# 20 NOV 2023

SARIMA, which stands for Seasonal AutoRegressive Integrated Moving Average, is an extension of the traditional ARIMA (AutoRegressive Integrated Moving Average) model in statistics. SARIMA is a powerful tool for modeling and forecasting time series data that exhibit both non-seasonal and seasonal patterns. It’s particularly useful when dealing with data that displays regular fluctuations or seasonality at fixed time intervals, such as monthly sales data, quarterly economic indicators, or daily temperature readings.

SARIMA incorporates the following components:

1. AutoRegressive (AR) Component: Like in ARIMA, the AR component models the relationship between the current observation and its previous values. It captures the serial correlation in the data and is denoted by the order ‘p.’
2. Integrated (I) Component: The integrated component deals with differencing the data to achieve stationarity, similar to ARIMA. The order of differencing is ‘d’ and is determined empirically.
3. Moving Average (MA) Component: The MA component accounts for the correlation between the current observation and past error terms, reducing the effect of random shocks on the data. It’s represented by the order ‘q.’
4. Seasonal AutoRegressive (SAR) Component: SARIMA introduces the seasonal AR component, denoted by ‘P,’ which models the relationship between the current observation and its previous seasonal observations at a fixed lag.
5. Seasonal Integrated (SI) Component: The seasonal integrated component, denoted by ‘D,’ captures the differencing needed to remove seasonality at the seasonal lag. This order is determined empirically like ‘d.’
6. Seasonal Moving Average (SMA) Component: Finally, the seasonal MA component, represented by ‘Q,’ accounts for the correlation between the current observation and past seasonal errors at the seasonal lag.

Selecting the appropriate orders for these components (p, d, q, P, D, Q) is a critical step in SARIMA modeling, usually guided by data analysis, visualization, and statistical criteria. Once determined, you can estimate the model parameters using methods like maximum likelihood estimation.

SARIMA models are versatile and capable of capturing complex seasonal patterns in time series data. They provide a robust framework for forecasting future values while considering both short-term and long-term dependencies. Software packages like Python’s statsmodels and R’s forecast library offer tools for implementing SARIMA models, making them accessible and widely used in various fields, including economics, finance, and climate science, among others, where seasonal patterns play a significant role in data analysis and prediction.