scholarly journals Approximating Correlation Matrices Using Stochastic Lie Group Methods

Mathematics ◽  
2021 ◽  
Vol 9 (1) ◽  
pp. 94
Author(s):  
Michelle Muniz ◽  
Matthias Ehrhardt ◽  
Michael Günther
Keyword(s):  
Lie Group ◽  
Orthogonal Group ◽  
Time Dependent ◽  
Financial Mathematics ◽  
Correlation Matrices ◽  
Special Orthogonal Group ◽  
Lie Group Integrators ◽  
Group Methods ◽  
Valid Time ◽  
Dependent Correlation

Specifying time-dependent correlation matrices is a problem that occurs in several important areas of finance and risk management. The goal of this work is to tackle this problem by applying techniques of geometric integration in financial mathematics, i.e., to combine two fields of numerical mathematics that have not been studied yet jointly. Based on isospectral flows we create valid time-dependent correlation matrices, so called correlation flows, by solving a stochastic differential equation (SDE) that evolves in the special orthogonal group. Since the geometric structure of the special orthogonal group needs to be preserved we use stochastic Lie group integrators to solve this SDE. An application example is presented to illustrate this novel methodology.

A new methodology to create valid time-dependent correlation matrices via isospectral flows

2020 ◽  
Vol 54 (2) ◽  
pp. 361-371
Author(s):  
Long Teng ◽  
Xueran Wu ◽  
Michael Günther ◽  
Matthias Ehrhardt
Keyword(s):  
Risk Management ◽  
Correlation Matrix ◽  
Time Dependent ◽  
Correlation Matrices ◽  
Know How ◽  
Initial Correlation ◽  
Valid Time ◽  
Dependent Correlation

In many areas of finance and of risk management it is interesting to know how to specify time-dependent correlation matrices. In this work we propose a new methodology to create valid time-dependent instantaneous correlation matrices, which we called correlation flows. In our methodology one needs only an initial correlation matrix to create these correlation flows based on isospectral flows. The tendency of the time-dependent matrices can be controlled by requirements. An application example is presented to illustrate our methodology.

A Study on Neural Learning on Manifold Foliations: The Case of the Lie Group SU(3)

2008 ◽  
Vol 20 (4) ◽  
pp. 1091-1117 ◽  
Author(s):  
Simone Fiori
Keyword(s):  
Lie Algebra ◽  
Lie Group ◽  
Orthogonal Group ◽  
Special Orthogonal Group ◽  
Neural Learning ◽  
Gradient Based ◽  
Special Case

Learning on differential manifolds may involve the optimization of a function of many parameters. In this letter, we deal with Riemannian-gradient-based optimization on a Lie group, namely, the group of unitary unimodular matrices SU(3). In this special case, subalgebras of the associated Lie algebra su(3) may be individuated by computing pair-wise Gell-Mann matrices commutators. Subalgebras generate subgroups of a Lie group, as well as manifold foliation. We show that the Riemannian gradient may be projected over tangent structures to foliation, giving rise to foliation gradients. Exponentiations of foliation gradients may be computed in closed forms, which closely resemble Rodriguez forms for the special orthogonal group SO(3). We thus compare optimization by Riemannian gradient and foliation gradients.

scholarly journals An Intrinsic Aggregation Model on the Special Orthogonal Group SO(3): Well-posedness and Collective Behaviours

2021 ◽  
Vol 31 (5) ◽  
Author(s):  
Razvan C. Fetecau ◽  
Seung-Yeal Ha ◽  
Hansol Park
Keyword(s):  
Orthogonal Group ◽  
Well Posedness ◽  
Aggregation Model ◽  
Special Orthogonal Group

scholarly journals Isometries of the special orthogonal group

2013 ◽  
Vol 439 (1) ◽  
pp. 174-188 ◽  
Author(s):  
Toshikazu Abe ◽  
Shigeki Akiyama ◽  
Osamu Hatori
Keyword(s):  
Orthogonal Group ◽  
Special Orthogonal Group

Vibrational relaxation dynamics in transient grating spectroscopy studied by rate equations based on time-dependent correlation function

2012 ◽  
Vol 21 (4) ◽  
pp. 043402 ◽  
Author(s):  
Guo-Yang Yu ◽  
Yun-Fei Song ◽  
Xing He ◽  
Xian-Xu Zheng ◽  
Duo-Wang Tan ◽  
...  
Keyword(s):  
Correlation Function ◽  
Vibrational Relaxation ◽  
Time Dependent ◽  
Rate Equations ◽  
Relaxation Dynamics ◽  
Transient Grating ◽  
Transient Grating Spectroscopy ◽  
Grating Spectroscopy ◽  
Dependent Correlation

Space-Time-Dependent Correlation Functions for One-Dimensional Heisenberg Spin Systems in a Lorentzian-Gaussian Approximation

1969 ◽  
Vol 185 (2) ◽  
pp. 854-859 ◽  
Author(s):  
D. G. McFadden ◽  
R. A. Tahir-Kheli ◽  
G. Bruce Taggart
Keyword(s):  
Correlation Functions ◽  
Gaussian Approximation ◽  
Space Time ◽  
Spin Systems ◽  
Time Dependent ◽  
Heisenberg Spin ◽  
One Dimensional ◽  
Dependent Correlation
Sign in / Sign up

Export Citation Format

Share Document