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Shannon’s entropy is given by - Sum over i P(i) ln(P(i)) where P(i) is the probability for the state i. The question is: What is the state i? In particular, different choices of the states in question lead to different entropies. Traditionally, the states associated with a gas with no degeneracy in energy levels are those for which energy E is distributed among N particles, with each distribution carrying the same weight. The number of permutations of a specific { n(ei) } set is: N!/ Product over i n(ei) where Sum over i n(ei) = N and Sum over i ei n(ei) = E (or Eaverage for a sharp distribution) may map into different physical configurations with different pressures. For example, an arrangement of three e1’s followed by 4 e2’s may not be the same physical situation as 1 e1, 2 e2s, 2 e1s 2 e2s. In other words, there may be a pressure related reason for choosing states. If each distribution carries the same weight, then ln(1/#arrangements) = ln(Probability of the average scenario of N particles). For N large, n(ei) = Np(ei) where p(ei) is single particle probability for independent particles. Then the average probability corresponds to: Product over i (p(ei)) [to the power of Np(ei) ]. Ln(average probability) = Sum over i p(ei) ln(p(ei)) + ln(N). -Sum over i p(ei) ln(p(ei) is then the Shannon’s entropy of the gas for no degeneracy in energy levels. The degeneracy in energy levels also seems to assume an average n(ei) for each ei as we discuss in this note (following (1)) i.e. considering boson and fermion cases where again the idea of pressure related states appears. In (2) it seems that a different grouping of probabilities is made to define physical states. (2) suggests that single particle situations which have the same number of n(e1), n(e2), n(e3).. etc i.e. specific sets { n(ei) } such that Sum n(ei) = N and Sum over i ei n(ei) = E(average) should be grouped together i.e. have their probabilities summed. Thus all permutations of the same set of n(ei)’s are grouped into one probability and this is used to compute Shannon’s entropy i.e. Probability = N!/ Product n(ei)! Product p(ei) [to the power n(ei)]. No degeneracy in energy levels is considered and ultimately Maxwell-Boltzmann distributions are used for single particles. This then leads to a different entropy than using the p(ei)’s corresponding to the average picture (i.e. average of all the equally weight distributions of E over N). Arguments are made that this entropy allows for -ln(N!) to disappear as T->0 as all particles drop to the ground state. For S=- ln(N!/ Product over i n(ei)!) , T→0 also means that the MB distribution no longer holds and that n(ground state)=N and so S also tends to 0.

For the case of physical quantities with units, such as mass, time, length etc, a change in the value of the quantity makes sense by itself in addition to a relative change. In other words, one does not need to know the mass of an object to understand what a change of 1 kg is. Probability, on the other hand, is a strictly relative quantity and so needs to be normalized. If it is not normalized, one needs to know relative probabilities. To say that an unnormalized probability changes by a value of 10 does not mean anything. Given that equations often solve for an unnormalized probability function, we argue that the notion of the relative probability should appear in equations. Even in the case of ln(P(y)), called information in information theory, for which there is no derivative, taking a derivative, say d/dy, immediately creates a relative probability derivative. Thus the form ln(P) is in keeping with the notion of a relative derivative. Here we examine the idea of a relative derivative and its association with information ln(P), Fisher information, quantum mechanics and the Fokker-Planck equation as well as classical statistical balance in the presence of a potential.

Green supply chain management considers the environmental effects of all activities related to the supply chain, from obtaining raw materials to the final delivery of finished goods. Selecting the right supplier is a critical decision in green supply chain management. We propose a fuzzy green supplier selection model for sustainable supply chains in reverse logistics. We define a novel hierarchical fuzzy best-worst method (HFBWM) to determine the importance weights of the green criteria and sub-criteria selected. The fuzzy extension of Shannon’s entropy, a more complex evaluation method, is also used to determine the criteria' weights, providing a reference comparison benchmark. Several hybrid models integrating both weighting techniques with fuzzy versions of complex proportional assessment (COPRAS), multi-objective optimization by ratio analysis plus the full multiplicative form (MULTIMOORA), and the technique for order of preference by similarity to ideal solution (TOPSIS) are designed to rank the suppliers based on their ability to recycle in reverse logistics. We aggregate these methods' ranking results through a consensus ranking model and illustrate the capacity of relatively simple methods such as fuzzy COPRAS and fuzzy MOORA to provide robust rankings highly correlated with those delivered by more complex techniques such as fuzzy MULTIMOORA. We also find that the ranking results obtained by these hybrid models are more consistent when HFBWM determines the weights. A case study in the asphalt manufacturing industry is presented to demonstrate the proposed methods' applicability and efficacy.

The Maxwell-Boltzmann MB distribution is often derived by maximizing the number of possible arrangements of particles with different energies subject to the constraint of a fixed number of particles and fixed energy. Using the canonical approximation that only average energy is conserved and using Stirling’s approximation for large number factorials leads to the Maxwell-Boltzmann distribution. The ln of total number of arrangements using Stirling’s approximation is linked to a math form called Shannon’s entropy i.e. - Sum over i f(ei) ln(f(ei)). Interestingly this same MB result may be obtained through reaction balance which makes no use of the idea of arrangements or entropy or maximization. It is concerned with including all possible physical reactions (for example two body reactions between all energy particles) subject to conservation of energy. In this sense it deals directly with the dynamics of the problem while the arrangement approach does not. The dynamical approach considers all possible reactions with notions of independence or lack of information and ensures the system remains unchanged. Why are they the same? We argue that it is because the “recipe” for the probability of two body reactions namely n(e1)n(e2) is of the same form as the leading factor of n(e1)! n(e2)!. In the case of quantum mechanics one may apply the dynamical approach to obtain the time-independent Schrodinger equation. In such a case one does not need to think in terms of “maximizing” any quantity although one considers all possible reactions with a potential V(x). a(p) factors in W(x) = wavefunction = Sum over p a(p)exp(ipx) are interrelated, but retain their identity after all possible interactions occur. This we argue is the idea for the eigenvalue equation. At the same time, Shannon’s entropy, which is directly linked to the notion of arrangements, is applied to a bound state and possibly linked to thermodynamics i.e. dE = TdS -PdV which may be applied to an adiabatic expansion. Shannon’s entropy is created from existing probabilities in this case which do not arise from explicitly maximizing entropy subject to constraints as the probabilities follow directly from the Schrodinger equation. One might ask why Shannon’s entropy has any relevance if it is the dynamics which determines the value of W(x) and a(p). It seems, however, that one may knock out a particle in a state p with probability P(p)= a*(p)a(p) or find a particle at x with probability P(x)=W*(x)W(x). These are classical type probabilities linked to the quantum dynamical probabilities a(p) and W(x). It seems Shannon’s entropy is only applied to P(x) and P(p) or a product P(x)P(p) which may give it physical meaning in a quantum bound state. We examine these ideas in this note.

Probiotic bacteria are microorganisms with beneficial effects on human health and are currently used in numerous food supplements. However, no selection process is able to effectively distinguish probiotics from non-probiotic organisms on the basis of their genomic characteristics. In the current study, four Machine Learning algorithms were employed to accurately identify probiotic bacteria based on their DNA characteristics. Although the prediction accuracies of all algorithms were excellent, the Neural Network returned the highest scores in all the evaluation metrics, managing to discriminate probiotics from non-probiotics with an accuracy greater than 90%. Interestingly, our analysis also highlighted the information content of the tRNA sequences as the most important feature in distinguishing the two groups of organisms probably because tRNAs have regulatory functions and might have allowed probiotics to evolve faster in the human gut environment. Through the methodology presented here, it was also possible to identify seven promising new probiotics that have a higher information content in their tRNA sequences compared to non-probiotics. In conclusion, we prove for the first time that Machine Learning methods can discriminate human probiotic from non-probiotic organisms underlining information within tRNA sequences as the most important genomic feature in distinguishing them.

In a previous note (1) we considered dE (thermal)= Sum over i ei dP(i) where P(i) is the probability to have ei. In other words, one has the constant vector {ei} forming a dot product with the differential operator { dP(i) }.We suggested one might wish to find a vector function { G(P(i) } and create the dot product S= - Sum over i G(P(i)) P(i) such that dS= - Sum over i G(P(i)) dP(i). For small changes {dP(i)} the vector { G(P(i))} is a constant of the system with G=ln the logarithm function as a solution. In this note we compare this picture with that of information theory for which one writes Poverall = Product over i P(i) [to the power NP(i)] where N is large. This approach also leads to the form of Shannon’s entropy i.e. one does not need to find the function ln - it appears automatically, but one needs a large N limit to formulate the problem whereas no such assumption is needed in (1). We consider the situation of P(i) → P(i) + dP(i). We ultimately try to link dE thermal to the change in the product probability of P(i)’s in a gas with dS being linked to the change in the product probability which itself is linked to the overall number of arrangements. We also consider how the form ln is associated with converting P(i)P(j) into a sum i.e. ln(P(i)) + ln(P(j)). For example, for an ideal gas one may consider two body elastic collisions. Assuming that P(i)P(j) =P(k)P(l) and given ei+ej=ek+el one may take the ln of the former and associate ln(P(i)) with -ei/T, where T is a constant. Thus one uses the idea of ln converting a product to a sum and the idea of conservation which in this particular case is conservation of energy. One may ask how this links with the picture of P(i)ln(P(i)) leading to a locally constant vector? We argue that one may consider two particles and consider a change in P(i) i.e. P(i)+dP(i). In such a case it is not ln(P(i))+ln(P(i)) which is linked with conservation, but one is interested in the conservation rule P(i)+P(j)=constant. We show that it is form P(i)ln(P(i)) which is then linked with a conservation law namely Sum over i dP(i)=0 where i=1,2 in this case. Thus there is a link between the derivation of the Maxwell-Boltzmann distribution strictly from P(i)P(j) and ei+ej=ek+el and Shannon’s entropy which is related to ln(P(i)), but is slightly different, namely P(i)ln(P(i)), in order to make use of conservation of probability.

In previous notes (1) we discussed the free particle classical action in the form A= -Et+px in which p,x and E and t are independent for the relative and nonrelativistic cases. This leads to a scenario in which one cannot follow a particle in time i.e. there does not exist an x(t) as in classical mechanics. Instead one may develop eigenvalue equations -i d/dx partial exp(iA) = p exp(iA) and id/dt partial exp(iA) = E exp(iA) for the free particle. The solution exp(-iEt+ipx) is similar to a classical wave (which is a different solution following form A=-Et+px) in that it describes a kind of probability resonance in all of space. If one introduces a potential V(x), this is localized and one needs to think in terms of impulse hit resonances like exp(ikx) creating V(x). In other words, both an ensemble of exp(ipx) representing the particle and V(x)=Sum over k Vk exp(ikx) are described in terms of this resonance type periodic probability which interferes to create local features. One might argue that this feature of V(x) drives the particle. In this note we wish to examine the coupled oscillator H= p1p1/2 + p2p2/2 + w1w1/2(x1x1+x2x2) + .5w2w2(x1-x2)(x1-x2) (with m=1). Transforming variables leads to an equation with a separable solution, but in y=x1+x2 and y1=x1-x2. In this note we try to examine this solution in terms of the above resonance picture and also suggest a possible overall Shannon’s entropy.

Shannon’s entropy - Sum over i P(i) ln(P(i)) where P(i) is the probability of i is often shown to follow from the large N limit of a calculation of the ln(arrangements) using factorials or in information theory from a large N scenario for which NP(i) is the number of times an outcome i appears for large N. In this case overall Probability = 1/ number of arrangements. In both cases, there seems to be an argument for maximizing Shannon’s entropy as it is linked to the number of arrangements. In (1), we argued that one may obtain Shannon’s entropy with no consideration of N or to arrangements, by seeking a function of P(i) which changes as: Sum over i f(P(i)) dP(i) just as dE thermal changes as Sum over n ei dP(en). In the case of entropy, i does not have to be en, it may be other variables such as p and x for a quantum eigenstate. The above consideration leads to S=-Sum over i P(i)ln(P(i)). There is, however, no reason for this quantity to be stationary if P(i) is changed subject to Sum over i dP(i) = 0. If ln(P(i)), however, is constant as in the case of fair coin or die, then S is automatically stationary i.e. dS=0 and one has: P(i)=constant. This may be formally obtained by maximizing S subject to Sum over i P(i) = 1. One may ask what happens in the case of a different constraint such as Eave= Sum over n en P(en). There may be a number of {P(en)} sets which yield the same Eave, but different values of S. We argue that a stationary S is one for which small dP(i) fluctuations do not change its value to first order and so this represents an equilibrium result. No use of large N is needed and no notion of arrangements. One still uses the idea of finding a stationary solution of S subject to constraints. This suggests that Shannon’s entropy should hold in systems without large N and that one may maximize (find stationary solutions) in such a system subject to its constraints.

For a classical ideal gas or single particle quantum system with energy levels {ei}, both average energy E=Sum over i ei P(ei/T) and Shannon’s entropy S= - Sum over i P(ei/T) ln(P(ei/T) change (for mechanical parameters kept fixed) in terms of d(P(ei/T)). In fact, in this case ln(P(ei/T)) appears as a “constant vector” in dS i.e. dS= - Sum over i ln(P(ei/T)) dP(ei/T). Both dE (mechanical parameter fixed) and dS change as dP(ei/T), thus there are two vectors {ei/T} and {ln(P(ei/T))} which represent constants over ei’s for a small range of dT where T is temperature. In this case, however, the two vectors are strongly linked as P(ei/T)=exp(-ei/T)/C(T) so ln(P) = -ei/T - ln(C). This is why the relationship E-TS = -T ln(C(T)) exists. For a quantum bound state one may write: e = Integral dp a(p)a(p) pp/2m + Integral dx V(x) W(x)W(x) where W(x)=Sum over p a(p)exp(ipx) is the wavefunction. Instead of having vectors of ei’s, one has two vectors, one over x and the other p (momenta) where a(p)a(p) = P(p) and P(x)=W(x)W(x) (bound states). Thus d e thermal = Integral dp pp/2m dPe(p) + Integral dx V(x) dPe(x) ((1)). In other words de thermal changes linearly in terms of dPe(p) and dPe(x) whereas for a classical ideal gas or many ei system it changes linearly in terms of dP(ei/T). Ei is the variable instead of x and p in that case. In other words {pp/2m} is a constant vector in p space and {V(x)} in x space. Mathematically Shannon’s entropy Sp = - Sum over p Pe(p) ln(Pe(p)) and Sx =- Sum over x Pe(x) ln(Pe(x)) also yield constant vectors in their respective spaces namely ln(Pe(p)) and ln(Pe(x)) which are called information. In particular dSp = - Sum over p ln(Pe(p)) dPe(p) then ln(Pe(p)) behaves like a constant vector within a region of small dPe(p) changes. Note: Sum over p dPe(p)=0. One may consider a set of single valued energy levels ei which are very close (imagine a particle in a box with infinite walls) as a region in which dPe(p) changes (or dPe(x) ones) are very small. In this region, ln(Pe(p)) and ln(Pe(x)), information, represents a constant vector statistically defining the small region of ei states. This we argue is the statistical significance of information ln(P). One may note that both de ((1)) adn dSp and dSx change linearly with dPe(p) and dPe(x), but the vectors {pp/2m} and {V(x)} over p and x space and ln(Pe(p)) and ln(Pe(x)) over the same space are constant vectors, but not equivalent except for the ground state of the quantum oscillator for which they become very similar because Pe(p)=exp(-app) and Pe(x)=exp(-xx/a) with a being proportional to 1/ e which plays the same role as temperature in the classical ideal gas case. For other quantum bound systems, the two sets of vectors are different, but the information vectors seem to give a kind of statistical identity to small regions containing very close ei levels for which dPe(p) and dPe(x) are small. Thus, though one may argue for the presence of ln(P) in Shannon’s entropy S=-Sum over i P(i)ln(P(i)) in many ways, it seems that ln(P(i)) is a constant vector in a certain space i for small dP(i) and gives a statistical identity to that small region.&nbsp

The von Neumann entropy definition is -Tr ( d ln(d)) and is linked to: = Tr (d A) ((1)). Here d is a matrix as is A i.e Aij = . As a result d is often not diagonal. It seems further that the idea of entropy is linked to an operator (2). In other words, one might have certain probabilities linked to the operator P momentum and a different set to X. For a diagonal d, von Neumann’s form is the same as Shannon’s entropy, but uses the eigenvalues of d. For a pure quantum bound state Wn(x), it seems that one considers the two operators X and P and writes two entropies using Shannon’s entropy form: - Integral dx W(x)W(x) ln(WW) and -Integral dp a(p)a(p) ln(a(p)a(p)) where W(x)=wavefunction= Sum over p a(p)exp(ipx). Here P(x)=W(x)W(x) and P(p)=a(p)a(p). For example in (3), one may see calculations of these “distinct” entropies for a particle in a box with infinite potential walls. X and P are linked to measurables and for a quantum bound system one cannot know exactly x and p at the same time. One may write < KE > for a bound state in two ways: Integral dp a(p)a(p) pp/2m or Integral dx W(x) (-1/2m d/dx dW/dx). Using Shannon’s entropy approach one obtains two different entropy values Sx and Sp. With respect to von Neumann’s entropy, it is zero for a pure state. Although one cannot know x and p at the same time (in terms of a measurement) that does not prevent one from writing x and p together (i.e. both known) in exp(ipx) for the wavefunction of a free particle. In this note we suggest that one might consider product probabilities linked to more than one operator when considering entropy. For example, for one may write Integral dx P(x) Integral dp pp/2m P(p/x) thus acknowledging that both x and p play a role. In this case, P(x)P(p/x) is the probability to use in Shannon’s entropy as we have shown in (4). In (3) it is suggested that P(x)P(p/x) be used in order to incorporate both variables x and p. Using P(x)P(p) in Shannon’s entropy yields the same overall entropy when integrated over x and p i.e. S=Sx+Sp. One reason that the sum is useful is that it eliminates parameters such as L (box length) for a particle in a box with infinite walls (or k the spring constant for an oscillator. This allows for a quantum adiabatic transformation of a pure state to be isentropic. This follows from a Fourier transformation i.e. W(x)=Sum over p a(p)exp(ipx). The two variables p and x scale with L in the opposite manner i.e. x/L and pL (5). Thus we argue that entropy is not necessarily linked to only one operator, but rather for a pure state is linked to variables which appear in a Fourier transform. Usually a Fourier transform is simply a mathematical procedure, but not in the case of the quantum pure state wavefunction where the variable p is actually momentum. In this note we try to investigate how the quantum relationship between p and x emerges. We argue that it exists at the level of the classical action for a free particle (relativistic or nonrelativistic) A=-Et+px where dA/dx partial =p and dA/dt partial = -E. Thus p and x are variable pairs in classical dynamics and this carries over to classical statistical phase space linked to p and x. We argue that a quantum dynamical link between p and x is also present in A i.e. -id/dx exp(iA) = p exp(iA). In other words p and x are linked dynamically in quantum mechanics as well (for a free particle) with exp(-iEt+ipx) being part of this link. The Fourier series form is already present in this solution as one may create a momentum distribution i.e. Sum over p a(p)exp(ipx). This depends on p and x and both affect any operator based on p and x (or one of these). Thus we expect that probabilities linked to both p and x should appear in an expression of entropy for a quantum bound state.

Shannon’s entropy is one of the building blocks of information theory and an essential aspect of Machine Learning (ML) methods (e.g., Random Forests). Yet, it is only finitely defined for distributions with fast decaying tails on a countable alphabet. The unboundedness of Shannon’s entropy over the general class of all distributions on an alphabet prevents its potential utility from being fully realized. To fill the void in the foundation of information theory, Zhang (2020) proposed generalized Shannon’s entropy, which is finitely defined everywhere. The plug-in estimator, adopted in almost all entropy-based ML method packages, is one of the most popular approaches to estimating Shannon’s entropy. The asymptotic distribution for Shannon’s entropy’s plug-in estimator was well studied in the existing literature. This paper studies the asymptotic properties for the plug-in estimator of generalized Shannon’s entropy on countable alphabets. The developed asymptotic properties require no assumptions on the original distribution. The proposed asymptotic properties allow for interval estimation and statistical tests with generalized Shannon’s entropy.

A very recent proposal of a set of entropy measures for spatial data, based on building pairs of realizations, allows to split the data heterogeneity that is usually assessed via Shannon's entropy into two components: spatial mutual information, identifying the role of space, and spatial residual entropy, measuring heterogeneity due to other sources. A further decomposition into partial terms deeply investigates the role of space at specific distance ranges. The present work proposes improvements to the method and adds relevant results proving that the new set of spatial entropies satisfies a list of desirable properties. We extend the methodology to sets of realizations greater than pairs. We also show that the approach is more general, better performing and more interpretable than the most popular proposals in the literature, thanks to the property of additivity and a new way of computing entropy that explicitly discards the order within sets. A novel procedure for building the necessary quantities for computations is also provided. A comparative study illustrates the superior performance of the new set of measures over representative spatial configurations. Practical questions are answered by means of a case study on land use data.

The Republic of Trinidad and Tobago is an archipelagic Small Island Developing State (SIDS), situated on the southern end of the chain of Caribbean islands. Several factors such as climate, topography, and hydrological characteristics increase its susceptibility and vulnerability to flooding which results in adverse socio-economic impacts. Many Caribbean islands, including Trinidad and Tobago lack a flood risk assessment tool which is essential for a proactive mitigation approach to floods, specifically in the Caribbean due to the incommensurate flooding events that occur because of the inherent characteristics of SIDS. This research focuses on the problem of flooding using susceptibility analysis, vulnerability analysis and risk assessment for the island of Trinidad, whilst also presenting a repeatable and appropriate methodology to assess these risks in regions that have similar characteristics to Trinidad. This is especially useful in Caribbean countries because of a lack of internal human capacity to support such efforts. Flood hazard indexes (FHI) and vulnerability indexes (VI) were generated for this study using subjective and objective weighting technique models to identify regions that are affected by flooding. These models were Analytical Hierarchy Process (AHP), Frequency Ratio (FR) and Shannon’s Entropy (SE). Comparative analyses of the three models were conducted to assess the efficacy and accuracy of each to determine which is most suitable. These were used to conduct a risk assessment to identify risks associated with each Regional Corporation of Trinidad. Results indicate that FR is the most accurate weighting technique model to assess flood susceptibility and risk assessment in Trinidad, with an Area Under the Curve (AUC) of 0.76 and 0.64 respectively. This study provides an understanding of the most appropriate weighting techniques that can be used in regions where there are challenges in accessing comprehensive data sets and limitations as it relates to access to advanced technology and technical expertise. The results also provide reasonably accurate outcomes that can assist in identifying priority areas where further quantitative assessments may be required and where mitigation and management efforts should be focused. This is critical for SIDS where vulnerability to flooding is high while access to financial and human resources is limited.

Motif discovery and characterization are important for gene regulation analysis. The lack of intuitive and integrative web servers impedes effective use of motifs. Here we describe Bipartite Motifs Learning (BML), a web server that provides a user-friendly portal for online discovery and analysis of sequence motifs, using high-throughput sequencing data as the input. BML utilizes both position weight matrix (PWM) and dinucleotide weight matrix (DWM), the latter of which enables the expression of the interdependencies of neighboring bases. With input parameters concerning the motifs are given, the BML achieves significantly higher accuracy than other available tools for motif finding. When no parameters are given by non-expert users, unlike other tools BML employs a learning method to identify motifs automatically and achieve accuracy comparable to the scenario where the parameters are set. The BML web server is freely available at http://motif.t-ridership.com/.

Online reviews are integral to consumer decision-making while purchasing products on an e-commerce platform. Extant literature has conclusively established the effects of various review and reviewer related predictors towards perceived helpfulness. However, background research is limited in addressing the following problem: how can readers interpret the topical summary of many helpful reviews that explain multiple themes and consecutively focus in-depth? To fill this gap, we drew upon Shannon's Entropy Theory and Dual Process Theory to propose a set of predictors using NLP and text mining to examine helpfulness. We created four predictors - review depth, review divergence, semantic entropy and keyword relevance to build our primary empirical models. We also reported interesting findings from the interaction effects of the reviewer's credibility, age of review, and review divergence. We also validated the robustness of our results across different product categories and higher thresholds of helpfulness votes. Our study contributes to the electronic commerce literature with relevant managerial and theoretical implications through these findings. © 2022 Elsevier B.V. 2-s2.0-85129723722

Diversity and dissimilarity within and between species assemblages have now been studied for more than half a century by community ecologists in relation to their connections with ecosystem functioning. However, a generalized framework that puts diversity and dissimilarity coefficients under the same formal umbrella is still lacking. In this paper, we show that generalized means represent an effective tool to develop a unifying formulation for the construction of a large array of parametric diversity and dissimilarity measures. These measures include some of the classical diversity coefficients, such as the Shannon entropy, the Gini-Simpson index or the parametric diversity of Patil and Taillie, together with a large number of dissimilarity coefficients of the Bray-Curtis family and can be further extended to the measurement of functional and phylogenetic differences within and between plots.

The necessity to progress towards sustainability has inspired modern researchers to examine the lubrication and cooling effects of vegetable oils on conventional metal cutting operations. Consequently, as an eco-friendly vegetable product, castor oil can be the right choice as Minimum quantity lubrication (MQL) base fluid. Nonetheless, the high viscosity of castor oil limits its flowability and restricts its industrial application. Conversely, palm oil possesses superior lubricity, as well as flowability characteristics. Hence, an attempt has been made to improve the lubrication behavior of castor oil. Here, six castor-palm mixtures (varying from 1:0.5&ndash, 1:3) were utilized as MQL-fluid, and the values of machining responses viz. average surface roughness, specific cutting energy, and tool wear were evaluated. Furthermore, an integrated Shannon&rsquo, s Entropy-based Technique for order preference by similarity to ideal solution (TOPSIS) framework was employed for selecting the most suitable volume ratio of castor-palm oil mixture. The rank provided by the TOPSIS method confirmed that 1:2 was the best volume ratio for castor-palm oil mixture. Afterward, a comparative analysis demonstrated that the best castor-palm volume fraction resulted in 8.262 and 16.146% lowering of surface roughness, 5.459 and 7.971% decrement of specific cutting energy, 2.445 and 3.155% drop in tool wear compared to that of castor and palm oil medium, respectively.

In the literature (e.g. (1)), the expression - density(x) ln(density(x)) is used as Shannon’s spatial entropy for a quantum system (bound or not). For the bound case, the correspondence principle is said to hold for high energy levels (2). In that case, quantum density = W(x)W(x) (W(x)=real wavefunction) is said to be equivalent to the classical density C/v(x) where v(x) the velocity is given by .5mv(x)v(x) + V(x) = E. Classical density yields a curve which passes through peak regions of quantum density humps. In the classical case, particle motion is completely determinate. C/v(x) as a density (probability) is proportional to dt, the amount of time a particle spends passing through a fixed length dx. If the particle moves quickly, it spends little time, and hence there is less time for a measurement (i.e one may miss the particle due to its speed). We argue in this note that such a scenario does not seem to be in keeping with the ideas of Shannon’s entropy where one has probabilities for certain mutually exclusive events at each ti. For example at each ti, one may toss a coin. If it is not “heads”, this means it is tails. For the classical particle in the correspondence principle, each dx region has its own probability proportional to dt = dx/v(x). Thus, each dx is a region of its own in terms of probability. Thus, if Ci measurements of the presence of the particle are made within dx of xi out of N trials, the probability that a measurement is made in dx at xi is Ci/N and the probability that it is not is 1-Ci/N. Thus, unlike Shannon’s picture, various arrangements do not have the same weight or probability. We investigate this scenario in more detail and argue that it does not follow the approach of Shannon’s entropy. We also try to connect with the quantum situation.

Urban sprawl has become a remarkable characteristic of urban development worldwide in the last decades. Urban sprawl refers to the extent of urbanization, which is a global phenomenon mainly driven by population growth and large scale migration. In developing countries like Iran, urban sprawl is taking its toll on the natural resources at an alarming pace. The purpose of this paper is to study urban growth and effective factors on them in the city of Urmia, Iran. We used quantitive data of the study area from the period between 1989 and 2007, and population censuses of Urmia. To measure the model of urban growth, Holderness and Shannon’s entropy were employed. The Urmia case is interesting for several reasons: first, it is a case of very fast urban growth even for a developing country; second, it illustrates how the fastest rates of urban sprawl may correspond to middle size cities rather than to large centers. Third, it portrays a land substitution process in which agricultural land is not the primary provider of urban land which is relatively rare in urban contexts, and fourth, it also illustrates how urban sprawl may also hide important internal land uses such as the presence of agricultural plots within urban boundaries.

In the literature, one often finds a quantum expression for Shannon’s entropy of the form: - (W*W) ln(W*W) ((1)) where W*W is spatial density i.e. W(x)= the wavefunction. This implies W*(x)W(x) is an independent probability. Given that W(x)= Sum over p a(p) exp(ipx), and W*W= Sum over p1, p2 a(p1)a(p2) exp(ip1 x) exp(ip2 x), one may see that for p1/-p1 in the latter with a(p)=a(-p) leads to cos(p1x) terms. These are positive and negative in different x regions. In fact, in many cases, the overall spatial density exhibits peaks and valleys suggesting that W*(x)W(x) is correlated, i.e. not independent we argue. Furthermore, the form ((1)) uses only position when there is an expectation value for < p*p> an , both functions of x, where p is momentum. Thus, the variance of p-ensemble average may change with x, yet position only is used in the computation of Shannon’s entropy, which we argue seems unusual. In a previous note (1), we suggest using a(1)exp(ip1 x) a(p2)exp(ip2 x) as the probability for Shannon’s entropy. This leads to a spatial form 2W x d/dx W which matches ((1)) for the ground state of the quantum oscillator case. For the ground state oscillator, W*W=C exp(-ax*x) i.e. a Gaussian. This suggests that for this case only, W*(x)W(x) represents an independent probability which may be used in Shannon’s entropy, we argue. We attempt to consider these issues in this note.

In a previous note (1), we argued that information= ln(Pi), where Pi is the probability for the ith letter in the 26 letter English alphabet to appear in a long message, may be obtained by considering “interactions” which leave Pi unchanged as the initial message undergoes interactions until it reaches a destination. ln(Pi) then naturally appears as a conserved quantity in the interactions which ensure that Pi does not change as one moves from source to destination. Thus, each letter contains a substance ln(Pi), called information, which is similar to the energy of a particle in a gas. An interaction may change one letter into another, but the overall information of a long message cannot change because otherwise Pi would change. In the literature (2), (3), there are descriptions of ln(Pi) as being associated with different arrangements of the same letters in a long message. In this note, we try to show how the conserved quantity ln(Pi) is linked to ideas of arrangements of letters.

This study is focused on developing an approach for spatial mapping of groundwater by considering four types of factors (geological, topographical, hydrological, and climatic factors), and by using different bivariate statistical models, such as frequency ratio (FR) and Shannon’s entropy (SE). The developed approach was applied in a fractured aquifer basin (Ameln Basin, Western Anti-Atlas, Morocco), to map the spatial variation of groundwater potential. Fifteen factors (15) influencing groundwater were considered in this study, including slope degree, slope aspect, elevation, topographic wetness index (TWI), slope length (LS), topographic position index (TPI), plane curvature, profile curvature, drainage density, lineament density, distance to rivers and fault network, normalized difference vegetation index (NDVI), lithology, and land surface temperature (LST). The potential maps produced were then classified into five classes to illustrate the spatial view of each potential class obtained. The predictive capacity of the frequency ratio and Shannon’s entropy models was determined using two different methods, the first one based on the use of flow data from 49 boreholes drilled in the study area, to test and statistically calibrate the predictive capacity of each model. The results show that the percentage of positive water points corresponds to the most productive areas (high water flow) (42.86% and 30.61% for the FR and SE models, respectively). On the other hand, the low water flows are consistent with the predicted unfavorable areas for hydrogeological prospecting (4.08% for the FR model and 6.12% for the SE model). Additionally, the second validation method involves the integration of 7200 Hz apparent resistivity data to identify conductive zones that are groundwater circulation zones. The interpretation of the geophysical results shows that the high-potential zones match with low apparent resistivity zones, and therefore promising targets for hydrogeological investigation. The FR and SE models have proved very efficient for hydrogeological mapping at a fractured basement area and suggest that the northern and southern part of the study area, specifically the two major fault zones (Ameln Valley in the north, and the Tighmi-Tifermit Valley in the south) has an adequate availability of groundwater, whereas the central part, covering the localities of Tarçouat, Boutabi, Tililan, and Ighalen, presents a scarcity of groundwater. The trend histogram of the evolution of positive water points according to each potentiality class obtained suggests that the FR model was more accurate than the SE model in predicting the potential groundwater areas. The results suggest that the proposed approach is very important for hydrogeological mapping of fractured aquifers, and the resulting maps can be helpful to managers and planners to generate groundwater development plans and attenuate the consequences of future drought.

This paper shows if and how the predictability and complexity of stock market data changed over the last half-century and what influence the M1 money supply has. We use three different machine learning algorithms, i.e., a stochastic gradient descent linear regression, a lasso regression, and an XGBoost tree regression, to test the predictability of two stock market indices, the Dow Jones Industrial Average and the NASDAQ (National Association of Securities Dealers Automated Quotations) Composite. In addition, all data under study are discussed in the context of a variety of measures of signal complexity. The results of this complexity analysis are then linked with the machine learning results to discover trends and correlations between predictability and complexity. Our results show a decrease in predictability and an increase in complexity for more recent years. We find a correlation between approximate entropy, sample entropy, and the predictability of the employed machine learning algorithms on the data under study. This link between the predictability of machine learning algorithms and the mentioned entropy measures has not been shown before. It should be considered when analyzing and predicting complex time series data, e.g., stock market data, to e.g., identify regions of increased predictability.

Study Region State of Odisha, a data-scarce tropical savanna region in eastern India. Study Focus This study evaluated the temporal variability in depth to groundwater (DTW) in the study region with heavily stressed aquifers during 1995–2015 using the modified Mann Kendall test. Subsequently, Shannon’s entropy assessed spatial variability in DTW and determined the dominant Hydrological, Geological, and Climatological (HGC) factor regulating the observed spatio-temporal variability taking land use/ land cover (LULC), geomorphology, lithology, topography, and rainfall as HGC factors. New Hydrological Insights The overall and seasonal trend analysis revealed that the study region possessed both rising and declining trends with a slightly higher percentage of wells with a rising trend. The spatial distribution of trends and the associated magnitude accentuated the unforeseen groundwater temporal variability and higher-order susceptibility of DTW to rising and declining trends. The marginal entropy revealed the higher-order spatial variability associated with deeper DTW and vice versa. Evaluation of the HGC factors revealed that LULC could explain the maximum variability in the DTW as a dominant HGC factor. It was found that the impact of LULC features on DTW variability is not straightforward, necessitating impact assessment studies in the location with significant to highly significant trends. This formulated approach can immensely contribute to the planning and management in attaining groundwater sustainability in data-scarce regions.

Entropy measures are standard tools in environmental and ecological sciences to describe the heterogeneity of data. This paper reviews a selection of spatial entropy indices, some of which are very recent, suitable to deal with spatial data on variables presenting a finite number of categories. A special focus is given on biodiversity data, but methods can be applied to any other environmental phenomena. The new R package SpatEntropy is here introduced to compute spatial entropy measures in practice. The extension from traditional entropy measures to their spatial version is a unique feature of the package, which is able to work with both areal and point data. A practical part is also presented, where two types of environmental data are considered, regarding trees biodiversity and urban expansion, respectively. The package SpatEntropy is run over these dataset and shown to represent a user-friendly and helpful tool for new package users.

As a response to the challenge of rapid pace of urbanization and lack of reliable data for environmental and urban planning, especially in the developing countries, this paper evaluates land use/cover change (LCLU) and urban spatial expansion, from 1987 to 2013, in the Qom, Iran, using satellite images, field observations, and socio-economic data. The supervised classification technique by maximum likelihood classifier has been employed to create a classified image and has been assessed based on Kappa index. The urban sprawl was also measured using Shannon’s entropy based on its primary spatial forms. To our knowledge, measuring urban sprawl based on its spatial forms would contribute to prioritizing policies and specific regulations in dealing with its dominant form. Finally, LCLU change and urban growth were simulated for 2022, using CA-Markov model. The results revealed that dramatic growth of built-up areas has led to a significant decrease in the area of agriculture, gardens and wasteland, from 1987 to 2013. The obtained relative entropy values have indicated that the Qom city has experienced increasing urban sprawling over the last three decades. The continuous linear and non-continuous linear developments along the major roads and highways are the dominant forms of sprawl in Qom city. The CA-Markov model estimated that this unsustainable trend will continue in the future and built-up areas will be increased by 10% by 2022 resulting in potential loss of 438.03 ha agriculture land, 638.37 ha wasteland, and 17.01 ha gardens. Those results indicated the necessity of appropriate policies and regulations particularly for limiting linear sprawl along the main roads.

Maxwell-Boltzmann (MB) distributions and even expressions of Shannon’s entropy emerged in the 1800s and early 1900s with Maxwell, Boltzmann and Gibbs and others. Later, in the 1980s, Tsallis suggested a non-MB distribution and many other such distributions have been developed since. It has been argued in (1) that conservation of energy in two body collisions and kinematics are all that are necessary for obtaining these new distributions i.e. the concept of maximization of entropy is not needed. Nevertheless, the “maximization” of Shannon’s entropy subject to a constraint on overall energy does yield the MB distribution. This begs the question: Why does an information theoretical approach lead to the same result as a physical two body elastic collision scheme? Secondly, why does information theory lead to the MB result and not to other distributions? (One should note that for other distributions, entropy functions have been created such that their maximization subject to constant energy and particle number yield the new distributions, but such entropy expressions seem to be mathematical constructs.)

In normative studies, which provide qualitative result to promote children's independent mobility (CIM), enough attention is not paid to the attitudes of children, parents and experts. Positive research that investigates CIM using measuring instruments, the researcher understanding of the study context and perceptual characteristics of children and parents is limited. The present study aims at investigating the variables of CIM using exploration and confirmatory methods, respectively. Analysis of 1896 behavioral places made possible discovering 9 selected ranges of independent mobility and 4 principles related to the promotion of CIM. Expert attitudes were assessed by Delphi technique and the exploratory principles were ranked by Shannon's entropy. Factor analysis was used to confirm the effect of 5 factors on improving CIM (N = 204). The housing type, number of floors, level of residency, street type, green-space, perceived safety and gender differences were variables related to CIM.

The graph entropies inspired by Shannon’s entropy concept become the information-theoretic quantities for measuring the structural information of graphs and complex networks. In this paper, we continue studying some new properties of the graph entropies based on information functionals involving vertex degrees. We prove the monotonicity of the graph entropies with respect to the power exponent. Considering only the maximum and minimum degrees of the ( n , m ) -graph, we obtain some upper and lower bounds for the degree-based graph entropy. These bounds have different performances to restrict the degree-based graph entropy in different kinds of graphs. Moreover the degree-based graph entropy can be estimated by these bounds.

Personalized information systems are information-filtering systems that endeavor to tailor information-exchange functionality to the specific interests of their users. The ability of these systems to profile users is, on the one hand, what enables such intelligent functionality, but on the other, the source of innumerable privacy risks. In this paper, we justify and interpret KL divergence as a criterion for quantifying the privacy of user profiles. Our criterion, which emerged from previous work in the domain of information retrieval, is here thoroughly examined by adopting the beautiful perspective of the method of types and large deviation theory, and under the assumption of two distinct adversary models. In particular, we first elaborate on the intimate connection between Jaynes' celebrated method of entropy maximization and the use of entropies and divergences as measures of privacy; and secondly, we interpret our privacy metric as false positives and negatives in a binary hypothesis testing. (C) 2013 Elsevier B.V. All rights reserved.

This paper presents a framework to identify critical nodes of a gas pipeline network. This framework calculates a set of metrics typical of the social network analysis considering the topological characteristics of the network. Such metrics are utilized as inputs and outputs of a (Data Envelopment Analysis) DEA model to generate a cross-efficiency index that identifies the most important nodes in the network. The framework was implemented to assess the US interstate gas network between 2013 and 2017 from both the demand and supply-side perspectives. Results emerging from the US gas network case suggest that different analysis perspectives should necessarily be considered to have a more in-depth and comprehensive view of the network capacity and performance.

Dominance and diversity are important characteristics for the description of communities. The most commonly used indices are Simpson's dominance indexand Shannon's and Simpson's indices of diversity. This paper uses the basic concepts of statistics as applied to community analysis to develop new dominance and diversity indices that will enable scientists to establish correlations among various indices. The present study proves that the variance of the number of individuals of different species in a sample can be used to calculateSimpson's dominance and diversity indices. New indices have been developed from the ratios ofthe variance to number of species, and the mean number of individuals per species in a quadrat. A wide range of data, varying from high dominance to high evenness, was simulated for 25 quadrats, with each quadrat having ten species and 100 individuals in different combinations. Variance and standard deviation-based indices were computed using the simulated data and were found to be highly correlated with Simpson's and Shannon's indices. The proposed indices will give both the dominance and diversity of a community on the same scale based on the same statistic. Another important contribution of the present study relates to the variance of a sample consisting of a single value. It has been proved that the variance of a sample having only one value is equal to the square of that value. The paper establishes a new link between diversity studies and statistics., Ecology; Environmental chemistry; Environmental geochemistry; Environmental impact assessment; Environmental risk assessment; Environmental toxicology; Diversity indices; Simpson's dominance; Shannon's entropy; Sample variance; Covariance; Binary information plots

Collaborative tagging is one of the most popular services available online, and it allows end user to loosely classify either online or offline resources based on their feedback, expressed in the form of free-text labels (i.e., tags). Although tags may not be per se sensitive information, the wide use of collaborative tagging services increases the risk of cross referencing, thereby seriously compromising user privacy. In this paper, we make a first contribution toward the development of a privacy-preserving collaborative tagging service, by showing how a specific privacy-enhancing technology, namely tag suppression, can be used to protect end-user privacy. Moreover, we analyze how our approach can affect the effectiveness of a policy-based collaborative tagging system that supports enhanced web access functionalities, like content filtering and discovery, based on preferences specified by end users.

A great challenge facing final manufacturers of final products and OEMs today is product variety management which is crucial for mass-customized manufacturing. Product variations in most industries tend to be very numerous and firms often find themselves under pressure to solve problems with over-inventories, planning and product assembly complexity. In this paper, we investigate and present an approach to creating all possible product configurations and variations based on a given number of base components and optional number of complementary components. The idea of determining all possible product architectures is to investigate the influence of product variety on complexity of assembly processes.

In this paper, a new method for specification of recurrence or transient of states in one and two dimensional simple random walk based on upper and lower bounds of {\it r}-combinations from a set of m elements $(C^{m}_{r})$ via binary entropy is introduced.

Using the concavity property of the log mapping and the weighted arithmetic mean - geometric mean inequality, we point out an analytic inequality for the logarithmic map and apply it for the Kullback-Leibler distance in Information Theory. Some applications for Shannon’s entropy are given as well.

The purpose of the paper is to introduce, using the known results concerning the entropy in product MV algebras, the concepts of mutual information and Kullback–Leibler divergence for the case of product MV algebras and examine algebraic properties of the proposed measures. In particular, a convexity of Kullback–Leibler divergence with respect to states in product MV algebras is proved, and chain rules for mutual information and Kullback–Leibler divergence are established. In addition, the data processing inequality for conditionally independent partitions in product MV algebras is proved.

In this paper, we study the Shannon's entropy and its applications in the regular exponential models.

Most ecological diversity indices summarize the information about the relative abundances of community species without reflecting taxonomic differences between species. Nevertheless, in the environmental conservation practice, data on species abundances are generally unknown. In such cases, to summarize the conservation value of a given site, so-called ‘biological diversity’ measures need to be used. Most of these measures are based on taxonomic relations among species and ignore species relative abundances. In a recent paper, Izsak and Papp suggest that the quadratic entropy index ( Q ) is the only diversity index used to date in the ecological practice that incorporates both species relative abundances and a measure of the pairwise taxonomic differences between species in the analyzed data set. I show here that a number of traditional ecological diversity measures can be generalized to take into account a taxonomic weighting factor. Since these new indices violate part of the mathematical properties that an index should meet to be termed an ecological diversity index, I defined this new family of indices ‘weak diversity indices’.

The degree-based network entropy which is inspired by Shannon’s entropy concept becomes the information-theoretic quantity for measuring the structural information of graphs and complex networks. In this paper, we study some properties of the degree-based network entropy. Firstly we develop a refinement of Jensen’s inequality. Next we present the new and more accurate upper bound and lower bound for the degree-based network entropy only using the order, the size, the maximum degree and minimum degree of a network. The bounds have desirable performance to restrict the entropy in different kinds of graphs. Finally, we show an application to structural complexity analysis of a computer network modeled by a connected graph.

International audience; This contribution deals with binary detection networks optimization using an entropybased criterion. The optimization of a detection component consists in applying a variable threshold on the likelihood ratio, which depends on a posteriori probabilities. A gradient algorithm is proposed to find this threshold. The optimization results of the detection component using entropy and Bayes’ criteria are compared. Assuming conditional independence of the observations, the optimization of a complex parallel distributed detection network, constituted by several detection components, can be made using a person-by-person technique, whatever the considered local criterion (Bayes’ or entropy-based criterion): a compromise between a good error probability and a good ROC curve is obtained.

In this paper, the Weibull-X family is proposed and some of its properties are discussed. A member of the Weibull-X family, the Weibull-logistic distribution, is defined and studied. Various properties of the Weibulllogistic distribution are obtained. The distribution is found to be unimodal and the shape can be symmetric, right skewed or left skewed. The structural analysis of the distribution in this paper includes limiting behavior, mode, quantiles, moments, skewness, kurtosis, Shannon’s entropy and order statistics. The method of maximum likelihood estimation is proposed for estimating the model parameters. A real data set is used to illustrate the application of the Weibull-logistic distribution.

This paper contains the solution of a functional equation which is a generalization of a functional equation arising in a characterization of directed divergence and inaccuracy.

Projecte realitzat en el marc d’un programa de mobilitat amb la University of Piraeus In recent times, a growing number of information retrieval applications are disposable, aiming to trace users’ online behavior and activities. One of the most popular social networks, which can be considered as a valuable source of information to this kind of applications, is Twitter. Aggregated data that derive from Twitter can show great power in delivering information related to users’ interests and preferences. The process of correlating information can result in the construction of comprehensive user profiles that may disclose detailed personal information and raise challenges to users’ privacy as well. Extracted behavioral patterns of users can be substantial to the development of personalization services, however, inevitably at the expense of users’ privacy. Although there are a number of privacy-enhancing technologies, which strive to mitigate many of these concerns, significant gaps remain regarding the privacy protection of users. In addition, it is essential to provide a comprehensive view on metrics which consist in quantifying privacy. Most of the efforts devoted to devise privacy metrics are quite limited, as they apply to concrete systems. The lack of suitable metrics is deterrent to the proper privacy evaluation. Therefore, even though proposed approaches have made meaningful contributions to the challenging privacy landscape, there still exists a certain ambiguity about their effectiveness and adjustment to different contexts. In this work, we present an effort towards the construction of user profiles, through the development process of an information retrieval application. We also tackle the privacy issues related to user profiling, as personal information contained in user profiles is disclosed. The last part of this thesis approaches the theme of quantifying user privacy by applying information-theoretic notions as measures of the privacy of user profiles.