### Abstract

Multiclass receiver operating characteristic (ROC) analysis has remained an open theoretical problem since the introduction of binary ROC analysis in the 1950s. Previously, we have developed a paradigm for three-class ROC analysis that extends and unifies decision theoretic, linear discriminant analysis, and probabilistic foundations of binary ROC analysis in a three-class paradigm. One critical element in this paradigm is the equal error utility (EEU) assumption. This assumption allows us to reduce the intrinsic space of the three-class ROC analysis (5-D hypersurface in 6-D hyperspace) to a 2-D surface in the 3-D space of true positive fractions (sensitivity space). In this work, we show that this 2-D ROC surface fully and uniquely provides a complete descriptor for the optimal performance of a system for a three-class classification task, i.e., the triplet of likelihood ratio distributions, assuming such a triplet exists. To be specific, we consider two classifiers that utilize likelihood ratios, and we assumed each classifier has a continuous and differentiable 2-D sensitivity-space ROC surface. Under these conditions, we proved that the classifiers have the same triplet of likelihood ratio distributions if and only if they have the same 2-D sensitivity-space ROC surfaces. As a result, the 2-D sensitivity surface contains complete information on the optimal three-class task performance for the corresponding likelihood ratio classifier.

Original language | English (US) |
---|---|

Article number | 5306177 |

Pages (from-to) | 206-215 |

Number of pages | 10 |

Journal | IEEE Transactions on Medical Imaging |

Volume | 29 |

Issue number | 1 |

DOIs | |

State | Published - Jan 2010 |

### Fingerprint

### Keywords

- Extended receiver operating characteristic (ROC) analysis
- ROC analysis
- Three-class ROC analysis

### ASJC Scopus subject areas

- Electrical and Electronic Engineering
- Computer Science Applications
- Radiological and Ultrasound Technology
- Software

### Cite this

*IEEE Transactions on Medical Imaging*,

*29*(1), 206-215. [5306177]. https://doi.org/10.1109/TMI.2009.2034516

**Three-class ROC analysistoward a general decision theoretic solution.** / He, Xin; Gallas, Brandon D.; Frey, Eric.

Research output: Contribution to journal › Article

*IEEE Transactions on Medical Imaging*, vol. 29, no. 1, 5306177, pp. 206-215. https://doi.org/10.1109/TMI.2009.2034516

}

TY - JOUR

T1 - Three-class ROC analysistoward a general decision theoretic solution

AU - He, Xin

AU - Gallas, Brandon D.

AU - Frey, Eric

PY - 2010/1

Y1 - 2010/1

N2 - Multiclass receiver operating characteristic (ROC) analysis has remained an open theoretical problem since the introduction of binary ROC analysis in the 1950s. Previously, we have developed a paradigm for three-class ROC analysis that extends and unifies decision theoretic, linear discriminant analysis, and probabilistic foundations of binary ROC analysis in a three-class paradigm. One critical element in this paradigm is the equal error utility (EEU) assumption. This assumption allows us to reduce the intrinsic space of the three-class ROC analysis (5-D hypersurface in 6-D hyperspace) to a 2-D surface in the 3-D space of true positive fractions (sensitivity space). In this work, we show that this 2-D ROC surface fully and uniquely provides a complete descriptor for the optimal performance of a system for a three-class classification task, i.e., the triplet of likelihood ratio distributions, assuming such a triplet exists. To be specific, we consider two classifiers that utilize likelihood ratios, and we assumed each classifier has a continuous and differentiable 2-D sensitivity-space ROC surface. Under these conditions, we proved that the classifiers have the same triplet of likelihood ratio distributions if and only if they have the same 2-D sensitivity-space ROC surfaces. As a result, the 2-D sensitivity surface contains complete information on the optimal three-class task performance for the corresponding likelihood ratio classifier.

AB - Multiclass receiver operating characteristic (ROC) analysis has remained an open theoretical problem since the introduction of binary ROC analysis in the 1950s. Previously, we have developed a paradigm for three-class ROC analysis that extends and unifies decision theoretic, linear discriminant analysis, and probabilistic foundations of binary ROC analysis in a three-class paradigm. One critical element in this paradigm is the equal error utility (EEU) assumption. This assumption allows us to reduce the intrinsic space of the three-class ROC analysis (5-D hypersurface in 6-D hyperspace) to a 2-D surface in the 3-D space of true positive fractions (sensitivity space). In this work, we show that this 2-D ROC surface fully and uniquely provides a complete descriptor for the optimal performance of a system for a three-class classification task, i.e., the triplet of likelihood ratio distributions, assuming such a triplet exists. To be specific, we consider two classifiers that utilize likelihood ratios, and we assumed each classifier has a continuous and differentiable 2-D sensitivity-space ROC surface. Under these conditions, we proved that the classifiers have the same triplet of likelihood ratio distributions if and only if they have the same 2-D sensitivity-space ROC surfaces. As a result, the 2-D sensitivity surface contains complete information on the optimal three-class task performance for the corresponding likelihood ratio classifier.

KW - Extended receiver operating characteristic (ROC) analysis

KW - ROC analysis

KW - Three-class ROC analysis

UR - http://www.scopus.com/inward/record.url?scp=73849120367&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=73849120367&partnerID=8YFLogxK

U2 - 10.1109/TMI.2009.2034516

DO - 10.1109/TMI.2009.2034516

M3 - Article

C2 - 19884079

AN - SCOPUS:73849120367

VL - 29

SP - 206

EP - 215

JO - IEEE Transactions on Medical Imaging

JF - IEEE Transactions on Medical Imaging

SN - 0278-0062

IS - 1

M1 - 5306177

ER -