Entropy information

Campbell [campj82], in his Grammatical Man Information, Entropy, Language and Life, was one of the first to assert the importance of adding information to our short list of fundamental descriptors of nature ... 

Information, /, is the difference between information entropies, viz the a priori information entropy, H, that characterizes the uncertainty before a message is obtained or an experiment is carried out, and the a posteriori information entropy, II, that remains afterwards ... 

In the case of m discrete phenomena x, (e.g., outputs of an experiment), the information entropy is calculated by means of the respective probabilities... 

On the other hand, the information entropy characterizing the uncertainty of a continuous random quantity with a probability density p(x) is given by... 

The IT approach provides a systematic estimate of the pn under the constraints of the available information, defined as the set prl] that maximizes an information entropy rj subject to the information constraints,... 

Maximizing the information entropy under the constraints of Eq. (9) leads to... 

The idea here was to examine which pair of techniques and individual columns could lead to the best separations in 2DLC. This is achievable by using ID separations and then comparing how the retention of each component varies across the separation space. Another innovation here was the use of IT-derived metrics such as information entropy, informational similarity, and the synentropy. As stated in this paper, The informational similarity of 2D chromatographic systems, H is a measure of global... 

Yet another indirect connection between momentum and coordinate space charge densities is derived via a quantity called the Shannon information entropy... 

The information entropy of a probability distribution is defined as S[p(] = — p, In ph where p, forms the set of probabilities of a distribution. For continuous probability distributions such as momentum densities, the information entropy is given by. S yj = - Jy(p) In y(p) d3p, with an analogous definition in position space... 

This so-called Hirshfeld scheme is particularly popular within the so-called conceptual density functional theory (DFT) [26,27], The weighting function, which identifies the AIM as one that is most similar to the isolated atom [28], has been shown to be directly derivable from information entropy [6,29-33]. Here again, the atoms do not... 

The generality of similarity as a basic concept throughout chemistry makes it the theory of MQS very useful nearly in any field of chemistry including for instance biological activity. Not only is similarity by itself an interesting subject, it also opens the path to many related issues such as complexity and more generally many concepts in information entropy measures [78]. 

If we examine the current geographical distribution of a mutation, it is hard to estimate the value of the population density n at the position and time where the mutation originates. It makes sense to treat n as a random variable selected from a certain probability density p n). The constraints imposed onp n) are the conservation of the normalization condition f p(n) dn = 1 and the range of variation, noc > n > 0. The maximum information entropy approach leads to a uniform distribution... 

The methods developed from either NN or IT undoubtedly reduce the requirements concerning the hardware that must be applied. In the NN approach it is sufficient to determine electrostatic potential, for which far less computer resources are required than for solving HFR equations supplemented with the energy of correlations, necessary for reliable calculation of interaction energy. Similarly, the calculation of information entropy (the IT case) based on the electron density is possible, and can be done with much shorter calculation time than in the case of HFR equations with the correlation energy. 

To a significant extent, the theoretical basis of modern communication theory arose from the work of Claude Shannon at Bell Labs. [80]. In these seminal works, the concept of the information entropy associated with an arbitrary signal arose. In 1981, Watanabe realised the close association between entropy minimization and pattern recognition [81]. An association between entropy minimization and the principle of simplicity is also recognized [82]. The basic mathematical form of signal... 

Legeza, O., Solyom, J. Optimizing the density-matrix renormalization group method using quantum information entropy. Phys. Rev. B 2003, 68(19), 195116. 

The maximum entropy method (MEM) is an information-theory-based technique that was first developed in the field of radioastronomy to enhance the information obtained from noisy data (Gull and Daniell 1978). The theory is based on the same equations that are the foundation of statistical thermodynamics. Both the statistical entropy and the information entropy deal with the most probable distribution. In the case of statistical thermodynamics, this is the distribution of the particles over position and momentum space ( phase space ), while in the case of information theory, the distribution of numerical quantities over the ensemble of pixels is considered. 

The maximum entropy method was first introduced into crystallography by Collins (1982), who, based on Eq. (5.47), expressed the information entropy of the electron density distribution as a sum over M grid points in the unit cell, using... 

We now assume that < t(0) i(0> is known and we want to guess the probability distribution P. We do this by maximizing the information entropy,65 SIf of this distribution... 

Maximizing the information entropy of a distribution gives in some sense the smoothest distribution consistent with our available information65 on this distribution. We have tested the information theory prediction of from for two different systems the Stockmayer and modified Stockmayer simulations of CO. We have already seen that these two systems represent two extreme forms of molecular rotational motion. In the Stockmayer simulation the molecules rotate essentially freely whereas in the modified Stockmayer simulation there is evidence for strongly hindered rotational motion. from the... 

There are relationships between thermodynamic and informational entropy. For example, the well-known Maxwell s demon, which reverses thermodynamic entropy with information, but getting that information exactly balances out the thermodynamic gain the demon would otherwise achieve. 

Renyi (78) generalized the concept of information entropy to measure different aspects of system homogeneity, and Alemanskin et al. (79), and Alemanskin and Manas-Zloczower (80) adopted the Renyi entropy for measuring mixing. Considering only statistically independent partition, Renyi determined that the information entropy can be replaced with the following single-variable function... 

Success expectation when using information entropy viewpoint... 

Concrete objects to discuss from the viewpoint of information entropy... 

The application of information entropy to other phenomena will be easy if the applications described above are sufficiently understood. 

The most important facet of the book is information entropy, which was selected by the author as a viewpoint for observing the phenomena. In any case, before going to the main subject, the gist of the concept1-4 and important matters of information entropy are explained in detail in the next section so that the rest of the contents of the book can be understood easily. 

This average amount of information is called as information entropy. In particular, when the system is a single event/phenomena, the information entropy is called self-entropy. For example, the self-entropy in the case of casting an unbiased die becomes... 

The unit of information entropy depends on the base of the logarithm as follows ... 

However, the unit need not be a cause of concern since the information entropy is generally discussed based on the relative values. 

Information entropy is named after entropy in thermodynamics since the equation of its definition, Eq. (1.3) is similar to that of entropy in thermodynamics. Information entropy is also termed negentropy because its formula has a negative sign. 

In the case that the events/phenomena in both the systems are non-exclusive (e.g., the case of ordering for wines and side dishes), the amount of uncertainty of the result in F-system under the condition that result a appears in F-system is expressed by the information entropy as... 

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