ARIMAXSARIMAX

ARIMAXSARIMAX refers to a class of time series models that extend the traditional ARIMA (AutoRegressive Integrated Moving Average) model by incorporating exogenous variables, hence the "X". SARIMAX stands for Seasonal AutoRegressive Integrated Moving Average with eXogenous variables. This model is capable of capturing both seasonal and non-seasonal patterns in a time series, as well as the influence of external factors that may affect the series.

The core of the SARIMAX model consists of three components: AR (Autoregressive), I (Integrated), and MA (Moving Average). The AR component models the relationship between an observation and a number of lagged observations. The I component accounts for differencing the time series to achieve stationarity, meaning the statistical properties of the series do not change over time. The MA component models the relationship between an observation and a residual error from a moving average model applied to lagged observations. The "S" prefix indicates that seasonal components are also modeled, which are similar to the non-seasonal components but apply to observations at specific seasonal lags. Finally, the "X" denotes the inclusion of one or more exogenous variables, which are external time series that are believed to influence the target time series. These exogenous variables are incorporated linearly into the model.

SARIMAX models are widely used in fields such as econometrics, finance, and environmental science for forecasting and understanding complex time series data where external influences are significant. The model's parameters are typically estimated using maximum likelihood estimation.