Dimensionindex

Dimensionindex is a term used to describe a scalar metric that summarizes the effective dimensionality of a dataset or feature space. It is intended to reflect how many directions in the data carry meaningful variation, rather than simply counting measured features.

Calculation of the dimensionindex often relies on the eigenvalue spectrum of the data’s covariance matrix. A common approach is the participation ratio, defined as PR = (sum of eigenvalues)^2 divided by the sum of squared eigenvalues. Other methods include counting the number of eigenvalues above a threshold or estimating intrinsic dimension through spectral gaps. In practice, the dimensionindex provides a single number that captures the balance between variance explained and noise across components.

Interpretation of the dimensionindex depends on context. For data with equal variance across all components, the index approaches the dimensionality of the feature space. For data concentrated along a smaller subspace, the index is lower and can indicate a lower intrinsic dimensionality. It is not a universal standard, but a useful heuristic in conjunction with visualization and domain knowledge.

Applications include guiding dimensionality reduction decisions, such as selecting the number of principal components, informing model complexity, and aiding anomaly detection by highlighting deviations in dimensionality. The concept is related to intrinsic dimension, fractal dimension, and other estimators that seek to quantify effective dimensionality beyond raw feature counts.

Limitations include sensitivity to scaling, noise, and sample size, and the fact that different definitions may yield different results. Users should consider the method’s assumptions and interpretation within their specific domain. See also intrinsic dimension, PCA, and dimensionality reduction.