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Non-seasonal ARIMA models are usually denoted ARIMA(p, d, q) where parameters p, d, q are non-negative integers: p is the order (number of time lags) of the autoregressive model, d is the degree of differencing (the number of times the data have had past values subtracted), and q is the order of the moving-average model.
The enhancement to ordinary ARMA models is as follows: Take the original data series and high-pass filter it with fractional differencing enough to make the result stationary, and remember the order d of this fractional difference, d usually between 0 and 1 ... possibly up to 2+ in more extreme cases.
MATLAB includes functions such as arma, ar and arx to estimate autoregressive, exogenous autoregressive and ARMAX models. See System Identification Toolbox and Econometrics Toolbox for details. Julia has community-driven packages that implement fitting with an ARMA model such as arma.jl.
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To estimate the total number of lags, use the Ljung–Box test until the value of these are less than, say, 10% significant. The Ljung–Box Q-statistic follows χ 2 {\displaystyle \chi ^{2}} distribution with n degrees of freedom if the squared residuals ϵ t 2 {\displaystyle \epsilon _{t}^{2}} are uncorrelated.
This is a form of superresolution. Its chief disadvantage is that it requires the number of components to be known in advance, so the original method cannot be used in more general cases. Methods exist for estimating the number of source components purely from statistical properties of the autocorrelation matrix.
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