

V Haggan and T Ozaki. Modeling nonlinear random vibrations using an amplitude dependent autoregressive time series model. Biometri-ka, 68 186-196, 1981. 

Autoregressive parameter, residual Mahalanobis angle MFC cost function at time k... 

The measurement at time t plus the time lag r is predictable on the basis of the autocorrelation coefficient, r(r), and the y value at time t. Here, e represents the random error. Note that this is a very simple model. Good predictions can only be expected for time series that really obey this simple autoregressive model. 

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... 

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