scholarly journals On singular values distribution of a matrix large auto-covariance in the ultra-dimensional regime

2015 ◽  
Vol 04 (04) ◽  
pp. 1550015
Author(s):  
Qinwen Wang ◽  
Jianfeng Yao
Keyword(s):  
Singular Values ◽  
Singular Value ◽  
Moment Condition ◽  
Fourth Moment ◽  
Limiting Behavior ◽  
The Moment ◽  
Fixed Positive Integer ◽  
The Right ◽  
Auto Covariance ◽  
Random Limit

Let [Formula: see text] be a sequence of independent real random vectors of [Formula: see text]-dimension and let [Formula: see text] be the lag-[Formula: see text] ([Formula: see text] is a fixed positive integer) auto-covariance matrix of [Formula: see text]. This paper investigates the limiting behavior of the singular values of [Formula: see text] under the so-called ultra-dimensional regime where [Formula: see text] and [Formula: see text] in a related way such that [Formula: see text]. First, we show that the singular value distribution of [Formula: see text], after a suitable normalization, converges to a non-random limit [Formula: see text] (quarter law) under the fourth-moment condition. Second, we establish the convergence of its largest singular value to the right edge of the support of [Formula: see text]. Both results are derived using the moment method.

scholarly journals Automatic Partition of Orbital Spaces Based on Singular Value Decomposition in the Context of Embedding Theories

2018 ◽  
Author(s):  
Daniel Claudino ◽  
Nicholas Mayhall
Keyword(s):  
Singular Values ◽  
Decanoic Acid ◽  
Molecular Orbitals ◽  
Singular Value ◽  
Mulliken Charge ◽  
Space Partitioning ◽  
Embedding Theory ◽  
Torsional Potential ◽  
Orbital Space ◽  
The Right

We present a simple approach for orbital space partitioning to be employed in the projection-based embedding theory developed by Goodpaster and coworkers [<i>J. Chem. Theory Comput</i>. 2012, 8, 2564]. Once the atoms are assigned to the desired subspaces, the molecular orbitals are projected onto the atomic orbitals centered on active atoms and then singular value decomposed. The right singular vectors are used to rotate the initial molecular orbitals, taking the largest gap in the singular values spectrum to define the most suitable partition of the occupied orbital space. This scheme is free from numerical parameters, contrary to the Mulliken charge threshold or the completeness criterion previously used. The performance of this new prescription is assessed in a test set of several distinct reactions, the deprotonation of decanoic acid, the torsional potential of a retinal derivative, and the critical points along the reaction coordinate of an example of the Menshutkin S<sub>N</sub>2 reaction inside a carbon nanotube.

Deriving source-time functions using principal component analysis

1989 ◽  
Vol 79 (3) ◽  
pp. 711-730
Author(s):  
D. W. Vasco
Keyword(s):  
Principal Component Analysis ◽  
Moment Tensor ◽  
Principal Component ◽  
Singular Values ◽  
Singular Value ◽  
Time Function ◽  
Basis Functions ◽  
Single Source ◽  
Source Time Function ◽  
The Moment

Abstract Factors such as source complexity, microseismic noise, and lateral heterogeneity all introduce nonuniqueness into the source-time function. The technique of principal component analysis is used to factor the moment tensor into a set of orthogonal source-time functions. This is accomplished through the singular value decomposition of the time-varying moment tensor. The adequacy of assuming a single source-time function may then be examined through the singular values of the decomposition. The F test can also be used to assess the significance of the various principal component basis functions. The set of significant basis functions can be used to test models of the source-time functions, including multiple sources. Application of this technique to the Harzer nuclear explosion indicated that a single source-time function was found to adequately explain the moment tensor. It consists of a single pulse appearing on the diagonal elements of the moment-rate tensor. The decomposition of the moment tensor for a deep teleseism in the Bonin Islands revealed three basis functions associated with relatively large singular values. The F test indicated that only two of the principal components were significant. The principal component associated with the largest singular value consists of a large pulse followed 16-sec later by a diminished pulse. The second principal component, a long-period oscillation, appears to be a manifestation of the poor resolution of the moment-rate tensor at low frequencies.

scholarly journals Automatic Partition of Orbital Spaces Based on Singular Value Decomposition in the Context of Embedding Theories

2018 ◽  
Author(s):  
Daniel Claudino ◽  
Nicholas Mayhall
Keyword(s):  
Singular Values ◽  
Decanoic Acid ◽  
Molecular Orbitals ◽  
Singular Value ◽  
Mulliken Charge ◽  
Space Partitioning ◽  
Embedding Theory ◽  
Torsional Potential ◽  
Orbital Space ◽  
The Right

We present a simple approach for orbital space partitioning to be employed in the projection-based embedding theory developed by Goodpaster and coworkers [<i>J. Chem. Theory Comput</i>. 2012, 8, 2564]. Once the atoms are assigned to the desired subspaces, the molecular orbitals are projected onto the atomic orbitals centered on active atoms and then singular value decomposed. The right singular vectors are used to rotate the initial molecular orbitals, taking the largest gap in the singular values spectrum to define the most suitable partition of the occupied orbital space. This scheme is free from numerical parameters, contrary to the Mulliken charge threshold or the completeness criterion previously used. The performance of this new prescription is assessed in a test set of several distinct reactions, the deprotonation of decanoic acid, the torsional potential of a retinal derivative, and the critical points along the reaction coordinate of an example of the Menshutkin S<sub>N</sub>2 reaction inside a carbon nanotube.

Time-Series Data Analysis Using Sliding Window Based SVD for Motion Evaluation

2017 ◽  
Vol 21 (7) ◽  
pp. 1240-1250
Author(s):  
Yinlai Jiang ◽  
Isao Hayashi ◽  
Shuoyu Wang ◽  
Kenji Ishida ◽  
◽  
...  
Keyword(s):  
Time Series ◽  
Time Series Data ◽  
Sliding Window ◽  
Singular Values ◽  
Singular Value ◽  
Series Data ◽  
Singular Vectors ◽  
Walking Difficulty ◽  
The Right ◽  
Value Decomposition

A method based on singular value decomposition (SVD) is proposed for extracting features from motion time-series data observed with various sensing systems. Matrices consisting of the sliding window (SW) subsets of time-series data are decomposed, yielding singular vectors as the patterns of the motion, and the singular values as a scalar, by which the corresponding singular vectors describe the matrices.The sliding window based singular value decomposition was applied to analyze acceleration during walking. Three levels of walking difficulty were simulated by restricting the right knee joint in the measurement. The accelerations of the middles of the shanks and the back of the waist were measured and normalized before the SW-SVD was performed.The results showed that the first singular values inferred from the acceleration data of the restricted side (the right shank) significantly related to the increase of the restriction among all the subjects while there were no common trends in the singular values of the left shank and the waist. The SW-SVD was suggested to be a reliable method to evaluate walking disability. Furthermore, a 2D visualization tool is proposed to provide intuitive information about walking difficulty which can be used in walking rehabilitation to monitor recovery.

scholarly journals Random non-Abelian G-circulant matrices. Spectrum of random convolution operators on large finite groups

2020 ◽  
pp. 2250002
Author(s):  
Radosław Adamczak
Keyword(s):  
Symmetric Groups ◽  
Singular Values ◽  
Singular Value ◽  
Circulant Matrices ◽  
Plancherel Measure ◽  
Convolution Operators ◽  
Limiting Distributions ◽  
Dihedral Groups ◽  
Limiting Behavior ◽  
Abelian Case

We analyze the limiting behavior of the eigenvalue and singular value distribution for random convolution operators on large (not necessarily Abelian) groups, extending the results by Meckes for the Abelian case. We show that for regular sequences of groups, the limiting distribution of eigenvalues (respectively singular values) is a mixture of eigenvalue (respectively singular value) distributions of Ginibre matrices with the directing measure being related to the limiting behavior of the Plancherel measure of the sequence of groups. In particular, for the sequence of symmetric groups, the limiting distributions are just the circular and quarter circular laws, whereas e.g. for the dihedral groups, the limiting distributions have unbounded supports but are different than in the Abelian case. We also prove that under additional assumptions on the sequence of groups (in particular, for symmetric groups of increasing order) families of stochastically independent random convolution operators converge in moments to free circular elements. Finally, in the Gaussian case, we provide Central Limit Theorems for linear eigenvalue statistics.

scholarly journals Automatic Partition of Orbital Spaces Based on Singular Value Decomposition in the Context of Embedding Theories

2018 ◽  
Author(s):  
Daniel Claudino ◽  
Nicholas Mayhall
Keyword(s):  
Singular Values ◽  
Decanoic Acid ◽  
Molecular Orbitals ◽  
Singular Value ◽  
Mulliken Charge ◽  
Space Partitioning ◽  
Embedding Theory ◽  
Torsional Potential ◽  
Orbital Space ◽  
The Right

We present a simple approach for orbital space partitioning to be employed in the projection-based embedding theory developed by Goodpaster and coworkers [<i>J. Chem. Theory Comput</i>. 2012, 8, 2564]. Once the atoms are assigned to the desired subspaces, the molecular orbitals are projected onto the atomic orbitals centered on active atoms and then singular value decomposed. The right singular vectors are used to rotate the initial molecular orbitals, taking the largest gap in the singular values spectrum to define the most suitable partition of the occupied orbital space. This scheme is free from numerical parameters, contrary to the Mulliken charge threshold or the completeness criterion previously used. The performance of this new prescription is assessed in a test set of several distinct reactions, the deprotonation of decanoic acid, the torsional potential of a retinal derivative, and the critical points along the reaction coordinate of an example of the Menshutkin S<sub>N</sub>2 reaction inside a carbon nanotube.

scholarly journals Random Toeplitz matrices: The condition number under high stochastic dependence

2021 ◽  
Author(s):  
Paulo Manrique-Mirón
Keyword(s):  
Generating Function ◽  
Condition Number ◽  
Toeplitz Matrix ◽  
Singular Values ◽  
Moment Generating Function ◽  
Circulant Matrix ◽  
Singular Value ◽  
The Matrix ◽  
Matrix Formula ◽  
The Moment

In this paper, we study the condition number of a random Toeplitz matrix. As a Toeplitz matrix is a diagonal constant matrix, its rows or columns cannot be stochastically independent. This situation does not permit us to use the classic strategies to analyze its minimum singular value when all the entries of a random matrix are stochastically independent. Using a circulant embedding as a decoupling technique, we break the stochastic dependence of the structure of the Toeplitz matrix and reduce the problem to analyze the extreme singular values of a random circulant matrix. A circulant matrix is, in fact, a particular case of a Toeplitz matrix, but with a more specific structure, where it is possible to obtain explicit formulas for its eigenvalues and also for its singular values. Among our results, we show the condition number of a non-symmetric random circulant matrix [Formula: see text] of dimension [Formula: see text] under the existence of the moment generating function of the random entries is [Formula: see text] with probability [Formula: see text] for any [Formula: see text], [Formula: see text]. Moreover, if the random entries only have the second moment, the condition number satisfies [Formula: see text] with probability [Formula: see text]. Also, we analyze the condition number of a random symmetric circulant matrix [Formula: see text]. For the condition number of a random (non-symmetric or symmetric) Toeplitz matrix [Formula: see text] we establish [Formula: see text], where [Formula: see text] is the minimum singular value of the matrix [Formula: see text]. The matrix [Formula: see text] is a random circulant matrix and [Formula: see text], where [Formula: see text] are deterministic matrices, [Formula: see text] indicates the conjugate transpose of [Formula: see text] and [Formula: see text] are random diagonal matrices. From random experiments, we conjecture that [Formula: see text] is well-conditioned if the moment generating function of the random entries of [Formula: see text] exists.

Would you Make the Right Decision? – Decision Making Biases in Economy - Related Dilemmas

2018 ◽  
Vol 9 (1) ◽  
pp. 59-66
Author(s):  
Zsuzsanna Gödör ◽  
Georgina Szabó
Keyword(s):  
Decision Making ◽  
Online Survey ◽  
Framing Effect ◽  
Economic Knowledge ◽  
Bank Account ◽  
Financial Decisions ◽  
Hungarian Population ◽  
The Moment ◽  
The Right ◽  
Decision Making Biases

Abstract As they say, money can’t buy happiness. However, the lack of it can make people’s lives much harder. From the moment we open our first bank account, we have to make lots of financial decisions in our life. Should I save some money or should I spend it? Is it a good idea to ask for a loan? How to invest my money? When we make such decisions, unfortunately we sometimes make mistakes, too. In this study, we selected seven common decision making biases - anchoring and adjustment, overconfidence, high optimism, the law of small numbers, framing effect, disposition effect and gambler’s fallacy – and tested them on the Hungarian population via an online survey. In the focus of our study was the question whether the presence of economic knowledge helps people make better decisions? The decision making biases found in literature mostly appeared in the sample as well. It proves that people do apply them when making decisions and in certain cases this could result in serious and costly errors. That’s why it would be absolutely important for people to learn about them, thus increasing their awareness and attention when making decisions. Furthermore, in our research we did find some connection between decisions and the knowledge of economics, people with some knowledge of economics opted for the better solution in bigger proportion

JUSTICE AND SOCIAL CHANGE FOR INCLUSIVE EDUCATION IN INDONESIA AND ASEAN FOR DISABLED PERSONS (STUDY ON PEOPLE AFFECTED BY LEPROSY)

2017 ◽  
Vol 1 (1) ◽  
pp. 1-5
Author(s):  
Nuah Perdamenta Tarigan ◽  
Christian Siregar ◽  
Simon Mangatur Tampubolon
Keyword(s):  
Inclusive Education ◽  
Local Governments ◽  
Disabled Persons ◽  
Equal Rights ◽  
Central Java ◽  
The People ◽  
Asean Countries ◽  
The Government ◽  
The Moment ◽  
The Right

Justice that has not existed and is apparent among the disabilities in Indonesia is very large and spread in the archipelago is very large, making the issue of equality is a very important thing especially with the publication of the Disability Act No. 8 of 2016 at the beginning of that year. Only a few provinces that understand properly and well on open and potential issues and issues will affect other areas including the increasingly growing number of elderly people in Indonesia due to the increasing welfare of the people. The government of DKI Jakarta, including the most concerned with disability, from the beginning has set a bold step to defend things related to disability, including local governments in Solo, Bali, Makassar and several other areas. Leprosy belonging to the disability community has a very tough marginalization, the disability that arises from leprosy quite a lot, reaches ten percent more and covers the poor areas of Indonesia, such as Nusa Tenggara Timur, Papua, South Sulawesi Provinces and even East Java and West Java and Central Java Provinces. If we compare again with the ASEAN countries we also do not miss the moment in ratifying the CRPD (Convention of Rights for People with Disability) into the Law of Disability No. 8 of 2016 which, although already published but still get rejections in some sections because do not provide proper empowerment and rights equality. The struggle is long and must be continued to build equal rights in all areas, not only health and welfare but also in the right of the right to receive continuous inclusive education.

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