scholarly journals Dirichlet Polynomials and Entropy

Entropy ◽  
2021 ◽  
Vol 23 (8) ◽  
pp. 1085
Author(s):  
David I. Spivak ◽  
Timothy Hosgood
Keyword(s):  
Probability Distribution ◽  
Shannon Entropy ◽  
Empirical Distribution ◽  
Geometric Mean ◽  
The Other ◽  
Cross Entropy ◽  
Dirichlet Polynomial ◽  
Area Formula ◽  
Additive Structure ◽  
Dirichlet Polynomials

A Dirichlet polynomial d in one variable y is a function of the form d(y)=anny+⋯+a22y+a11y+a00y for some n,a0,…,an∈N. We will show how to think of a Dirichlet polynomial as a set-theoretic bundle, and thus as an empirical distribution. We can then consider the Shannon entropy H(d) of the corresponding probability distribution, and we define its length (or, classically, its perplexity) by L(d)=2H(d). On the other hand, we will define a rig homomorphism h:Dir→Rect from the rig of Dirichlet polynomials to the so-called rectangle rig, whose underlying set is R⩾0×R⩾0 and whose additive structure involves the weighted geometric mean; we write h(d)=(A(d),W(d)), and call the two components area and width (respectively). The main result of this paper is the following: the rectangle-area formula A(d)=L(d)W(d) holds for any Dirichlet polynomial d. In other words, the entropy of an empirical distribution can be calculated entirely in terms of the homomorphism h applied to its corresponding Dirichlet polynomial. We also show that similar results hold for the cross entropy.

Assessment of the exposure of swimmers to microbiological contaminants in fresh waters

1997 ◽  
Vol 35 (11-12) ◽  
pp. 157-163 ◽  
Author(s):  
G. J. Medema ◽  
I. A. van Asperen ◽  
A. H. Havelaar
Keyword(s):  
Geometric Mean ◽  
Microbiological Quality ◽  
The Other ◽  
Pollution Level ◽  
Intestinal Infection ◽  
E Coli ◽  
Faecal Pollution ◽  
Run Off ◽  
Thermotolerant Coliforms ◽  
Mean Concentration

As part of a prospective cohort study among triathletes to determine a relationship between the microbiological quality of fresh bathing water and the risk of acquiring an intestinal infection, the exposure of the triathletes to microbiological contaminants was assessed. Waters were collected at seven triathlons (swimming course 1–1.5km) held in the summer of 1993 and 1994 to have a range of water qualities. All were influenced by sewage effluents, most also by agricultural run-off. Samples were collected several weeks before the event to establish a sampling programme (1993) and during the actual exposure of the triathletes (1993 and 1994) and examined for thermotolerant coliforms alone (samples preceding the event) and for E. coli, faecal enterococci, Staphylococcus aureus, F-specific RNAphages, enteroviruses (1993 and 1994) and for thermophilic Campylobacter, Salmonella, Aeromonas, Plesiomonas shigelloides and Pseudomonas aeruginosa (1993). The samples taken in the weeks before the exposure showed significant differences in thermotolerant coliform concentration between locations, depths and times. Also during swimmer exposure, significant differences occurred in microorganism levels at the different sampling points over the swimming course. As the triathletes swam as a group, they were exposed to approximately the same water at the same time. The geometric mean concentration was used to characterise each site. In the epidemiological study, the risk of an intestinal infection correlated with the concentration of thermotolerant coliforms and E. coli but not with the other parameters. The geometric mean concentration of thermotolerant coliforms at the triathlons ranged from 11–330/100mL and 54–1,200/100mL E. coli. Ranking of the seven sites by faecal pollution level, based on the geometric mean concentration of a faecal indicator, resulted in a different ranking for each indicator. At the fresh water sites studied, only the ratio between the geometric mean density of E. coli and thermotolerant coliforms was constant. The ratio between the other parameters related to faecal pollution (faecal enterococci, F-specific RNA phages, enteroviruses) varied considerably. Water quality standards relating to faecal pollution can only be based on parameters that show a significant correlation with risk of intestinal illness.

scholarly journals Biases Associated with Four Stream Substrate Samplers

1991 ◽  
Vol 48 (10) ◽  
pp. 1882-1886 ◽  
Author(s):  
Michael K. Young ◽  
Wayne A. Hubert ◽  
Thomas A. Wesche
Keyword(s):  
Particle Size ◽  
Geometric Mean ◽  
The Other ◽  
Substrate Composition ◽  
Mean Particle Size ◽  
Core Samples ◽  
Freeze Core ◽  
Single Probe ◽  
Core Sampler ◽  
Stream Substrate

We compared samples collected from 10 substrates of various compositions with a single-probe freeze-core sampler, a triple-probe freeze-core sampler, a McNeil sampler, and a shovel. The accuracy with which these devices sampled particles larger than 50 mm in diameter varied; they were oversampled by the freeze-core devices, sampled in proportion to their availability by a shovel, and sampled inconsistently by the McNeil sampler. The geometric mean particle size and variance of single-probe freeze-core samples consistently exceeded those of samples collected with the other devices. Most sample means also exceeded the test substrate means. By excluding the proportions of particles larger than 50 mm in diameter in our analyses, we found that proportions of several particles sizes in samples collected by different methods differed significantly from the actual proportions in test substrates. There were few differences between the single- and triple-probe freeze-core samples or between McNeil and shovel samples. All four samplers were biased, but the McNeil sampler most frequently produced samples that approximated the true substrate composition.

scholarly journals Some Order Preserving Inequalities for Cross Entropy and Kullback–Leibler Divergence

Entropy ◽  
2018 ◽  
Vol 20 (12) ◽  
pp. 959 ◽  
Author(s):  
Mateu Sbert ◽  
Min Chen ◽  
Jordi Poch ◽  
Anton Bardera
Keyword(s):  
Machine Learning ◽  
Information Theory ◽  
Geometric Mean ◽  
Cross Entropy ◽  
Rearrangement Inequality ◽  
Numerical Examples ◽  
Leibler Divergence ◽  
Recent Theorem ◽  
The Difference ◽  
New Inequalities

Cross entropy and Kullback–Leibler (K-L) divergence are fundamental quantities of information theory, and they are widely used in many fields. Since cross entropy is the negated logarithm of likelihood, minimizing cross entropy is equivalent to maximizing likelihood, and thus, cross entropy is applied for optimization in machine learning. K-L divergence also stands independently as a commonly used metric for measuring the difference between two distributions. In this paper, we introduce new inequalities regarding cross entropy and K-L divergence by using the fact that cross entropy is the negated logarithm of the weighted geometric mean. We first apply the well-known rearrangement inequality, followed by a recent theorem on weighted Kolmogorov means, and, finally, we introduce a new theorem that directly applies to inequalities between K-L divergences. To illustrate our results, we show numerical examples of distributions.

scholarly journals Evaluating Different Methods for Determining the Velocity-Dip Position over the Entire Cross Section and at the Centerline of a Rectangular Open Channel

Entropy ◽  
2020 ◽  
Vol 22 (6) ◽  
pp. 605
Author(s):  
Zhongfan Zhu ◽  
Pengfei Hei ◽  
Jie Dou ◽  
Dingzhi Peng
Keyword(s):  
Experimental Data ◽  
Cross Section ◽  
Shannon Entropy ◽  
Open Channel ◽  
Lateral Distribution ◽  
Tsallis Entropy ◽  
The Other ◽  
Deterministic Models ◽  
Entire Cross Section ◽  
General Index

The velocity profile of an open channel is an important research topic in the context of open channel hydraulics; in particular, the velocity-dip position has drawn the attention of hydraulic scientists. In this study, analytical expressions for the velocity-dip position over the entire cross section and at the centerline of a rectangular open channel are derived by adopting probability methods based on the Tsallis and general index entropy theories. Two kinds of derived entropy-based expressions have the same mathematical form as a function of the lateral distance from the sidewall of the channel or of the aspect ratio of the channel. Furthermore, for the velocity-dip position over the entire cross section of the rectangular open channel, the derived expressions are compared with each other, as well as with two existing deterministic models and the existing Shannon entropy-based expression, using fifteen experimental datasets from the literature. An error analysis shows that the model of Yang et al. and the Tsallis entropy-based expression predict the lateral distribution of the velocity-dip position better than the other proposed models. For the velocity-dip position at the centerline of the rectangular open channel, six existing conventional models, the derived Tsallis and general index entropy-based expressions, and the existing Shannon entropy-based models are tested against twenty-one experimental datasets from the literature. The results show that the model of Kundu and the Shannon entropy-based expression have superior prediction accuracy with respect to experimental data compared with other models. With the exception of these models, the Tsallis entropy-based expression has the highest correlation coefficient value and the lowest root mean square error value for experimental data among the other models. This study indicates that the Tsallis entropy could be a good addition to existing deterministic models for predicting the lateral distribution of the velocity-dip position of rectangular open channel flow. This work also shows the potential of entropy-based expressions, the Shannon entropy and the Tsallis entropy in particular, to predict the velocity-dip position at the centerline of both narrow and wide rectangular open channels.

scholarly journals The Evolutionary Status of the Secondaries of Cataclysmic Binaries

1983 ◽  
Vol 72 ◽  
pp. 257-262
Author(s):  
H. Ritter
Keyword(s):  
Probability Distribution ◽  
Main Sequence ◽  
The Other ◽  
Initial Mass ◽  
Evolutionary Status ◽  
Mass Ratio ◽  
Main Sequence Stars ◽  
Cataclysmic Binaries ◽  
Low Mass ◽  
Orbital Periods

ABSTRACTIt is shown that the secondary components of cataclysmic binaries with orbital periods of less than ~10 hours are indistinguishable from ordinary low-mass main-sequence stars and that, therefore, they are essentially unevolved. On the other hand, it is shown that, depending on the mass ratio of the progenitor system, the secondary of a cataclysmic binary could be significantly evolved. The fact that nevertheless most of the observed secondaries are essentially unevolved can be accounted for by assuming that the probability distribution for the initial mass ratio is not strongly peaked towards unity mass ratio.

scholarly journals Haemagglutination-inhibiting (HI) antibodies against four prototype strains of influenza A virus in different age groups

1979 ◽  
Vol 82 (2) ◽  
pp. 225-230 ◽  
Author(s):  
A. L. Terzin ◽  
S. Djurišić ◽  
B. Vuković ◽  
V. Vujkov
Keyword(s):  
Influenza Virus ◽  
Hong Kong ◽  
New Jersey ◽  
Antibody Response ◽  
Influenza A ◽  
Geometric Mean ◽  
Age Groups ◽  
The Other ◽  
Younger Age ◽  
Lower Antibody Response

SUMMARYSera of 197 apparently well persons were tested for residual haemagglutination-inhibiting antibodies against live Hong Kong/68, A/FM/47 and A/PR/34 strains. Sera of 62 well persons, regularly exposed to contacts with swine, were tested against an inactivated A/New Jersey/76 antigen.Those born some time before and during a certain influenza era showed a significantly greater proportion of homologous residual titres against the subtype prevailing in that influenza era, than those born after the termination of the same era.In each of the seven age groups tested both the percentage of positives and the geometric mean titres were usually highest against the Hong Kong strain (representing the most recent era); the next highest were those against the FM1 strain and the lowest were those against the PR8 strain (representing the most distant of these three influenza eras).The serological involvement of donors exposed to regular contacts with swine was relatively stronger against the New Jersey antigen than the response of other serum donors shown against the other three, more recent, prototypes of influenza virus A. The oldest age groups showed significantly lower antibody response against the PR8, FM1 and Hong Kong strains (but not against the New Jersey antigen) than the next one or two of the younger age groups.

On the wind resource in Algeria: Probability distributions evaluation

2020 ◽  
pp. 095765092097588
Author(s):  
Houdayfa Ounis ◽  
Nawel Aries
Keyword(s):  
Wind Speed ◽  
Probability Distribution ◽  
Probability Distributions ◽  
Resource Assessment ◽  
Descriptive Statistics ◽  
The Other ◽  
Wind Speeds ◽  
Wind Resource Assessment ◽  
The Mean ◽  
Wind Resource

The present study aims to present a contribution to the wind resource assessment in Algeria using ERA-Interim reanalysis. Firstly, the ERA-Interim reanalysis 10 m wind speed data are considered for the elaboration of the mean annual 10 m wind speed map for a period starting from 01-01-2000 to 31-12-2017. Moreover, the present study intends to highlight the importance of the descriptive statistics other than the mean in wind resource assessment. On the other hand, this study aims also to select the proper probability distribution for the wind resource assessment in Algeria. Therefore, nine probability distributions were considered, namely: Weibull, Gamma, Inverse Gaussian, Log Normal, Gumbel, Generalized Extreme Value (GEV), Nakagami, Generalized Logistic and Pearson III. Furthermore, in combination with the distribution, three parameter estimation methods were considered, namely, Method of Moment, Maximum Likelihood Method and L-Moment Method. The study showed that Algeria has several wind behaviours due to the diversified topographic, geographic and climatic properties. Moreover, the annual mean 10 m wind speed map showed that the wind speed varies from 2.3 to 5.3 m/s, where 73% of the wind speeds are above 3 m/s. The map also showed that the Algerian Sahara is windiest region, while, the northern fringe envelopes the lowest wind speeds. In addition, it has been shown that the study of the mean wind speeds for the evaluation of the wind potential alone is not enough, and other descriptive statistics must be considered. On the other hand, among the nine considered distribution, it appears that the GEV is the most appropriate probability distribution. Whereas, the Weibull distribution showed its performance only in regions with high wind speeds, which, implies that this probability distribution should not be generalized in the study of the wind speed in Algeria.

Aging Population, Retirement, and Risk Taking: Reply

2020 ◽  
Vol 66 (6) ◽  
pp. 2796-2799
Author(s):  
Haim Levy
Keyword(s):  
Empirical Evidence ◽  
Risk Taking ◽  
Geometric Mean ◽  
Aging Population ◽  
The Other ◽  
Investment Horizon ◽  
Technical Error ◽  
Long Run ◽  
Original Theorem

In theorem 1 given in my paper, “Aging Population, Retirement, and Risk Taking” [Levy H (2016a) Aging population, retirement, and risk taking. Management Sci. 62(5):1415–1430.], there is indeed a technical error. Yet, adding one condition to the theorem (which can be added in two alternate ways) is sufficient to ensure the dominance of stocks over bonds in the very long run. For the commonly employed preferences, the empirical evidence conforms with the claim given in my original theorem 1, asserting that the portfolio with the higher geometric mean (stocks) dominates the other portfolio under consideration (bonds) as the investment horizon increases indefinitely. Thus, as advocated in my paper, stocks dominate bonds for investors with typical preferences who save for retirement. This paper was accepted by Karl Deither, finance.

Fractional dynamics of allometry

2012 ◽  
Vol 15 (1) ◽  
Author(s):  
Bruce West ◽  
Damien West
Keyword(s):  
Fractal Geometry ◽  
Empirical Distribution ◽  
Linear Regression Analysis ◽  
The Other ◽  
Fractional Dynamics ◽  
Data Sets ◽  
Fractal Statistics ◽  
Metabolic Data ◽  
Detailed Application ◽  
Dynamic Variables

AbstractAllometry relations (ARs) in physiology are nearly two hundred years old. In general if X ij is a measure of the size of the i th member of a complex host network from species j and Y ij is a property of a complex subnetwork embedded within the host network an intraspecies AR exists between the two when Y ij = aX ijb. We emphasize that the reductionist models of AR interpret X ij and Y ij as dynamic variables, albeit the ARs themselves are explicitly time independent. On the other hand, the phenomenological models of AR are based on the statistical analysis of data and interpret 〈X i〉 and 〈Y i〉 as averages over an ensemble of individuals to yields the interspecies AR 〈Y i〉 = a〈X i〉b. Modern explanations of AR begin with the application of fractal geometry and fractal statistics to scaling phenomena. The detailed application of fractal geometry to the explanation of intraspecies ARs is a little over a decade old and although well received it has not been universally accepted. An alternate perspective is given by the interspecies AR based on linear regression analysis of fluctuating data sets. We emphasize that the intraspecies and interspecies ARs are not the same and show that the interspecies AR can only be derived from the intraspecies one for a narrow distribution of fluctuations. This condition is not satisfied by metabolic data as is shown separately for aviary and mammal data sets. The empirical distribution of metabolic allometry coefficients is shown herein to be Pareto in form. A number of reductionist arguments conclude that the allometry exponent is universal, however herein we derive a deterministic relation between the allometry exponent and the allometry coefficient using the fractional calculus. The co-variation relation violates the universality assumption. We conclude that the interspecies physiologic AR is entailed by the scaling behavior of the probability density, which is derived using the fractional probability calculus.

scholarly journals The risks of mixing dependency lengths from sequences of different length

Glottotheory ◽  
2014 ◽  
Vol 5 (2) ◽  
Author(s):  
Ramon Ferrer-i-Cancho ◽  
Haitao Liu
Keyword(s):  
Null Hypothesis ◽  
Empirical Distribution ◽  
A Priori ◽  
Sentence Length ◽  
The Other ◽  
Language Research ◽  
Varying Length ◽  
Syntactic Dependency ◽  
Better Than

AbstractMixing dependency lengths from sequences of different length is a common practice in language research. However, the empirical distribution of dependency lengths of sentences of the same length differs from that of sentences of varying length. The distribution of dependency lengths depends on sentence length for real sentences and also under the null hypothesis that dependencies connect vertices located in random positions of the sequence. This suggests that certain results, such as the distribution of syntactic dependency lengths mixing dependencies from sentences of varying length, could be a mere consequence of that mixing. Furthermore, differences in the global averages of dependency length (mixing lengths from sentences of varying length) for two different languages do not simply imply a priori that one language optimizes dependency lengths better than the other because those differences could be due to differences in the distribution of sentence lengths and other factors.

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