Time Series Analysis: Trend, Seasonality, and Forecasting

A time series is a sequence of observations measured at regular time intervals (daily, monthly, quarterly). Most real-world time series contain a combination of components: (1) Trend (T): the long-run direction of the series — overall increase or decrease over time. (2) Seasonality (S): regular, repeating patterns within a fixed period (monthly retail sales spike in December, flu cases peak in winter). Seasonality has a known, fixed frequency. (3) Cyclical variation (C): longer-term oscillations not tied to a fixed calendar period (business cycles of 3–7 years). (4) Irregular/Random variation (I): unpredictable noise remaining after other components are removed. Classical additive decomposition model: Y_t = T_t + S_t + C_t + I_t — appropriate when seasonal variation is roughly constant in absolute magnitude. Multiplicative decomposition: Y_t = T_t × S_t × C_t × I_t — appropriate when seasonal variation increases proportionally with the trend level (common in economic data). Decomposition is used diagnostically and for seasonal adjustment: removing the seasonal component (seasonally adjusted series) allows comparison of underlying trends across months with different baseline levels. Moving average: a simple trend estimation by averaging consecutive observations over a window (e.g., 12-month centered moving average for monthly data removes seasonality and isolates trend).