2,197 research outputs found
Exact multi-parameter persistent homology of time-series data: one-dimensional reduction of multi-parameter persistence theory
In various applications of data classification and clustering problems,
multi-parameter analysis is effective and crucial because data are usually
defined in multi-parametric space. Multi-parameter persistent homology, an
extension of persistent homology of one-parameter data analysis, has been
developed for topological data analysis (TDA). Although it is conceptually
attractive, multi-parameter persistent homology still has challenges in theory
and practical applications. In this study, we consider time-series data and its
classification and clustering problems using multi-parameter persistent
homology. We develop a multi-parameter filtration method based on Fourier
decomposition and provide an exact formula and its interpretation of
one-dimensional reduction of multi-parameter persistent homology. The exact
formula implies that the one-dimensional reduction of multi-parameter
persistent homology of the given time-series data is equivalent to choosing
diagonal ray (standard ray) in the multi-parameter filtration space. For this,
we first consider the continuousization of time-series data based on Fourier
decomposition towards the construction of the exact persistent barcode formula
for the Vietoris-Rips complex of the point cloud generated by sliding window
embedding. The proposed method is highly efficient even if the sliding window
embedding dimension and the length of time-series data are large because the
method precomputes the exact barcode and the computational complexity is as low
as the fast Fourier transformation of . Further the proposed
method provides a way of finding different topological inferences by trying
different rays in the filtration space in no time.Comment: 29 page
A multi-domain hybrid method for head-on collision of black holes in particle limit
A hybrid method is developed based on the spectral and finite-difference
methods for solving the inhomogeneous Zerilli equation in time-domain. The
developed hybrid method decomposes the domain into the spectral and
finite-difference domains. The singular source term is located in the spectral
domain while the solution in the region without the singular term is
approximated by the higher-order finite-difference method.
The spectral domain is also split into multi-domains and the
finite-difference domain is placed as the boundary domain. Due to the global
nature of the spectral method, a multi-domain method composed of the spectral
domains only does not yield the proper power-law decay unless the range of the
computational domain is large. The finite-difference domain helps reduce
boundary effects due to the truncation of the computational domain. The
multi-domain approach with the finite-difference boundary domain method reduces
the computational costs significantly and also yields the proper power-law
decay.
Stable and accurate interface conditions between the finite-difference and
spectral domains and the spectral and spectral domains are derived. For the
singular source term, we use both the Gaussian model with various values of
full width at half maximum and a localized discrete -function. The
discrete -function was generalized to adopt the Gauss-Lobatto
collocation points of the spectral domain.
The gravitational waveforms are measured. Numerical results show that the
developed hybrid method accurately yields the quasi-normal modes and the
power-law decay profile. The numerical results also show that the power-law
decay profile is less sensitive to the shape of the regularized
-function for the Gaussian model than expected. The Gaussian model also
yields better results than the localized discrete -function.Comment: 25 pages; published version (IJMPC
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