scholarly journals The Ordering of Shannon Entropies for the Multivariate Distributions and Distributions of Eigenvalues

Entropy ◽  
2019 ◽  
Vol 21 (2) ◽  
pp. 201
Author(s):  
Ming-Tien Tsai ◽  
Feng-Ju Hsu ◽  
Chia-Hsuan Tsai
Keyword(s):  
General Population ◽  
Covariance Matrix ◽  
Shannon Entropy ◽  
Distribution Functions ◽  
Multivariate Distributions ◽  
Convex Ordering ◽  
Totally Positive ◽  
Entropy Measures ◽  
Set Up ◽  
Entropy Inequalities

In this paper, we prove the Shannon entropy inequalities for the multivariate distributions via the notion of convex ordering of two multivariate distributions. We further characterize the multivariate totally positive of order 2 ( M T P 2 ) property of the distribution functions of eigenvalues of both central Wishart and central MANOVA models, and of both noncentral Wishart and noncentral MANOVA models under the general population covariance matrix set-up. These results can be directly applied to both the comparisons of two Shannon entropy measures and the power monotonicity problem for the MANOVA problem.

scholarly journals Parton distributions with theory uncertainties: general formalism and first phenomenological studies

2019 ◽  
Vol 79 (11) ◽  
Author(s):  
Rabah Abdul Khalek ◽  
Richard D. Ball ◽  
Stefano Carrazza ◽  
Stefano Forte ◽  
Tommaso Giani ◽  
...  
Keyword(s):  
Covariance Matrix ◽  
Cross Sections ◽  
Distribution Functions ◽  
Matrix Formalism ◽  
Parton Distribution Functions ◽  
Central Values ◽  
Wide Range ◽  
Algebraic Framework ◽  
Set Up ◽  
The Impact

AbstractWe formulate a general approach to the inclusion of theoretical uncertainties, specifically those related to the missing higher order uncertainty (MHOU), in the determination of parton distribution functions (PDFs). We demonstrate how, under quite generic assumptions, theory uncertainties can be included as an extra contribution to the covariance matrix when determining PDFs from data. We then review, clarify, and systematize the use of renormalization and factorization scale variations as a means to estimate MHOUs consistently in deep inelastic and hadronic processes. We define a set of prescriptions for constructing a theory covariance matrix using scale variations, which can be used in global fits of data from a wide range of different processes, based on choosing a set of independent scale variations suitably correlated within and across processes. We set up an algebraic framework for the choice and validation of an optimal prescription by comparing the estimate of MHOU encoded in the next-to-leading order (NLO) theory covariance matrix to the observed shifts between NLO and NNLO predictions. We perform a NLO PDF determination which includes the MHOU, assess the impact of the inclusion of MHOUs on the PDF central values and uncertainties, and validate the results by comparison to the known shift between NLO and NNLO PDFs. We finally study the impact of the inclusion of MHOUs in a global PDF determination on LHC cross-sections, and provide guidelines for their use in precision phenomenology. In addition, we also compare the results based on the theory covariance matrix formalism to those obtained by performing PDF determinations based on different scale choices.

scholarly journals Multivariate Distributions in Hydrology and River Regulation

1976 ◽  
Vol 7 (5) ◽  
pp. 265-280
Author(s):  
N.A. Kartvelishvili ◽  
L.T. Gottschalk
Keyword(s):  
Markov Process ◽  
Finite Difference ◽  
Difference Scheme ◽  
River Runoff ◽  
Finite Difference Scheme ◽  
River Regulation ◽  
Distribution Functions ◽  
Multivariate Distributions ◽  
Model Order

It is assumed that the river runoff process can be approximated by a Markov process. The process is thus described by M distribution functions: Fn (qt, t ; qt-1; t-1;…;qt-n, t-n), t ≡ 1, 2, …, M where M is the number of time intervals within the year, n - the order of the Markov process and qt, in general, is a vector representing runoff at several sites in a river or neighbouring rivers. Fundamental hypothesis of relations between multivariate distributions and corresponding marginal distributions is given. A finite difference scheme for multisite and multilag generation of river runoff is derived. The derivation is based on the multivariate normal distribution. Different methods for determination of the order of the finite difference scheme are discussed as well as the influence of model order and method of parameter estimation on properties of the model.

The Palomar-Leiden Survey

1971 ◽  
Vol 12 ◽  
pp. 183-186
Author(s):  
C.J. Van Houten
Keyword(s):  
Semimajor Axis ◽  
Continuous Extension ◽  
Distribution Functions ◽  
Set Up

The Palomar-Leiden survey (PLS) was set up as an extension to fainter magnitudes of the McDonald survey. The latter is, therefore, the more important survey as far as asteroid statistics are concerned. The main result of the PLS is that no clearcut differences exist between the fainter asteroids found in this survey and the numbered asteroids in the distribution functions of eccentricity, inclination, and semimajor axis and that the statistical relations found in the McDonald survey have a continuous extension in the PLS material. I would, therefore, propose not to summarize the results of the PLS, which would appear to be a tedious job, but to give here some new results that should properly have been included in the publication, but, for reasons of pressures to publish as soon as possible, were not.

scholarly journals Covariance Matrix Estimation under Total Positivity for Portfolio Selection*

2020 ◽  
Author(s):  
Raj Agrawal ◽  
Uma Roy ◽  
Caroline Uhler
Keyword(s):  
Covariance Matrix ◽  
Total Positivity ◽  
General Purpose ◽  
Shrinkage Estimators ◽  
Positive Dependence ◽  
Matrix Estimation ◽  
Sample Covariance ◽  
Totally Positive ◽  
Markowitz Portfolio ◽  
Previous State

Abstract Selecting the optimal Markowitz portfolio depends on estimating the covariance matrix of the returns of N assets from T periods of historical data. Problematically, N is typically of the same order as T, which makes the sample covariance matrix estimator perform poorly, both empirically and theoretically. While various other general-purpose covariance matrix estimators have been introduced in the financial economics and statistics literature for dealing with the high dimensionality of this problem, we here propose an estimator that exploits the fact that assets are typically positively dependent. This is achieved by imposing that the joint distribution of returns be multivariate totally positive of order 2 (MTP2). This constraint on the covariance matrix not only enforces positive dependence among the assets but also regularizes the covariance matrix, leading to desirable statistical properties such as sparsity. Based on stock market data spanning 30 years, we show that estimating the covariance matrix under MTP2 outperforms previous state-of-the-art methods including shrinkage estimators and factor models.

Recurrence Plot based entropies and their ability to detect transitions

2020 ◽  
Author(s):  
K. Hauke Kraemer ◽  
Norbert Marwan ◽  
Karoline Wiesner ◽  
Jürgen Kurths
Keyword(s):  
Time Series ◽  
Shannon Entropy ◽  
Recurrence Plot ◽  
Climate Dynamics ◽  
Value Distribution ◽  
Tipping Points ◽  
Recurrence Plots ◽  
The Past ◽  
Entropy Measures ◽  
Regime Transitions

<p>Many dynamical processes in Earth Sciences are the product of many interacting components and have often limited predictability, not least because they can exhibit regime transitions (e.g. tipping points).To quantify complexity, entropy measures such as the Shannon entropy of the value distribution are widely used. Amongst other more sophisticated ideas, a number of entropy measures based on recurrence plots have been suggested. Because different structures, e.g. diagonal lines, of the recurrence plot are used for the estimation of probabilities, these entropy measures represent different aspects of the analyzed system and, thus, behave differently. In the past, this fact has led to difficulties in interpreting and understanding those measures. We review the definitions, the motivation and interpretation of these entropy measures, compare their differences and discuss some of the pitfalls when using them.</p><p>Finally, we illustrate their potential in an application on paleoclimate time series. Using the presented entropy measures, changes and transitions in the climate dynamics in the past can be identified and interpreted.</p>

scholarly journals Improved Approximation of the Sum of Random Vectors by the Skew Normal Distribution

2014 ◽  
Vol 51 (2) ◽  
pp. 466-482 ◽  
Author(s):  
Marcus C. Christiansen ◽  
Nicola Loperfido
Keyword(s):  
Normal Distribution ◽  
Normal Approximation ◽  
Distribution Functions ◽  
Skew Normal Distribution ◽  
Random Vectors ◽  
Multivariate Distributions ◽  
Uniform Distance ◽  
Special Case ◽  
Independent Identically Distributed ◽  
Skew Normal

We study the properties of the multivariate skew normal distribution as an approximation to the distribution of the sum of n independent, identically distributed random vectors. More precisely, we establish conditions ensuring that the uniform distance between the two distribution functions converges to 0 at a rate of n-2/3. The advantage over the corresponding normal approximation is particularly relevant when the summands are skewed and n is small, as illustrated for the special case of exponentially distributed random variables. Applications to some well-known multivariate distributions are also discussed.

scholarly journals Immunological Characterization of HIV and SARS-CoV-2 Coinfected Young Individuals

Cells ◽  
2021 ◽  
Vol 10 (11) ◽  
pp. 3187
Author(s):  
Claudia Vanetti ◽  
Daria Trabattoni ◽  
Marta Stracuzzi ◽  
Antonella Amendola ◽  
Clara Fappani ◽  
...  
Keyword(s):  
Immune Response ◽  
General Population ◽  
Crucial Role ◽  
Rna Virus ◽  
Hiv Positive ◽  
Specific Antibodies ◽  
Increased Risk ◽  
Set Up

While the risk of SARS-CoV-2 infection and/or COVID-19 disease progression in the general population has been largely assessed, its impact on HIV-positive individuals remains unclear. We present clinical and immunological data collected in a cohort of HIV-infected young individuals during the first wave of COVID-19 pandemic. SARS-CoV-2 RNA, virus-specific antibodies, as well as the expression of factors involved in the anti-viral immune response were analyzed. Moreover, we set up an in vitro coinfection assay to study the mechanisms correlated to the coinfection process. Our results did not show any increased risk of severe COVID-19 in HIV-positive young individuals. In those subjects who contracted SARS-CoV-2 infection, an increase in IL-10 expression and production was observed. Furthermore, in the in vitro coinfection assay, we revealed a reduction in SARS-CoV-2 replication associated to an upregulation of IL-10. We speculate that IL-10 could play a crucial role in the course of SARS-CoV-2 infection in HIV-positive individuals. These results might help defining clinical management of HIV/SARS-CoV-2 co-infected young individuals, or putative indications for vaccination schedules in this population.

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