We have a Source distribution, $P_{\text{source}}$, and a Target distribution, $P_{\text{target}}$:

• Source distribution: $(X^{\text{source}}, Y^{\text{source}}) ∼ P_{\text{source}}$

• Target distribution: $(X^{\text{target}}, Y^{\text{target}}) ∼ P_{\text{target}}$

Covariate drifts happen when:

$$P_{\text{target}}(Y \mid X) = P_{\text{source}}(Y \mid X) \quad \text{but} \quad P_{\text{target}}(X) \ne P_{\text{source}}(X)$$

• Maximum Mean Discrepancy Two-Sample Test - MMD Test

from source.mmd import MMD_test
"""
x_before (np.ndarray): First sample of shape (n, d) from source distribution.
x_after (np.ndarray): Second sample of shape (m, d) from reference distribution.
n_permutations (int): Number of permutations for the permutation test.
sigma (float): Bandwidth parameter for the Gaussian kernel.
"""
sigma = 1.0
n_permutations=1000
mmd_statistic, mmd_perms, pval = MMD_test(x_before, x_after, sigma, n_permutations=n_permutations)
print(f"MMD Statistic: {mmd}, p-value: {pval}")

• Log-Likelihood Ratio Test - LLR Test

from source.ratio import LLR_test
"""
x_before (np.ndarray): First sample of shape (n, d) from source distribution.
x_after (np.ndarray): Second sample of shape (m, d) from reference distribution.
bandwidth (float): Bandwidth parameter for KDE.
n_permutations (int): Number of permutations for the permutation test. Default is 1000.
"""
bandwidth = 0.5
n_permutations=1000
llr_statistic, llr_perms, p_value = LLR_test(x_before, x_after, bandwidth=bandwidth, n_permutations=n_permutations)
print(f'LLR Statistic: {llr_statistic}, p-value: {p_value}')

• Data stream with simulated mean drifts

• Drifts detected with MOVING reference window with LLR-test

Useful to build adaptive learning models in streaming environments. The learning model is updated or rebuilt as soon as a drift-event is detected.

• Drifts detected with FIXED reference window with LLR-test

Useful to monitor automated systems and identify the full duration of the concept drift. The reference period should be representative.

0. Offline layer: define the Reference Component

Using data collected offline, perform the following steps:

• 1. Define the reference distribution: select a fixed portion of the offline data to construct a stable covariate distribution representing normal condition.
• 2. Simulate streaming data via batch sampling: from the remaining offline data, draw multiple batches to simulate streaming behavior. For each batch, compute the statistic of interest.
• 3. Model null distribution: Aggregate the statistics to form a distribution (i.e. Null hypothesis) that captures the natural variability of the statistic under normal conditions. This distribution serves as a reference and is passed to the streaming layer for real-time monitoring.

1. Streaming layer: define Monitoring component

For each incoming batch of data in streaming:

• 1. Compare the current batch against the reference distribution by computing the statistic of interest.
• 2. Verify where the computed statistic fall within the null distribution derived in the offline layer. If the statistic exceeds a predifined threshold (e.g, quantile(1-$\alpha$)), flag the batch as a potential drift event and trigger an alert.