Detecting Stationarity in Time Series

### What is Time Series Stationarity?

Time series stationarity refers to the property where statistical characteristics (such as mean, variance, and autocorrelation structure) remain constant over time. Stationarity is a fundamental assumption in many time series analyses (like ARIMA modeling) because only stationary data can ensure that the model's predictive capability remains unaffected by temporal changes. In code implementation, stationarity testing typically involves checking if these statistical properties show significant variation across different time segments.

### How to Test for Stationarity?

Common methods for detecting time series stationality include:

Visual Inspection: Plotting the time series trajectory along with rolling mean and variance calculations can provide initial insights into trends or seasonality. In Python, this can be implemented using matplotlib with rolling statistics computed via pandas.DataFrame.rolling().

ADF Test (Augmented Dickey-Fuller Test): A statistical hypothesis testing method that examines whether a time series contains a unit root (non-stationarity). If the p-value falls below the significance level (e.g., 0.05), we reject the null hypothesis (non-stationary) and conclude stationarity. Implementation typically uses statsmodels.tsa.stattools.adfuller() function which returns test statistics and critical values.

KPSS Test: Complementary to ADF test, with the null hypothesis assuming stationarity. Discordant results between ADF and KPSS may indicate trend-stationarity or difference-stationarity. The kpss() function in statsmodels provides this implementation with automatic lag selection.

ACF/PACF Analysis: Stationary data typically exhibits rapid decay in autocorrelation functions, while non-stationary data shows slow decay patterns. The statsmodels.tsa.stattools.acf() and pacf() functions help visualize these characteristics with confidence intervals.

### Relationship Between Stationarity Testing and Chaos Detection

Traditional chaos detection methods (like Lyapunov exponents, fractal dimensions) typically analyze long-term behaviors of nonlinear dynamical systems, while stationarity testing focuses on statistical stability. In some cases, non-stationarity may originate from dynamic changes in chaotic systems, making stationarity testing a valuable supplementary approach for chaos analysis. For example, a seemingly random time series might exhibit non-stationarity due to underlying chaotic mechanisms rather than noise. Combining stationarity tests with chaos analysis methods (such as using nolds package for Lyapunov exponent calculation) provides comprehensive data characterization.

Overall, stationarity testing serves as a foundational step in time series analysis. While not directly equivalent to chaos detection, it can function as a reference indicator for chaotic system analysis, particularly when implementing hybrid approaches that integrate statistical tests with dynamical system characterization.