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The traditional test for the presence of first-order autocorrelation is the Durbin–Watson statistic or, if the explanatory variables include a lagged dependent variable, Durbin's h statistic. The Durbin-Watson can be linearly mapped however to the Pearson correlation between values and their lags. [12]
The first order correlation function, measured at the same time and position gives us the intensity i.e. () (,) =. The classical nth order normalized correlation function is defined by dividing the n-th order correlation function by all corresponding intensities:
Correlogram example from 400-point sample of a first-order autoregressive process with 0.75 correlation of adjacent points, along with the 95% confidence intervals (plotted about the correlation estimates in black and about zero in red), as calculated by the equations in this section.
Durbin and Watson (1950, 1951) applied this statistic to the residuals from least squares regressions, and developed bounds tests for the null hypothesis that the errors are serially uncorrelated against the alternative that they follow a first order autoregressive process. Note that the distribution of this test statistic does not depend on ...
An alternative formulation is in terms of the autocorrelation function. The AR parameters are determined by the first p+1 elements () of the autocorrelation function. The full autocorrelation function can then be derived by recursively calculating [8]
Plotting the partial autocorrelation function and drawing the lines of the confidence interval is a common way to analyze the order of an AR model. To evaluate the order, one examines the plot to find the lag after which the partial autocorrelations are all within the confidence interval. This lag is determined to likely be the AR model's order ...
Visual comparison of convolution, cross-correlation and autocorrelation.. A correlation function is a function that gives the statistical correlation between random variables, contingent on the spatial or temporal distance between those variables. [1]
The second order autocorrelation curve is generated from the intensity trace as follows: (;) = (+) where g 2 (q;τ) is the autocorrelation function at a particular wave vector, q, and delay time, τ, and I is the