ML Interview Q Series: In what scenarios would correlation-based distances be advantageous when performing k-Means clustering?

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Comprehensive Explanation

Correlation-based distance focuses on the similarity of the shape of data vectors rather than their magnitude. Instead of relying on absolute numerical differences (as in Euclidean distance), correlation-based measures capture how two data points change relative to each other. This approach becomes beneficial when the main interest is in the pattern or trend across features rather than their raw values.

When using a correlation-based distance, the distance between two vectors u and v might be defined by 1 minus their correlation coefficient. For example, the Pearson correlation-based distance can be represented by the following key formula:

where:

• u and v are data vectors (for instance, feature vectors of two observations).

• Corr(u, v) typically denotes the Pearson correlation between these two vectors.

• d(u, v) = 1 - Corr(u, v) ensures that vectors with a perfect positive correlation (Corr = 1) have distance 0, while vectors with a perfect negative correlation (Corr = -1) have distance 2, and uncorrelated vectors (Corr = 0) have distance 1.

In many clustering applications, the magnitude of the features might be less important than the overall shape of how these features vary together. For example, in gene expression studies, two genes might differ in their expression levels but follow very similar patterns across multiple conditions. If we only used Euclidean distance, the absolute scale differences might dominate the distance. On the other hand, correlation-based distance would emphasize their similar pattern.

k-Means clustering generally uses Euclidean distance in its standard form. However, it can be modified or adapted to accommodate other distance measures, including correlation-based ones. There are two main considerations:

• The centroid recomputation step: When we move away from Euclidean space, the standard mean may not be the direct minimizer of the within-cluster variance in correlation space. Several adaptations, such as transforming vectors to a space where correlation is more naturally handled, are sometimes employed.

• The objective function: k-Means tries to minimize the sum of squared distances within each cluster. If you change the distance metric, you effectively alter what “minimizing within-cluster distance” means. In correlation space, you aim to group items that share strong correlation patterns.

Correlation-based distances help in scenarios involving scale differences, multiplicative factors, or contexts where the exact values are less relevant than whether features rise and fall together. However, if absolute differences in magnitude matter, correlation-based metrics might obscure meaningful distinctions. Additionally, correlation is sensitive to outliers, so if extreme values exist, they might significantly affect the correlation measure.

Practical Implementation Details

While the built-in implementations of k-Means in many libraries (like scikit-learn in Python) default to Euclidean distance, you can still implement correlation-based approaches:

import numpy as np
from sklearn.cluster import KMeans
from sklearn.metrics import pairwise_distances

def correlation_distance(X):
    # X is assumed to be a 2D array: (number_of_samples, number_of_features)
    # Convert correlation to distance
    # We will compute a correlation matrix, then transform to distance
    corr = np.corrcoef(X)
    # correlation-based distance is (1 - corr)
    dist = 1 - corr
    return dist

# Suppose we have data X
X = np.random.rand(100, 5)

# Compute correlation distance matrix
dist_matrix = correlation_distance(X.T)  # Corr is typically computed across samples, adjust if needed.

# KMeans in scikit-learn uses Euclidean, but we can do a workaround or use libraries that support custom distances.
# As a rough approach, you can manually define cluster assignments by computing pairwise distances yourself,
# or explore specialized packages that allow custom distance metrics.

This snippet demonstrates how to compute a correlation distance matrix, but customizing k-Means to use this distance usually involves manual iterative updates or specialized libraries. Another approach is to transform the data so that Euclidean distances in the transformed space reflect correlation-based distances in the original space, though this can be more involved.

Potential Follow-Up Questions

How does correlation distance affect the centroid computation in k-Means?

When using correlation-based distance, the mean of a cluster may no longer serve as the best centroid that minimizes the within-cluster correlation-based distance. In standard Euclidean space, the arithmetic mean is the minimizer of the sum of squared distances. Under correlation metrics, the optimal “centroid” might require iterative or alternative methods, sometimes referred to as “k-medoids-like approaches,” or specialized algorithms such as kernel k-Means that can work with arbitrary distances.

Why might you prefer correlation distance for time series data?

Time series data often involves focus on patterns over time, such as trends, periodicity, or consistent rises and falls. Two time series might have very different amplitudes yet exhibit the same pattern (e.g., both see spikes at the same points in time). Correlation-based distance groups these together based on pattern similarity. However, if the amplitude or absolute differences matter (for example, we really care about how big the spikes are), correlation-based distance may not be as useful.

In what cases is correlation-based distance not suitable?

If the magnitude or absolute scale of data points is an important aspect of your clustering objective, correlation-based distance can hide these differences. Moreover, correlation measures can be heavily influenced by outliers, so if your data is noisy or prone to extreme values, you might want robust alternatives (e.g., rank correlations or robust distance metrics).

How does one handle the objective function if the algorithm is adapted to correlation distance?

Standard k-Means optimizes a sum of squared Euclidean distances. If you shift to correlation, you must revise the objective to minimize the sum of correlation-based distances. This might alter not only the distance calculations but also the update rules for centroids. This leads some practitioners to prefer methods like k-Medoids (Partitioning Around Medoids), which can natively incorporate arbitrary distance measures more gracefully.

How do you normalize data when correlation distance is used?

Correlation inherently ignores the mean and variance, effectively normalizing the data on a per-vector basis. Additional global normalization may not be strictly necessary, but in practice, you might still standardize each feature so that outliers have less impact on the correlation computation. Alternatively, you can apply robust scaling methods (like median-based scaling) if outliers are a major concern.