📂Statistical Analysis

Fractional ARIMA Model (FARIMA)

Model ^{1 2}

Let the Hurst index be represented by $H$. For white noise $e_{t}$ and the backshift operator $B$, and for the 차분 레벨 $d = H - 1/2$, $$ \left( 1 - \sum_{i=1}^{p} \phi_{i} B^{i} \right) \left( 1 - B \right)^{d} Y_{t} = \left( 1 + \sum_{i=1}^{q} \theta_{i} B^{i} \right) e_{t} $$ the $\left\{ Y_{t} \right\}_{t \in \mathbb{N}}$ defined as above is called the $\left( p , d , q \right)$-order fractional ARIMA model $\text{FARIMA} \left( p , d , q \right)$.

Explanation

ARIMA model: $$ \nabla^{d} Y_{t} := \sum_{i = 1}^{p} \phi_{i} \nabla^{d} Y_{t-i} + e_{t} - \sum_{i = 1}^{q} \theta_{i} e_{t-i} $$

Binomial series: $$ \begin{align*} (1 + x )^{\alpha} =& \sum_{k=0}^{\infty} \binom{\alpha}{k} x^{k} \\ =& 1 + \alpha x + \dfrac{\alpha (\alpha-1)}{2!}x^{2} + \dfrac{\alpha (\alpha-1)(\alpha-2)}{3!}x^{3} + \cdots \end{align*} $$

The fractional ARIMA model is, as a generalization of the ARIMA model, an application of the formal series expansion of $\nabla^{d} = \left( 1 - B \right)^{d}$ for real $d \in \mathbb{R}$. $$ \left( 1 - B \right)^{d} = 1 - d B + {\frac{ d (d-1) }{ 2! }} B^{2} - {\frac{ d (d-1) (d-2) }{ 3! }} B^{3} + \cdots $$

The parameters $p$ of the autoregressive process $\text{AR}(p)$ are determined to approximate the infinite series induced by $d$, and can provide a way forward when differencing for stationarity is problematic.

It goes without saying that, in practice, implementing the model is effectively equivalent to including many autoregressive terms, as the formulas suggest. The advantage of ARFIMA is said to be its strength with long-memory; indeed, when long time-series data are involved this is inevitable. The downside, of course, is the increased computational burden, and, secondarily, that the model becomes harder to interpret.