Autoregressive model

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. 

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). 

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

Non-linearity with Memory. The AR-MNL model is clearly somewhat restrictive in that most distortion mechanisms will involve memory. For example an amplifier with a non-linear output stage will probably have feedback so that the memoryless non-linearity will be included within a feedback loop and the overall system could not be modelled as a memory less non-linearity. The general NARMA model incorporates memory but its use imposes a number of analytical problems. A special case of the NARMA model is the NAR (Non-linear AutoRegressive) model in which the current output x[w] is a non-linear function of only past values of output and the present input s [n ]. Under these conditions equation 4.27 becomes ... 

The autoregression model for a relationship between two variables x t) and y(i) and autocorrelated errors is as follows ... 

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. 

Consider an autocorrelated process described by an autoregressive model AR p),... 

AutoRegressive model with eXogenous inputs (ARX). A special case of ARMAX is obtained by letting C q) = 1 ric = 0). 

Dynamics. I. Numerical Integration of the Generalized Langevin Equation through Autoregressive Modeling of the Memory Function. 

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. 

ZHAN, Y. M. JARDINE, A. K. (2005) Adaptive autoregressive modeling of non-stationary vibration signals under distinct gear states. Part 1 modeling. Journal of Sound and Vibration, 286 (3), pp. 429—450. 

After obtaining daily TRP ratios for all stocks in the portfolio, we calculate the 1 month s average TRP ratio for each asset. Then, we compare the current 1 month TRP ratio to the long-term TRP ratio mean, which was obtained by the autoregressive model (ARl). 

Autoregressive Model in this model, all polynomials except the Ap z " )-polynomial are assumed to have zero order. This gives a model of the form... 

It is called an autoregressive model because the model solely depends on the past values of the process itself, that is, Eq. (5.28) can be written as... 

The pure seasonal autoregressive model is similarly defined but solely considers the seasonal component given hy Ap(z ). 

Autoregressive model, AR(p) Exponential decay p significant peaks... 

Estimating the model parameter values is in general performed using one of two methods the method of moments leading to the Yule-Walker equations or the maximum-likelihood method. Although the Yule-Walker equations are simpler, they only provide an efficient estimator for autoregressive models. Also, the Yule-Walker equations are useful for estimating the partial autocorrelation function. Least-squares estimates are also possible, but they are difficult to solve analytically due to the complex nature of the models. 

Yule-Walker Equations for Estimating an Autoregressive Model... 

It can be noted that, irrespective of the approach taken, these equations will generally have to be solved numerically using some form of an optimisation algorithm. The required initial guess can be obtained based on either the Yule-Walker parameter estimates or some other approaches. For autoregressive models, a closed-form solution to the above equations is available. The final result is identical to the Yule-Walker parameter estimates for an autoregressive model. 

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