Autoregressive moving average model

Dina statistik, modél autoregressive moving average (ARMA) nyaéta aplikasi tipikal nu dipaké dina data deret waktu.

Anggap urang boga dua deret waktu, x₁, x₂, x₃, ..., jeung y₁, y₂, y₃, .... Deret x sacara konvensional teu bisa "sacara pasti" di-prediksi ku pangaruh atawa parobahan y. Urang diharéokeun keur ngira-ngira y_t. Lamun modél prediksi ngan miboga watesan x, modél disebut modél moving average (MA). Lamun modél prediksi ngan miboga watesan y, modél disebut modél autoregressive (AR). Lamun prediski miboga duanana watesan boh x sarta y terms, modél disebut modél autoregressive moving average (ARMA).

Model moving average

Lambang MA(q) hartina modél moving average mibanda watesan q. modél MA(q) bisa dituliskeun

$y_t=x_t+{\theta }_1{x}_{t-1}+\cdots +{\theta }_q{x}_{t-q}$

keur sababraha koefisien θ₁, ..., θ_q. modél moving average modél mangrupa hal penting dina finite impulse response filter nu mibanda sawangan tambahan dina éta tempat.

Model Autoregressive

Lambang AR(p) hartina modél autoregressive mibanda watesa p. modél AR(p) bisa dituliskeun

$y_t={\varphi }_1{y}_{t-1}+\cdots +{\varphi }_p{y}_{t-p}$

keur sababaraha koefisien φ₁, ..., φ_p. modél autoregressive modél mangrupa hal penting dina infinite impulse response filter nu mibanda sawangan tambahan dina éta tempat.

Model Autoregressive moving average

Lambang ARMA(p, q) hartina modél mibanda watesan p autoregressive sarta watesan q moving average. Ieu modél mangrupa jumlah tina modél AR jeung MA,

$y_t={\varphi }_1{y}_{t-1}+\cdots +{\varphi }_p{y}_{t-p}+x_t+{\theta }_1{x}_{t-1}+\cdots +{\theta }_q{x}_{t-q}$

Generalisasi

Kawengku kana y_t dina nilai x atawa y saméméhna dianggap bakal linier iwal dina kasus husus. Lamun dependen nonlinéar, modél sacara husus disebut nonlinear moving average (NMA), nonlinear autoregressive (NAR), atawa modél nonlinear autoregressive moving average (NARMA) .

modél autoregressive moving average modéls bisa digeneralisir maké cara séjén. Tempo ogé modél autoregressive conditional heteroskedasticity (ARCH) sarta modél autoregressive integrated moving average (ARIMA).

Rujukan

• George E.P. Box and F.M. Jenkins. Time Series Analysis: Forecasting and Control, second edition. Oakland, CA: Holden-Day, 1976.