- Autoregressive fractionally integrated moving average
In

statistics ,**autoregressive fractionally integrated moving average**models aretime series models that generalizeARIMA ("autoregressive integrated moving average") models by allowing non-integer values of the differencingparameter and are useful in modeling time series with long memory.In an

ARIMA model, the "integrated" part of the model includes the differencing operator, in terms of the backspace operator "B", as an integer power of (1 − "B"). For example :$(1-B)^2=1-2B+B^2\; ,,$where:$B^2X\_t=X\_\{t-2\}\; ,.$In a "fractional" model, the power is allowed to be fractional, with the meaning of the term identified using the following formal expansion:$(1-B)^d=1-dB+frac\{d(d-1)\}\{2\}B^2\; +cdots\; ,.$**References*** C. W. J. Granger and R. Joyeux. "An introduction to long-memory time series and fractional differencing", Journal of Time Series Analysis, 1980.

* J. R. M. Hosking. "Fractional differencing", Biometrika 68(1):165-176, 1981.

* P. M. Robinson. "Time Series With Long Memory", Oxford University Press 2003.

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