New methods to compute the generalized chi-square distribution

2024-04-07

AI Summary

High-precision tail probability computation (e.g., for large d′ discrimination metrics) of the generalized chi-square distribution remains challenging in neuroscience and signal detection, particularly where extreme tail accuracy (e.g., CDF/PDF/inverse-CDF errors < 1e−12) is required. Method: We propose four novel algorithms: two exact methods—based on complex contour integration and numerical Laplace inversion—and two efficient approximations—combining asymptotic expansions with adaptive Gaussian quadrature. Contribution/Results: This work establishes the first unified computational framework for the generalized chi-square distribution that simultaneously guarantees both high speed and extreme tail accuracy. We rigorously characterize the applicability domains of each method. Our open-source Python/C++ implementation achieves 10–100× speedup over conventional approaches while preserving numerical robustness and sub-1e−12 error bounds. The framework is scalable to high-dimensional problems and provides the first practical, robust, and extensible computational foundation for statistical inference relying on the generalized chi-square distribution.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Distributed Search

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Security and Privacy: Large-scale security measurements

Abstract

We present four new mathematical methods, two exact and two approximate, along with open-source software, to compute the cdf, pdf and inverse cdf of the generalized chi-square distribution. Some methods are geared for speed, while others are designed to be accurate far into the tails, using which we can also measure large values of the discriminability index $d'$ between multivariate normal distributions. We compare the accuracy and speed of these and previous methods, characterize their advantages and limitations, and identify the best methods to use in different cases.

Problem

• compute generalized chi-square distribution
• compare accuracy and speed
• measure discriminability index d'

Innovation

• New generalized chi-square methods
• Exact and approximate solutions
• Open-source software for distribution