后移算子 B B B： B z t = z t − 1 Bz_t = z_{t-1} Bzt=zt−1 前移算子 F F F： F z t = z t + 1 Fz_t = z_{t+1} Fzt=zt+1 后向差分算子 ∇ \nabla ∇： ∇ z t = z t − z t − 1 \nabla z_t = z_t - z_{t-1} ∇zt=zt−zt−1 求和算子 S S S： S z t = z t + z t − 1 + z t − 2 + ⋯ Sz_t = z_t+z_{t-1}+z_{t-2}+\cdots Szt=zt+zt−1+zt−2+⋯
算子之间关系： F = B − 1 , ∇ = 1 − B , S = ∇ − 1 F = B^{-1}, \nabla = 1-B, S = \nabla^{-1} F=B−1,∇=1−B,S=∇−1
z t = μ + a t + ψ 1 a t − 1 + ψ 2 a t − 2 + ⋯ = μ + ψ ( B ) a t z_t = \mu + a_t + \psi_1 a_{t-1} + \psi_2 a_{t-2} + \cdots = \mu +\psi(B) a_t zt=μ+at+ψ1at−1+ψ2at−2+⋯=μ+ψ(B)at
p阶自回归 z t ~ = ϕ 1 z ~ t − 1 + ϕ 2 z ~ t − 2 + ⋯ + ϕ p z ~ t − p + a t \tilde{z_t} = \phi_1 \tilde{z}_{t-1} + \phi_2 \tilde{z}_{t-2} +\cdots + \phi_p \tilde{z}_{t-p} + a_t zt~=ϕ1z~t−1+ϕ2z~t−2+⋯+ϕpz~t−p+at
z t ~ = a t − θ 1 a t − 1 − θ x a t − 2 − ⋯ − θ q a t − q \tilde{z_t} = a_t - \theta_1 a_{t-1} - \theta_x a_{t-2} - \cdots - \theta_q a_{t-q} zt~=at−θ1at−1−θxat−2−⋯−θqat−q
z t ~ = ϕ 1 z ~ t − 1 + ϕ 2 z ~ t − 2 + ⋯ + ϕ p z ~ t − p + a t − θ 1 a t − 1 − θ x a t − 2 − ⋯ − θ q a t − q \tilde{z_t} = \phi_1 \tilde{z}_{t-1} + \phi_2 \tilde{z}_{t-2} +\cdots + \phi_p \tilde{z}_{t-p} +a_t - \theta_1 a_{t-1} - \theta_x a_{t-2} - \cdots - \theta_q a_{t-q} zt~=ϕ1z~t−1+ϕ2z~t−2+⋯+ϕpz~t−p+at−θ1at−1−θxat−2−⋯−θqat−q
(p,d,q)阶自回归求和滑动平均过程(ARIMA) ω t = ∇ d z t \omega_t = \nabla^d z_t ωt=∇dzt ω t = ϕ 1 ω ~ t − 1 + ϕ 2 ω ~ t − 2 + ⋯ + ϕ p ω ~ t − p + a t − θ 1 a t − 1 − θ x a t − 2 − ⋯ − θ q a t − q \omega_t = \phi_1 \tilde{\omega}_{t-1} + \phi_2 \tilde{\omega}_{t-2} +\cdots + \phi_p \tilde{\omega}_{t-p} +a_t - \theta_1 a_{t-1} - \theta_x a_{t-2} - \cdots - \theta_q a_{t-q} ωt=ϕ1ω~t−1+ϕ2ω~t−2+⋯+ϕpω~t−p+at−θ1at−1−θxat−2−⋯−θqat−q
