While monitoring individual features for drift using univariate statistical tests like Kolmogorov-Smirnov provides a basic check, it often fails to capture the full picture. Production data rarely changes one feature at a time. More commonly, the relationships between features shift, altering the joint distribution of the data even if the marginal distributions of individual features appear stable. Relying solely on univariate tests can lead to a false sense of security, missing significant drifts that degrade model performance.

Consider a model predicting loan defaults based on income and debt_level. The individual distributions of income and debt might remain similar over time. However, if the correlation changes, perhaps higher-income individuals start taking on proportionally more debt than before, the model's understanding of risk based on the training data's correlations becomes outdated. This change in the joint distribution is multivariate drift, and detecting it requires techniques that look beyond single features.

The Challenge of High Dimensions

Directly comparing high-dimensional probability distributions is computationally expensive and statistically challenging due to the "curse of dimensionality." As the number of features ( $d$ ) increases, the volume of the feature space grows exponentially, making the data points increasingly sparse. This sparsity makes it difficult to estimate density accurately or apply traditional statistical tests reliably. Multivariate drift detection methods aim to overcome this by summarizing the high-dimensional distribution or focusing on specific aspects sensitive to change.

Using Distance Metrics: Mahalanobis Distance

One approach is to use distance metrics that account for the correlation structure of the data. The Mahalanobis distance is a prominent example. Unlike Euclidean distance, which treats all dimensions equally, Mahalanobis distance measures the distance between a point and a distribution's center (mean), scaled by the data's covariance.

For a point $x$ and a distribution with mean $\mu$ and covariance matrix $\Sigma$ , the squared Mahalanobis distance is:

D_M^2(x, \mu, \Sigma) = (x - \mu)^T \Sigma^{-1} (x - \mu)

In the context of drift detection, we compare a target (production) dataset to a reference (training) dataset. We can compute the Mahalanobis distance of each point in the target dataset relative to the reference distribution's mean ( $\mu_{ref}$ ) and covariance ( $\Sigma_{ref}$ ).

The distribution of these distances provides insight into drift. If the target data follows the same distribution as the reference data, the squared Mahalanobis distances should approximately follow a Chi-squared ( $\chi^2$ ) distribution with $d$ degrees of freedom (where $d$ is the number of features), assuming the data is multivariate normal.

A common approach is:

Calculate $\mu_{ref}$ and $\Sigma_{ref}$ from the reference data. Ensure $\Sigma_{ref}$ is invertible (handle multicollinearity if necessary, e.g., using regularization or dimensionality reduction first).
For a window of incoming target data, calculate $D_M^2(x_{target}, \mu_{ref}, \Sigma_{ref})$ for each point $x_{target}$ .
Compare the distribution of these calculated distances to the expected $\chi^2_d$ distribution using a statistical test (e.g., KS test). Alternatively, monitor the mean or specific quantiles of the $D_M^2$ values over time and trigger an alert if they deviate significantly from expected values based on the $\chi^2_d$ distribution.

Advantages:

Accounts for linear correlations between features.
Provides a single statistic summarizing multivariate distance.

Disadvantages:

Assumes data is approximately multivariate normal for the $\chi^2$ test interpretation.
Sensitive to the accuracy of the estimated covariance matrix ( $\Sigma_{ref}$ ), which requires sufficient data.
Requires the covariance matrix to be invertible.
Less effective at capturing non-linear relationships.

The marginal distributions (projections onto Feature 1 or Feature 2 axes) for the Reference (blue) and Target (orange) datasets might appear similar. However, the correlation structure has clearly changed, indicating multivariate drift. A univariate test on each feature might miss this shift.

Dimensionality Reduction Approaches

Another strategy is to first reduce the dimensionality of the data and then apply drift detection methods (including univariate ones) in the lower-dimensional space. The idea is that significant changes in the high-dimensional structure will manifest as changes in the lower-dimensional representation.

Principal Component Analysis (PCA) is a common choice.

Fit PCA on the reference data to find the principal components (directions of maximum variance).
Project both the reference and target data onto the first few principal components.
Apply univariate drift detection tests (e.g., KS test, population stability index) to the distributions of the data along each selected principal component.

Alternatively, monitor the principal components themselves. A significant change in the data distribution might alter the directions of maximum variance or the amount of variance explained by each component. Comparing the PCA eigenspectrum (eigenvalues) between the reference and target data can reveal such structural changes.

Advantages:

Reduces computational complexity by working in lower dimensions.
Can capture linear correlation changes reflected in principal components.

Disadvantages:

Information loss during dimensionality reduction might obscure certain types of drift.
The effectiveness depends heavily on the chosen dimensionality reduction technique and the number of components retained. PCA specifically focuses on variance, not necessarily on the directions most sensitive to drift.

Other Multivariate Techniques

Comparing Covariance Matrices Directly: Methods exist to directly compare the reference covariance matrix $\Sigma_{ref}$ with a target covariance matrix $\Sigma_{target}$ calculated on a window of recent data. This can involve calculating matrix distances (e.g., Frobenius norm $||\Sigma_{ref} - \Sigma_{target}||_F$ ) or statistical tests based on likelihood ratios under assumptions like multivariate normality. This directly targets changes in the linear relationships between features.
Domain Classifiers (Adversarial Validation): As briefly mentioned in the chapter introduction, training a classification model to distinguish between reference data (label 0) and target data (label 1) is a powerful, model-agnostic technique. If the classifier achieves high accuracy (e.g., AUC significantly greater than 0.5), it indicates that the two datasets are distinguishable, meaning drift has occurred. The features the classifier relies on most heavily can also help diagnose the nature of the drift. This approach is explored in detail later in the "Using Adversarial Validation for Drift Assessment" section.

Choosing and Implementing

The best multivariate drift detection method depends on factors like:

Dimensionality: Mahalanobis distance becomes less reliable in very high dimensions without sufficient data or regularization. Dimensionality reduction or domain classifiers might be more suitable.
Data Volume: Estimating covariance matrices or training domain classifiers requires a reasonable amount of data in both reference and target windows.
Computational Resources: Some methods, like direct covariance comparison or complex classifiers, can be computationally intensive.
Interpretability: Distance metrics provide a single drift score. Domain classifiers or PCA-based methods can sometimes offer more insight into which features or relationships are drifting.
Assumptions: Be mindful of underlying assumptions, such as multivariate normality for certain statistical interpretations of Mahalanobis distance or covariance comparison tests.

In practice, you might employ multiple methods. For instance, using Mahalanobis distance for a quick check on overall distributional shift, supplemented by a domain classifier run periodically or when the distance metric flags potential drift, to get a more robust assessment and better interpretability. The hands-on exercise later in this chapter will provide practical experience implementing one of these techniques.