LDRI - Log Det Ratio Index

The Log Det Ratio Index (LDRI) is an internal clustering evaluation metric [39]. It is a logarithmic variant of the Det-Ratio Index (DRI), scaled by the total number of observations.

Like DRI, it measures the ratio between the determinant of the total scatter matrix (overall data dispersion) and the determinant of the pooled within-cluster scatter matrix (internal cluster dispersion). The logarithm smooths out extreme volume differences, and scaling by the number of observations makes the metric comparable across datasets of different sizes.

\[\text{LDRI} = N \log \left( \frac{|T|}{|W|} \right)\]

Where:

  • \(N\) is the total number of observations (data points).

  • \(|T|\) is the determinant of the Total Scatter Matrix \(T\).

  • \(|W|\) is the determinant of the pooled Within-Cluster Scatter Matrix \(W\)

Handling Edge Cases (Finite Values)

The calculation of LDRI is susceptible to two mathematical edge cases:

  1. Singular Matrix: If the within-cluster scatter matrix is singular (e.g., highly collinear features), its determinant \(|W|\) is 0, causing a division by zero error.

  2. Negative Ratio: In rare numerical instabilities, if the ratio \(|T|/|W| \le 0\), the natural logarithm becomes undefined.

  • force_finite (bool): If True, the function catches these undefined mathematical operations and returns a safe, finite number instead of raising a ValueError. Default is True.

  • finite_value (float): The specific fallback value returned when force_finite=True and the calculation fails. Because a larger score is better for LDRI, the default fallback is a large negative penalty value (-1e10).

Properties

  • Best possible score: +inf (Larger value is better, indicating extremely tight and well-separated clusters).

  • Worst possible score: 0.0 (Since \(|T| \ge |W|\), the minimum valid mathematical ratio is 1, and \(\log(1) = 0\). However, the penalty score falls back to the defined negative value).

  • Range: [0.0, +inf) (or defined penalty bound).

Example Usage

from permetrics.clustering import ClusteringMetric
import numpy as np

# ==============================================================================
# SCENARIO 1: Normal Clustering Evaluation
# ==============================================================================
print("--- 1. BASIC LOG DET RATIO INDEX EXAMPLE ---")

X_data = np.array([[1, 2, 1], [1, 4, 2], [1, 0, 1], [10, 2, 8], [10, 4, 9], [10, 0, 7]])
y_pred_labels = np.array([0, 0, 0, 1, 1, 1])

cm = ClusteringMetric(X=X_data, y_pred=y_pred_labels)
ldri_score = cm.LDRI()
print(f"Log Det Ratio Index: {ldri_score}")

# ==============================================================================
# SCENARIO 2: Edge Case with Singular Matrix (|W| = 0)
# ==============================================================================
print("\n--- 2. EDGE CASE (SINGULAR MATRIX) EXAMPLE ---")

# Highly collinear data where the within-cluster determinant will evaluate to 0
X_singular = np.array([[1, 1], [2, 2], [3, 3], [10, 10], [11, 11], [12, 12]])
cm_singular = ClusteringMetric(X=X_singular, y_pred=y_pred_labels)

# Returns the penalty finite_value (-1e10) instead of crashing
ldri_safe = cm_singular.LDRI(force_finite=True, finite_value=-1e10)
print(f"LDRI with Singular Matrix (Safe Mode): {ldri_safe}")