SAP HANA Predictive Analysis Library (PAL)

The ARIMA (p,d,q) model is used to model non-seasonal data. To treat with seasonal data, a seasonal ARIMA
(SARIMA) model, written as ARIMA (p,d,q)(P,D,Q) s , is introduced. The definition of SARIMA is as follows:
Where
CSS Estimation
An ARMA (p, q) model can also be expressed as:
Y t =φ 0 +φ 1 Y t-1 +φ 2 Y t-2 +…+φ p Y t-p +…+ε t +θ 1 ε t-1 +θ 2 ε t-2 +…+θ q ε t-q
Where .
Let r = max (p, q), y i , i = 1, 2, …, N denote the observed series data, with the length N,
Y r+1 =φ 0 +φ 1 Y r +φ 2 Y r-1 +…+φ p Y r-p+1 +…+ε r+1 +θ 1 ε r +θ 2 ε r-1 +…+θ q ε r-q+1
Conditional on Y r =y r , Y r-1 =y r-1 , …, Y r-p+1 =y r-p+1 , ε r =ε r-1 =…= ε r-q+1 =0 we have
Y r+1 ~N((φ 0 +φ 1 Y r +φ 2 Y r-1 +…+φ p Y r-p+1 ), σ 2 )
Where the sequence {ε r+1 , ε r+2 , …, ε N } can be calculated by iterations:
ε t =Y t ‒ϕ 0 ‒ϕ 1 Y t-1 ‒ …‒ ϕ p Y t-p ‒θ 1 ε t-1 ‒θ 2 ε t-2 ‒…‒θ q ε t-q , t = r+1, r+2, …, N
Therefore the log likelihood is
L(φ,θ,σ 2 )=logf(Y N ,Y N-1 ,…,Y r+1 |y r ,…,y 1 ,ε r =ε r-1 =…=ε r-q+1 =0,φ,θ,σ 2 )
MLE Estimation
Kalman filtering is applied to calculate MLE. An ARMA (p, q) model can be expressed as a Kalman state space
model.
Where
322 P U B L I C
SAP HANA Predictive Analysis Library (PAL)
PAL Functions