scholarly journals Refined Young Inequality and Its Application to Divergences

Entropy ◽  
2021 ◽  
Vol 23 (5) ◽  
pp. 514
Author(s):  
Shigeru Furuichi ◽  
Nicuşor Minculete
Keyword(s):  
Geometric Mean ◽  
Arithmetic Mean ◽  
Young Inequality ◽  
Rényi Divergence ◽  
Weighted Arithmetic ◽  
The Difference

We give bounds on the difference between the weighted arithmetic mean and the weighted geometric mean. These imply refined Young inequalities and the reverses of the Young inequality. We also studied some properties on the difference between the weighted arithmetic mean and the weighted geometric mean. Applying the newly obtained inequalities, we show some results on the Tsallis divergence, the Rényi divergence, the Jeffreys–Tsallis divergence and the Jensen–Shannon–Tsallis divergence.

scholarly journals The Weighted Arithmetic Mean–Geometric Mean Inequality is Equivalent to the Hölder Inequality

Symmetry ◽  
2018 ◽  
Vol 10 (9) ◽  
pp. 380 ◽  
Author(s):  
Yongtao Li ◽  
Xian-Ming Gu ◽  
Jianxing Zhao
Keyword(s):  
Geometric Mean ◽  
Arithmetic Mean ◽  
Hölder Inequality ◽  
Power Mean ◽  
Weighted Arithmetic ◽  
Mathematical Equivalence ◽  
Power Mean Inequality ◽  
Mathematical Relations

In the current note, we investigate the mathematical relations among the weighted arithmetic mean–geometric mean (AM–GM) inequality, the Hölder inequality and the weighted power-mean inequality. Meanwhile, the proofs of mathematical equivalence among the weighted AM–GM inequality, the weighted power-mean inequality and the Hölder inequality are fully achieved. The new results are more generalized than those of previous studies.

Geospatial Suitability Indices Toolbox (GSI Toolbox)

2021 ◽  
Author(s):  
Christina Saltus ◽  
Todd Swannack ◽  
S. McKay
Keyword(s):  
Habitat Suitability ◽  
Geometric Mean ◽  
Arithmetic Mean ◽  
Limiting Factor ◽  
Ecological Processes ◽  
Weighted Arithmetic ◽  
Suitability Index ◽  
Multiple Parameter ◽  
Spatially Distributed

Habitat suitability models are widely adopted in ecosystem management and restoration, where these index models are used to assess environmental impacts and benefits based on the quantity and quality of a given habitat. Many spatially distributed ecological processes require application of suitability models within a geographic information system (GIS). Here, we present a geospatial toolbox for assessing habitat suitability. The Geospatial Suitability Indices (GSI) toolbox was developed in ArcGIS Pro 2.7 using the Python® 3.7 programming language and is available for use on the local desktop in the Windows 10 environment. Two main tools comprise the GSI toolbox. First, the Suitability Index Calculator tool uses thematic or continuous geospatial raster layers to calculate parameter suitability indices based on user-specified habitat relationships. Second, the Overall Suitability Index Calculator combines multiple parameter suitability indices into one overarching index using one or more options, including: arithmetic mean, weighted arithmetic mean, geometric mean, and minimum limiting factor. The resultant output is a raster layer representing habitat suitability values from 0.0 to 1.0, where zero is unsuitable habitat and one is ideal suitability. This report documents the model purpose and development as well as provides a user’s guide for the GSI toolbox.

The Risk Premium on Ordinary Shares

1995 ◽  
Vol 1 (2) ◽  
pp. 251-330 ◽  
Author(s):  
A.D. Wilkie
Keyword(s):  
Risk Premium ◽  
Geometric Mean ◽  
Arithmetic Mean ◽  
Short Term ◽  
Arithmetic Means ◽  
Price Inflation ◽  
Geometric Means ◽  
True Model ◽  
The Difference

ABSTRACTThe risk premium on ordinary shares is investigated, by studying the total returns on ordinary shares, and on both long-term and short-term fixed-interest investments over the period 1919 to 1994, and by analysing the various components of that return. The total returns on ordinary shares exceeded those on fixed-interest investments by over 5% p.a. on a geometric mean basis and by over 7% p.a. on an arithmetic mean basis, but it is argued that these figures are misleading, because most of the difference can be accounted for by the fact that price inflation turned out to be about 4.5% p.a. over the period, whereas investors had been expecting zero inflation.Quotations from contemporary authors are brought forward to demonstrate what contemporary attitudes were. Simulations are used along with the Wilkie stochastic asset model to show what the results would be if investors make various assumptions about the future, but the true model turns out to be different from what they expected. The differences between geometric means of the data and arithmetic means are shown to correspond to differences between using medians or means of the distribution of future returns, and it is suggested that, for discounting purposes, medians are the better measure.

scholarly journals Comparison of different ways of computing grades in continuous assessment into the final grade

2017 ◽  
Vol 8 ◽  
pp. 1
Author(s):  
Juan A. Marin-Garcia ◽  
Julien Maheut ◽  
Julio J. Garcia Sabater
Keyword(s):  
Learning Outcomes ◽  
Geometric Mean ◽  
Arithmetic Mean ◽  
Harmonic Mean ◽  
Alternative Methods ◽  
Continuous Assessment ◽  
Final Grade ◽  
Weighted Arithmetic ◽  
Student’S Performance ◽  
The Subject

<p>We present the results of comparing various ways of calculating students' final grades from continuous assessment grades. Traditionally the weighted arithmetic mean has been used and we compare this method with other alternatives: arithmetic mean, geometric mean, harmonic mean and multiplication of the percentage of overcoming of each activi-ty. Our objective is to verify, if any of the alternative methods, agree with the student’s performance proposed by the teacher of the subject, further discriminating the grade be-tween high and low learning outcomes and reducing the number of approved opportunists.</p><p> </p><p>[Comparación del efecto de diferentes modos de agregar las califica-ciones de evaluación continua en la nota final]</p>

scholarly journals Controlling for temporal discounting shifts rats from geometric to human-like arithmetic bisection

2018 ◽  
Author(s):  
Charles D Kopec ◽  
Carlos D Brody
Keyword(s):  
Human Subjects ◽  
Geometric Mean ◽  
Temporal Discounting ◽  
Behavioral Model ◽  
Arithmetic Mean ◽  
Bisection Task ◽  
Temporal Bisection ◽  
Bisection Point ◽  
The Difference ◽  
Temporal Bisection Task

AbstractHow our brains measure the passage of time is still largely open for debate. One behavioral task commonly used to study how durations are perceived is the Temporal Bisection Task, in which subjects categorize time durations as either “short” or “long.” The duration equally likely to be categorized as short or long is known as the bisection point. It has been consistently demonstrated that for humans, the bisection point is near the arithmetic mean of the longest and shortest durations the subject was trained on. In contrast, for non-human subjects it has been consistently found near the geometric mean. This difference implies that humans may process or represent temporal durations differently than other species. Here we present a behavioral model that reconciles the differences by demonstrating that rats’ performance on this task is driven not only by their noisy estimates of duration, but also by the temporally-discounted value of future rewards. The model correctly predicts shifts in the bisection point induced by unequal rewards and explains otherwise-paradoxical psychometric reversals documented three decades ago. Furthermore, as predicted by the model, we found that modifying the Temporal Bisection Task to eliminate the temporally-discounted reward component shifted the rats’ bisection point from the geometric mean to the arithmetic mean, thus bringing the rat results into line with the human results. We therefore propose that humans and rats (and perhaps other non-human subjects as well) process temporal information similarly, and that the difference between them in the Temporal Bisection Task may be simply due to rats weighing temporal discounting of future rewards more strongly than humans.

A New Way to View the Magnitude of the Difference between the Arithmetic Mean and the Geometric Mean and the Difference between the Slopes When a Continuous Dependent Variable Is Expressed in Raw Form Versus Logged Form

2017 ◽  
Vol 47 (1) ◽  
pp. 165-181
Author(s):  
Leo A. Goodman
Keyword(s):  
Present Note ◽  
Geometric Mean ◽  
Arithmetic Mean ◽  
Form Versus ◽  
Geometric Means ◽  
Dependent Variables ◽  
The Difference ◽  
Group 2 ◽  
Multiplicative Models ◽  
Group 1

In Trond Petersen’s “Multiplicative Models for Continuous Dependent Variables: Estimation on Unlogged versus Logged Form” (this volume, pp. 113–164), the following phenomenon is noted: With the arithmetic and geometric means of a nonnegative quantitative variable measured for two groups (say, groups 1 and 2), it is possible for the arithmetic mean [Formula: see text] for group 1 to be larger than the arithmetic mean [Formula: see text] for group 2 and the geometric mean [Formula: see text] for group 1 to be smaller than the geometric mean [Formula: see text] for group 2. In the present note, some new formulas are introduced that will help to make clear when this phenomenon will occur and when it will not occur. The phenomenon considered here is of interest in situations where, for example, as noted in Petersen’s paper, results obtained using the raw form for a continuous dependent variable are compared with the corresponding results obtained using the logged form for the dependent variable. In addition, the formulas that are introduced in the present note also provide a new way to view the magnitude of the difference between the arithmetic mean and the geometric mean.

scholarly journals Teaching Business Statistics: Some Useful Relationships

2020 ◽  
Vol 15 (5) ◽  
pp. 73
Author(s):  
Philip F. Rice ◽  
Chris Brune
Keyword(s):  
Geometric Mean ◽  
Joint Probability ◽  
Arithmetic Mean ◽  
Simple Arithmetic ◽  
Weighted Arithmetic ◽  
Multiplication Rule ◽  
Business Statistics ◽  
Statistics Course ◽  
Compound Interest ◽  
The Relationship

The purpose of this paper is to suggest an instructional approach in the introductory business statistics course that utilizes relationships between separately introduced topics. The paper will explore three &ldquo;useful relationships&rdquo; that can assist classroom instruction: (1) the relationship between the simple arithmetic mean, the weighted arithmetic mean, and the expected value of a discrete probability distribution; (2) the relationship between the use of the multiplication rule to calculate the joint probability associated with two events, use of tree diagrams, and the use of the binomial and hypergeometric distributions; and (3) the relationship between the geometric mean and the compound interest rate. Each discussion includes detailed examples of calculations to demonstrate the relationships.

scholarly journals Economic Security in Post-Soviet Countries: Level of Ensuring and Development Trends

2021 ◽  
Vol 22 (2) ◽  
pp. 86-101
Author(s):  
Ganna Kozachenko ◽  
◽  
Igor Andrushchenko ◽  
Yuriy Pogorelov ◽  
Larysa Gerasymenko ◽  
...  
Keyword(s):  
Comparative Analysis ◽  
Geometric Mean ◽  
National Level ◽  
Economic Security ◽  
Arithmetic Mean ◽  
Security Studies ◽  
Economic Systems ◽  
Economic Potential ◽  
Weighted Arithmetic ◽  
Low Levels

At the national level of economic security studies, a special place has alway s b e lon ged t o t he estimating side of the issue. Estimations of state economic security serve as input data for the determinat ion of directions and ways of further security provision. At the same time, such estimations should be considered not only as a result of a certain methodology application in a certain co un try b ut a lso i n t h e c o nte xt o f comparing the economic security estimations across a set of countries. The aim of the article is to determin e the level of ensuring economic security in post-Soviet countries and recognize patterns, ri sk s, a n d t h rea ts that affect the future development of state economic security. For comparative analysis of economic security, Ukraine, Azerbaijan, Kazakhstan, Moldova, Georgia, and period 2016-2020 have been selected. The methodological basis of the study included the followingmethods: comparative economic studies, methods of summation, arithmetic mean, weighted arithmetic, geometric mean, rating; international index systems we re used as a basis for comparative analysis. Using the suggested methodology of estimation allowed obta ini ng results that characterize level real economic security, is lower than average. None of the analyzed countri es has managed to maintain an acceptable level of economic security. The determined levels allow u s t o st a te that the economic systems of the analyzed post-Soviet countries demonstrat e h i gh p erc ep ti ve ne ss t o t h e actualization of various threats. And this perceptiveness, in its turn, leads to various negative changes in t h e economic systems of these countries. The high perceptiveness of the economic systems in the analyze d p o st-Soviet countries to the actualization of various threats can be explained by the c h an gin g q u ali ty o f t he ir economic potential, low levels of their innovativeness, and also the lack of proper condi ti ons t o a p ply t h e innovations

scholarly journals Multiattribute Decision-Making Problems in terms of the Weighted Mean Operation of Two Aggregation Operators of Orthopair Z-Numbers

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Mailing Zhao ◽  
Jun Ye
Keyword(s):  
Decision Making ◽  
Comparative Analysis ◽  
Geometric Mean ◽  
Real Life ◽  
Arithmetic Mean ◽  
Aggregation Operators ◽  
Weighted Mean ◽  
Reliability Measure ◽  
Reliability Level ◽  
Weighted Arithmetic

The Z number defined by Zadeh can depict the fuzzy restriction/value and reliability measure by an ordered pair of fuzzy values to strengthen the reliability of the fuzzy restriction/value. However, there exist truth and falsehood Z-numbers in real life. Thus, the Z number cannot reflect both. To indicate both, this study presents an orthopair Z-number (OZN) set to depict truth and falsehood values (intuitionistic fuzzy values) and their reliability levels in uncertain and incomplete cases. Next, we define the operations, score and accuracy functions, and sorting rules of OZNs. Further, the OZN weighted arithmetic mean (OZNWAM) and OZN weighted geometric mean (OZNWGM) operators are proposed based on the operations of OZNs. According to the weighted mean operation of the OZNWAM and OZNWGM operators, a multiattribute decision-making (MADM) model is established in the case of OZNs. Lastly, a numerical example is presented to reflect the flexibility and rationality of the presented MADM model. Comparative analysis indicates that the presented MADM model can indicate its superiority in the reliability and flexibility of decision results. Meanwhile, the resulting advantage of this study is that the presented MADM model can strengthen the reliability level of orthopair fuzzy values and make the decision results more reliable and flexible.

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