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.

Autocorrelation And Stationarity Tests

Tool	Description
ACF	Compute autocorrelation values across lags with optional confidence intervals and Ljung-Box statistics.
ACOVF	Estimate autocovariance values of a time series across lags.
ADFULLER	Run the Augmented Dickey-Fuller unit-root test for stationarity diagnostics.
CCF	Compute cross-correlation between two time series across nonnegative lags.
CCOVF	Estimate cross-covariance values between two time series across lags.
KPSS	Run the KPSS stationarity test under level or trend null hypotheses.
PACF	Compute partial autocorrelation values across lags for lag-order diagnostics.
Q_STAT	Compute Ljung-Box Q statistics and p-values from autocorrelation coefficients.
RURTEST	Run the range unit-root test as an alternative stationarity diagnostic.
ZIVOT_ANDREWS	Run the Zivot-Andrews unit-root test allowing one endogenous structural break.

Decomposition And Seasonality

Tool	Description
DETREND	Remove linear or constant trend from input data.
MSTL	Perform multi-seasonal STL decomposition on a time series.
PERIODOGRAM	Estimate the power spectral density of a time series using a periodogram.
SEASDECOMP	Decompose a time series into trend, seasonal, and residual components.
STL	Perform STL decomposition of a univariate time series.
WELCH	Estimate the power spectral density of a time series using Welch’s method.

Forecasting Models

Tool	Description
ARIMA_FORECAST	Fit an ARIMA model and return out-of-sample forecasts.
ARMA_ORDER_IC	Select ARMA order using an information criterion.
HANNAN_RISSANEN	Estimate ARMA parameters using the Hannan-Rissanen procedure.
HOLT_FORECAST	Fit Holt trend exponential smoothing and return forecasts.
HW_FORECAST	Fit Holt-Winters exponential smoothing and return forecasts.
INNOVATIONS_MLE	Estimate SARIMA parameters using innovations maximum likelihood.
SARIMAX_FORECAST	Fit a SARIMAX model and return out-of-sample forecasts.
SES_FORECAST	Fit simple exponential smoothing and return forecasts.

Moving Averages

Tool	Description
EMA_LFILTER	Compute an exponential moving average using recursive linear filtering.
EMA_PERIOD	Compute an exponential moving average using a period-derived smoothing constant.
SMA_CONV	Compute a simple moving average using discrete convolution with a uniform window.
SMA_CUMSUM	Compute a simple moving average using cumulative-sum differencing.
WINMA_CONV	Compute a weighted moving average by convolving data with a user-defined weight window.
WMA	Compute a rolling weighted moving average using user-supplied weights.