Entropic Kullback‐Leibler type distance measures for quantum distributions

Laguna, Humberto G.; Salazar, Saúl J. C.; Sagar, Robin P.

doi:10.1002/qua.25984

Cited by 9 publications

(5 citation statements)

References 52 publications

(56 reference statements)

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“…In cases where the distributions share the same angular parts and are normalized, these cancel in the logarithmic argument and the equations reduce to an integral over the radial parts of the wave functions,

\begin{matrix} lefttrue \\ KL (ρ_{1}, ρ_{2}) = \int_{0}^{r_{c}} r^{2} {(||, R_{n_{1} l_{1}} ((), r))}^{2} \ln [\frac{{(||, R_{n_{1} l_{1}} ((), r))}^{2}}{{(||, R_{n_{2} l_{2}} ((), r))}^{2}}] dr, \\ KL (π_{1}, π_{2}) = \int_{0}^{\infty} p^{2} {(||, {\overset{false˜}{R}}_{n_{1} l_{1}} ((), p))}^{2} \ln [\frac{{(||, {\overset{false˜}{R}}_{n_{1} l_{1}} ((), p))}^{2}}{{(||, {\overset{false˜}{R}}_{n_{2} l_{2}} ((), p))}^{2}}] dp . \end{matrix}

These measures are well defined if the density in the denominator of the logarithmic argument, known as the reference density, shares the same interior nodal structure with the density in the numerator …”

Section: Computational Detailsmentioning

confidence: 99%

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Shannon‐information entropy sum in the confined hydrogenic atom

Salazar

Laguna

Prasad

et al. 2020

Int J of Quantum Chemistry

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The appearance of critical points in the Shannon entropy sum as a function of confinement radius, in ground and excited state confined hydrogenic systems, is discussed. We illustrate that the Coulomb potential in tandem with the hard sphere confinement are responsible for these points. The positions of these points are observed to vary with the intensity of the potential. The effects of the Coulomb potential on the system are further probed, by examining the differences between the densities of the confined atom and those of the particle confined in a spherical box, for the same confinement radius. These differences are quantified by using Kullback-Leibler and cumulative residual Kullback-Leibler distance measures from information theory. These measures detect that the effects of the Coulomb potential are squeezed out of the system as the confinement radius decreases. That is, the confined atom densities resemble the particle in a box ones, for smaller confinement radii. Furthermore, the critical points in the entropy sum lie in the same regions where there are changes in the distance measures, as the atom behaves more particle in a spherical box-like. The analysis is further complemented by examination of the derivative of the entropy sum with respect to confinement radius. This study illustrates the inhomogeneity in the magnitudes of the derivatives of the entropy sum components and their dependence on the Coulomb potential. A link between the derivative and the entropic force is also illustrated and discussed. Similar behaviors are observed when the virial ratio is compared to the entropic power one, as a function of confinement radius. K E Y W O R D S confined hydrogen atom, Kullback-Leibler distance measures, quantum uncertainties, Shannon entropy sum

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\begin{matrix} lefttrue \\ KL (ρ_{1}, ρ_{2}) = \int_{0}^{r_{c}} r^{2} {(||, R_{n_{1} l_{1}} ((), r))}^{2} \ln [\frac{{(||, R_{n_{1} l_{1}} ((), r))}^{2}}{{(||, R_{n_{2} l_{2}} ((), r))}^{2}}] dr, \\ KL (π_{1}, π_{2}) = \int_{0}^{\infty} p^{2} {(||, {\overset{false˜}{R}}_{n_{1} l_{1}} ((), p))}^{2} \ln [\frac{{(||, {\overset{false˜}{R}}_{n_{1} l_{1}} ((), p))}^{2}}{{(||, {\overset{false˜}{R}}_{n_{2} l_{2}} ((), p))}^{2}}] dp . \end{matrix}

Section: Computational Detailsmentioning

confidence: 99%

“…The cumulative residual Kullback‐Leibler (CRKL) measure, based on the survival (cumulative residual) densities, has been introduced to address the situation when the reference density possesses nodal structure which is not present in the other density …”

Section: Computational Detailsmentioning

confidence: 99%

Shannon‐information entropy sum in the confined hydrogenic atom

Salazar

Laguna

Prasad

et al. 2020

Int J of Quantum Chemistry

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“…The K-L distance derived from cross-entropy reflects the similarity between the two vectors (Kullbac and Leibler, 1951). At present, the K-L distance has been employed in a wide range of fields (Laguna et al, 2019;Sun and Dong, 2019).…”

Section: Introductionmentioning

confidence: 99%

A novel generalized grey target decision method with index and weight both containing mixed types of data

Tian

Yue

2021

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PurposeThis paper is to propose a novel generalized grey target decision method (GGTDM) with index and weight both containing mixed types of data.Design/methodology/approachThe decision-making steps of the proposed approach are as follows. First, all mixed attribute values of alternatives and weights are transformed into binary connection numbers and also comprised two-tuple (determinacy, uncertainty) numbers. Then, the two-tuple (determinacy, uncertainty) numbers of target center indices are calculated. Next, the certain weights are determined by the Gini–Simpson (G–S) index-based method. Following this, the comprehensive-weighted Kullback–Leibler distances (CWKLDs) of all alternatives and the target center are obtained. Finally, the alternative ranking relies on the CWKLD considering the smaller value as the better option.FindingsThe certain weights determined by the improved Gini–Simpson index (IGSI) based method are more accurate in compared with that by the proximity-based method and the weight function method. The discrimination ability of alternatives ranking of the proposed approach is stronger than that of the compared comprehensive-weighted proximity (CWP) based method and comprehensive-weighted Gini–Simpson index (CWGSI) based method.Research limitations/implicationsThe proposed method fulfills the decision-making task relying on CWKLD, which solves the uncertain measurement from the viewpoint of entropy.Originality/valueThe proposed approach adopts the IGSI to transform uncertain weights into certain ones and takes the CWKLD as the basis for the decision-making.

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“…As such, the current study considers several alternative scoring functions that use SQR and compares how sensitive they are in quantifying the quality of a pdf estimate. Other types of information measures that use cumulative relative entropy [17] or residual cumulative Kullback-Leibler information [18,19] are possible. However, these alternatives are outside the scope of this study, which focuses on leveraging SQR properties.…”

Section: Introductionmentioning

confidence: 99%

Universal Sample Size Invariant Measures for Uncertainty Quantification in Density Estimation

Farmer

Merino

Gray

et al. 2019

Entropy

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Previously, we developed a high throughput non-parametric maximum entropy method (PLOS ONE, 13(5): e0196937, 2018) that employs a log-likelihood scoring function to characterize uncertainty in trial probability density estimates through a scaled quantile residual (SQR). The SQR for the true probability density has universal sample size invariant properties equivalent to sampled uniform random data (SURD). Alternative scoring functions are considered that include the Anderson-Darling test. Scoring function effectiveness is evaluated using receiver operator characteristics to quantify efficacy in discriminating SURD from decoy-SURD, and by comparing overall performance characteristics during density estimation across a diverse test set of known probability distributions.

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Entropic Kullback‐Leibler type distance measures for quantum distributions

Cited by 9 publications

References 52 publications

Shannon‐information entropy sum in the confined hydrogenic atom

Shannon‐information entropy sum in the confined hydrogenic atom

A novel generalized grey target decision method with index and weight both containing mixed types of data

Universal Sample Size Invariant Measures for Uncertainty Quantification in Density Estimation

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