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Random orthogonal matrices and the Cayley transform

Publication ,  Journal Article
Jauch, M; Hoff, PD; Dunson, DB
Published in: Bernoulli
January 1, 2020

Random orthogonal matrices play an important role in probability and statistics, arising in multivariate analysis, directional statistics, and models of physical systems, among other areas. Calculations involving random orthogonal matrices are complicated by their constrained support. Accordingly, we parametrize the Stiefel and Grassmann manifolds, represented as subsets of orthogonal matrices, in terms of Euclidean parameters using the Cayley transform. We derive the necessary Jacobian terms for change of variables formulas. Given a density defined on the Stiefel or Grassmann manifold, these allow us to specify the corresponding density for the Euclidean parameters, and vice versa. As an application, we present a Markov chain Monte Carlo approach to simulating from distributions on the Stiefel and Grassmann manifolds. Finally, we establish that the Euclidean parameters corresponding to a uniform orthogonal matrix can be approximated asymptotically by independent normals. This result contributes to the growing literature on normal approximations to the entries of random orthogonal matrices or transformations thereof.

Duke Scholars

Published In

Bernoulli

DOI

ISSN

1350-7265

Publication Date

January 1, 2020

Volume

26

Issue

2

Start / End Page

1560 / 1586

Related Subject Headings

  • Statistics & Probability
  • 4905 Statistics
  • 1403 Econometrics
  • 0104 Statistics
 

Citation

APA
Chicago
ICMJE
MLA
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Jauch, M., Hoff, P. D., & Dunson, D. B. (2020). Random orthogonal matrices and the Cayley transform. Bernoulli, 26(2), 1560–1586. https://doi.org/10.3150/19-BEJ1176
Jauch, M., P. D. Hoff, and D. B. Dunson. “Random orthogonal matrices and the Cayley transform.” Bernoulli 26, no. 2 (January 1, 2020): 1560–86. https://doi.org/10.3150/19-BEJ1176.
Jauch M, Hoff PD, Dunson DB. Random orthogonal matrices and the Cayley transform. Bernoulli. 2020 Jan 1;26(2):1560–86.
Jauch, M., et al. “Random orthogonal matrices and the Cayley transform.” Bernoulli, vol. 26, no. 2, Jan. 2020, pp. 1560–86. Scopus, doi:10.3150/19-BEJ1176.
Jauch M, Hoff PD, Dunson DB. Random orthogonal matrices and the Cayley transform. Bernoulli. 2020 Jan 1;26(2):1560–1586.

Published In

Bernoulli

DOI

ISSN

1350-7265

Publication Date

January 1, 2020

Volume

26

Issue

2

Start / End Page

1560 / 1586

Related Subject Headings

  • Statistics & Probability
  • 4905 Statistics
  • 1403 Econometrics
  • 0104 Statistics