Granular Entropy Research Articles

Granularity self-information based uncertainty measure for feature selection and robust classification

Information entropy theory has been widely studied and successfully applied to machine learning and data mining. The fuzzy entropy and neighborhood entropy theories have been rapidly developed and widely used in uncertainty measure. In this paper, a granularity self-information theory is first proposed to measure uncertainty robustly. The theory improves the shortcomings of neighborhood self-information in measuring sample uncertainty by combining with data distributions. Then, granularity entropy theory is put forward and fully explained. With the proposed theories, a novel feature selection algorithm and a robust classification algorithm are designed and validated with some experiments. The experimental results show the designed algorithms have good performance. This indicates the efficacy of granularity self-information and granularity entropy for evaluating samples and features.

Using Knowledge Granularity Entropy to Measure Eco-Environmental Impacts of Land Cover Changes in ASEAN from 2001 to 2020

The eco-environment is the basis for the political, economic and social development of any nation or group of nations, such as the Association of Southeast Asian Nations (ASEAN). There is an inseparable association between land cover change (LCC) and eco-environmental change. The effects of the regional eco-environment are seen in the spatial and temporal variations in the eco-environment of different land cover types (LCTs). The Remote Sensing Ecological Index (RSEI), which is based on information granulation and spatial information granules, is used in this study to characterize the ecosystem. This issue is solved by breaking down the intricate ecosystem structure into straightforward spatially granular computational units; this approach greatly reduces the complexity of spatial data computation. The main contributions of this study are as follows: (1) A model based on the concept of “granularity entropy” of the RSEI values of the different LCTs has been proposed by condensing and combining the spatial information granules. This model seeks to evaluate the stability or change of the eco-environment over time. (2) The main LCC factors for the decline in eco-environmental quality in ASEAN from 2001 to 2020 are the interval changes in eco-environmental indicator values caused by the decrease in forest area and the increase in grassland area; climate alteration is also a significant contributor to changes in regional eco-environmental quality.

“Absolute zero” temperature in a vertically vibrated granular system

In thermodynamics, absolute zero is the coldest temperature and will never be reached because the thermal motion of microscopic particles never ceases. However, this situation could be changed for a collection of macroscopic particles. In the present paper, an experimental study was performed to explore the possible lowest temperature in a vertically vibrated granular system. It was found that the granular “absolute zero” temperature appears when the vibrating intensity is adjusted to about 4.6 times the gravitational acceleration. At this temperature, the macroscopic particles are arranged closely and behave like a rigid body without relative motions during the vibration. Near the absolute zero, inelastic collisions and energy transfer are responsible for the variation of granular temperature with time and vibrating parameters. Interestingly, the temperature variation reveals that the vibrated macroscopic particles are neither a crystal nor an amorphous system. This study introduces the granular entropy, including entropy generation and entropy flow, to describe the order of the vibrated particles. The entropy change could be illustrated by the temperature profiles of the granular system and its outside. It was also found that, unlike microscopic particles, which could maintain their entropy at a constant temperature, a granular system necessarily behaves completely disordered unless the system achieves the absolute zero granular temperature.

Temporal and spatial variations of eco-environment in Association of Southeast Asian Nations from 2000 to 2021 based on information granulation

As a multinational organization with rapid economic development, the eco-environment of Association of Southeast Asian Nations (ASEAN) will be an important factor in determining the success or failure of the future development of the region. This study uses information granulation based on granular computing and spatial distribution characteristics of the remote sensing ecological index (RSEI) to granulate indicator. According to the different influences of different countries in ASEAN, a new scale-up evaluation method of RSEI is proposed, which considers the influence of countries and establishes the weight by scale transformation knowledge granularity entropy. The RSEI research results of ASEAN from 2000 to 2021 show that 1) from 2000 to 2021, the eco-environmental quality of ASEAN was good. The areas with better RSEI accounted for 40.3566%, while the poor areas accounted for 32.6194%. The areas with extremely poor and excellent remote sensing ecology accounted for less. 2) From 2000 to 2021, the remote sensing eco-environment of ASEAN showed a certain volatility, the quality of the remote sensing eco-environment was declining, and the spatial differentiation was increasing. 3) The spatial distribution of the remote sensing eco-environmental quality of the Malay Peninsula is better than that of the Indochina Peninsula. 4) The order of impact of ASEAN remote sensing eco-environmental indicators is heat, dryness, greenness and humidity.

Finding the grain size distribution that produces the densest arrangement in frictional sphere packings: Revisiting and rediscovering the century-old Fuller and Thompson distribution

By means of discrete-element methods, we investigate the joint effects of the grain size distribution (GSD) and contact friction on the structure of three-dimensional samples composed of spherical grains. Specifically, we compress these systems isotropically until jamming and then analyze their structure in terms of density, connectivity, coefficients of uniformity and curvature, and parameters of grading entropy. Our study focuses on power-law GSDs and particularly on the Fuller and Thompson distribution, proposed over a century ago. First, we show that, among the set of GSDs investigated, this particular distribution produces the densest and best-connected systems, falsifying a conjecture recently posed in the literature. Second, we find that the jamming packing fraction can be accurately predicted as a function of simple descriptors of the GSD, but among these descriptors the granular entropy concept proves to be the most useful. This allows for an alternative interpretation of both jamming and grading entropy concepts. Finally, we compare the Fuller and Thompson distribution with two well-known GSDs: that of the Apollonian sphere packing and that towards which granular systems evolve after intensive grain fracturing. Surprisingly, we find that these three GSDs are practically coincident in the limit of large size spans, despite having been introduced or discovered in different scientific contexts (i.e., engineering, mathematics, and earth sciences, respectively).

Thermodynamic modeling of stress–strain behavior of saturated sand considering temperature effect

Based on the thermodynamic process of granular matter, considering the influence of temperature on energy dissipation at the mesoscopic level, a thermodynamic constitutive model, which can describe the thermal properties of saturated sand, is proposed in this paper. The model adopts the idea of double entropy theory, which introduces the concept of granular entropy to reflect the energy dissipation at the mesoscopic level. The energy dissipative structure composition and the related migration coefficient relationships are determined theoretically. Combined with the thermodynamic differential equation and the conservation and entropy increase equations, the thermodynamic identity is established; then, the mathematical expression of effective stress, plastic deformation, and granular temperature motion including the temperature effect are deduced accordingly. The relationship between the mesoscopic energy dissipation process caused by temperature change and macro-mechanical behavior is established by means of migration coefficient relationships and the energy function model. In addition, the dilatancy equation considering the state parameters is combined with the model parameters, so the stress–strain behavior of saturated sand considering the temperature effect can be described. The ability of the model to describe the thermal properties of saturated sand is verified by comparing the simulation results with the heating-induced volume change and triaxial undrained shear test results.

An algorithm acceleration framework for correlation-based feature selection

Repeated calculations lead to a sharp increase in the time of correlation-based feature selection. Incremental iteration has been applied in some algorithms to improve the efficiency. However, the computational efficiency of correlation has generally be ignored. An algorithm acceleration framework for correlation-based feature selection (AFCFS) is proposed. In AFCFS, the criterion of the feature selection will be analyzed and reconstructed based on entropy granularity, and the algorithm structure will also be adjusted accordingly. Specifically, all repeated part of calculation will be saved in mapping tables and can be accessed in next time directly, so as to further reduce the calculation repetition rate and improve the efficiency. The experimental results show that AFCFS can greatly reduce the cost time of these algorithms, and keep the corresponding classification accuracy basically unchanged.

Shaping Distribution Identification of Phase Rotated Probabilistically Shaped Signals With Radius-Based Expectation Maximization

A radius-based expectation maximization (R-EM) algorithm is proposed for shaping distribution identification (SDI) with phase rotated probabilistically shaped (PS) signals in a flexible transceiver. Since the proposed SDI algorithm is independent of carrier phase, shaping distribution (SD)-aware algorithms for frequency offset estimation and carrier phase recovery can be implemented after SDI to fully leverage the shaping gain. To remove the impact of the carrier phase on the calculation of posterior probability, the channel is modeled as a function of the radius rather than the complex symbol as in the conventional EM SDI. An approximation method is employed to analytically derive the parameter updating equation. A low-complexity initialization step is also proposed for the R-EM algorithm. The performance of the proposed algorithm is investigated numerically via Monte-Carlo simulation of a polarization division multiplexing PS-64QAM system. The optimization of the R-EM algorithm and the impacts of the shaping granularity, the block length and the signal-to-noise ratio (SNR) estimation accuracy on the success rates are analyzed. In the case of ideal SNR estimation, high success rates (≥99.6%) are achieved at operation points (i.e., SNR threshold at the normalized generalized mutual information of 0.9) with an entropy granularity of 0.21 bit/symbol. For a coarser entropy granularity (0.42 bit/symbol), even under a SNR estimation error up to ±1 dB, the proposed algorithm provides robust performance (≥99.6% success rate) at operation points. The results indicate that the R-EM algorithm is a promising SDI method for elastic optical networks with PS signals.

Modeling the Hydro-Mechanical Coupling Behavior of Unsaturated Geotechnical Materials Based on Non-Equilibrium Thermodynamic Theory

A new hydro-mechanical model for unsaturated geotechnical materials based on the non-equilibrium thermodynamic theory is presented in this paper. Common concepts, such as yield criterion and flow rules, are not involved in the constitutive relationships, and are replaced with the thermodynamic concepts of granular temperature, granular entropy, migration coefficients, and energy functions. The dissipation system and the migration coefficient relationships are theoretically determined, and the constitutive relations of non-elastic deformation and granular temperature are obtained by dissipation relations and thermodynamic identity. Thus, the relationship between dissipation mechanism and macro mechanical behavior can be established by migration coefficients and energy functions. The model can reflect the complex hydro-mechanical coupling behavior of unsaturated geotechnical materials subjected to various mechanical paths. The validity of the model is verified by comparing the modeling results with experimental data, and reasonable agreement is achieved.

Joint Probabilistic-Nyquist Pulse Shaping for an LDPC-Coded 8-PAM Signal in DWDM Data Center Communications

M-ary pulse-amplitude modulation (PAM) meets the requirements of data center communication because of its simplicity, but coarse entropy granularity cannot meet the dynamic bandwidth demands, and there is a large capacity gap between uniform formats and the Shannon limit. The dense wavelength division multiplexing (DWDM) system is widely used to increase the channel capacity, but low spectral efficiency of the intensity modulation/direct detection (IM/DD) solution restricts the throughput of the modern DWDM data center networks. Probabilistic shaping distribution is a good candidate to offer us a fine entropy granularity and efficiently reduce the gap to the Shannon limit, and Nyquist pulse shaping is widely used to increase the spectral efficiency. We aim toward the joint usage of probabilistic shaping and Nyquist pulse shaping with low-density parity-check (LDPC) coding to improve the bit error rate (BER) performance of 8-PAM signal transmission. We optimized the code rate of the LDPC code and compared different Nyquist pulse shaping parameters using simulations and experiments. We achieved a 0.43 dB gain using Nyquist pulse shaping, and a 1.1 dB gain using probabilistic shaping, while the joint use of probabilistic shaping and Nyquist pulse shaping achieved a 1.27 dB gain, which offers an excellent improvement without upgrading the transceivers.

QoS Routing Algorithm Based on Entropy Granularity in the Network Transmission

Abstract With the rapid development of computer networks, more hosts are connected to the Internet where they could communicate with each other. The need for network service has exceeded the service capacity of the network, and the Quality of Service (QoS) is gradually declining. Based on existing Shortest Path First (SPF) algorithm, this paper proposes a new QoS required transmission path approach by considering the overhead balance of network resources. This paper uses the entropy granularity as the main line in the application of routing protocols. Firstly, it researches the optimization of routing algorithms for network load balancing resources, routing algorithms based on link traffic distributing weights, link weight optimization based on adaptive genetic algorithm and computational intelligence based on entropy granularity theory. This research proposes a method to apply entropy granularity to Open Shortest Path First (OSPF) routing, including the implementation of the method. After that, a case study is presented by using some examples.

Granular Partition and Concept Lattice Division Based on Quotient Space

In this paper, we investigate the relationship between the concept lattice and quotient space by granularity. A new framework of knowledge representation - granular quotient space - is constructed and it demonstrates that concept lattice classing is linked to quotient space. The covering of the formal context is firstly given based on this granule, then the granular concept lattice model and its construction are discussed on the sub-context which is formed by the granular classification set. We analyze knowledge reduction and give the description of granular entropy techniques, including some novel formulas. Lastly, a concept lattice constructing algorithm is proposed based on multi-granular feature selection in quotient space. Examples and experiments show that the algorithm can obtain a minimal reduct and is much more efficient than classical incremental concept formation methods.

Measurement of general granules

The measurement of granules is often considered the most foundational problem in granular computing and it has been studied in many fields but in relative isolation. This paper attempts to explore the measurement for general granules, whether they are granules based on equivalence or general relations, whether they are macrocosmic or microcosmic granules, and whether they are granules on a finite or an infinite set. For a finite set we discuss the measurement of granules corresponding to equivalence relations and introduce definitions of granularity, fineness and granular entropy. In order to distinguish those granules with the same granularity, fineness and granular entropy we present conditional granularity, conditional fineness and conditional granular entropy and obtain the necessary and sufficient condition which can judge one granule as coarser or finer than another. While the conditional granularity is a subsethood measure which satisfies monotonic properties. What’s more, we introduce relation matrix to measure the granules, and it not only can be easily generalized to measure the granules on general relations but also connects granular computing to graph theory. For an infinite set we can define different granularities in different questions, but we suggest the basic idea of granularity, that is, the granularity of a granule is the sum of the relative information contents of its atomic granules.

A fully coupled THM model based on a non-equilibrium thermodynamic approach and its application

Summary In this study, a fully coupled thermo-hydro-mechanical (THM) theoretical model, called the TSINGHUA-THERMOSOIL model, which is applicable in saturated soils, was developed based on a non-equilibrium thermodynamic approach (see Groot S, Mazur P. Non-Equilibrium thermodynamics. Amsterdam: North-Holland Pub. Co., 1962). This model simplifies the mathematical description of a THM coupling system to the modeling of a group of migration coefficients and energy functions by means of the non-equilibrium thermodynamic theory. The important concept of granular entropy was introduced for describing the granular fluctuation and the resulting transient elasticity. Therefore, non-elastic mechanical constitutive relations that do not require such concepts as yield surface and flow rule are obtained. The physical process of bound pore water transforming into free water when the temperature rises was taken into consideration for describing the phenomenon of the volumetric compression of normally consolidated and slightly over-consolidated clays under non-isothermal conditions. The behavior of thermal consolidation and drained shear tests at different temperatures for saturated Kaolin clay with different over-consolidation ratios (OCR) were simulated using this model. The effect of the OCR value, the heating rate and temperature cycling on the thermal consolidation and the effect of the temperature on the shear strength were analyzed and compared with existing experimental data. The results showed that the model proposed in this paper can accurately describe the thermal consolidation and the basic law of shearing for saturated soil, which preliminarily proves the validity of the model. Copyright © 2016 John Wiley & Sons, Ltd.

A thermodynamic constitutive model for undrained monotonic and cyclic shear behavior of saturated soils

Based on a non-equilibrium thermodynamic approach, this paper has established a constitutive model for saturated soils different from the classical elasto-plastic model and other thermodynamics-based models. It does not require concepts such as yield surface and flow rule. Instead, through a quantitative description of the energy in the reversible and irreversible processes in materials, the constitutive relations have been obtained theoretically. Using the elastic potential energy function, the stress is expressed as a function of elastic strain. By introducing the granular entropy and locked energy concepts, the non-elastic strain evolution is determined according to the non-equilibrium thermodynamics. The model can describe many basic mechanical properties of saturated soils under monotonic and cyclic shear loadings, such as the stress state boundary, the critical state and especially the cyclic mechanical behavior. Based on this model, the paper simulates the cyclic undrained triaxial shear tests for saturated clays and analyzes the impact of factors that include cyclic stress ratio, OCR value and cyclic stress frequency. The simulation results are compared with experimental results in order to validate the model.

Devising a protocol-related statistical mechanics framework for granular materials.

Devising a statistical mechanics framework for jammed granular materials is a challenging task as those systems do not share some important properties required to characterize them with statistical thermodynamics tools. In a recent paper [Asenjo et al. Phys. Rev. Lett. 112, 098002 (2014)], a new definition of a granular entropy, which puts the protocol used to generate the packings at its roots, has been proposed. Following up these results, it is shown that the protocol used in Asenjo et al. can be recast as a canonical ensemble with a particular value of the temperature. Signature of gaussianity for large system sizes strongly suggests an asymptotic equivalence with a corresponding microcanonical ensemble where jammed states with certain basin volumes are sampled uniformly. We argue that this microcanonical ensemble is not Edwards's microcanonical ensemble and generalize this argument to other protocols.

Numerical Calculation of Granular Entropy

We present numerical simulations that allow us to compute the number of ways in which N particles can pack into a given volume V. Our technique modifies the method of Xu, Frenkel, and Liu [Phys. Rev. Lett. 106, 245502 (2011)] and outperforms existing direct enumeration methods by more than 200 orders of magnitude. We use our approach to study the system size dependence of the number of distinct packings of a system of up to 128 polydisperse soft disks. We show that, even though granular particles are distinguishable, we have to include a factor 1=N! to ensure that the entropy does not change when exchanging particles between systems in the same macroscopic state. Our simulations provide strong evidence that the packing entropy, when properly defined, is extensive. As different packings are created with unequal probabilities, it is natural to express the packing entropy as S = − Σ(i)p(i) ln pi − lnN!, where pi denotes the probability to generate the ith packing. We can compute this quantity reliably and it is also extensive. The granular entropy thus (re)defined, while distinct from the one proposed by Edwards [J. Phys. Condens. Matter 2, SA63 (1990)], does have all the properties Edwards assumed.

Concept Granule-Based Granular Lattice and Application in Knowledge Retrieval

In this paper, from the viewpoint of granular computing, information granule and concept granule are introduced to information systems, and then structure of concept granules and their distance, concept granular entropy and significance degree are presented. The concept of concept granule-based granular lattice is proposed in information systems. The representation and operation rules of complete granular lattice are given. Moreover, some important properties and propositions about these concepts presented above are discussed as well, whose performances are shown through an illustrative example. Finally, a conceptual framework of knowledge retrieval with granular lattice using concept granules may help to understand the nature of granules and granular hierarchical structures, and further enlarge the application areas of granular computing.

The other entropy

Granular materials are not in thermodynamic equilibrium. Nevertheless, concepts borrowed from equilibrium statistical mechanics are extensively used to characterise granular systems. One often-invoked concept is the so-called granular entropy. In this paper, we consider what it would take to compute granular entropy numerically. Our focus is the analogy (or, more accurately, lack thereof) between statistical mechanical ensembles, as used in Monte Carlo simulations of thermal systems, and preparation protocols used in the generation of granular packings. Attempts to phrase the description of granular media in ‘thermal’ Monte Carlo language reveal some awkward problems. Whilst some of these problems had been reported in the existing literature, the present perspective yields new insights.

Simulation of Nonisothermal Consolidation of Saturated Soils Based on a Thermodynamic Model

Based on the nonequilibrium thermodynamics, a thermo-hydro-mechanical coupling model for saturated soils is established, including a constitutive model without such concepts as yield surface and flow rule. An elastic potential energy density function is defined to derive a hyperelastic relation among the effective stress, the elastic strain, and the dry density. The classical linear non-equilibrium thermodynamic theory is employed to quantitatively describe the unrecoverable energy processes like the nonelastic deformation development in materials by the concepts of dissipative force and dissipative flow. In particular the granular fluctuation, which represents the kinetic energy fluctuation and elastic potential energy fluctuation at particulate scale caused by the irregular mutual movement between particles, is introduced in the model and described by the concept of granular entropy. Using this model, the nonisothermal consolidation of saturated clays under cyclic thermal loadings is simulated in this paper to validate the model. The results show that the nonisothermal consolidation is heavily OCR dependent and unrecoverable.