From Bareiss' Algorithm to the Stable Computation of Partial Correlations

Abstract

This paper presents the derivation of a new algorithm for the stable computation of sample partial correlation coefficients. The authors start with Bareiss' algorithm for the solution of linear system of equations with (nonsymmetric) Toeplitz coefficient matrix and show how to generalize it to matrices that are not Toeplitz. The so generalized Bareiss algorithm computes the LU and UL factorizations of those matrices whose contiguous principal submatrices are all non-singular. For symmetric positive-definite matrices A, which naturally satisfy this condition, the normalized version of Bareiss' algorithm is just the Hyperbolic Cholesky algorithm, which computes the upper and lower triangular Cholesky factors U and L of A by means of 2 x 2 hyperbolic rotations. Guided by the data flow graph of the Hyperbolic Cholesky algorithm, we show that there exists one sequence of 2 x 2 orthogonal rotations that effects the transformation from U to L such that the sines of these rotations equal the hyperbolic tangents of the hyperbolic rotations. From the connection to the Hyperbolic Cholesky algorithm it also follows that, if A is a sample covariance matrix, the sines ar sample partial correlation coefficients.

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Document Details

Document Type
Technical Report
Publication Date
May 01, 1988
Accession Number
ADA198696

Entities

People

  • Ilse C. Ipsen
  • Jean-marc Delosme

Organizations

  • Yale University

Tags

Communities of Interest

  • Air Platforms

DTIC Thesaurus Topics

  • Algorithms
  • Band Structures
  • Coefficients
  • Computations
  • Computer Science
  • Decomposition
  • Electrical Engineering
  • Elimination
  • Energy Bands
  • Linear Systems
  • Military Research
  • Numbers
  • Parallel Computing
  • Random Variables
  • Rotation
  • Square Roots
  • Theorems

Readers

  • Graph Algorithms and Convex Optimization.
  • Operations Research
  • Statistical inference.