Stationarity Test

The Augmented Dickey-Fuller (ADF) test is the most common unit root test for stationarity. It assesses the null hypothesis that a time series has a unit root (i.e., is non-stationary). A low p-value (typically <0.05) leads to rejecting the null, indicating the series is stationary.

• Mechanism: It fits a regression model to the differenced series with lagged difference terms to correct for higher-order autocorrelation.
• Key Output: A test statistic and a p-value. The more negative the test statistic, the stronger the rejection of non-stationarity.
• Use Case: The standard first test for checking if a series needs differencing to become stationary for ARIMA modeling.

The Kwiatkowski-Phillips-Schmidt-Shin (KPSS) test inverts the hypothesis of the ADF test. Its null hypothesis is that the series is stationary (around a deterministic trend). A test statistic exceeding a critical value rejects stationarity.

• Mechanism: It decomposes the series into a deterministic trend, a random walk, and a stationary error, testing the variance of the random walk component.
• Complementary Use: Used alongside the ADF test for a robust conclusion. An ideal result is ADF rejects non-stationarity (p<0.05) and KPSS does not reject stationarity (p>0.05).
• Trend Handling: Has variants for testing stationarity around a level versus around a deterministic trend.

The Phillips-Perron (PP) test is another unit root test similar to the Dickey-Fuller test but uses non-parametric statistical methods to account for serial correlation and heteroskedasticity in the error terms.

• Key Difference: Unlike the ADF test, which adds lagged difference terms to the model, the PP test modifies the test statistic directly. This makes it more robust to a wide range of error specifications.
• Advantage: It is often more powerful than the basic Dickey-Fuller test when the error process is complex.
• Consideration: While robust, it can be less reliable in finite samples compared to the ADF test with appropriately chosen lags.

The Zivot-Andrews test is a unit root test that allows for a single structural break in the series at an unknown point in time. This is critical because a structural break can make a stationary series appear non-stationary to standard tests.

• Mechanism: It tests the null hypothesis of a unit root with drift, against the alternative of a trend-stationary process with a one-time break in the intercept, trend, or both.
• Primary Use: Diagnosing whether an apparent unit root is actually caused by a sudden shift in the series' mean or growth rate (e.g., a policy change, market crash).
• Drift Detection Link: A detected structural break is a definitive form of drift, signaling a permanent change in the data-generating process.

While not a direct test of series stationarity, the Ljung-Box test is applied to the residuals of a time series model (like ARIMA) to check for remaining autocorrelation. Its failure is a key diagnostic for inadequate stationarity transformation.

• Purpose: Tests the null hypothesis that the residuals are independently distributed (no autocorrelation).
• Application in Stationarity: After differencing a series to achieve stationarity, the Ljung-Box test on the resulting series' autocorrelations helps verify if the differencing was sufficient. Significant p-values indicate remaining structure and potential non-stationarity.
• Lag Parameter: The test is applied over a set number of lags (e.g., 10, 20) to check for long-range dependence.

Autocorrelation Function (ACF) and Partial Autocorrelation Function (PACF) plots are essential visual diagnostics for stationarity before formal testing.

• Stationary Series ACF: The ACF of a stationary series decays to zero relatively quickly. A slow, linear decay suggests a unit root (non-stationarity).
• Differencing: If the ACF decays slowly, differencing the series and re-plotting the ACF is the standard remedy. A stationary series after differencing will show an ACF that cuts off or decays rapidly.
• PACF Role: The PACF helps identify the order of an autoregressive (AR) process, which is only valid for stationary series. A sharp cutoff in the PACF suggests the AR order.
• Practical First Step: Always plot the series, its ACF, and PACF to guide the choice and interpretation of formal statistical tests.

A change in the statistical properties of the target variable (the concept a model aims to predict) over time in relation to the input features. This degrades a model's predictive performance. Key characteristics:

• The relationship P(Y|X) between inputs and outputs changes.
• Often requires labeled data for detection.
• A stationarity test on model error rates can be an indirect indicator of concept drift.

A change in the distribution of the input features P(X) between training and deployment, while the relationship P(Y|X) remains stable. Key characteristics:

• Can be detected on unlabeled data.
• Stationarity tests (like ADF) are directly applied to individual feature streams to detect non-stationarity.
• A primary cause of model performance decay even if the underlying concept is stable.

The identification of specific points in a time series where its statistical properties undergo a significant, often abrupt, change. Key characteristics:

• Focuses on locating the when of a shift.
• Methods include CUSUM and the Page-Hinkley Test.
• Stationarity tests (e.g., ADF) can signal a change but do not precisely localize it; they assess an entire series.

A statistical method for determining if two data samples originate from the same underlying distribution. Forms the basis for batch drift detection. Common tests include:

• Kolmogorov-Smirnov Test: Compares empirical cumulative distribution functions.
• Maximum Mean Discrepancy (MMD): A kernel-based test.
• Stationarity tests like ADF are a specialized form for time series, testing a single sample against the property of stationarity.

A method of quality control using statistical techniques like control charts to monitor a process. Adapted for model monitoring. Key tools:

• Control Charts: Plot a metric (e.g., prediction error) over time with control limits.
• CUSUM Charts: A specific type for detecting small mean shifts.
• Stationarity is an underlying assumption for many SPC methods; non-stationarity violates control limit assumptions.

The task of identifying input data that falls outside the distribution of a model's training data. A key component of unsupervised drift detection. Relation to stationarity:

• A stream of OOD detection scores (e.g., likelihood) should be stationary under normal operation.
• A stationarity test on this score can signal when novel, anomalous data patterns are becoming systematic, indicating drift.