Zipf Mandelbrot Research Articles

N-convexity and weighted majorization with applications to f-divergences and Zipf–Mandelbrot law

n-convexity and weighted majorization with applications to f-divergences and Zipf–Mandelbrot law

Characteristics and Driving Mechanisms of Understory Vegetation Diversity Patterns in Central and Southern China

Large-scale forest restoration projects significantly reduce the net rates of forest loss. However, as a key component of forest restoration, planted forests have failed to restore biodiversity. China has implemented a large-scale afforestation program, which includes pure planted forests in particular, leading to various changes in ecosystem processes. Despite this, a comprehensive analysis of understory vegetation diversity patterns in these pure planted forests is still lacking. This study aimed to analyze the data on understory vegetation diversity from three typical pure and natural forest ecosystems of Hunan ecological forests to reveal their diversity patterns. The results revealed no significant difference in the understory diversity index between natural and pure forest types, although natural forests had a bigger species pool. The Zipf–Mandelbrot model was a better fit for species abundance distribution. The fitted results suggested that both environmental filtering and neutral processes affected the species abundance distribution and pure understory communities during restoration succession. Natural forests had the most stable understory diversity structure, whereas pure Phyllostachys heterocycla (Carr.) Mitford forests had the least stable structure. Multivariate regression tree analysis identified indicator species for each community. The gradient boosting model indicated that isothermality and slope direction were the most important factors affecting diversity. The β-diversity analysis showed that community establishment in the four forest types was affected via different mechanisms. The findings of this study have significant implications for understanding the impact of afforestation on the mechanisms for maintaining diversity.

New bounds of Popoviciu's difference via weighted Hadamard type inequalities with applications in information theory

We generalize Popoviciu's inequality utilizing the theory of 4‐convex function under the effect of new Green functions. We formulate result for power means and quasi means. Also, we give applications in information theory by giving new estimations of generalized Csiszár divergence, Rényi divergence, Shannon entropy, Kullback–Leibler divergence, divergence, Zipf–Mandelbrot entropy, and Hybrid Zipf–Mandelbrot entropy.

Species-Abundance Models for the Early Postfire Succession of Subalpine Shrub Grassland

Fire is one of the principal factors influencing subalpine ecosystem succession. Species numbers and plant compositions are used to determine postfire disturbance, vegetation, structural change, and succession. Ecologists also integrate species diversity and mathematical models to enable researchers to obtain increasingly detailed insights into habitats during post-disturbance restoration processes. This study employed five species-abundance models, namely the niche preemption model, the broken-stick model, the log-normal model, the Zipf model, and the Zipf–Mandelbrot model, to perform fitting analysis on the abundance data of postfire species coverage in shrub grasslands near 369 Hut at Xue Mountain in Shei-Pa National Park, Taiwan. We performed the logarithmic transformation on plant-coverage areas for each period of postfire shrub-grassland succession, and then, based on histograms drawn for species–coverage distribution modes, the test results consistently showed normal distributions (p < 0.05). Species-coverage histograms measuring various periods showed that there were comparatively higher numbers of common species during postfire succession and that the numbers of rare species progressively increased. The fitting results of the five species-abundance models showed that although the most suitable abundance models for each period of postfire succession varied, the majority of these periods demonstrated decent fitting with respect to the Zipf–Mandelbrot model. These findings showed that fuel consumption provided nutrients in a manner that facilitated postfire regeneration. Moreover, dominant species, such as Yushania niitakayamensis, and Miscanthus transmorrisonensis, did not fully occupy growing spaces and resource availabilities; consequently, seeded species were able to grow.

Fundamental limits to economic development in developing and underdeveloped countries imposed by global hierarchy

Despite great leaps in science, technology, and human civilization since World War II, many countries in the world have been stuck in “middle income” traps or remained hopelessly poor. To understand this reality,we introduce a new metric, the revealed comparative wealth (RCW), and develop a theory of global hierarchical order (GHO), which is characterized by a Zipf–Mandelbrot law (ZML) for the yearly ranked RCW for all the countries in the world. Major factors contributing to the existence of GHO have been identified. Various types of behavior, including equilibria characterized by largely constant RCW, strongly coupled advanced economies modulated by business cycles, strongly competing developing and underdeveloped economies opposite-phase locked to advanced economies, and steadily growing or decaying economies synergistically interact to create ZML. The mathematical structure of ZML imposes fundamental limits to world economic development and sheds light on why “middle income” traps have persisted.

Generalizations of Levinson type inequalities via new Green functions with applications to information theory

In this paper, four new Green functions are used to generalize Levinson-type inequalities for the class of 3-convex functions. The f-divergence, Renyi entropy, Renyi divergence, Shannon entropy, and the Zipf–Mandelbrot law are also used to apply the main results to information theory.

Estimations of the Jensen Gap and Their Applications Based on 6-Convexity

The main purpose of this manuscript is to present some new estimations of the Jensen gap in a discrete sense along with their applications. The proposed estimations for the Jensen gap are provided with the help of the notion of 6-convex functions. Some numerical experiments are performed to determine the significance and correctness of the intended estimates. Several outcomes of the main results are discussed for the Hölder inequality and the power and quasi-arithmetic means. Furthermore, some applications are presented in information theory, which provide some bounds for the divergences, Bhattacharyya coefficient, Shannon entropy, and Zipf–Mandelbrot entropy.

Random sampling of the Zipf–Mandelbrot distribution as a representation of vocabulary growth

We develop a discrete model of type-token dynamics based on random type selection from the Zipf–Mandelbrot probability distribution, with a view to examining the relationships between the constants of Zipf’s and Heaps’ laws. Analysis of items randomly selected items from the Standardised Project Gutenberg Corpus (SPGC) reveal a significant low-frequency “droop” in the β-slope of the types vs. frequency distribution, inconsistent with the model when vocabulary is unlimited: when a finite vocabulary limit is imposed, optimal parameter selection allows the droop to be reproduced. We adjust the parameters of both the limited and unlimited vocabulary models to obtain optimal agreement with the vocabulary growth curves: the limited vocabulary model usually yields the best optimised agreement, but a sizeable minority of items are better represented by an unlimited vocabulary. While the optimised Zipf α indices correlate strongly with the corresponding values obtained directly from document statistics, the former are generally larger than the latter (though this is partially explained by the distorting effect of large values of the Mandelbrot parameter m). The β indices optimised from the limited vocabulary model are also compared with their directly measured equivalents, showing significant positive correlation. The relationship between optimised α and β agrees plausibly with the well-known continuum model, though the degree of agreement depends on how β is defined. The experiments yield repeatable results from each of three 100-item samples, demonstrating the statistical significance of the experiments.

Species-Abundance Distribution Patterns of Plant Communities in the Gurbantünggüt Desert, China

It is important to study the species-abundance distribution pattern in a community to reveal the mechanism of community assembly. Six abundance models (log-normal distribution model, Zipf model, Zipf–Mandelbrot model, broken stick model, niche preemption model, and Volkov model) were used to fit the species-abundance distribution pattern of six scales (10 m × 10 m, 20 m × 20 m, 40 m ×40 m, 60 m × 60 m, 80 m × 80 m, 100 m × 100 m) in fixed, semifixed, and mobile sand dunes in the Gurbantünggüt Desert, respectively. The best-fitting model was determined using the K-S test, the Chi-square test, and the Akaike information criterion. The results showed that the values of soil salinity, nutrients, water content, Shannon–Wiener diversity index (H′), Pielou evenness index (E), and Simpson index (D) were ranked in all three habitats as fixed dunes > semifixed dunes > mobile dunes. The rank curves span a narrow range on the horizontal axis at scales of 10 m × 10 m and 20 m × 20 m, and species richness is minimal. As the scale increases, the span range of the curve gradually increases, and species richness becomes higher at scales of 40 m × 40 m, 60 m × 60 m, 80 m × 80 m, and 100 m × 100 m. At the 10 m × 10 m and 20 m × 20 m scales, the broken stick model fits best in the three dune habitats. At the 40 m × 40 m and 60 m × 60 m scales, the niche preemption model fits best in the three dune habitats. At the 80 m × 80 m and 100 m × 100 m scales, the Volkov neutral model fits best in the fixed and semifixed dune habitats, and the niche preemption model fits best in the mobile dune habitats. In fixed, semifixed, and mobile dunes, both niche and neutral processes played important roles in community construction, reflecting the manifestation of the community niche-neutral continuum.

Generalized cyclic Jensen and information inequalities

We present Levinson’s type generalizations of cyclic refinements of Jensen’s inequality by employing recent class of functions that further characterize and extend the family of 3-convex functions. We get monotonic cyclic Jensen’s inequalities and particularly the renowned Jensen’s inequality for 3-convex functions at a point (f∈κ1c(I)). As an applications in information theory, we first introduce new Csiszár type cyclic divergence functional for 3-convex functions and establish cyclic-Kullback–Leibler and Hellinger distances. We give monotonicity of cyclic divergence functionals which enable us to construct monotonic Shannon, Relative and Zipf–Mandelbrot entropies.

Estimations of the Slater Gap via Convexity and Its Applications in Information Theory

Convexity has played a prodigious role in various areas of science through its properties and behavior. Convexity has booked record developments in the field of mathematical inequalities in the recent few years. The Slater inequality is one of the inequalities which has been acquired with the help of convexity. In this note, we obtain some estimations for the Slater gap while dealing with the notion of convexity in an extensive manner. We acquire the deliberated estimations by utilizing the definition of convex function, Jensen’s inequality for concave functions, and triangular, power mean, and Hölder inequalities. We discuss several consequences of the main results in terms of inequalities for the power means. Moreover, by utilizing the main results, we give estimations for the Csiszár and Kullback–Leibler divergences, Shannon entropy, and the Bhattacharyya coefficient. Furthermore, we present some estimations for the Zipf–Mandelbrot entropy as additional applications of the acquired results. The perception and approaches adopted in this note may pretend more research in this direction.

Patterns and Driving Factors of Diversity in the Shrub Community in Central and Southern China

Climate, topography, and human activities are known to influence plant diversity. In the present study, species-abundance distribution (SAD) patterns of the shrub community were fitted, and the mechanism of contribution of 22 driving factors was assessed. The results showed that the α-diversity index exhibited no significant differences between artificial disturbance and the natural community. The Zipf and Zipf–Mandelbrot models were found to exhibit a good SAD fitting of the communities, thereby exhibiting a different diversity structure. It was observed that the SAD followed more than one rule, and the Zipf–Mandelbrot model was better than other models. The gradient boosting model indicated that precipitation in the wettest month, annual precipitation, and slope direction showed the strongest impact on plant richness. The indicator species of the artificial disturbance and natural community were identified from a multiple regression tree. Furthermore, an increase in species diversity was observed with a rise in latitude, exhibiting a single-peaked curve with increased altitude. β-diversity analysis indicated that both habitat filtering and the neutral effect influenced the establishment of the natural community, while the establishment of the artificial disturbance community was only affected by habitat filtering. Our study provides a better understanding of the ecological process of the maintenance of shrub-community diversity.

Improvements of Slater’s Inequality by Means of 4-Convexity and Its Applications

In 2021, Ullah et al., introduced a new approach for the derivation of results for Jensen’s inequality. The purpose of this article, is to use the same technique and to derive improvements of Slater’s inequality. The planned improvements are demonstrated in both discrete as well as in integral versions. The quoted results allow us to provide relationships for the power means. Moreover, with the help of established results, we present some estimates for the Csiszár and Kullback–Leibler divergences, Shannon entropy, and Bhattacharyya coefficient. In addition, we discuss some additional applications of the main results for the Zipf–Mandelbrot entropy.

On the Converse Jensen-Type Inequality for Generalized f-Divergences and Zipf–Mandelbrot Law

Motivated by some recent investigations about the sharpness of the Jensen inequality, this paper deals with the sharpness of the converse of the Jensen inequality. These results are then used for deriving new inequalities for different types of generalized f-divergences. As divergences measure the differences between probability distributions, these new inequalities are then applied on the Zipf–Mandelbrot law as a special kind of a probability distribution.

NDN Content Store and Caching Policies: Performance Evaluation

Among various factors contributing to performance of named data networking (NDN), the organization of caching is a key factor and has benefited from intense studies by the networking research community. The performed studies aimed at (1) finding the best strategy to adopt for content caching; (2) specifying the best location, and number of content stores (CS) in the network; and (3) defining the best cache replacement policy. Accessing and comparing the performance of the proposed solutions is as essential as the development of the proposals themselves. The present work aims at evaluating and comparing the behavior of four caching policies (i.e., random, least recently used (LRU), least frequently used (LFU), and first in first out (FIFO)) applied to NDN. Several network scenarios are used for simulation (2 topologies, varying the percentage of nodes of the content stores (5–100), 1 and 10 producers, 32 and 41 consumers). Five metrics are considered for the performance evaluation: cache hit ratio (CHR), network traffic, retrieval delay, interest re-transmissions, and the number of upstream hops. The content request follows the Zipf–Mandelbrot distribution (with skewness factor α=1.1 and α=0.75). LFU presents better performance in all considered metrics, except on the NDN testbed, with 41 consumers, 1 producer and a content request rate of 100 packets/s. For the level of content store from 50% to 100%, LRU presents a notably higher performance. Although the network behavior is similar for both skewness factors, when α=0.75, the CHR is significantly reduced, as expected.

An Information Theoretic Approach to Symbolic Learning in Synthetic Languages.

An important aspect of using entropy-based models and proposed “synthetic languages”, is the seemingly simple task of knowing how to identify the probabilistic symbols. If the system has discrete features, then this task may be trivial; however, for observed analog behaviors described by continuous values, this raises the question of how we should determine such symbols. This task of symbolization extends the concept of scalar and vector quantization to consider explicit linguistic properties. Unlike previous quantization algorithms where the aim is primarily data compression and fidelity, the goal in this case is to produce a symbolic output sequence which incorporates some linguistic properties and hence is useful in forming language-based models. Hence, in this paper, we present methods for symbolization which take into account such properties in the form of probabilistic constraints. In particular, we propose new symbolization algorithms which constrain the symbols to have a Zipf–Mandelbrot–Li distribution which approximates the behavior of language elements. We introduce a novel constrained EM algorithm which is shown to effectively learn to produce symbols which approximate a Zipfian distribution. We demonstrate the efficacy of the proposed approaches on some examples using real world data in different tasks, including the translation of animal behavior into a possible human language understandable equivalent.

Statistical tests for text homogeneity: using forward and backward processes of numbers of different words

The processes of growth in the number of diverse words in a text, when reading in the forward and backward directions, are studied in this article. Based upon the statistics achieved from the difference between these two processes, we construct a statistical test. This statistical test is used for text homogeneity checks. The elementary model states that words in a text are selected from some dictionary independent of each other according to the Zipf–Mandelbrot law. P-values of the statistical test are calculated based on the elementary probabilistic model using the asymptotic normality of corresponding statistics. At last but not least, this statistical test is applied for the analysis of homogeneity of sequences of sonnets.

Entropy Estimation Using a Linguistic Zipf-Mandelbrot-Li Model for Natural Sequences.

Entropy estimation faces numerous challenges when applied to various real-world problems. Our interest is in divergence and entropy estimation algorithms which are capable of rapid estimation for natural sequence data such as human and synthetic languages. This typically requires a large amount of data; however, we propose a new approach which is based on a new rank-based analytic Zipf–Mandelbrot–Li probabilistic model. Unlike previous approaches, which do not consider the nature of the probability distribution in relation to language; here, we introduce a novel analytic Zipfian model which includes linguistic constraints. This provides more accurate distributions for natural sequences such as natural or synthetic emergent languages. Results are given which indicates the performance of the proposed ZML model. We derive an entropy estimation method which incorporates the linguistic constraint-based Zipf–Mandelbrot–Li into a new non-equiprobable coincidence counting algorithm which is shown to be effective for tasks such as entropy rate estimation with limited data.

Lerch distribution based on maximum nonsymmetric entropy principle: Application to Conway’s game of life cellular automaton

Conway’s Game of Life (GoL) is a biologically inspired computational model which can approach the behavior of complex natural phenomena such as the evolution of ecological communities and populations. The GoL frequency distribution of events on log-log scale has been proved to satisfy the power-law scaling. In this work, GoL is connected to the entropy concept through the maximum nonsymmetric entropy (MaxNSEnt) principle. In particular, the nonsymmetric entropy is maximized to lead to a general Zipf’s law under the special auxiliary information parameters based on Hurwitz–Lerch Zeta function. The Lerch distribution is then generated, where the Zipf, Zipf–Mandelbrot, Good and Zeta distributions are analyzed as particular cases. In addition, the Zeta distribution is linked to the famous golden number. For GoL simulations, the Good distribution presented the best performance in log-log linear regression models for individual cell population, whose exponents were far from the golden number. This result suggests that individual cell population decays slower than a hypothetical slope equal to a (fast decaying) negative golden number.

Zipf’s law for cosmic structures: How large are the greatest structures in the universe?

The statistical characterization of the distribution of visible matter in the universe is a central problem in modern cosmology. In this respect, a crucial question still lacking a definitive answer concerns how large the greatest structures in the universe are. This point is closely related to whether or not such a distribution can be approximated as being homogeneous on large enough scales. Here we assess this problem by considering the size distribution of superclusters of galaxies and by leveraging the properties of Zipf–Mandelbrot law, providing a novel approach which complements the standard analysis based on the correlation functions. We find that galaxy superclusters are well described by a pure Zipf’s law with no deviations and this implies that all the catalogs currently available are not sufficiently large to spot a truncation in the power-law behavior. This finding provides evidence that structures larger than the greatest superclusters already observed are expected to be found when deeper redshift surveys will be completed. As a consequence, the scale beyond which galaxy distribution crossovers toward homogeneity, if any, should increase accordingly.