📈 ARIMA (AutoRegressive Integrated Moving Average)

ARIMA (AutoRegressive Integrated Moving Average)#

A classic model combining autoregression, differencing and moving-average terms for forecasting.

Important

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What it is#

ARIMA (AutoRegressive Integrated Moving Average) is a classical statistical model for time-series forecasting that predicts future values from a series’ own past. It fuses three ideas — autoregression (AR), integration / differencing (I) and moving average (MA) — and describes a series by its autocorrelations rather than by explicit trend or seasonality. A model is written ARIMA(p, d, q).

The three parameters#

Each letter is one parameter. p is the AR order — how many lagged past values the current value is regressed on. d is the differencing order — how many times the series is differenced to make it stationary (removing trend). q is the MA order — how many past error terms feed the forecast. In backshift form,

\[\phi_p(B)\,(1 - B)^d\, y_t = \theta_q(B)\,\varepsilon_t,\]

where \(B\) is the backshift operator (\(B y_t = y_{t-1}\)), \(\phi_p\) and \(\theta_q\) are the AR and MA polynomials, and \((1 - B)^d\) applies the differencing. Setting parameters to zero recovers the simpler AR, MA and ARMA models.

Building one (Box-Jenkins)#

The Box-Jenkins recipe has three stages. First, make the series stationary by differencing, checked with a unit-root test such as the Dickey-Fuller test. Next, pick p and q: the ACF (autocorrelation function) guides q, the PACF (partial autocorrelation function) guides p, and among candidates you choose the one with the lowest AIC (or BIC). Finally, validate the residuals — they should be uncorrelated white noise; if not, revisit the orders.

Strengths, limits, and SARIMA#

ARIMA is flexible and interpretable for linear, univariate series — finance, demand, sales — and gives stable longer-term forecasts. But it assumes a linear autocorrelation structure, requires stationarity, and struggles with non-linear patterns where LSTMs or Transformers do better. For periodic data, the SARIMA extension adds seasonal terms, written \(\text{ARIMA}(p, d, q)(P, D, Q)_m\) with season length \(m\).

Theme: Signal Processing & Time Series · All terminology

Hint

Mind map — connected ideas

Prophet — Time Series Forecasting by Facebook (Meta) · LSTM — Long Short-Term Memory Networks · Time Series · Bayesian Time Series · Seasonality · Temporal autocorrelation (Serial Correlation)

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