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Starting with a new formulation for the mutual information (MI) between a pair of events, this paper derives alternative upper bounds and extends those to the case of two discrete random variables. Normalized mutual information (NMI) measures are then obtained from those bounds, emphasizing the use of least upper bounds. Conditional NMI measures are also derived for three different events and three different random variables. Since the MI formulation for a pair of events is always nonnegative, it can properly be extended to include weighted MI and NMI measures for pairs of events or for random variables that are analogous to the well-known weighted entropy. This weighted MI is generalized to the case of continuous random variables. Such weighted measures have the advantage over previously proposed measures of always being nonnegative. A simple transformation is derived for the NMI, such that the transformed measures have the value-validity property necessary for making various appropriate comparisons between values of those measures. A numerical example is provided.

Richness and evenness, two important components of diversity, have been the subject of numerous studies exploring their potential dependence or lack thereof. The results have been contradictory and inconclusive, but tending to indicate only a low (positive or negative) correlation. While such reported studies have been based on particular data sets and species abundance distributions, the present article provides the results of a study using randomly generated abundance distributions and hence more generalizable findings and valid statistical results. The results reveal no statistically significant correlation between richness and evenness based on such random sample of abundance distributions and on four well‐known measures of diversity, including Simpson's indices and the entropy index. Of the two diversity components, evenness is found to have the strongest influence on diversity, but for numbers‐equivalent or effective‐number formulations, richness tends to be the most influential diversity component. For analyzing the tradeoff between richness and evenness for any given diversity measure and abundance distribution, the richness‐evenness curve is introduced as a new tool for diversity analysis. Richness and evenness are two important components of any measure of diversity. This article presents a simple graphical method showing the tradeoff between those two components of a diversity measure. Real biological data are used as an illustration.

The Theil index is one of the most popular indices of economic inequality, one reason for which is no doubt due to its convenient additive decomposition property. One of its weaknesses, however, is its lack of any intuitively meaningful interpretations. Another, and more serious, limitation of Theil’s index, as argued in this paper, is its lack of the value-validity property. That is, this index does not meet a particular condition based on metric distances between income-share distributions required in order for the range of potential index values to provide true, realistic, and valid representations of the economic inequality characteristic. After outlining the value-validity condition, this paper derives a simple transformation of Theil’s index that meets this condition to a high degree of approximation. Randomly generated income-share distributions are used to demonstrate and verify the validity of the corrected index. The new index formulation, which is simply a power function of Theil’s index, can then be used to make appropriate and reliable representations of absolute and relative difference comparisons of economic inequalities.

As a measure of randomness or uncertainty, the Boltzmann–Shannon entropy H has become one of the most widely used summary measures of a variety of attributes (characteristics) in different disciplines. This paper points out an often overlooked limitation of H: comparisons between differences in H-values are not valid. An alternative entropy H K is introduced as a preferred member of a new family of entropies for which difference comparisons are proved to be valid by satisfying a given value-validity condition. The H K is shown to have the appropriate properties for a randomness (uncertainty) measure, including a close linear relationship to a measurement criterion based on the Euclidean distance between probability distributions. This last point is demonstrated by means of computer generated random distributions. The results are also compared with those of another member of the entropy family. A statistical inference procedure for the entropy H K is formulated.

When dealing with nominal categorical data, it is often desirable to know the degree of association or dependence between the categorical variables. While there is literally no limit to the number of alternative association measures that have been proposed over the years, they all yield greatly varying, contradictory, and unreliable results due to their lack of an important property: value validity. After discussing the value-validity property, this paper introduces a new measure of association (dependence) based on the mean Euclidean distance between probability distributions, one being a distribution under independence. Both the asymmetric form, when one variable can be considered as the explanatory (independent) variable and one as the response (dependent) variable, and the symmetric form of the measure are introduced. Particular emphasis is given to the important 2 × 2 case when each variable has two categories, but the general case of any number of categories is also covered. Besides having the value-validity property, the new measure has all the prerequisites of a good association measure. Comparisons are made with the well-known Goodman–Kruskal lambda and tau measures. Statistical inference procedure for the new measure is also derived and numerical examples are provided.

The Herfindahl-Hirschman index (HHI) appears to be the most widely used index of market or industrial concentration. It is a summary measure that indicates the degree of competition, market power, and efficiency within a market or an industry. The HHI is also used by government agencies when evaluating potential violation of antitrust laws and regulations. As emphasized in this paper, and in spite of its several desirable properties, HHI has one serious imitation: it lacks the value-validity property. Lacking this property, caution has to be exercised when using HHI in order to avoid invalid and misleading results and conclusions. A corrected index is developed as a simple reformulation of HHI. Since this new index formulation meets the conditions imposed by the value-validity property to a high degree of approximation, and has other desirable properties comparable to those of HHI, the corrected index can safely be used to make various types of comparisons that are true and valid representations of market (industry) concentration. Numerical data are provided to support and exemplify the use of the corrected index.

Various properties have been advocated for biological evenness indices, with some properties being clearly desirable while others appear questionable. With a focus on such properties, this paper makes a distinction between properties that are clearly necessary and those that appear to be unnecessary or even inappropriate. Based on Euclidean distances as a criterion, conditions are introduced in order for an index to provide valid, true, and realistic representations of the evenness characteristic (attribute) from species abundance distributions. Without such value-validity property, it is argued that a measure or index provides only limited information about the evenness and results in misleading interpretations and evenness comparisons and incorrect results and conclusions. Among the overabundant variety of evenness indices, each of which is typically derived by rescaling a diversity measure to the interval from 0 to 1 and thereby controlling or adjusting for the species richness, most are found to lack the value-validity property and some lack the property of strict Schur-concavity. The most popular entropy-based index reveals an especially poor performance with a substantial overstatement of the evenness characteristic or a large positive value bias. One evenness index emerges as the preferred one, satisfying all properties and conditions. This index is based directly on Euclidean distances between relevant species abundance distributions and has an intuitively meaningful interpretation in terms of relative distances between distributions. The value validity of the indices is assessed by using a recently introduced probability distribution and from the use of computer-generated distributions with randomly varying species richness and probability (proportion) components.

PurposeTo investigate the contribution of demographic trends in countries’ age and gender composition to value set validity and obsolescence.MethodsTime-trade off (TTO) valuation data from 3 EQ-5D-3L value sets of 20 years or older from the United Kingdom, Japan, and the United States were re-analyzed using Bayesian heteroskedastic Tobit models with sex and age group-specific scale parameters. Original value sets were obtained by weighting the original preference structures with the countries’ original demographic composition at the time of the data collection. Updated value sets were created using the original preference structure weighted using the countries’ most recent demographic composition. The differences between the original and updated value sets were monitored and compared based on 95% credible intervals.ResultsThe gender and age composition of the investigated countries changed in all 3 countries over time. The modelled health state preferences also depended on the respondents’ gender and age. However, the overall impact of this demographic change on the investigated value sets was negligeable in all 3 countries and this finding was robust to accounting for the impact of ethnicity trends in the United States.ConclusionValue sets may become redundant and obsolete for various reasons, but demographic change was not identified as a contributing factor.

The use of lean tools and techniques to reduce waste from the workflow has been prominently gaining popularity in the construction industry worldwide. Last Planner System (LPS) is one such distinguished tool used by construction majors. Much research suggests that LPS plays a significant role in improving collaboration among project team members and the firm's organizational culture. This paper reports the intermediating effect of organizational culture on construction projects that implement the Last Planner System and also identifies the effect of LPS on the project's operational performance. Various survey items pertaining to each variable were identified and validated through Content Validation. All the established items were then compiled into a questionnaire, and multiple data were collected from a variety of projects. The data were analysed using bivariate correlation and multiple regression analysis. Additionally, the \"organizational culture assessment instrument\" was utilized to compare the organizational cultures of various construction sites. It was discovered that the Last Planner System had a positive but negligible relationship with the operational performance of the project, and the organizational culture. It was observed that all the projects implementing LPS had a balance culture. The contribution of this work is the knowledge that implementation of the Last Planner System is not necessarily enhanced by culture to a great extent. However, it has a positive impact on both the operational performance of the project and the organizational culture.

In this paper, polynomial equations with real coefficients and in one variable were considered which contained a small, positive but specified and fixed parameter$ \\varepsilon_0 \\neq 0 $ . By using the classical asymptotic method, roots of the polynomial equations have been constructed in the literature, which were proved to be valid for sufficiently small$ \\varepsilon $ -values (or equivalently for$ \\varepsilon \\to 0 $ ). In this paper, it was assumed that for some or all roots of a polynomial equation, the first few terms in a Taylor or Laurent series in a small parameter depending on$ \\varepsilon $exist and can be constructed. We also assumed that at least two approximations$ x_1(\\varepsilon) $and$ x_2(\\varepsilon) $for the real roots exist and can be constructed. For a complex root, we assumed that at least two real approximations$ a_1(\\varepsilon) $and$ a_2(\\varepsilon) $for the real part of this root, and that at least two real approximations$ b_1(\\varepsilon) $and$ b_2(\\varepsilon) $for the imaginary part of this root, exist and can be constructed. Usually it was not clear whether for$ \\varepsilon = \\varepsilon_0 $the approximations were valid or not. It was shown in this paper how the classical asymptotic method in combination with the bisection method could be used to prove how accurate the constructed approximations of the roots were for a given interval in$ \\varepsilon $(usually including the specified and fixed value$ \\varepsilon_0 \\neq 0 $ ). The method was illustrated by studying a polynomial equation of degree five with a small but fixed parameter$ \\varepsilon_0 = 0.1 $ . It was shown how (absolute and relative) error estimates for the real and imaginary parts of the roots could be obtained for all values of the small parameter in the interval$ (0, \\varepsilon_0] $ .