Decompose Time Series Data

Time Series Decomposition Expert explains how to separate temporal data into trend, seasonal, cyclical, and residual components. It covers additive and multiplicative models, classical moving-average decomposition, STL, X-13ARIMA-SEATS, empirical mode decomposition, ensemble EMD, wavelets, diagnostic assessment, visualization, and method selection. The goal is to make patterns interpretable and useful for analysis.

Use decomposition when analyzing trend and recurring patterns or distinguishing systematic behavior from residual variation. Classical decomposition and STL are presented for regular seasonality; X-13 is described for economic or business series requiring trading-day adjustment; EMD and EEMD address nonlinear or nonstationary data; wavelets support multi-resolution analysis. Consider trend-only models when seasonal strength is low, and robust methods when noise is high.

Inputs include a time-indexed series, its frequency or period, model choice, seasonal smoother, trend settings, and any domain context. A decomposition can return the observed series, trend, seasonal component, and residual or irregular component. Additive models express the series as trend plus seasonal plus residual; multiplicative models combine those components by multiplication. STL offers a robust option and flexible seasonal and trend parameters. An adaptive example infers frequency and selects seasonal lengths for daily, hourly, monthly, or quarterly data.

The classical example calls statsmodels.tsa.seasonal.seasonal_decompose(ts, model="additive", period=12); the period must reflect the series cadence. The adaptive STL example uses STL(ts, seasonal=seasonal_length, robust=True) and can derive seasonal_length from the inferred observation frequency.

X-13 examples include trading-day adjustment, automatic outlier handling, and extending the series for seasonal factors, with a fallback to STL when the external software is unavailable. EMD examples return intrinsic mode functions and a residue; EEMD adds an ensemble for noise robustness. Wavelet examples use discrete transforms to separate a low-frequency approximation from detail components, and continuous transforms to produce coefficients, frequencies, and scales.

The examples use Python, pandas, NumPy, statsmodels, PyEMD, PyWavelets, SciPy, and Matplotlib. Diagnostic code calculates reconstruction mean squared and absolute errors, tests residual autocorrelation with Ljung-Box, checks normality with Shapiro, and estimates trend and seasonal strength. Visualization arranges the observed series, trend, seasonality, and residuals in aligned panels.

Parameter guidance suggests odd STL seasonal lengths, often 7-15 for strong seasonality; db4 for general wavelet use, haar for sharp changes, and coif2 for smooth data. For EMD, it suggests starting max_imf around data length divided by ten, then adjusting based on meaningful components. Always assess residual diagnostics and use domain knowledge to judge whether components are interpretable.

Analysts, statisticians, forecasting practitioners, and engineers working with temporal data can use these methods to explore seasonality and signal structure before building further analyses or models.