Stationarity

The expected value of the series does not depend on time. The data fluctuates around a fixed horizontal level rather than exhibiting a long-term upward or downward trajectory.

• A non-stationary series with a trend will show a mean that systematically increases or decreases over different time windows.
• Differencing is the primary remediation technique: subtracting the previous observation from the current one to remove the trend component.
• Mathematically, ( E[Y_t] = \mu ) for all time points ( t ).

The dispersion of the series around its mean, known as homoscedasticity, remains uniform across all time periods. The amplitude of fluctuations should not systematically expand or contract.

• A series exhibiting heteroscedasticity—where volatility clusters or grows over time—violates this condition.
• Remediation often involves applying a power transformation, such as a logarithm or Box-Cox transformation, to stabilize the variance.
• Visual diagnosis: the vertical range of oscillations should appear consistent from the beginning to the end of the series.

The covariance between two observations depends only on the lag separating them, not on the specific point in time at which they are measured.

• This means the internal structure of the series—how today's value relates to yesterday's—is stable.
• The autocorrelation function (ACF) plot will show a rapid decay to zero for a stationary series, whereas a non-stationary series exhibits a slow, persistent decay.
• This property ensures that the learned patterns from historical data remain valid for forecasting future periods.

A strictly stationary series must not contain deterministic, repeating patterns tied to fixed calendar periods. Seasonal effects introduce a predictable, time-dependent structure that violates the constant-mean assumption.

• Seasonality can be detected through distinct spikes at seasonal lags (e.g., lag 12 for monthly data) in the ACF plot.
• Seasonal differencing—subtracting the value from the same period in the previous cycle—is used to remove this component.
• While a series can be made stationary through transformations, the raw data itself is considered non-stationary if seasonality is present.

A stationary series exhibits a mean-reverting behavior: deviations from the long-run average are temporary, and the series tends to be pulled back toward its mean.

• This contrasts sharply with a random walk, where shocks have a permanent effect and the series can drift arbitrarily far from any historical level.
• Mean reversion is a critical property for trading and inventory models, as it implies that extreme values are transient and forecasts should regress toward the historical average.
• The Augmented Dickey-Fuller (ADF) test formally evaluates this property by testing the null hypothesis that a unit root is present.

The influence of a past shock on the current value decays geometrically and eventually becomes negligible. The series has a finite, integrable memory.

• This is mathematically expressed as the sum of the autocovariances being finite, a condition known as ergodicity for the second moments.
• Practically, this means that events from the distant past do not permanently alter the current dynamics of the series.
• This property allows models to be estimated reliably from a finite sample of historical data, as the effective sample size is proportional to the length of the series.

A statistical technique that deconstructs a time series into its constituent components: trend, seasonality, and residual noise. Decomposition is often a precursor to achieving stationarity, as isolating and removing a deterministic trend or seasonal pattern is a primary method of transformation. Classical decomposition models can be additive (where components are summed) or multiplicative (where components are multiplied), with the choice depending on whether the seasonal amplitude varies with the level of the series.

A measure of the linear relationship between a time series and a lagged version of itself. The autocorrelation function (ACF) plots the correlation coefficient for various lags, revealing the internal structure of the data. A stationary series will typically exhibit an ACF that decays to zero rapidly, while a non-stationary series shows a slow, persistent decay. This makes the ACF a primary visual diagnostic tool for assessing stationarity before model selection.

A transformation technique used to stabilize the mean of a time series by computing the differences between consecutive observations. First-order differencing (y't = yt - yt-1) can remove a linear trend, while seasonal differencing (y't = yt - yt-m) removes periodic patterns. The order of differencing required to achieve stationarity is a key hyperparameter in ARIMA models, denoted by the 'd' term in ARIMA(p,d,q).

A formal statistical hypothesis test used to objectively determine if a unit root is present in a time series. The null hypothesis states that a unit root is present (the series is non-stationary), while the alternative hypothesis states that the series is stationary. A p-value below a chosen significance threshold (e.g., 0.05) leads to rejecting the null hypothesis, providing statistical evidence that the series is stationary and ready for modeling.

In production, a model's assumption of stationarity can be broken by drift. Data Drift is a change in the statistical properties of the input features (e.g., a shift in the mean of demand). Concept Drift is a change in the fundamental relationship between the features and the target variable. If a series was rendered stationary during training but the underlying data-generating process changes, the model's performance will silently degrade, requiring retraining.