tidsseriedecomposition

Tidsseriedecomposition, often referred to in English as time series decomposition, is a set of statistical techniques used to separate a time series into interpretable components that describe different sources of variation over time. The main components are trend, seasonality, and irregular or residual variation. In some formulations a cycle term may be included.

Two common formulations are additive and multiplicative models. In an additive model the observed value at time t is Y_t = T_t + S_t + e_t, where T_t is the trend, S_t is the seasonal component, and e_t is the irregular component. In a multiplicative model Y_t = T_t × S_t × e_t; here the components interact proportionally. The choice between additive and multiplicative forms depends on whether the seasonal fluctuations are roughly constant in magnitude or proportional to the level of the series.

Classical decomposition uses moving averages to estimate the trend component, then seasonal effects are inferred from the residuals by averaging across periods of the same season. Modern approaches such as STL (Seasonal and Trend decomposition using Loess) allow the seasonal and trend components to change over time and are more robust to outliers. Other methods include SEATS and X-12-ARIMA-SEATS, which are designed for more complex seasonal patterns and forecasting contexts.

Applications include seasonality adjustment for forecasting, removing predictable patterns to analyze underlying trends, and improving model fitting in economics, finance, and demand planning. Limitations include nonstationary patterns, changing seasonal shapes, outliers, and calendar effects. Decomposition results depend on the chosen model and data frequency; in some series the components are not clearly separable, and interpretation should consider context.