Autoregressive Integrated Moving Average (ARIMA) is a popular time series analysis and forecasting method that combines autoregression, differencing, and moving averages. ARIMA models are used to understand the underlying structure in a time series and make predictions about future points in the series.

The term “ARIMA” stands for:

1. **Autoregressive (AR):** The model includes terms that measure the relationship between an observation and its past values. The “p” parameter represents the order of the autoregressive component, indicating how many past values are used.

2. **Integrated (I):** The model includes differencing of the time series data. Differencing is the process of subtracting the previous observation from the current one to make the series stationary. The “d” parameter represents the order of differencing.

3. **Moving Average (MA):** The model includes terms that relate to the past forecast errors. The “q” parameter represents the order of the moving average component, indicating how many past forecast errors are used.

The general form of an ARIMA model is denoted as ARIMA(p, d, q). The model can be expressed mathematically as:

\[ Y_t’ = c + \phi_1 \cdot Y_{t-1}’ + \phi_2 \cdot Y_{t-2}’ + \ldots + \phi_p \cdot Y_{t-p}’ + \theta_1 \cdot \varepsilon_{t-1} + \theta_2 \cdot \varepsilon_{t-2} + \ldots + \theta_q \cdot \varepsilon_{t-q} + \varepsilon_t \]

where:

– \( Y_t \) is the observed time series,

– \( Y_t’ \) is the differenced time series,

– \( c \) is a constant term,

– \( \phi_1, \phi_2, \ldots, \phi_p \) are the autoregressive coefficients,

– \( \theta_1, \theta_2, \ldots, \theta_q \) are the moving average coefficients,

– \( \varepsilon_t \) is the error term at time \(t\),

– \( p \) is the order of the autoregressive component,

– \( d \) is the order of differencing,

– \( q \) is the order of the moving average component.

The steps involved in building an ARIMA model typically include:

1. **Identifying Stationarity:** Check whether the time series is stationary. If not, apply differencing until stationarity is achieved.

2. **Choosing Parameters (\(p, d, q\)):** Determine the order of autoregressive (\(p\)), differencing (\(d\)), and moving average (\(q\)) components based on the autocorrelation function (ACF) and partial autocorrelation function (PACF) of the differenced series.

3. **Estimating Coefficients:** Use statistical methods to estimate the coefficients of the ARIMA model.

4. **Model Evaluation:** Evaluate the model’s performance using appropriate metrics and diagnostics.

ARIMA models are widely used for time series forecasting in various fields, including finance, economics, and environmental science. They provide a flexible framework for capturing different patterns and trends in time-dependent data.