Shannon information and receiver operating characteristic analysis for multiclass classification in imaging

Area of Science:

• Information theory
• Statistical decision theory
• Medical imaging analysis

Background:

• Receiver Operating Characteristic (ROC) analysis is widely used for binary classification evaluation.
• Extending ROC analysis to multiclass problems presents challenges in interpretation and application.
• Shannon information offers a theoretical framework for quantifying information in communication systems.

Purpose of the Study:

• To establish a mathematical relationship between Shannon information and ROC analysis for multiclass classification.
• To introduce and define a Shannon Information ROC (SIROC) hypersurface.
• To explore the geometric properties and interrelations of ROC and SIROC hypersurfaces.

Main Methods:

• Derivation of Shannon information as a function of class prior probabilities based on the minimum probability of error for an ideal observer.
• Mathematical transformation linking the standard ROC hypersurface to the SIROC hypersurface.
• Analysis of convexity and geometric properties using Legendre transforms.

Main Results:

• Shannon information for multiclass classification is directly determined by the minimum probability of error, which is a function of prior probabilities.
• A novel SIROC hypersurface is defined and shown to be mathematically determined by the ROC hypersurface via a non-local integral transform.
• Both ROC and SIROC hypersurfaces exhibit convexity and related geometric properties.

Conclusions:

• A clear mathematical link exists between Shannon information and ROC analysis in multiclass imaging classification.
• The SIROC hypersurface provides a new perspective for evaluating classification performance, derived from the established ROC hypersurface.
• The geometric properties of these hypersurfaces offer deeper insights into the underlying classification decision processes.