Time Series

Overview

Time series analysis studies observations indexed in time order and models how current values depend on past behavior, seasonality, shocks, and noise. It is a core method in forecasting, monitoring, and signal interpretation across finance, operations, IoT, and scientific measurement. The foundational ideas are summarized in the Time series literature and then specialized for stationarity testing, decomposition, smoothing, and model-based prediction.

The unifying concepts are serial dependence, stationarity, trend/seasonality decomposition, and stochastic forecasting models. Autocorrelation diagnostics quantify dependence across lags, while stationarity tests assess whether distributional properties remain stable over time. Decomposition methods separate signals into components such as trend T_t, seasonal effects S_t, and residuals R_t, often expressed as y_t = T_t + S_t + R_t. Forecasting frameworks then model dynamics directly, including ARIMA-family equations such as \phi(L)(1-L)^d y_t = \theta(L)\varepsilon_t.

Implementation in this category is primarily backed by statsmodels for econometric and forecasting routines, with complementary signal-processing utilities from SciPy and numerical primitives from NumPy. This combination provides statistically grounded inference and efficient vectorized computation.

For dependence diagnostics and unit-root testing, ACF, ACOVF, PACF, CCF, and CCOVF measure auto/cross-structure across lags and help identify candidate model orders. Q_STAT adds Ljung-Box significance checks for residual whiteness. Formal stationarity decisions are supported by ADFULLER, KPSS, RURTEST, and ZIVOT_ANDREWS, including break-aware testing for structural shifts.

For signal preparation and seasonal structure analysis, DETREND removes linear or constant drift prior to downstream modeling, while SEASDECOMP, STL, and MSTL separate observed series into trend, seasonal, and remainder components under single- or multi-seasonal assumptions. In frequency-domain workflows, PERIODOGRAM and WELCH estimate spectral density to reveal dominant cycles and periodic drivers.

For model-driven prediction, ARMA_ORDER_IC and HANNAN_RISSANEN support ARMA order/parameter estimation, and INNOVATIONS_MLE provides likelihood-based SARIMA parameter fitting. Forecast generation is handled by ARIMA_FORECAST and SARIMAX_FORECAST for autoregressive integrated models with optional seasonal and exogenous structure. Smoother level/trend-seasonality forecasting is available through SES_FORECAST, HOLT_FORECAST, and HW_FORECAST.

For rolling smoothing and feature engineering, EMA_LFILTER and EMA_PERIOD compute exponential moving averages with direct- or period-based smoothing control, while SMA_CONV and SMA_CUMSUM provide two efficient implementations of simple moving averages. Weighted variants WINMA_CONV and WMA emphasize recent or user-prioritized observations and are common in trading signals, KPI trend dashboards, and preprocessing.