Stationarity

/ˌsteɪ.ʃəˈnɛr.ɪ.ti/

noun … “when time stops twisting the rules of a system.”

Stationarity is a property of a Time Series or stochastic process where statistical characteristics—such as the mean, variance, and autocorrelation—remain constant over time. A stationary series exhibits no systematic trends or seasonality, meaning its probabilistic behavior is invariant under time shifts. This property is essential for many time-series analyses and forecasting models, as it ensures that relationships learned from historical data are valid for predicting future behavior.

There are different forms of Stationarity. Strict stationarity requires that the joint distribution of any subset of observations is identical regardless of shifts in time. Weak (or wide-sense) stationarity is a more practical criterion, requiring only that the mean and autocovariance between observations depend solely on the lag between them, not the absolute time. Weak stationarity is sufficient for most statistical modeling, including methods like ARIMA and spectral analysis.

Stationarity intersects with several key concepts in time-series analysis. It is assessed through Autocorrelation functions, statistical tests (e.g., Augmented Dickey-Fuller), and visual inspection of rolling statistics. Achieving stationarity is often necessary before applying models such as AR, MA, ARMA, or Linear Regression on temporal data. Non-stationary series can be transformed using differencing, detrending, or seasonal adjustments to stabilize mean and variance.

Example conceptual workflow for verifying and achieving stationarity:

collect time-series dataset
plot series to observe trends and variance
compute rolling mean and variance to detect changes over time
apply statistical tests for stationarity
if non-stationary, perform differencing or detrending
reassess until statistical properties are approximately constant

Intuitively, Stationarity is like a calm lake where ripples occur but the overall water level and pattern remain steady over time. It provides a reliable foundation for analysis, allowing the underlying structure of data to be understood and future behavior to be forecast with confidence.