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Volume 23
Issue 11
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Entropy, Volume 23, Issue 11 (November 2021) – 185 articles

Cover Story (view full-size image): Different arguments have led to supposing that the deep origin of phase transitions has to be identified with suitable topological changes of potential related to submanifolds of onfiguration space of a physical system. An important step forward for this approach was achieved with two theorems stating that, for a wide class of physical systems, phase transitions should necessarily stem from topological changes of energy level submanifolds of the phase space. However, the sufficiency conditions are still a wide-open question. In this study, a first important step forward was performed in this direction; in fact, a differential equation was worked out which describes how entropy varies as a function of total energy.View this paper
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35 pages, 16121 KiB  
Article
Optical Channel Selection Avoiding DIPP in DSB-RFoF Fronthaul Interface
by Zbigniew Zakrzewski
Entropy 2021, 23(11), 1554; https://doi.org/10.3390/e23111554 - 22 Nov 2021
Cited by 7 | Viewed by 2433
Abstract
The paper presents a method of selecting an optical channel for transporting the double-sideband radio-frequency-over-fiber (DSB-RFoF) radio signal over the optical fronthaul path, avoiding the dispersion-induced power penalty (DIPP) phenomenon. The presented method complements the possibilities of a short-range optical network working in [...] Read more.
The paper presents a method of selecting an optical channel for transporting the double-sideband radio-frequency-over-fiber (DSB-RFoF) radio signal over the optical fronthaul path, avoiding the dispersion-induced power penalty (DIPP) phenomenon. The presented method complements the possibilities of a short-range optical network working in the flexible dense wavelength division multiplexing (DWDM) format, where chromatic dispersion compensation is not applied. As part of the study, calculations were made that indicate the limitations of the proposed method and allow for the development of an algorithm for effective optical channel selection in the presence of the DIPP phenomenon experienced in the optical link working in the intensity modulation–direct detection (IM-DD) technique. Calculations were made for three types of single-mode optical fibers and for selected microwave radio carriers that are used in current systems or will be used in next-generation wireless communication systems. In order to verify the calculations and theoretical considerations, a computer simulation was performed for two types of optical fibers and for two selected radio carriers. In the modulated radio signal, the cyclic-prefix orthogonal frequency division multiplexing (CP-OFDM) format and the 5G numerology were used. Full article
(This article belongs to the Special Issue Advances in Computer Recognition, Image Processing and Communications, Selected Papers from CORES 2021 and IP&C 2021)
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Figure 1
<p>Functional splits proposed by 3GPP [<a href="#B13-entropy-23-01554" class="html-bibr">13</a>] in NG-RAN with an example of IF/RF extensions proposed by the author for A-RoF functions introduced into the distributed unit (DU) and the radio unit (RU) (green blocks and options) [<a href="#B21-entropy-23-01554" class="html-bibr">21</a>]. Optical BH/MH/FH and their maximal links could be realized in mobile 5G systems [<a href="#B40-entropy-23-01554" class="html-bibr">40</a>].</p> Full article ">Figure 2
<p>Modeled attenuation characteristics of single-mode optical fibers in range of single-modality limited by the cut-off wavelength [<a href="#B41-entropy-23-01554" class="html-bibr">41</a>,<a href="#B42-entropy-23-01554" class="html-bibr">42</a>,<a href="#B43-entropy-23-01554" class="html-bibr">43</a>].</p> Full article ">Figure 3
<p>Modeled characteristics of the chromatic dispersion coefficient of the single-mode optical fibers in range of single-modality limited by the cut-off wavelength [<a href="#B41-entropy-23-01554" class="html-bibr">41</a>,<a href="#B42-entropy-23-01554" class="html-bibr">42</a>,<a href="#B43-entropy-23-01554" class="html-bibr">43</a>].</p> Full article ">Figure 4
<p>Example of the optical fronthaul path assembled with use of different standards of the single-mode optical fibers.</p> Full article ">Figure 5
<p>Averaged characteristics of the chromatic dispersion coefficients for the exemplary optical fronthaul paths.</p> Full article ">Figure 6
<p>Dispersion induced power penalty—carrier-to-interference ratio obtained after propagation over G.652D or G.657A single-mode fiber and direct detection in photodetector: (<b>a</b>) calculated values for the five selected RF carriers as a function of the optical fronthaul path length; (<b>b</b>) calculated values for the five selected optical fronthaul path lengths as a function of the RF carrier.</p> Full article ">Figure 7
<p>Dispersion induced power penalty—carrier-to-interference ratio obtained after propagation over G.655D single-mode non-zero dispersion-shifted fiber and direct detection in photodetector: (<b>a</b>) calculated values for the five selected RF carriers as a function of the optical fronthaul path length; (<b>b</b>) calculated values for the five selected optical fronthaul path lengths as a function of the RF carrier.</p> Full article ">Figure 8
<p>Dispersion-induced power penalty—carrier-to-interference ratio obtained after propagation over G.655E single-mode non-zero dispersion shifted fiber and direct detection in photodetector: (<b>a</b>) calculated values for the five selected RF carriers as a function of the optical fronthaul path length; (<b>b</b>) calculated values for the five selected optical fronthaul path lengths as a function of the RF carrier.</p> Full article ">Figure 9
<p>The results of the dispersion-induced power penalty calculations, taking into account the 3 dB and 10 dB thresholds, obtained in the range of the S, C and L optical bands for the fronthaul path based on the G.652D or G.657A optical fiber of variable length: (<b>a</b>) <math display="inline"><semantics> <mrow> <msub> <mi>f</mi> <mrow> <mi>R</mi> <mi>F</mi> </mrow> </msub> <mo>=</mo> <mn>12</mn> <mrow> <mo> </mo> <mi>GHz</mi> </mrow> </mrow> </semantics></math>; (<b>b</b>) <math display="inline"><semantics> <mrow> <msub> <mi>f</mi> <mrow> <mi>R</mi> <mi>F</mi> </mrow> </msub> <mo>=</mo> <mn>28</mn> <mrow> <mo> </mo> <mi>GHz</mi> </mrow> </mrow> </semantics></math>; (<b>c</b>) <math display="inline"><semantics> <mrow> <msub> <mi>f</mi> <mrow> <mi>R</mi> <mi>F</mi> </mrow> </msub> <mo>=</mo> <mn>60</mn> <mrow> <mo> </mo> <mi>GHz</mi> </mrow> </mrow> </semantics></math>; (<b>d</b>) <math display="inline"><semantics> <mrow> <msub> <mi>f</mi> <mrow> <mi>R</mi> <mi>F</mi> </mrow> </msub> <mo>=</mo> <mn>84</mn> <mrow> <mo> </mo> <mi>GHz</mi> </mrow> </mrow> </semantics></math>.</p> Full article ">Figure 10
<p>The results of the dispersion-induced power penalty calculations, taking into account the 3 dB and 10 dB thresholds, obtained in the range of the S, C and L optical bands for the fronthaul path based on the G.655D optical fiber of variable length: (<b>a</b>) <math display="inline"><semantics> <mrow> <msub> <mi>f</mi> <mrow> <mi>R</mi> <mi>F</mi> </mrow> </msub> <mo>=</mo> <mn>12</mn> <mrow> <mo> </mo> <mi>GHz</mi> </mrow> </mrow> </semantics></math>; (<b>b</b>) <math display="inline"><semantics> <mrow> <msub> <mi>f</mi> <mrow> <mi>R</mi> <mi>F</mi> </mrow> </msub> <mo>=</mo> <mn>28</mn> <mrow> <mo> </mo> <mi>GHz</mi> </mrow> </mrow> </semantics></math>; (<b>c</b>) <math display="inline"><semantics> <mrow> <msub> <mi>f</mi> <mrow> <mi>R</mi> <mi>F</mi> </mrow> </msub> <mo>=</mo> <mn>60</mn> <mrow> <mo> </mo> <mi>GHz</mi> </mrow> </mrow> </semantics></math>; (<b>d</b>) <math display="inline"><semantics> <mrow> <msub> <mi>f</mi> <mrow> <mi>R</mi> <mi>F</mi> </mrow> </msub> <mo>=</mo> <mn>84</mn> <mrow> <mo> </mo> <mi>GHz</mi> </mrow> </mrow> </semantics></math>.</p> Full article ">Figure 11
<p>The results of the dispersion-induced power penalty calculations, taking into account the 3 dB and 10 dB thresholds, obtained in the range of the S, C and L optical bands for the fronthaul path based on the G.655E optical fiber of variable length: (<b>a</b>) <math display="inline"><semantics> <mrow> <msub> <mi>f</mi> <mrow> <mi>R</mi> <mi>F</mi> </mrow> </msub> <mo>=</mo> <mn>12</mn> <mrow> <mo> </mo> <mi>GHz</mi> </mrow> </mrow> </semantics></math>; (<b>b</b>) <math display="inline"><semantics> <mrow> <msub> <mi>f</mi> <mrow> <mi>R</mi> <mi>F</mi> </mrow> </msub> <mo>=</mo> <mn>28</mn> <mrow> <mo> </mo> <mi>GHz</mi> </mrow> </mrow> </semantics></math>; (<b>c</b>) <math display="inline"><semantics> <mrow> <msub> <mi>f</mi> <mrow> <mi>R</mi> <mi>F</mi> </mrow> </msub> <mo>=</mo> <mn>60</mn> <mrow> <mo> </mo> <mi>GHz</mi> </mrow> </mrow> </semantics></math>; (<b>d</b>) <math display="inline"><semantics> <mrow> <msub> <mi>f</mi> <mrow> <mi>R</mi> <mi>F</mi> </mrow> </msub> <mo>=</mo> <mn>84</mn> <mrow> <mo> </mo> <mi>GHz</mi> </mrow> </mrow> </semantics></math>.</p> Full article ">Figure 12
<p>DIPP-CIR as a function of the optical wavelength for selected radio carrier frequencies and the 20 km optical path created on the basis of the G.652D or G.657A optical fiber: (<b>a</b>) calculation results without selection; (<b>b</b>) 3 dB cut-off optical access ranges; (<b>c</b>) 10 dB cut-off optical access ranges.</p> Full article ">Figure 13
<p>DIPP-CIR as a function of the optical wavelength for selected radio carrier frequencies and the 20 km optical path created on the basis of the G.655D optical fiber: (<b>a</b>) calculation results without selection; (<b>b</b>) 3 dB cut-off optical access ranges; (<b>c</b>) 10 dB cut-off optical access ranges.</p> Full article ">Figure 14
<p>DIPP-CIR as a function of the optical wavelength for selected radio carrier frequencies and the 20 km optical path created on the basis of the G.655E optical fiber: (<b>a</b>) calculation results without selection; (<b>b</b>) 3 dB cut-off optical access ranges; (<b>c</b>) 10 dB cut-off optical access ranges.</p> Full article ">Figure 15
<p>Differential DIPP-CIR as a function of the optical wavelength for selected radio carrier frequencies and the 20 km optical path created on the basis of the G.652D or G.657A optical fiber, and for selected radio channel widths: (<b>a</b>) 50 MHz; (<b>b</b>) 100 MHz; (<b>c</b>) 200 MHz; (<b>d</b>) 400 MHz.</p> Full article ">Figure 16
<p>Differential DIPP-CIR as a function of the optical wavelength for selected radio carrier frequencies and the 20 km optical path created on the basis of the G.655D optical fiber, and for selected radio channel widths: (<b>a</b>) 50 MHz; (<b>b</b>) 100 MHz; (<b>c</b>) 200 MHz; (<b>d</b>) 400 MHz.</p> Full article ">Figure 17
<p>Differential DIPP-CIR as a function of the optical wavelength for selected radio carrier frequencies and the 20 km optical path created on the basis of the G.655E optical fiber, and for selected radio channel widths: (<b>a</b>) 50 MHz; (<b>b</b>) 100 MHz; (<b>c</b>) 200 MHz; (<b>d</b>) 400 MHz.</p> Full article ">Figure 18
<p>Calculation results of relative sideband delay in the optical channel (length of optical path/link is equal to 20 km and the radio channel frequency bandwidth for CP-OFDM modulation format is equal to 100 MHz).</p> Full article ">Figure 19
<p>Optical channel selection algorithm based on DIPP-CIR calculations at two decision thresholds.</p> Full article ">Figure 20
<p>Simulation diagram prepared on the VPIphotonics Design Suite 11.1 platform, presenting three fronthaul parts, configured for downlink transmission: BBU + DU cloud side, fiber-optic path/link and AAU/O-RRH as an antenna side.</p> Full article ">Figure 21
<p>Example optical power spectrum of RFoF signal (<span class="html-italic">f</span><sub>RF</sub> = 60 GHz, G.655D fiber, optical channel no. 26 (195.7 THz), 4096 subcarriers, <span class="html-italic">µ</span> = 2): (<b>a</b>) inserted into the optical single-mode fiber/path, (<b>b</b>) at the photodetector input.</p> Full article ">Figure 22
<p>Example radio power spectrum of RF signal (<span class="html-italic">f</span><sub>RF</sub> = 60 GHz, 4096 subcarriers, <span class="html-italic">µ</span> = 2, 1.835 Gbps, Δ<span class="html-italic">f</span><sub>RF</sub> = 247.7 MHz): (<b>a</b>) input signal; (<b>b</b>) output signal transported over G.655D fiber in optical channel no. 26 (195.7 THz); (<b>c</b>) output signal transported over G.655D fiber in optical channel no. 14 (194.5 THz).</p> Full article ">Figure 23
<p>Example constellations of 256-QAM signal (<span class="html-italic">f</span><sub>RF</sub> = 60 GHz, G.655D fiber, <span class="html-italic">µ</span> = 2): (<b>a</b>) demodulated output signal transported over G.655D fiber in optical channel no. 26 (195.7 THz); (<b>b</b>) demodulated output signal transported over G.655D fiber in optical channel no. 14 (194.5 THz).</p> Full article ">Figure 24
<p>Example radio power spectrum of RF signal (<span class="html-italic">f</span><sub>RF</sub> = 60 GHz, 4096 subcarriers, <span class="html-italic">µ</span> = 3, <span class="html-italic">R<sub>b</sub></span> = 2.752 Gbps, Δ<span class="html-italic">f</span><sub>RF</sub> = 491.5 MHz): (<b>a</b>) input signal; (<b>b</b>) output signal transported over G.655D fiber in optical channel no. 26 (195.7 THz); (<b>c</b>) output signal transported over G.655D fiber in optical channel no. 14 (194.5 THz).</p> Full article ">Figure 25
<p>Example constellations of 64-QAM signal (<span class="html-italic">f</span><sub>RF</sub> = 60 GHz, G.655D fiber, <span class="html-italic">µ</span> = 3): (<b>a</b>) demodulated output signal transported over G.655D fiber in optical channel no. 26 (195.7 THz); (<b>b</b>) demodulated output signal transported over G.655D fiber in optical channel no. 14 (194.5 THz).</p> Full article ">Figure 26
<p>Simulation results presenting SER as a quality parameter of the selected CP-OFDM signals transported over the G.652D/G.657A single-mode fiber fronthaul (<span class="html-italic">L</span> = 20 km): (<b>a</b>) for radio carrier <span class="html-italic">f<sub>RF</sub></span> = 60 GHz and the calculated optical 10 dB subband: 193.1630–194.7505 THz; (<b>b</b>) for radio carrier <span class="html-italic">f<sub>RF</sub></span> = 28 GHz and the calculated optical 10 dB subband: 190.0130–197.2505 THz.</p> Full article ">Figure 27
<p>Simulation results presenting SER as a quality parameter of the selected CP-OFDM signals transported over the G.655D non-zero dispersion-shifted single-mode fiber fronthaul (<span class="html-italic">L</span> = 20 km): (<b>a</b>) for radio carrier <span class="html-italic">f<sub>RF</sub></span> = 60 GHz and the calculated optical 10 dB subband: 194.5130–196.7255 THz; (<b>b</b>) for radio carrier <span class="html-italic">f<sub>RF</sub></span> = 28 GHz and the calculated optical 10 dB subband: 184.4880–191.1068 THz.</p> Full article ">
22 pages, 1442 KiB  
Article
A Comparative Study of Functional Connectivity Measures for Brain Network Analysis in the Context of AD Detection with EEG
by Majd Abazid, Nesma Houmani, Jerome Boudy, Bernadette Dorizzi, Jean Mariani and Kiyoka Kinugawa
Entropy 2021, 23(11), 1553; https://doi.org/10.3390/e23111553 - 22 Nov 2021
Cited by 12 | Viewed by 3495
Abstract
This work addresses brain network analysis considering different clinical severity stages of cognitive dysfunction, based on resting-state electroencephalography (EEG). We use a cohort acquired in real-life clinical conditions, which contains EEG data of subjective cognitive impairment (SCI) patients, mild cognitive impairment (MCI) patients, [...] Read more.
This work addresses brain network analysis considering different clinical severity stages of cognitive dysfunction, based on resting-state electroencephalography (EEG). We use a cohort acquired in real-life clinical conditions, which contains EEG data of subjective cognitive impairment (SCI) patients, mild cognitive impairment (MCI) patients, and Alzheimer’s disease (AD) patients. We propose to exploit an epoch-based entropy measure to quantify the connectivity links in the networks. This entropy measure relies on a refined statistical modeling of EEG signals with Hidden Markov Models, which allow a better estimation of the spatiotemporal characteristics of EEG signals. We also propose to conduct a comparative study by considering three other measures largely used in the literature: phase lag index, coherence, and mutual information. We calculated such measures at different frequency bands and computed different local graph parameters considering different proportional threshold values for a binary network analysis. After applying a feature selection procedure to determine the most relevant features for classification performance with a linear Support Vector Machine algorithm, our study demonstrates the effectiveness of the statistical entropy measure for analyzing the brain network in patients with different stages of cognitive dysfunction. Full article
(This article belongs to the Special Issue Entropy: The Scientific Tool of the 21st Century)
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Figure 1

Figure 1
<p>Position of the 30 electrodes used for EEG recording (marked in color).</p> Full article ">Figure 2
<p>HMM modeling of an EEG signal with <span class="html-italic">N</span> states.</p> Full article ">Figure 3
<p>Illustration of multichannel (<span class="html-italic">D</span> = 2, <span class="html-italic">N</span> = 6) EEG signal modeling with HMM.</p> Full article ">Figure 4
<p>The global ranking of the four connectivity measures in terms of accuracy considering all the graph parameters and class comparisons together.</p> Full article ">Figure 5
<p>The average SVM posterior probability that one person is classified into the positive class for the four connectivity measure and the five graph parameters, when comparing: (<b>a</b>) SCI vs. AD, (<b>b</b>) SCI vs. MCI and (<b>c</b>) AD vs. MCI.</p> Full article ">
15 pages, 895 KiB  
Article
The Downlink Performance for Cell-Free Massive MIMO with Instantaneous CSI in Slowly Time-Varying Channels
by Tongzhou Han and Danfeng Zhao
Entropy 2021, 23(11), 1552; https://doi.org/10.3390/e23111552 - 22 Nov 2021
Cited by 1 | Viewed by 2239
Abstract
In centralized massive multiple-input multiple-output (MIMO) systems, the channel hardening phenomenon can occur, in which the channel behaves as almost fully deterministic as the number of antennas increases. Nevertheless, in a cell-free massive MIMO system, the channel is less deterministic. In this paper, [...] Read more.
In centralized massive multiple-input multiple-output (MIMO) systems, the channel hardening phenomenon can occur, in which the channel behaves as almost fully deterministic as the number of antennas increases. Nevertheless, in a cell-free massive MIMO system, the channel is less deterministic. In this paper, we propose using instantaneous channel state information (CSI) instead of statistical CSI to obtain the power control coefficient in cell-free massive MIMO. Access points (APs) and user equipment (UE) have sufficient time to obtain instantaneous CSI in a slowly time-varying channel environment. We derive the achievable downlink rate under instantaneous CSI for frequency division duplex (FDD) cell-free massive MIMO systems and apply the results to the power control coefficients. For FDD systems, quantized channel coefficients are proposed to reduce feedback overhead. The simulation results show that the spectral efficiency performance when using instantaneous CSI is approximately three times higher than that achieved using statistical CSI. Full article
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Figure 1

Figure 1
<p>In a cell-free massive MIMO, APs and UEs are equipped with one antenna.</p> Full article ">Figure 2
<p>In FDD system, uplink and downlink channels are different, and CSIs are estimated separately.</p> Full article ">Figure 3
<p>The pilot is in a block within a <math display="inline"><semantics> <msub> <mi>τ</mi> <mi>c</mi> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>B</mi> <mi>c</mi> </msub> </semantics></math>, and in different blocks, pilot will be multiplexing.</p> Full article ">Figure 4
<p>Cumulative distribution function of the achievable per-user rates for different schemes with max–min fair power allocations for <math display="inline"><semantics> <mrow> <mi>M</mi> <mo>=</mo> <mn>100</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>K</mi> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math>. <math display="inline"><semantics> <mrow> <msub> <mi>τ</mi> <mi>p</mi> </msub> <mo>=</mo> <mn>100</mn> </mrow> </semantics></math> for the proposed FDD cell-free massive MIMO (cf-mMIMO), and <math display="inline"><semantics> <mrow> <msub> <mi>τ</mi> <mi>p</mi> </msub> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math> for the TDD cell-free massive MIMO, small cell and massive MIMO (mMIMO).</p> Full article ">Figure 5
<p>Cumulative distribution function of the achievable per-user rates for different numbers of quantization bits with max–min fair power allocations for <math display="inline"><semantics> <mrow> <mi>M</mi> <mo>=</mo> <mn>100</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>K</mi> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math>.</p> Full article ">Figure 6
<p>Cumulative distribution function of the achievable per-user rates for different APs with max–min fair power allocations for <math display="inline"><semantics> <mrow> <mi>M</mi> <mo>=</mo> <mn>60</mn> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <mn>150</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>K</mi> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math>. The number of quantization bits is 5 bits.</p> Full article ">Figure 7
<p>Cumulative distribution function of the achievable per-user rates for different UE with max-min fair power allocations for <math display="inline"><semantics> <mrow> <mi>M</mi> <mo>=</mo> <mn>100</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>K</mi> <mo>=</mo> <mn>5</mn> <mo>,</mo> <mn>10</mn> <mo>,</mo> <mn>20</mn> <mo>,</mo> <mn>30</mn> <mo>,</mo> <mn>40</mn> </mrow> </semantics></math>. The number of quantization bits is 5 bits.</p> Full article ">Figure 8
<p>Cumulative distribution function of the achievable per-user rates for different scenarios with max-min fair power allocations for <math display="inline"><semantics> <mrow> <mi>M</mi> <mo>=</mo> <mn>100</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>K</mi> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math>. The number of quantization bits is 1, 5, and 10 bits.</p> Full article ">
29 pages, 7592 KiB  
Article
Performances of Transcritical Power Cycles with CO2-Based Mixtures for the Waste Heat Recovery of ICE
by Jinghang Liu, Aofang Yu, Xinxing Lin, Wen Su and Shaoduan Ou
Entropy 2021, 23(11), 1551; https://doi.org/10.3390/e23111551 - 21 Nov 2021
Cited by 13 | Viewed by 2130
Abstract
In the waste heat recovery of the internal combustion engine (ICE), the transcritical CO2 power cycle still faces the high operation pressure and difficulty in condensation. To overcome these challenges, CO2 is mixed with organic fluids to form zeotropic mixtures. Thus, [...] Read more.
In the waste heat recovery of the internal combustion engine (ICE), the transcritical CO2 power cycle still faces the high operation pressure and difficulty in condensation. To overcome these challenges, CO2 is mixed with organic fluids to form zeotropic mixtures. Thus, in this work, five organic fluids, namely R290, R600a, R600, R601a, and R601, are mixed with CO2. Mixture performance in the waste heat recovery of ICE is evaluated, based on two transcritical power cycles, namely the recuperative cycle and split cycle. The results show that the split cycle always has better performance than the recuperative cycle. Under design conditions, CO2/R290(0.3/0.7) has the best performance in the split cycle. The corresponding net work and cycle efficiency are respectively 21.05 kW and 20.44%. Furthermore, effects of key parameters such as turbine inlet temperature, turbine inlet pressure, and split ratio on the cycle performance are studied. With the increase of turbine inlet temperature, the net works of the recuperative cycle and split cycle firstly increase and then decrease. There exist peak values of net work in both cycles. Meanwhile, the net work of the split cycle firstly increases and then decreases with the increase of the split ratio. Thereafter, with the target of maximizing net work, these key parameters are optimized at different mass fractions of CO2. The optimization results show that CO2/R600 obtains the highest net work of 27.43 kW at the CO2 mass fraction 0.9 in the split cycle. Full article
(This article belongs to the Special Issue Supercritical Fluids for Thermal Energy Applications)
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<p>Systematic diagram of the recuperative cycle.</p> Full article ">Figure 2
<p>Systematic diagram of the split cycle.</p> Full article ">Figure 3
<p>T-s diagram of the recuperative cycle under design conditions.</p> Full article ">Figure 4
<p>T-s diagram of the split cycle under design conditions.</p> Full article ">Figure 5
<p>Thermodynamic calculation routine of the recuperative cycle.</p> Full article ">Figure 6
<p>Thermodynamic calculation routine of the split cycle.</p> Full article ">Figure 7
<p>Flow diagram of GA for the system parameter optimization.</p> Full article ">Figure 8
<p>Critical temperature of mixtures under different mass fractions of CO<sub>2</sub>.</p> Full article ">Figure 9
<p>Critical pressure of mixtures under different mass fractions of CO<sub>2</sub>.</p> Full article ">Figure 10
<p>Temperature glide of mixtures under different mass fractions of CO<sub>2</sub>.</p> Full article ">Figure 11
<p>Effect of turbine inlet temperature on net work: (<b>a</b>) recuperative cycle, (<b>b</b>) split cycle.</p> Full article ">Figure 12
<p>Effect of turbine inlet temperature on cycle efficiency: (<b>a</b>) recuperative cycle, (<b>b</b>) split cycle.</p> Full article ">Figure 13
<p>Effect of turbine inlet temperature on recovery efficiency: (<b>a</b>) recuperative cycle, (<b>b</b>) split cycle.</p> Full article ">Figure 14
<p>Effect of turbine inlet pressure on net work: (<b>a</b>) recuperative cycle, (<b>b</b>) split cycle.</p> Full article ">Figure 15
<p>Effect of turbine inlet pressure on cycle efficiency: (<b>a</b>) recuperative cycle, (<b>b</b>) split cycle.</p> Full article ">Figure 16
<p>Effect of turbine inlet pressure on recovery efficiency: (<b>a</b>) recuperative cycle, (<b>b</b>) split cycle.</p> Full article ">Figure 17
<p>Effect of condensation temperature on net work: (<b>a</b>) recuperative cycle, (<b>b</b>) split cycle.</p> Full article ">Figure 18
<p>Effect of condensation temperature on cycle efficiency: (<b>a</b>) recuperative cycle, (<b>b</b>) split cycle.</p> Full article ">Figure 19
<p>Effect of condensation temperature on recovery efficiency: (<b>a</b>) recuperative cycle, (<b>b</b>) split cycle.</p> Full article ">Figure 20
<p>Effect of split ratio on net work in the split cycle.</p> Full article ">Figure 21
<p>Effect of split ratio on cycle efficiency in the split cycle.</p> Full article ">Figure 22
<p>Effect of split ratio on recovery efficiency in the split cycle.</p> Full article ">Figure 23
<p>Net work of CO<sub>2</sub>-based mixtures at different CO<sub>2</sub> mass fractions: (<b>a</b>) recuperative cycle, (<b>b</b>) split cycle.</p> Full article ">Figure 24
<p>Exergy efficiency of CO<sub>2</sub>-based mixtures at different CO<sub>2</sub> mass fractions: (<b>a</b>) recuperative cycle, (<b>b</b>) split cycle.</p> Full article ">Figure 25
<p>Exergy destruction of CO<sub>2</sub>-based mixtures at different CO<sub>2</sub> mass fractions: (<b>a</b>) recuperative cycle, (<b>b</b>) split cycle.</p> Full article ">Figure 26
<p>Optimal net work of recuperative and split cycles at different CO<sub>2</sub> mass fractions.</p> Full article ">
15 pages, 2565 KiB  
Article
An Improved K-Means Algorithm Based on Evidence Distance
by Ailin Zhu, Zexi Hua, Yu Shi, Yongchuan Tang and Lingwei Miao
Entropy 2021, 23(11), 1550; https://doi.org/10.3390/e23111550 - 21 Nov 2021
Cited by 11 | Viewed by 4544
Abstract
The main influencing factors of the clustering effect of the k-means algorithm are the selection of the initial clustering center and the distance measurement between the sample points. The traditional k-mean algorithm uses Euclidean distance to measure the distance between sample points, thus [...] Read more.
The main influencing factors of the clustering effect of the k-means algorithm are the selection of the initial clustering center and the distance measurement between the sample points. The traditional k-mean algorithm uses Euclidean distance to measure the distance between sample points, thus it suffers from low differentiation of attributes between sample points and is prone to local optimal solutions. For this feature, this paper proposes an improved k-means algorithm based on evidence distance. Firstly, the attribute values of sample points are modelled as the basic probability assignment (BPA) of sample points. Then, the traditional Euclidean distance is replaced by the evidence distance for measuring the distance between sample points, and finally k-means clustering is carried out using UCI data. Experimental comparisons are made with the traditional k-means algorithm, the k-means algorithm based on the aggregation distance parameter, and the Gaussian mixture model. The experimental results show that the improved k-means algorithm based on evidence distance proposed in this paper has a better clustering effect and the convergence of the algorithm is also better. Full article
(This article belongs to the Special Issue Methods in Artificial Intelligence and Information Processing)
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<p>The algorithm flowchart of the improved k-means algorithm.</p> Full article ">Figure 2
<p>Adjusted rand index (iris).</p> Full article ">Figure 3
<p>Silhouette Coefficient (iris).</p> Full article ">Figure 4
<p>Number of iterations (iris).</p> Full article ">Figure 5
<p>Adjusted rand index (validation data set).</p> Full article ">Figure 6
<p>Silhouette coefficient (validation data set).</p> Full article ">Figure 7
<p>Number of iterations (validation data set).</p> Full article ">Figure 8
<p>Data set parameters.</p> Full article ">Figure 9
<p>Adjusted rand index.</p> Full article ">Figure 10
<p>Silhouette coefficient.</p> Full article ">Figure 11
<p>Number of iterations.</p> Full article ">Figure 12
<p>Algorithm runtime.</p> Full article ">
19 pages, 465 KiB  
Article
Age of Information of Parallel Server Systems with Energy Harvesting
by Josu Doncel
Entropy 2021, 23(11), 1549; https://doi.org/10.3390/e23111549 - 21 Nov 2021
Viewed by 1622
Abstract
Motivated by current communication networks in which users can choose different transmission channels to operate and also by the recent growth of renewable energy sources, we study the average Age of Information of a status update system that is formed by two parallel [...] Read more.
Motivated by current communication networks in which users can choose different transmission channels to operate and also by the recent growth of renewable energy sources, we study the average Age of Information of a status update system that is formed by two parallel homogeneous servers and such that there is an energy source that feeds the system following a random process. An update, after getting service, is delivered to the monitor if there is energy in a battery. However, if the battery is empty, the status update is lost. We allow preemption of updates in service and we assume Poisson generation times of status updates and exponential service times. We show that the average Age of Information can be characterized by solving a system with eight linear equations. Then, we show that, when the arrival rate to both servers is large, the average Age of Information is one divided by the sum of the service rates of the servers. We also perform a numerical analysis to compare the performance of our model with that of a single server with energy harvesting and to study in detail the aforementioned convergence result. Full article
(This article belongs to the Special Issue Age of Information: Concept, Metric and Tool for Network Control)
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<p>An example of <math display="inline"><semantics> <mrow> <mi mathvariant="sans-serif">Δ</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math>.</p> Full article ">Figure 2
<p>The energy harvesting model with two parallel data queues and a single energy queue. Energy packets are depicted with gray and data packets with white.</p> Full article ">Figure 3
<p>Average Age of Information of the three systems under comparison when <math display="inline"><semantics> <mi>λ</mi> </semantics></math> changes from <math display="inline"><semantics> <mrow> <mn>0.1</mn> </mrow> </semantics></math> to <math display="inline"><semantics> <msup> <mn>10</mn> <mn>3</mn> </msup> </semantics></math>. <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>1000</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>μ</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>.</p> Full article ">Figure 4
<p>Average Age of Information of the three systems under comparison when <math display="inline"><semantics> <mi>λ</mi> </semantics></math> changes from <math display="inline"><semantics> <mrow> <mn>0.1</mn> </mrow> </semantics></math> to <math display="inline"><semantics> <msup> <mn>10</mn> <mn>3</mn> </msup> </semantics></math>. <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>μ</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>.</p> Full article ">Figure 5
<p>Average Age of Information of the three systems under comparison when <math display="inline"><semantics> <mi>λ</mi> </semantics></math> changes from <math display="inline"><semantics> <mrow> <mn>0.1</mn> </mrow> </semantics></math> to <math display="inline"><semantics> <msup> <mn>10</mn> <mn>3</mn> </msup> </semantics></math>. <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>μ</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>.</p> Full article ">Figure 6
<p>Average Age of Information with respect to <math display="inline"><semantics> <mi>α</mi> </semantics></math> for different values of <math display="inline"><semantics> <mi>λ</mi> </semantics></math>, when <math display="inline"><semantics> <mi>α</mi> </semantics></math> varies from <math display="inline"><semantics> <mrow> <mn>0.1</mn> </mrow> </semantics></math> to <math display="inline"><semantics> <msup> <mn>10</mn> <mn>3</mn> </msup> </semantics></math>. <math display="inline"><semantics> <mrow> <mi>μ</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>p</mi> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>.</p> Full article ">Figure 7
<p>Average Age of Information with respect to <math display="inline"><semantics> <mi>α</mi> </semantics></math> for different values of <math display="inline"><semantics> <mi>λ</mi> </semantics></math>, when <math display="inline"><semantics> <mi>α</mi> </semantics></math> varies from <math display="inline"><semantics> <mrow> <mn>0.1</mn> </mrow> </semantics></math> to <math display="inline"><semantics> <msup> <mn>10</mn> <mn>3</mn> </msup> </semantics></math>. <math display="inline"><semantics> <mrow> <mi>μ</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>p</mi> <mo>=</mo> <mn>0.45</mn> </mrow> </semantics></math>.</p> Full article ">Figure 8
<p>Average Age of Information with respect to <span class="html-italic">p</span> for different values of <math display="inline"><semantics> <mi>λ</mi> </semantics></math> and <math display="inline"><semantics> <mi>μ</mi> </semantics></math>, when <span class="html-italic">p</span> varies from <math display="inline"><semantics> <mrow> <mn>0.01</mn> </mrow> </semantics></math> to <math display="inline"><semantics> <mrow> <mn>0.99</mn> </mrow> </semantics></math>. <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>.</p> Full article ">Figure A1
<p>The SHS Markov chain for the model with two parallel data queues and a battery.</p> Full article ">
9 pages, 269 KiB  
Article
Geometric Analysis of a System with Chemical Interactions
by Dmitry Gromov and Alexander Toikka
Entropy 2021, 23(11), 1548; https://doi.org/10.3390/e23111548 - 21 Nov 2021
Cited by 2 | Viewed by 1753
Abstract
In this paper, we present some initial results aimed at defining a framework for the analysis of thermodynamic systems with additional restrictions imposed on the intensive parameters. Specifically, for the case of chemical reactions, we considered the states of constant affinity that form [...] Read more.
In this paper, we present some initial results aimed at defining a framework for the analysis of thermodynamic systems with additional restrictions imposed on the intensive parameters. Specifically, for the case of chemical reactions, we considered the states of constant affinity that form isoffine submanifolds of the thermodynamic phase space. Wer discuss the problem of extending the previously obtained stability conditions to the considered class of systems. Full article
(This article belongs to the Special Issue Geometric Structure of Thermodynamics: Theory and Applications)
19 pages, 1577 KiB  
Article
Hierarchical Classification of Event-Related Potentials for the Recognition of Gender Differences in the Attention Task
by Karina Maciejewska and Wojciech Froelich
Entropy 2021, 23(11), 1547; https://doi.org/10.3390/e23111547 - 20 Nov 2021
Cited by 3 | Viewed by 2162
Abstract
Research on the functioning of human cognition has been a crucial problem studied for years. Electroencephalography (EEG) classification methods may serve as a precious tool for understanding the temporal dynamics of human brain activity, and the purpose of such an approach is to [...] Read more.
Research on the functioning of human cognition has been a crucial problem studied for years. Electroencephalography (EEG) classification methods may serve as a precious tool for understanding the temporal dynamics of human brain activity, and the purpose of such an approach is to increase the statistical power of the differences between conditions that are too weak to be detected using standard EEG methods. Following that line of research, in this paper, we focus on recognizing gender differences in the functioning of the human brain in the attention task. For that purpose, we gathered, analyzed, and finally classified event-related potentials (ERPs). We propose a hierarchical approach, in which the electrophysiological signal preprocessing is combined with the classification method, enriched with a segmentation step, which creates a full line of electrophysiological signal classification during an attention task. This approach allowed us to detect differences between men and women in the P3 waveform, an ERP component related to attention, which were not observed using standard ERP analysis. The results provide evidence for the high effectiveness of the proposed method, which outperformed a traditional statistical analysis approach. This is a step towards understanding neuronal differences between men’s and women’s brains during cognition, aiming to reduce the misdiagnosis and adverse side effects in underrepresented women groups in health and biomedical research. Full article
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<p>Event-related potentials (ERPs) signal pre-processing workflow: (<b>a</b>) raw electroencephalography (EEG) signal, (<b>b</b>) filtered EEG signal, (<b>c</b>) EEG signal after blink correction, (<b>d</b>) EEG signal after baseline correction, epoching and artifact detection, (<b>e</b>) averaged ERPs to target, and standard stimuli, (<b>f</b>) averaged ERP target-standard difference waveform. Please note the reversed polarity of the signal in this figure.</p> Full article ">Figure 2
<p>Epoched data specification (<b>a</b>) and segment specification (<b>b</b>). Panel (<b>a</b>) represents datasets for each person: from <math display="inline"><semantics> <msub> <mi>t</mi> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> </msub> </semantics></math> to <math display="inline"><semantics> <msub> <mi>t</mi> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> </msub> </semantics></math> timepoints and for d electrodes. Panel (<b>b</b>) represents the exemplary k-th segment.</p> Full article ">Figure 3
<p>Grand averaged target-standard difference ERPs for men (blue) and women (orange) at an exemplification electrode site Pz (midline parietal). The bottom time scale represents the length of the epoch used for data averaging (−100 to 1000 ms with −100–0 ms baseline). The top time scale represents the number of time segments (1–22) corresponding to 50-ms time bins.</p> Full article ">Figure 4
<p>Classification accuracies for each analyzed classifier: (<b>a</b>) Near-Neighbor (kNN), (<b>b</b>) Naive Bayes (NB), (<b>c</b>) Random Forrest (RF), and (<b>d</b>) Support Vector Machine (SVM), plotted for all segments of the epoched ERP signal.</p> Full article ">Figure 5
<p>A distribution of classification rates (<math display="inline"><semantics> <mi>γ</mi> </semantics></math>) for segment 9 (300–350 ms) and segment 16 (600–650 ms). The red horizontal line shows the chance level (0.5).</p> Full article ">Figure 6
<p>Individual ERP difference waveform from subject s4 (red) compared with averaged ERP difference waveforms for men (blue) women (orange) at electrode site P4 (right parietal).</p> Full article ">
48 pages, 5412 KiB  
Article
Winsorization for Robust Bayesian Neural Networks
by Somya Sharma and Snigdhansu Chatterjee
Entropy 2021, 23(11), 1546; https://doi.org/10.3390/e23111546 - 20 Nov 2021
Cited by 12 | Viewed by 3619
Abstract
With the advent of big data and the popularity of black-box deep learning methods, it is imperative to address the robustness of neural networks to noise and outliers. We propose the use of Winsorization to recover model performances when the data may have [...] Read more.
With the advent of big data and the popularity of black-box deep learning methods, it is imperative to address the robustness of neural networks to noise and outliers. We propose the use of Winsorization to recover model performances when the data may have outliers and other aberrant observations. We provide a comparative analysis of several probabilistic artificial intelligence and machine learning techniques for supervised learning case studies. Broadly, Winsorization is a versatile technique for accounting for outliers in data. However, different probabilistic machine learning techniques have different levels of efficiency when used on outlier-prone data, with or without Winsorization. We notice that Gaussian processes are extremely vulnerable to outliers, while deep learning techniques in general are more robust. Full article
(This article belongs to the Special Issue Probabilistic Methods for Deep Learning)
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<p>Crop yield predictions. X-axis shows arbitrary county indices which are sorted by the observed yield in ascending order. Y-axis represents the yield value. Black points are the observed yield. Navy blue line is the mean prediction and light blue points are the predictions in several individual runs.</p> Full article ">Figure 2
<p>Crop yield predictions for Minnesota and Illinois. Sub-plot (<b>f</b>) shows us the legend. Darker blue shade represents lower yield predictions and lighter shade represents higher yield predictions. Methods for better predictive performance (concrete dropout, mixture density network, and exact gp) are able to correctly predict the whole range of observed yield. Flipout and VGP-based Bayesian neural networks are unable to predict well especially in Minnesota counties.</p> Full article ">Figure 3
<p>Crop yield predictions. X-axis shows arbitrary county indices which are sorted by the observed yield in ascending order. Y-axis represents the yield value. Black points are the observed yield. Navy blue line is the mean prediction and epistemic uncertainty estimates is shown in turquoise.</p> Full article ">Figure 4
<p>Winsorization results from 0 to 25 percentile limits on crop yield dataset. Mean Squared Error is shown on the y-axis and the Winsorization limits are shown on the x-axis. Different lines represent different methods: Concrete dropout is shown as blue dashed line, exact GP is shown as green dotted line, mixture density network with 2 components is shown in red solid line, and mixture density network with 4 components is shown in magenta dashed-dotted line. As Winsorization limit increases on the training set, the model performance in terms of mean squared error for the untouched test set is shown in the sub-plots.</p> Full article ">Figure 5
<p>Winsorization results from 0 to 25 percentile limits on California housing dataset. Mean Squared Error is shown on the y-axis and the Winsorization limits are shown on the x-axis. Different lines represent different methods: Concrete dropout is shown as blue dashed line, exact GP is shown as green dotted line, mixture density network with 2 components is shown in red solid line, and mixture density network with 4 components is shown in magenta dashed-dotted line. As Winsorization limit increases in the training dataset, the model performance in terms of mean squared error for the untouched test set is shown in the picture.</p> Full article ">Figure 6
<p>Winsorization results from 0 to 25 percentile limits on Mauna dataset. Mean Squared Error is shown on the y-axis and the Winsorization limits are shown on the x-axis. Different lines represent different methods: Concrete dropout is shown as blue dashed line, exact GP is shown as green dotted line, mixture density network with 2 components is shown in red solid line, and mixture density network with 4 components is shown in magenta dashed-dotted line. As Winsorization limit increases in the training set, the model performance in terms of mean squared error in the untouched test set is shown in the sub-plots.</p> Full article ">Figure 7
<p>Winsorization results from 0 to 25 percentile limits on forest fires dataset. Mean Squared Error is shown on the y-axis and the Winsorization limits are shown on the x-axis. Different lines represent different methods: Concrete dropout is shown as blue dashed line, exact GP is shown as green dotted line, mixture density network with 2 components is shown in red solid line, and mixture density network with 4 components is shown in magenta dashed-dotted line. As Winsorization limit increases in the training set, the model performance in terms of Mean Squared Error in the untouched test set is show in the sub-plots.</p> Full article ">Figure 7 Cont.
<p>Winsorization results from 0 to 25 percentile limits on forest fires dataset. Mean Squared Error is shown on the y-axis and the Winsorization limits are shown on the x-axis. Different lines represent different methods: Concrete dropout is shown as blue dashed line, exact GP is shown as green dotted line, mixture density network with 2 components is shown in red solid line, and mixture density network with 4 components is shown in magenta dashed-dotted line. As Winsorization limit increases in the training set, the model performance in terms of Mean Squared Error in the untouched test set is show in the sub-plots.</p> Full article ">Figure 8
<p>Winsorization results from 0 to 25 percentile limits on GDSC dataset. Mean Squared Error is shown on the y-axis and the Winsorization limits are shown on the x-axis. Different lines represent different methods: Concrete dropout is shown as blue dashed line, exact GP is shown as green dotted line, mixture density network with 2 components is shown in red solid line, and mixture density network with 4 components is shown in magenta dashed-dotted line. As Winsorization limit increases in the training set, the model performance in terms of mean squared error in the untouched test set is shown in the sub-plots.</p> Full article ">Figure 9
<p>Relative efficiencies (REs) of Winsorized MSE with non-Winsorized MSE for different noise sites. The black dashed line represents an RE of one. RE values greater than one represent improvement in performance with Winsorized training and validation data and vice versa. (<b>a</b>) RE for noise free case in crop yield data. (<b>b</b>) RE for noise in target in crop yield data. (<b>c</b>) RE for noise in features in crop yield data. (<b>d</b>) RE for noise in target and features in crop yield data. (<b>e</b>) RE for noise free case in California data. (<b>f</b>) RE for noise in target in California data. (<b>g</b>) RE for noise in features in California data. (<b>h</b>) RE for noise in target and features in California data. (<b>i</b>) RE for noise free case in GDSC data. (<b>j</b>) RE for noise in target in GDSC data. (<b>k</b>) RE for noise in features in GDSC data. (<b>l</b>) RE for noise in target and features in GDSC data. (<b>m</b>) RE for noise free case in forest fires data. (<b>n</b>) RE for noise in target in forest fires data. (<b>o</b>) RE for noise in features in forest fires data. (<b>p</b>) RE for noise in target and features in forest fires data. (<b>q</b>) RE for noise free case in Mauna data. (<b>r</b>) RE for noise in target in Mauna data. (<b>s</b>) RE for noise in features in Mauna data. (<b>t</b>) RE for noise in target and features in Mauna data.</p> Full article ">Figure 9 Cont.
<p>Relative efficiencies (REs) of Winsorized MSE with non-Winsorized MSE for different noise sites. The black dashed line represents an RE of one. RE values greater than one represent improvement in performance with Winsorized training and validation data and vice versa. (<b>a</b>) RE for noise free case in crop yield data. (<b>b</b>) RE for noise in target in crop yield data. (<b>c</b>) RE for noise in features in crop yield data. (<b>d</b>) RE for noise in target and features in crop yield data. (<b>e</b>) RE for noise free case in California data. (<b>f</b>) RE for noise in target in California data. (<b>g</b>) RE for noise in features in California data. (<b>h</b>) RE for noise in target and features in California data. (<b>i</b>) RE for noise free case in GDSC data. (<b>j</b>) RE for noise in target in GDSC data. (<b>k</b>) RE for noise in features in GDSC data. (<b>l</b>) RE for noise in target and features in GDSC data. (<b>m</b>) RE for noise free case in forest fires data. (<b>n</b>) RE for noise in target in forest fires data. (<b>o</b>) RE for noise in features in forest fires data. (<b>p</b>) RE for noise in target and features in forest fires data. (<b>q</b>) RE for noise free case in Mauna data. (<b>r</b>) RE for noise in target in Mauna data. (<b>s</b>) RE for noise in features in Mauna data. (<b>t</b>) RE for noise in target and features in Mauna data.</p> Full article ">Figure 10
<p>Crop yield dataset result: Relative Efficiencies (RE) comparing performance of Winsorized results with standard Cauchy noise in the features with original performance on noise free data without Winsorization. Black dashed line represents RE of one. REs above one represent improvement in performance due to Winsorization on contaminated data.</p> Full article ">Figure 11
<p>Summarizing Winsorization results: The subplots show average of evaluation metrics over all methodologies used for cases when artificial perturbation is introduced in the datasets. The MSE and Median AE plot legends also convey the mean and standard error of the evaluation metric in the respective sub-plots for each dataset.</p> Full article ">Figure 12
<p>Apart from predictive performance in terms of accurate prediction, the precision can also be compared in terms of uncertainty estimates. On the y-axis, we measure the average standard error in prediction. (<b>a</b>) Uncertainty estimate for noise free case in crop yield data. (<b>b</b>) Uncertainty estimate for noise in target in crop yield data. (<b>c</b>) Uncertainty estimate for noise in features in crop yield data. (<b>d</b>) Uncertainty estimate for noise in target and features in crop yield data. (<b>e</b>) Uncertainty estimate for noise free case in California data. (<b>f</b>) Uncertainty estimate for noise in target in California data. (<b>g</b>) Uncertainty estimate for noise in features in California data. (<b>h</b>) Uncertainty estimate for noise in target and features in California data. (<b>i</b>) Uncertainty estimate for noise free case in GDSC data. (<b>j</b>) RE for noise in target in GDSC data. (<b>k</b>) Uncertainty estimate for noise in features in GDSC data. (<b>l</b>) Uncertainty estimate for noise in target and features in GDSC data. (<b>m</b>) Uncertainty estimate for noise free case in forest fires data. (<b>n</b>) Uncertainty estimate for noise in target in forest fires data. (<b>o</b>) Uncertainty estimate for noise in features in forest fires data. (<b>p</b>) Uncertainty estimate for noise in target and features in forest fires data. (<b>q</b>) Uncertainty estimate for noise free case in Mauna data. (<b>r</b>) Uncertainty estimate for noise in target in Mauna data. (<b>s</b>) Uncertainty estimate for noise in features in Mauna data. (<b>t</b>) Uncertainty estimate for noise in target and features in Mauna data.</p> Full article ">Figure 12 Cont.
<p>Apart from predictive performance in terms of accurate prediction, the precision can also be compared in terms of uncertainty estimates. On the y-axis, we measure the average standard error in prediction. (<b>a</b>) Uncertainty estimate for noise free case in crop yield data. (<b>b</b>) Uncertainty estimate for noise in target in crop yield data. (<b>c</b>) Uncertainty estimate for noise in features in crop yield data. (<b>d</b>) Uncertainty estimate for noise in target and features in crop yield data. (<b>e</b>) Uncertainty estimate for noise free case in California data. (<b>f</b>) Uncertainty estimate for noise in target in California data. (<b>g</b>) Uncertainty estimate for noise in features in California data. (<b>h</b>) Uncertainty estimate for noise in target and features in California data. (<b>i</b>) Uncertainty estimate for noise free case in GDSC data. (<b>j</b>) RE for noise in target in GDSC data. (<b>k</b>) Uncertainty estimate for noise in features in GDSC data. (<b>l</b>) Uncertainty estimate for noise in target and features in GDSC data. (<b>m</b>) Uncertainty estimate for noise free case in forest fires data. (<b>n</b>) Uncertainty estimate for noise in target in forest fires data. (<b>o</b>) Uncertainty estimate for noise in features in forest fires data. (<b>p</b>) Uncertainty estimate for noise in target and features in forest fires data. (<b>q</b>) Uncertainty estimate for noise free case in Mauna data. (<b>r</b>) Uncertainty estimate for noise in target in Mauna data. (<b>s</b>) Uncertainty estimate for noise in features in Mauna data. (<b>t</b>) Uncertainty estimate for noise in target and features in Mauna data.</p> Full article ">Figure A1
<p>Winsorization results from 0 to 25 percentile limits on Mauna dataset. Mean Squared Error is shown on the y-axis and the Winsorization limits are shown on the x-axis. Different lines represent different methods: Concrete dropout is shown as blue dashed line, exact GP is shown as green dotted line, mixture density network with 2 components is shown in red solid line, and mixture density network with 4 components is shown in magenta dashed-dotted line. As Winsorization limit increases in the training set, the model performance in terms of Mean Squared Error in the untouched test set is shown in the sub-plots.</p> Full article ">Figure A2
<p>Crop yield predictions. X-axis shows arbitrary county indices which are sorted by the observed yield in ascending order. Y-axis represents the yield value. Black points are the observed yield. Navy blue line is the mean prediction, and aleatoric uncertainty estimates are shown in turquoise.</p> Full article ">Figure A2 Cont.
<p>Crop yield predictions. X-axis shows arbitrary county indices which are sorted by the observed yield in ascending order. Y-axis represents the yield value. Black points are the observed yield. Navy blue line is the mean prediction, and aleatoric uncertainty estimates are shown in turquoise.</p> Full article ">Figure A3
<p>Crop yield predictions. X-axis shows arbitrary county indices which are sorted by the observed yield in ascending order. Y-axis represents the yield value. Black points are the observed yield. Navy blue line is the mean prediction and point predictions are shown in turquoise.</p> Full article ">
15 pages, 761 KiB  
Article
Conditional Deep Gaussian Processes: Multi-Fidelity Kernel Learning
by Chi-Ken Lu and Patrick Shafto
Entropy 2021, 23(11), 1545; https://doi.org/10.3390/e23111545 - 20 Nov 2021
Cited by 3 | Viewed by 3041
Abstract
Deep Gaussian Processes (DGPs) were proposed as an expressive Bayesian model capable of a mathematically grounded estimation of uncertainty. The expressivity of DPGs results from not only the compositional character but the distribution propagation within the hierarchy. Recently, it was pointed out that [...] Read more.
Deep Gaussian Processes (DGPs) were proposed as an expressive Bayesian model capable of a mathematically grounded estimation of uncertainty. The expressivity of DPGs results from not only the compositional character but the distribution propagation within the hierarchy. Recently, it was pointed out that the hierarchical structure of DGP well suited modeling the multi-fidelity regression, in which one is provided sparse observations with high precision and plenty of low fidelity observations. We propose the conditional DGP model in which the latent GPs are directly supported by the fixed lower fidelity data. Then the moment matching method is applied to approximate the marginal prior of conditional DGP with a GP. The obtained effective kernels are implicit functions of the lower-fidelity data, manifesting the expressivity contributed by distribution propagation within the hierarchy. The hyperparameters are learned via optimizing the approximate marginal likelihood. Experiments with synthetic and high dimensional data show comparable performance against other multi-fidelity regression methods, variational inference, and multi-output GP. We conclude that, with the low fidelity data and the hierarchical DGP structure, the effective kernel encodes the inductive bias for true function allowing the compositional freedom. Full article
(This article belongs to the Special Issue Probabilistic Methods for Deep Learning)
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Figure 1

Figure 1
<p>Sampling random functions from the approximate marginal prior <math display="inline"><semantics> <mrow> <mi>q</mi> <mo>(</mo> <mi mathvariant="bold">f</mi> <mo>)</mo> </mrow> </semantics></math> which carries the effective kernel in Equation (<a href="#FD12-entropy-23-01545" class="html-disp-formula">12</a>). The low fidelity data <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="bold">X</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi mathvariant="bold">y</mi> <mn>1</mn> </msub> </mrow> </semantics></math>, marked by the cross symbols, and the low fidelity function <math display="inline"><semantics> <mrow> <msub> <mi>f</mi> <mn>1</mn> </msub> <mrow> <mo>|</mo> </mrow> <msub> <mi mathvariant="bold">X</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi mathvariant="bold">y</mi> <mn>1</mn> </msub> </mrow> </semantics></math> and the uncertainty are shown in the top (noiseless) and the third (noisy) rows. Top row: the uncertainty in <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="bold">X</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi mathvariant="bold">y</mi> <mn>1</mn> </msub> </mrow> </semantics></math> is negligible so <math display="inline"><semantics> <msub> <mi>f</mi> <mn>1</mn> </msub> </semantics></math> is nearly a deterministic function, so the effective kernels are basically kernels with warped input. The corresponding samples from <span class="html-italic">q</span> are shown in the second row. Third row: the noise in <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="bold">X</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi mathvariant="bold">y</mi> <mn>1</mn> </msub> </mrow> </semantics></math> generates the samples in bottom row which carry additional high-frequency signals due to the non-stationary <math display="inline"><semantics> <msup> <mi>δ</mi> <mn>2</mn> </msup> </semantics></math> in Equation (<a href="#FD12-entropy-23-01545" class="html-disp-formula">12</a>).</p> Full article ">Figure 2
<p>Multi-fidelity regression with 30 observations (not shown) of low fidelity <math display="inline"><semantics> <mrow> <msub> <mi>f</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mo form="prefix">sin</mo> <mn>8</mn> <mi>π</mi> <mi>x</mi> </mrow> </semantics></math> and 10 observations (red dots) from the target function, <math display="inline"><semantics> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mi>x</mi> <mo>−</mo> <msqrt> <mn>2</mn> </msqrt> <mo>)</mo> </mrow> <msubsup> <mi>f</mi> <mn>1</mn> <mn>2</mn> </msubsup> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> (shown in red dashed line). Only the target prediction (solid dark) and associated uncertainty (shaded) are shown. Top row: (<b>a</b>) AR1, (<b>b</b>) LCM, (<b>c</b>) NARGP. Bottom row: (<b>d</b>) DEEP-MF, (<b>e</b>) MF-DGP, (<b>f</b>) our model with SE[SE] kernel.</p> Full article ">Figure 3
<p>Multi-fidelity regression on the low-level true function, <math display="inline"><semantics> <mrow> <mi>h</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> <mo>=</mo> <mo form="prefix">cos</mo> <mn>15</mn> <mi>x</mi> </mrow> </semantics></math>, with 30 observations and high-level one, <math display="inline"><semantics> <mrow> <mi>f</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> <mo>=</mo> <mi>x</mi> <mo form="prefix">exp</mo> <mo>[</mo> <mi>h</mi> <mo>(</mo> <mn>2</mn> <mi>x</mi> <mo>−</mo> <mn>0.2</mn> <mo>)</mo> <mo>]</mo> <mo>−</mo> <mn>1</mn> </mrow> </semantics></math>, with 15 observations. Top row: (<b>a</b>) AR1, (<b>b</b>) LCM, and (<b>c</b>) NARGP. Bottom row: (<b>d</b>) DEEP-MF, (<b>e</b>) MF-DGP, and (<b>f</b>) Our method with SE[SE] kernel.</p> Full article ">Figure 4
<p>Demonstration of compositional freedom and effects of uncertainty in low fidelity function <math display="inline"><semantics> <msub> <mi>f</mi> <mn>1</mn> </msub> </semantics></math> on the target function inference. Given the same high fidelity observations of target function, four different sets of observations of <math display="inline"><semantics> <mrow> <msub> <mi>f</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>x</mi> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>f</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mo form="prefix">tanh</mo> <mi>x</mi> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>f</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mo form="prefix">sin</mo> <mn>4</mn> <mi>π</mi> <mi>x</mi> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <msub> <mi>f</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mo form="prefix">sin</mo> <mn>8</mn> <mi>π</mi> <mi>x</mi> </mrow> </semantics></math> are employed as low fidelity data in inferring the target function. In panel (<b>a</b>), the low fidelity data are noiseless observations of the four functions. The true target function is partially outside the model confidence for the first two cases. In panel (<b>b</b>), the low fidelity data are noisy observations of the same four functions. Now the first three cases result in the inferred function outside the model confidence. The effect of uncertainty in low fidelity is most dramatic when comparing the third subplots in (<b>a</b>,<b>b</b>).</p> Full article ">Figure 5
<p>Denoising regression with 30 high-noise and 15 low-noise observations from the target function <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>=</mo> <mrow> <mo>(</mo> <mi>x</mi> <mo>−</mo> <msqrt> <mn>2</mn> </msqrt> <mo>)</mo> </mrow> <msup> <mo form="prefix">sin</mo> <mn>2</mn> </msup> <mn>8</mn> <mi>π</mi> <mi>x</mi> </mrow> </semantics></math> (red dashed line). The uncertainty is reduced in the GP learning with the SE[SE] kernels.</p> Full article ">Figure A1
<p>Comparison between the joint learning (<b>left</b>) and the sequential learning with Algorithm 1 (<b>right</b>). The same 10 training data are shown by the red dots. The joint learning algorithm results in a log marginal likelihood <math display="inline"><semantics> <mrow> <mn>1.65</mn> </mrow> </semantics></math> while the alternative one <math display="inline"><semantics> <mrow> <mn>2.64</mn> </mrow> </semantics></math>. The hyperparameters are <math display="inline"><semantics> <mrow> <mo>{</mo> <msub> <mi>σ</mi> <mrow> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mn>3.3</mn> <mo>,</mo> <mn>1.24</mn> <mo>)</mo> </mrow> <mo>,</mo> <msub> <mo>ℓ</mo> <mrow> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mn>0.12</mn> <mo>,</mo> <mn>1.40</mn> <mo>)</mo> </mrow> <mo>}</mo> </mrow> </semantics></math> (<b>left</b>) and <math display="inline"><semantics> <mrow> <mo>{</mo> <msub> <mi>σ</mi> <mrow> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mn>1.22</mn> <mo>,</mo> <mn>1.65</mn> <mo>)</mo> </mrow> <mo>,</mo> <msub> <mo>ℓ</mo> <mrow> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mn>0.08</mn> <mo>,</mo> <mn>0.98</mn> <mo>)</mo> </mrow> <mo>}</mo> </mrow> </semantics></math> (<b>right</b>).</p> Full article ">
37 pages, 1335 KiB  
Article
Intrinsic Entropy of Squeezed Quantum Fields and Nonequilibrium Quantum Dynamics of Cosmological Perturbations
by Jen-Tsung Hsiang and Bei-Lok Hu
Entropy 2021, 23(11), 1544; https://doi.org/10.3390/e23111544 - 20 Nov 2021
Cited by 11 | Viewed by 2154
Abstract
Density contrasts in the universe are governed by scalar cosmological perturbations which, when expressed in terms of gauge-invariant variables, contain a classical component from scalar metric perturbations and a quantum component from inflaton field fluctuations. It has long been known that the effect [...] Read more.
Density contrasts in the universe are governed by scalar cosmological perturbations which, when expressed in terms of gauge-invariant variables, contain a classical component from scalar metric perturbations and a quantum component from inflaton field fluctuations. It has long been known that the effect of cosmological expansion on a quantum field amounts to squeezing. Thus, the entropy of cosmological perturbations can be studied by treating them in the framework of squeezed quantum systems. Entropy of a free quantum field is a seemingly simple yet subtle issue. In this paper, different from previous treatments, we tackle this issue with a fully developed nonequilibrium quantum field theory formalism for such systems. We compute the covariance matrix elements of the parametric quantum field and solve for the evolution of the density matrix elements and the Wigner functions, and, from them, derive the von Neumann entropy. We then show explicitly why the entropy for the squeezed yet closed system is zero, but is proportional to the particle number produced upon coarse-graining out the correlation between the particle pairs. We also construct the bridge between our quantum field-theoretic results and those using the probability distribution of classical stochastic fields by earlier authors, preserving some important quantum properties, such as entanglement and coherence, of the quantum field. Full article
(This article belongs to the Special Issue Entropy Measures and Applications in Astrophysics)
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Figure 1

Figure 1
<p>The time variations of the particle number <inline-formula> <mml:math id="mm520" display="block"> <mml:semantics> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:semantics> </mml:math> </inline-formula> in (<bold>a</bold>), and coherence <inline-formula> <mml:math id="mm521" display="block"> <mml:semantics> <mml:mrow> <mml:mi>C</mml:mi> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:semantics> </mml:math> </inline-formula> in (<bold>b</bold>) for the parametric oscillator having the frequency modulation given by (<xref ref-type="disp-formula" rid="FD123-entropy-23-01544">123</xref>). We choose <inline-formula> <mml:math id="mm522" display="block"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="fraktur">R</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:semantics> </mml:math> </inline-formula> and <inline-formula> <mml:math id="mm523" display="block"> <mml:semantics> <mml:mrow> <mml:mi>ϖ</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0.5</mml:mn></mml:mrow> </mml:semantics> </mml:math> </inline-formula>. In (<bold>b</bold>) the blue dashed curve represents the real part of <inline-formula> <mml:math id="mm524" display="block"> <mml:semantics> <mml:mrow> <mml:mi>C</mml:mi> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:semantics> </mml:math> </inline-formula> and the orange solid curve denotes the imaginary part.</p> Full article ">Figure 2
<p>The time variations of the particle number <inline-formula> <mml:math id="mm525" display="block"> <mml:semantics> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:semantics> </mml:math> </inline-formula> in (<bold>a</bold>) and coherence <inline-formula> <mml:math id="mm526" display="block"> <mml:semantics> <mml:mrow> <mml:mi>C</mml:mi> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:semantics> </mml:math> </inline-formula> in (<bold>b</bold>) for the parametric oscillator having the frequency modulation given by (<xref ref-type="disp-formula" rid="FD124-entropy-23-01544">124</xref>). We choose <inline-formula> <mml:math id="mm527" display="block"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="fraktur">R</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:semantics> </mml:math> </inline-formula> and <inline-formula> <mml:math id="mm528" display="block"> <mml:semantics> <mml:mrow> <mml:mi>ϖ</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0.1</mml:mn></mml:mrow> </mml:semantics> </mml:math> </inline-formula>. In (<bold>b</bold>) the blue dashed curve represents the real part of <inline-formula> <mml:math id="mm529" display="block"> <mml:semantics> <mml:mrow> <mml:mi>C</mml:mi> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:semantics> </mml:math> </inline-formula> and the orange solid curve denotes the imaginary part.</p> Full article ">Figure 3
<p>The time variations of the particle number <inline-formula> <mml:math id="mm530" display="block"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>N</mml:mi> <mml:mi>k</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:semantics> </mml:math> </inline-formula> in (<bold>a</bold>) and coherence <inline-formula> <mml:math id="mm531" display="block"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>k</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:semantics> </mml:math> </inline-formula> in (<bold>b</bold>) for the Mukhanov-Sasaki variable <italic>u</italic> in the mode <inline-formula> <mml:math id="mm532" display="block"> <mml:semantics> <mml:mi mathvariant="bold-italic">k</mml:mi> </mml:semantics> </mml:math> </inline-formula> in de Sitter spacetime. Here, the contribution of the slow-roll parameters in (<xref ref-type="disp-formula" rid="FD17-entropy-23-01544">17</xref>) is ignored. The particle number <inline-formula> <mml:math id="mm533" display="block"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>N</mml:mi> <mml:mi>k</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:semantics> </mml:math> </inline-formula> monotonically increase with the time, but the coherence (here we only show the real part of <inline-formula> <mml:math id="mm534" display="block"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>k</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:semantics> </mml:math> </inline-formula>) oscillates with a larger amplitude at a rate depending on the mode, until the perturbations cross the horizon at time <inline-formula> <mml:math id="mm535" display="block"> <mml:semantics> <mml:mrow> <mml:mi>η</mml:mi> <mml:mo>∼</mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mi>k</mml:mi> </mml:mfrac> </mml:mrow> </mml:semantics> </mml:math> </inline-formula>, after which it becomes non-oscillating. We choose <inline-formula> <mml:math id="mm536" display="block"> <mml:semantics> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:semantics> </mml:math> </inline-formula> for the blue solid curve and <inline-formula> <mml:math id="mm537" display="block"> <mml:semantics> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>=</mml:mo> <mml:mn>5</mml:mn> </mml:mrow> </mml:semantics> </mml:math> </inline-formula> for the orange dashed curve.</p> Full article ">
8 pages, 249 KiB  
Article
Assumption-Free Derivation of the Bell-Type Criteria of Contextuality/Nonlocality
by Ehtibar N. Dzhafarov
Entropy 2021, 23(11), 1543; https://doi.org/10.3390/e23111543 - 19 Nov 2021
Cited by 7 | Viewed by 2026
Abstract
Bell-type criteria of contextuality/nonlocality can be derived without any falsifiable assumptions, such as context-independent mapping (or local causality), free choice, or no-fine-tuning. This is achieved by deriving Bell-type criteria for inconsistently connected systems (i.e., those with disturbance/signaling), based on the generalized definition of [...] Read more.
Bell-type criteria of contextuality/nonlocality can be derived without any falsifiable assumptions, such as context-independent mapping (or local causality), free choice, or no-fine-tuning. This is achieved by deriving Bell-type criteria for inconsistently connected systems (i.e., those with disturbance/signaling), based on the generalized definition of contextuality in the contextuality-by-default approach, and then specializing these criteria to consistently connected systems. Full article
39 pages, 1720 KiB  
Review
Role-Aware Information Spread in Online Social Networks
by Alon Bartal and Kathleen M. Jagodnik
Entropy 2021, 23(11), 1542; https://doi.org/10.3390/e23111542 - 19 Nov 2021
Cited by 7 | Viewed by 4813
Abstract
Understanding the complex process of information spread in online social networks (OSNs) enables the efficient maximization/minimization of the spread of useful/harmful information. Users assume various roles based on their behaviors while engaging with information in these OSNs. Recent reviews on information spread in [...] Read more.
Understanding the complex process of information spread in online social networks (OSNs) enables the efficient maximization/minimization of the spread of useful/harmful information. Users assume various roles based on their behaviors while engaging with information in these OSNs. Recent reviews on information spread in OSNs have focused on algorithms and challenges for modeling the local node-to-node cascading paths of viral information. However, they neglected to analyze non-viral information with low reach size that can also spread globally beyond OSN edges (links) via non-neighbors through, for example, pushed information via content recommendation algorithms. Previous reviews have also not fully considered user roles in the spread of information. To address these gaps, we: (i) provide a comprehensive survey of the latest studies on role-aware information spread in OSNs, also addressing the different temporal spreading patterns of viral and non-viral information; (ii) survey modeling approaches that consider structural, non-structural, and hybrid features, and provide a taxonomy of these approaches; (iii) review software platforms for the analysis and visualization of role-aware information spread in OSNs; and (iv) describe how information spread models enable useful applications in OSNs such as detecting influential users. We conclude by highlighting future research directions for studying information spread in OSNs, accounting for dynamic user roles. Full article
(This article belongs to the Special Issue Role-Aware Analysis of Complex Networks)
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Figure 1

Figure 1
<p>Illustration of two types of networks with the same set of nodes and different edges (links): (<b>a</b>) a social network in which dashed edges represent social relationships among users (e.g., Facebook friendships), and (<b>b</b>) a directed interaction network laid over the social network, in which solid edges represent user interactions (e.g., user <math display="inline"><semantics> <msub> <mi>v</mi> <mn>4</mn> </msub> </semantics></math> retweeted a message originated by <math display="inline"><semantics> <msub> <mi>v</mi> <mn>3</mn> </msub> </semantics></math> and another message originated by <math display="inline"><semantics> <msub> <mi>v</mi> <mn>5</mn> </msub> </semantics></math>).</p> Full article ">Figure 2
<p>High-level taxonomy of analysis strategies for information spread in online social networks. Grey panels group research approaches for role-aware analysis (left panel), and the associated models (right panel). Refer to <a href="#sec4-entropy-23-01542" class="html-sec">Section 4</a> for a list of studies exemplifying structural models, non-structural models, hybrid models, and models employing external information.</p> Full article ">Figure 3
<p>An illustration of simple and complex contagion in a social network. (<b>a</b>) Simple contagion: node <span class="html-italic">A</span> was infected by a disease after exposure to a single infected node (colored in gray). (<b>b</b>) Complex contagion: node <span class="html-italic">A</span> adopted a product (a smartphone) after being exposed by three nodes (colored in gray) who adopted the product.</p> Full article ">Figure 4
<p>An illustration of the models Susceptible-Infected (SI) [<a href="#B187-entropy-23-01542" class="html-bibr">187</a>], Susceptible-Infected-Susceptible (SIS) [<a href="#B187-entropy-23-01542" class="html-bibr">187</a>], Susceptible-Infected-Recovered (SIR) [<a href="#B188-entropy-23-01542" class="html-bibr">188</a>], Susceptible-Infected-Recovered-for-Susceptible (SIRS) where immunity lasts for only a short period of time [<a href="#B187-entropy-23-01542" class="html-bibr">187</a>], Susceptible-Exposed-Infected-Recovered (SEIR) [<a href="#B189-entropy-23-01542" class="html-bibr">189</a>], the Linear Threshold model (LTM) for influence maximization [<a href="#B37-entropy-23-01542" class="html-bibr">37</a>], and the Independent Cascade model (IC) [<a href="#B190-entropy-23-01542" class="html-bibr">190</a>]. In the LTM, a node is exposed to its neighbors, and if the number of infected neighbors exceeds a threshold, the exposed node is infected. In the IC model, each infected node stays active during one time step only and tries to infect its susceptible neighbors with a certain probability. The attempts are independent random events. A susceptible node that was infected will attempt to infect its neighbors at the next time step.</p> Full article ">Figure 5
<p>A directed social network <math display="inline"><semantics> <mrow> <mi>G</mi> <mo>(</mo> <mi>V</mi> <mo>,</mo> <mi>E</mi> <mo>)</mo> </mrow> </semantics></math> (e.g., Twitter Following–Followee relationships) with a directed interaction network <math display="inline"><semantics> <mrow> <msub> <mi>G</mi> <mrow> <mi>T</mi> <mi>w</mi> </mrow> </msub> <mrow> <mo>(</mo> <msub> <mi>V</mi> <mrow> <mi>T</mi> <mi>w</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>E</mi> <mrow> <mi>T</mi> <mi>w</mi> </mrow> </msub> <mo>)</mo> </mrow> </mrow> </semantics></math> (e.g., retweets). The interaction network at time <math display="inline"><semantics> <msub> <mi>t</mi> <mn>1</mn> </msub> </semantics></math> contains the set of nodes <math display="inline"><semantics> <mrow> <msub> <mi>V</mi> <mrow> <mi>T</mi> <mi>w</mi> </mrow> </msub> <mo>=</mo> <mrow> <mo>{</mo> <msub> <mi>v</mi> <mn>0</mn> </msub> <mo>,</mo> <msub> <mi>v</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>v</mi> <mn>2</mn> </msub> <mo>}</mo> </mrow> </mrow> </semantics></math>, and the social network contains the set of nodes <math display="inline"><semantics> <mrow> <mi>V</mi> <mo>=</mo> <mo>{</mo> <msub> <mi>v</mi> <mn>0</mn> </msub> <mo>,</mo> <msub> <mi>v</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>v</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mi>v</mi> <mn>3</mn> </msub> </mrow> </semantics></math>}. In <span class="html-italic">G</span>, node <math display="inline"><semantics> <msub> <mi>v</mi> <mn>2</mn> </msub> </semantics></math> follows node <math display="inline"><semantics> <msub> <mi>v</mi> <mn>0</mn> </msub> </semantics></math>, indicated by a dashed edge (link). Thus, <math display="inline"><semantics> <msub> <mi>v</mi> <mn>0</mn> </msub> </semantics></math> exposes <math display="inline"><semantics> <msub> <mi>v</mi> <mn>2</mn> </msub> </semantics></math> to information. <math display="inline"><semantics> <msub> <mi>v</mi> <mn>2</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>v</mi> <mn>1</mn> </msub> </semantics></math> retweeted <math display="inline"><semantics> <msub> <mi>v</mi> <mn>0</mn> </msub> </semantics></math>’s original tweet at time <math display="inline"><semantics> <mrow> <mi>t</mi> <mn>1</mn> </mrow> </semantics></math>, indicated by two solid edges.</p> Full article ">Figure 6
<p>An illustration of local and global topic influence. Local and global topic influence of a set of tweets on the same topic <math display="inline"><semantics> <mrow> <mi>W</mi> <mo>=</mo> <mo>{</mo> <msub> <mi>w</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>w</mi> <mn>2</mn> </msub> <mo>}</mo> </mrow> </semantics></math> that were posted by user <math display="inline"><semantics> <msub> <mi>v</mi> <mn>0</mn> </msub> </semantics></math> at times <math display="inline"><semantics> <msub> <mi>t</mi> <mn>0</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>t</mi> <mn>1</mn> </msub> </semantics></math> respectively. (<b>a</b>) Local topic influence: user <math display="inline"><semantics> <msub> <mi>v</mi> <mn>1</mn> </msub> </semantics></math> who follows user <math display="inline"><semantics> <msub> <mi>v</mi> <mn>0</mn> </msub> </semantics></math> retweets <math display="inline"><semantics> <msub> <mi>w</mi> <mn>1</mn> </msub> </semantics></math> at time <math display="inline"><semantics> <msub> <mi>t</mi> <mn>1</mn> </msub> </semantics></math>. Another example of local topic influence occurs after user <math display="inline"><semantics> <msub> <mi>v</mi> <mn>2</mn> </msub> </semantics></math> was exposed to <math display="inline"><semantics> <msub> <mi>w</mi> <mn>1</mn> </msub> </semantics></math> at <math display="inline"><semantics> <msub> <mi>t</mi> <mn>1</mn> </msub> </semantics></math> by user <math display="inline"><semantics> <msub> <mi>v</mi> <mn>1</mn> </msub> </semantics></math> whom s/he follows. Then, user <math display="inline"><semantics> <msub> <mi>v</mi> <mn>2</mn> </msub> </semantics></math> retweets tweet <math display="inline"><semantics> <msub> <mi>w</mi> <mn>2</mn> </msub> </semantics></math> on the same topic as <math display="inline"><semantics> <msub> <mi>w</mi> <mn>1</mn> </msub> </semantics></math>. (<b>b</b>) Global topic influence: user <math display="inline"><semantics> <msub> <mi>v</mi> <mn>2</mn> </msub> </semantics></math> retweets <math display="inline"><semantics> <msub> <mi>w</mi> <mn>2</mn> </msub> </semantics></math> before any of the users who are followed by <math display="inline"><semantics> <msub> <mi>v</mi> <mn>2</mn> </msub> </semantics></math> retweeted/posted tweet from <span class="html-italic">W</span>.</p> Full article ">
17 pages, 2703 KiB  
Article
An Approach to Growth Delimitation of Straight Line Segment Classifiers Based on a Minimum Bounding Box
by Rosario Medina-Rodríguez, César Beltrán-Castañón and Ronaldo Fumio Hashimoto
Entropy 2021, 23(11), 1541; https://doi.org/10.3390/e23111541 - 19 Nov 2021
Viewed by 1913
Abstract
Several supervised machine learning algorithms focused on binary classification for solving daily problems can be found in the literature. The straight-line segment classifier stands out for its low complexity and competitiveness, compared to well-knownconventional classifiers. This binary classifier is based on distances between [...] Read more.
Several supervised machine learning algorithms focused on binary classification for solving daily problems can be found in the literature. The straight-line segment classifier stands out for its low complexity and competitiveness, compared to well-knownconventional classifiers. This binary classifier is based on distances between points and two labeled sets of straight-line segments. Its training phase consists of finding the placement of labeled straight-line segment extremities (and consequently, their lengths) which gives the minimum mean square error. However, during the training phase, the straight-line segment lengths can grow significantly, giving a negative impact on the classification rate. Therefore, this paper proposes an approach for adjusting the placements of labeled straight-line segment extremities to build reliable classifiers in a constrained search space (tuned by a scale factor parameter) in order to restrict their lengths. Ten artificial and eight datasets from the UCI Machine Learning Repository were used to prove that our approach shows promising results, compared to other classifiers. We conclude that this classifier can be used in industry for decision-making problems, due to the straightforward interpretation and classification rates. Full article
(This article belongs to the Special Issue Machine Learning Ecosystems: Opportunities and Threats)
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<p>Example of a separable problem in 2D space. Decision boundaries obtained by (<b>a</b>) k-NN classifier; (<b>b</b>) LVQ, prototypes are marked by larger symbols; (<b>c</b>) NFL, feature line spaces drawn from 5 points. Figures extracted from (<b>a</b>,<b>b</b>) [<a href="#B23-entropy-23-01541" class="html-bibr">23</a>] and (<b>c</b>) [<a href="#B12-entropy-23-01541" class="html-bibr">12</a>].</p> Full article ">Figure 2
<p>Representation of the distance between a point <span class="html-italic">x</span> and a straight-line segment with extremities <span class="html-italic">p</span> and <span class="html-italic">q</span> <math display="inline"><semantics> <mrow> <mo>∈</mo> <msup> <mi mathvariant="double-struck">R</mi> <mrow> <mi>d</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> </semantics></math>.</p> Full article ">Figure 3
<p>Representation of two sets of straight-line segments in color red and blue.</p> Full article ">Figure 4
<p>Training algorithm steps: placing (<b>a</b>–<b>d</b>) and tuning (<b>e</b>–<b>f</b>). Modified from [<a href="#B15-entropy-23-01541" class="html-bibr">15</a>].</p> Full article ">Figure 5
<p>Representation of the initial (dashed line) and final (solid line) positions of the straight-line segments for the S-Shape distribution using 2-2 straight-line segments per class. In the plots, the red color lines belong to <math display="inline"><semantics> <mrow> <mi>c</mi> <mi>l</mi> <mi>a</mi> <mi>s</mi> <msub> <mi>s</mi> <mn>1</mn> </msub> </mrow> </semantics></math> and the ones in black, to <math display="inline"><semantics> <mrow> <mi>c</mi> <mi>l</mi> <mi>a</mi> <mi>s</mi> <msub> <mi>s</mi> <mn>0</mn> </msub> </mrow> </semantics></math>.</p> Full article ">Figure 6
<p>Distance comparison between one fixed point X (in purple) and three straight-line segments (i) in blue, a short length and far from point X; (ii) in green, a short length and close to the point X; and, (iii) in red, long length and far from point X.</p> Full article ">Figure 7
<p>Proposed bounding boxes for restricting the straight-line segments’ growing space, using scale factors (<math display="inline"><semantics> <mrow> <mi>m</mi> <mi>b</mi> <mi>b</mi> <mo>_</mo> <mi>f</mi> <mi>a</mi> <mi>c</mi> <mi>t</mi> <mi>o</mi> <mi>r</mi> </mrow> </semantics></math>): 0.0, 2.0 and 4.0, respectively.</p> Full article ">Figure 8
<p>Artificial dataset types used in our research. On the first row: (<b>i</b>) F-Shape, (<b>ii</b>) S-Shape, (<b>iii</b>) Simple-Shape, (<b>iv</b>) X-Shape, and (<b>v</b>) Blobs. On the second row: (<b>vi</b>) Blobs with Noise, (<b>vii</b>) Circles, (<b>viii</b>) Gaussian, (<b>ix</b>) Imbalanced, and (<b>x</b>) Moon. On each graph, <math display="inline"><semantics> <mrow> <mi>c</mi> <mi>l</mi> <mi>a</mi> <mi>s</mi> <msub> <mi>s</mi> <mn>0</mn> </msub> </mrow> </semantics></math> is represented by gray color and <math display="inline"><semantics> <mrow> <mi>c</mi> <mi>l</mi> <mi>a</mi> <mi>s</mi> <msub> <mi>s</mi> <mn>1</mn> </msub> </mrow> </semantics></math> by blue color.</p> Full article ">Figure 9
<p>Number of iterations at gradient descent algorithm stops when using different bounding box scale factors.</p> Full article ">Figure 10
<p>A 3D visualization of the final straight-line segment positions obtained after applying the training algorithm, using one straight-line segment per class. Each column represents a different value of <math display="inline"><semantics> <mrow> <mi>m</mi> <mi>b</mi> <mi>b</mi> <mo>_</mo> <mi>f</mi> <mi>a</mi> <mi>c</mi> <mi>t</mi> <mi>o</mi> <mi>r</mi> </mrow> </semantics></math> = <math display="inline"><semantics> <mrow> <mo>{</mo> <mn>0</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>4</mn> <mo>}</mo> </mrow> </semantics></math> with accuracies of 73.35%, 99.9% and 99.9%, respectively.</p> Full article ">Figure 11
<p>Correct classification at test phase for distributions S-Shape and Simple-Shape using from 1 to 5 straight line segments per class and 0, 1, 2, 4, 6 as bounding box scale factors.</p> Full article ">Figure 12
<p>Classification errors at testing phase for the Simple-Shape distribution using <math display="inline"><semantics> <mrow> <mo>{</mo> <mn>1.0</mn> <mo>,</mo> <mn>1.5</mn> <mo>,</mo> <mn>2.0</mn> <mo>}</mo> </mrow> </semantics></math> as scale factors for the bounding box.</p> Full article ">
17 pages, 3458 KiB  
Article
Entropy-Based Shear Stress Distribution in Open Channel for All Types of Flow Using Experimental Data
by Yeon-Moon Choo, Hae-Seong Jeon and Jong-Cheol Seo
Entropy 2021, 23(11), 1540; https://doi.org/10.3390/e23111540 - 19 Nov 2021
Viewed by 1721
Abstract
Korean river design standards set general design standards for rivers and river-related projects in Korea, which systematize the technologies and methods involved in river-related projects. This includes measurement methods for parts necessary for river design, but does not include information on shear stress. [...] Read more.
Korean river design standards set general design standards for rivers and river-related projects in Korea, which systematize the technologies and methods involved in river-related projects. This includes measurement methods for parts necessary for river design, but does not include information on shear stress. Shear stress is one of the factors necessary for river design and operation. Shear stress is one of the most important hydraulic factors used in the fields of water, especially for artificial channel design. Shear stress is calculated from the frictional force caused by viscosity and fluctuating fluid velocity. Current methods are based on past calculations, but factors such as boundary shear stress or energy gradient are difficult to actually measure or estimate. The point velocity throughout the entire cross-section is needed to calculate the velocity gradient. In other words, the current Korean river design standards use tractive force and critical tractive force instead of shear stress because it is more difficult to calculate the shear stress in the current method. However, it is difficult to calculate the exact value due to the limitations of the formula to obtain the river factor called the tractive force. In addition, tractive force has limitations that use an empirically identified base value for use in practice. This paper focuses on the modeling of shear-stress distribution in open channel turbulent flow using entropy theory. In addition, this study suggests a shear stress distribution formula, which can easily be used in practice after calculating the river-specific factor T. The tractive force and critical tractive force in the Korean river design standards should be modified by the shear stress obtained by the proposed shear stress distribution method. The present study therefore focuses on the modeling of shear stress distribution in an open channel turbulent flow using entropy theory. The shear stress distribution model is tested using a wide range of forty-two experimental runs collected from the literature. Then, an error analysis is performed to further evaluate the accuracy of the proposed model. The results reveal a correlation coefficient of approximately 0.95–0.99, indicating that the proposed method can estimate shear-stress distribution accurately. Based on this, the results of the distribution of shear stress after calculating the river-specific factors show a correlation coefficient of about 0.86 to 0.98, which suggests that the equation can be applied in practice. Full article
(This article belongs to the Section Information Theory, Probability and Statistics)
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<p><math display="inline"><semantics> <mi>ξ</mi> </semantics></math>-<math display="inline"><semantics> <mrow> <mi>η</mi> </mrow> </semantics></math> coordinates in an open channel flow (Chiu [<a href="#B25-entropy-23-01540" class="html-bibr">25</a>,<a href="#B38-entropy-23-01540" class="html-bibr">38</a>]).</p> Full article ">Figure 1 Cont.
<p><math display="inline"><semantics> <mi>ξ</mi> </semantics></math>-<math display="inline"><semantics> <mrow> <mi>η</mi> </mrow> </semantics></math> coordinates in an open channel flow (Chiu [<a href="#B25-entropy-23-01540" class="html-bibr">25</a>,<a href="#B38-entropy-23-01540" class="html-bibr">38</a>]).</p> Full article ">Figure 2
<p>Parameter estimation flowchart.</p> Full article ">Figure 3
<p>Verification of the proposed shear stress distribution model with six uniform flows.</p> Full article ">Figure 4
<p>Verification of the proposed shear-stress distribution model with six accelerating non-uniform flows.</p> Full article ">Figure 5
<p>Verification of the proposed shear-stress distribution model with six decelerating non-uniform flows.</p> Full article ">Figure 5 Cont.
<p>Verification of the proposed shear-stress distribution model with six decelerating non-uniform flows.</p> Full article ">Figure 6
<p>Verification of the proposed shear-stress distribution model with unsteady flow of one slope (S-25-931).</p> Full article ">Figure 7
<p>Equilibrium <math display="inline"><semantics> <mrow> <mo>∅</mo> <mrow> <mo>(</mo> <msup> <mi>T</mi> <mo>′</mo> </msup> <mo>)</mo> </mrow> </mrow> </semantics></math> in steady flow.</p> Full article ">Figure 8
<p>Equilibrium <math display="inline"><semantics> <mrow> <mo>∅</mo> <mrow> <mo>(</mo> <msup> <mi>T</mi> <mo>′</mo> </msup> <mo>)</mo> </mrow> </mrow> </semantics></math> in unsteady flow.</p> Full article ">
12 pages, 273 KiB  
Article
First Integrals of Shear-Free Fluids and Complexity
by Sfundo C. Gumede, Keshlan S. Govinder and Sunil D. Maharaj
Entropy 2021, 23(11), 1539; https://doi.org/10.3390/e23111539 - 19 Nov 2021
Cited by 5 | Viewed by 1629
Abstract
A single master equation governs the behaviour of shear-free neutral perfect fluid distributions arising in gravity theories. In this paper, we study the integrability of yxx=f(x)y2, find new solutions, and generate a new [...] Read more.
A single master equation governs the behaviour of shear-free neutral perfect fluid distributions arising in gravity theories. In this paper, we study the integrability of yxx=f(x)y2, find new solutions, and generate a new first integral. The first integral is subject to an integrability condition which is an integral equation which restricts the function f(x). We find that the integrability condition can be written as a third order differential equation whose solution can be expressed in terms of elementary functions and elliptic integrals. The solution of the integrability condition is generally given parametrically. A particular form of f(x)∼1x51−1x−15/7 which corresponds to repeated roots of a cubic equation is given explicitly, which is a new result. Our investigation demonstrates that complexity of a self-gravitating shear-free fluid is related to the existence of a first integral, and this may be extendable to general matter distributions. Full article
(This article belongs to the Special Issue Complexity of Self-Gravitating Systems)
10 pages, 366 KiB  
Article
Tight and Scalable Side-Channel Attack Evaluations through Asymptotically Optimal Massey-like Inequalities on Guessing Entropy
by Andrei Tănăsescu, Marios O. Choudary, Olivier Rioul and Pantelimon George Popescu
Entropy 2021, 23(11), 1538; https://doi.org/10.3390/e23111538 - 18 Nov 2021
Cited by 6 | Viewed by 2074
Abstract
The bounds presented at CHES 2017 based on Massey’s guessing entropy represent the most scalable side-channel security evaluation method to date. In this paper, we present an improvement of this method, by determining the asymptotically optimal Massey-like inequality and then further refining it [...] Read more.
The bounds presented at CHES 2017 based on Massey’s guessing entropy represent the most scalable side-channel security evaluation method to date. In this paper, we present an improvement of this method, by determining the asymptotically optimal Massey-like inequality and then further refining it for finite support distributions. The impact of these results is highlighted for side-channel attack evaluations, demonstrating the improvements over the CHES 2017 bounds. Full article
(This article belongs to the Special Issue Types of Entropies and Divergences with Their Applications)
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<p>Bounds for the simulated (<b>left</b>) and real (<b>right</b>) datasets, when targeting a single subkey byte. These are averaged results over 100 experiments.</p> Full article ">Figure 2
<p>Bounds for the simulated (<b>left</b>) and real (<b>right</b>) datasets, when targeting two subkey bytes. These are averaged results over 100 experiments.</p> Full article ">Figure 3
<p>Bounds for the simulated (<b>left</b>) and real (<b>right</b>) datasets, when targeting all the 16 AES key bytes. These are averaged results over 100 experiments.</p> Full article ">
16 pages, 4943 KiB  
Article
Target Classification Method of Tactile Perception Data with Deep Learning
by Xingxing Zhang, Shaobo Li, Jing Yang, Qiang Bai, Yang Wang, Mingming Shen, Ruiqiang Pu and Qisong Song
Entropy 2021, 23(11), 1537; https://doi.org/10.3390/e23111537 - 18 Nov 2021
Cited by 6 | Viewed by 2540
Abstract
In order to improve the accuracy of manipulator operation, it is necessary to install a tactile sensor on the manipulator to obtain tactile information and accurately classify a target. However, with the increase in the uncertainty and complexity of tactile sensing data characteristics, [...] Read more.
In order to improve the accuracy of manipulator operation, it is necessary to install a tactile sensor on the manipulator to obtain tactile information and accurately classify a target. However, with the increase in the uncertainty and complexity of tactile sensing data characteristics, and the continuous development of tactile sensors, typical machine-learning algorithms often cannot solve the problem of target classification of pure tactile data. Here, we propose a new model by combining a convolutional neural network and a residual network, named ResNet10-v1. We optimized the convolutional kernel, hyperparameters, and loss function of the model, and further improved the accuracy of target classification through the K-means clustering method. We verified the feasibility and effectiveness of the proposed method through a large number of experiments. We expect to further improve the generalization ability of this method and provide an important reference for the research in the field of tactile perception classification. Full article
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<p>Proposed ResNet10-v1 structure.</p> Full article ">Figure 2
<p>Different kinds of convolutional kernels.</p> Full article ">Figure 3
<p>Principle of K-means algorithm.</p> Full article ">Figure 4
<p>Convolutional layer principle.</p> Full article ">Figure 5
<p>Dataset objects [<a href="#B14-entropy-23-01537" class="html-bibr">14</a>]; 26 targets used in our experiments.</p> Full article ">Figure 6
<p>Tactile maps obtained when tactile glove grabs different targets. (<b>a</b>) Cup; (<b>b</b>) tennis ball; (<b>c</b>) cola can.</p> Full article ">Figure 7
<p>Result comparison of base learning rate optimization.</p> Full article ">Figure 8
<p>Result comparison of epoch optimization.</p> Full article ">Figure 9
<p>Result comparison of Batch_size optimization.</p> Full article ">Figure 10
<p>Result comparison dropout optimization.</p> Full article ">Figure 11
<p>Optimization result comparison chart of different capture method datasets.</p> Full article ">Figure 12
<p>Comparison of accuracy classification prediction of the model before and after optimization.</p> Full article ">Figure 13
<p>Comparison of ResNet10-v1, ResNet18, and ResNet50 model classification prediction accuracy.</p> Full article ">
13 pages, 988 KiB  
Article
Bert-Enhanced Text Graph Neural Network for Classification
by Yiping Yang and Xiaohui Cui
Entropy 2021, 23(11), 1536; https://doi.org/10.3390/e23111536 - 18 Nov 2021
Cited by 20 | Viewed by 5511
Abstract
Text classification is a fundamental research direction, aims to assign tags to text units. Recently, graph neural networks (GNN) have exhibited some excellent properties in textual information processing. Furthermore, the pre-trained language model also realized promising effects in many tasks. However, many text [...] Read more.
Text classification is a fundamental research direction, aims to assign tags to text units. Recently, graph neural networks (GNN) have exhibited some excellent properties in textual information processing. Furthermore, the pre-trained language model also realized promising effects in many tasks. However, many text processing methods cannot model a single text unit’s structure or ignore the semantic features. To solve these problems and comprehensively utilize the text’s structure information and semantic information, we propose a Bert-Enhanced text Graph Neural Network model (BEGNN). For each text, we construct a text graph separately according to the co-occurrence relationship of words and use GNN to extract text features. Moreover, we employ Bert to extract semantic features. The former part can take into account the structural information, and the latter can focus on modeling the semantic information. Finally, we interact and aggregate these two features of different granularity to get a more effective representation. Experiments on standard datasets demonstrate the effectiveness of BEGNN. Full article
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<p>The architecture of BEGNN. (<b>a</b>) The input document. (<b>b</b>) Graph construction and graph neural network based feature extraction. (<b>c</b>) Bert based feature extraction. (<b>d</b>) Interactive feature aggregation. (<b>e</b>) Fully-connected layer.</p> Full article ">Figure 2
<p>The graph constructed for a document with five words.</p> Full article ">Figure 3
<p>Co-attention layer.</p> Full article ">Figure 4
<p>Ablation study of the text graph and the co-attention modules of the model.</p> Full article ">
17 pages, 6237 KiB  
Article
Modulo Periodic Poisson Stable Solutions of Quasilinear Differential Equations
by Marat Akhmet, Madina Tleubergenova and Akylbek Zhamanshin
Entropy 2021, 23(11), 1535; https://doi.org/10.3390/e23111535 - 18 Nov 2021
Cited by 16 | Viewed by 1895
Abstract
In this paper, modulo periodic Poisson stable functions have been newly introduced. Quasilinear differential equations with modulo periodic Poisson stable coefficients are under investigation. The existence and uniqueness of asymptotically stable modulo periodic Poisson stable solutions have been proved. Numerical simulations, which illustrate [...] Read more.
In this paper, modulo periodic Poisson stable functions have been newly introduced. Quasilinear differential equations with modulo periodic Poisson stable coefficients are under investigation. The existence and uniqueness of asymptotically stable modulo periodic Poisson stable solutions have been proved. Numerical simulations, which illustrate the theoretical results are provided. Full article
(This article belongs to the Special Issue Nonlinear Dynamics and Entropy of Complex Systems: Advances and Perspectives II)
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<p>Coordinates of the solution <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math> of system (9) with initial values <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> <mo>=</mo> <mn>2.5</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> <mo>=</mo> <mn>1.5</mn> <mo>,</mo> </mrow> </semantics></math> which asymptotically converge to the coordinates of the <math display="inline"><semantics> <mrow> <mi>M</mi> <mi>P</mi> <mi>P</mi> <mi>S</mi> </mrow> </semantics></math> solution <math display="inline"><semantics> <mrow> <mi>z</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math> of the system.</p> Full article ">Figure 2
<p>The trajectory of the solution <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math> of the Equation (9), which asymptotically approaches the MPPS solution <math display="inline"><semantics> <mrow> <mi>z</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math> of the system.</p> Full article ">Figure 3
<p>Coordinates of the solution <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> <mo>,</mo> </mrow> </semantics></math> with initial values <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> <mo>=</mo> <mn>1</mn> <mo>,</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>3</mn> </msub> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> <mo>=</mo> <mn>1</mn> <mo>,</mo> </mrow> </semantics></math> which asymptotically converge to the coordinates of the <math display="inline"><semantics> <mrow> <mi>M</mi> <mi>P</mi> <mi>P</mi> <mi>S</mi> </mrow> </semantics></math> solution of system (10).</p> Full article ">Figure 4
<p>The trajectory of the solution, <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> <mo>,</mo> </mrow> </semantics></math> of Equation (10), which asymptotically approaches the MPPS solution of the equation.</p> Full article ">Figure 5
<p>The coordinates of the solution <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> <mo>,</mo> </mrow> </semantics></math> with <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> <mo>=</mo> <mn>1</mn> <mo>,</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> <mo>=</mo> <mn>1</mn> <mo>,</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>3</mn> </msub> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> <mo>=</mo> <mn>1</mn> <mo>,</mo> </mrow> </semantics></math> which is asymptotic for the Poisson stable solution of the system (19).</p> Full article ">Figure 6
<p>The trajectory of the solution <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> <mo>,</mo> </mrow> </semantics></math> with <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> <mo>=</mo> <mn>1</mn> <mo>,</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> <mo>=</mo> <mn>1</mn> <mo>,</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>3</mn> </msub> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> <mo>=</mo> <mn>1</mn> <mo>,</mo> </mrow> </semantics></math> which illustrates the Poisson stability of the system (19).</p> Full article ">
27 pages, 1813 KiB  
Article
Sequential Learning of Principal Curves: Summarizing Data Streams on the Fly
by Le Li and Benjamin Guedj
Entropy 2021, 23(11), 1534; https://doi.org/10.3390/e23111534 - 18 Nov 2021
Cited by 2 | Viewed by 2379
Abstract
When confronted with massive data streams, summarizing data with dimension reduction methods such as PCA raises theoretical and algorithmic pitfalls. A principal curve acts as a nonlinear generalization of PCA, and the present paper proposes a novel algorithm to automatically and sequentially learn [...] Read more.
When confronted with massive data streams, summarizing data with dimension reduction methods such as PCA raises theoretical and algorithmic pitfalls. A principal curve acts as a nonlinear generalization of PCA, and the present paper proposes a novel algorithm to automatically and sequentially learn principal curves from data streams. We show that our procedure is supported by regret bounds with optimal sublinear remainder terms. A greedy local search implementation (called slpc, for sequential learning principal curves) that incorporates both sleeping experts and multi-armed bandit ingredients is presented, along with its regret computation and performance on synthetic and real-life data. Full article
(This article belongs to the Special Issue Approximate Bayesian Inference)
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Figure 1

Figure 1
<p>A principal curve.</p> Full article ">Figure 2
<p>A principal curve and projections of data onto it.</p> Full article ">Figure 3
<p>Principal curves with different numbers (<span class="html-italic">k</span>) of segments. (<b>a</b>) A too small <span class="html-italic">k</span>. (<b>b</b>) Right <span class="html-italic">k</span>. (<b>c</b>) A too large <span class="html-italic">k</span>.</p> Full article ">Figure 4
<p>An example of a lattice <math display="inline"><semantics> <msub> <mo>Γ</mo> <mi>δ</mi> </msub> </semantics></math> in <math display="inline"><semantics> <msup> <mi mathvariant="double-struck">R</mi> <mn>2</mn> </msup> </semantics></math> with <math display="inline"><semantics> <mrow> <mi>δ</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> (spacing between blue points) and <math display="inline"><semantics> <mrow> <mi>B</mi> <mo>(</mo> <mn>0</mn> <mo>,</mo> <mn>10</mn> <mo>)</mo> </mrow> </semantics></math> (black circle). The red polygonal line is composed of vertices in <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="script">Q</mi> <mi>δ</mi> </msub> <mo>=</mo> <mi>B</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>,</mo> <mn>10</mn> <mo>)</mo> </mrow> <mspace width="3.33333pt"/> <mo>∩</mo> <mspace width="3.33333pt"/> <msub> <mo>Γ</mo> <mi>δ</mi> </msub> </mrow> </semantics></math>.</p> Full article ">Figure 5
<p>An example of a Voronoi partition.</p> Full article ">Figure 6
<p>Synthetic data. Black dots represent data <math display="inline"><semantics> <msub> <mi>x</mi> <mrow> <mn>1</mn> <mo>:</mo> <mi>t</mi> </mrow> </msub> </semantics></math>. The red point is the new observation <math display="inline"><semantics> <msub> <mi>x</mi> <mrow> <mi>t</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </semantics></math>. <tt>princurve</tt> (solid red) and <tt>slpc</tt> (solid green). (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>150</mn> </mrow> </semantics></math>, <tt>princurve</tt>. (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>450</mn> </mrow> </semantics></math>, <tt>princurve</tt>. (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>150</mn> </mrow> </semantics></math>, <tt>incremental SCMS</tt>. (<b>d</b>) <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>450</mn> </mrow> </semantics></math>, <tt>incremental SCMS</tt>. (<b>e</b>) <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>150</mn> </mrow> </semantics></math>, <tt>slpc</tt>. (<b>f</b>) <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>450</mn> </mrow> </semantics></math>, <tt>slpc</tt>.</p> Full article ">Figure 7
<p>Mean estimation of regret and per-round regret of <tt>slpc</tt> with respect to time round <span class="html-italic">t</span>, for the horizon <math display="inline"><semantics> <mrow> <mi>T</mi> <mo>=</mo> <mn>500</mn> </mrow> </semantics></math>. (<b>a</b>) Mean estimation of the regret of <tt>slpc</tt> over 20 trials (black line) and a bisection line (green) with respect to time round <span class="html-italic">t</span>. (<b>b</b>) Per-round of estimated regret of <tt>slpc</tt> with respect to <span class="html-italic">t</span>.</p> Full article ">Figure 8
<p>Synthetic data. Zooming in: how a new data point impacts the principal curve only locally. (<b>a</b>) At time <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>97</mn> </mrow> </semantics></math>. (<b>b</b>) And at time <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>98</mn> </mrow> </semantics></math>.</p> Full article ">Figure 9
<p><tt>slpc</tt> (green line) on synthetic high dimensional data from different perspectives. Black dots represent recordings <math display="inline"><semantics> <msub> <mi>x</mi> <mrow> <mn>1</mn> <mo>:</mo> <mn>99</mn> </mrow> </msub> </semantics></math>; the red dot is the new recording <math display="inline"><semantics> <msub> <mi>x</mi> <mn>200</mn> </msub> </semantics></math>. (<b>a</b>) <tt>slpc</tt>, <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>199</mn> </mrow> </semantics></math>, 1st and 2nd coordinates. (<b>b</b>) <tt>slpc</tt>, <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>199</mn> </mrow> </semantics></math>, 3th and 5th coordinates. (<b>c</b>) <tt>slpc</tt>, <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>199</mn> </mrow> </semantics></math>, 4th and 6th coordinates.</p> Full article ">Figure 10
<p>Seismic data. Black dots represent seismic recordings <math display="inline"><semantics> <msub> <mi>x</mi> <mrow> <mn>1</mn> <mo>:</mo> <mi>t</mi> </mrow> </msub> </semantics></math>; the red dot is the new recording <math display="inline"><semantics> <msub> <mi>x</mi> <mrow> <mi>t</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </semantics></math>. (<b>a</b>) <tt>princurve</tt>, <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>100</mn> </mrow> </semantics></math>. (<b>b</b>) <tt>princurve</tt>, <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>125</mn> </mrow> </semantics></math>. (<b>c</b>) <tt>incremental SCMS</tt>, <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>100</mn> </mrow> </semantics></math>. (<b>d</b>) <tt>incremental SCMS</tt>, <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>125</mn> </mrow> </semantics></math>. (<b>e</b>) <tt>slpc</tt>, <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>100</mn> </mrow> </semantics></math>. (<b>f</b>) <tt>slpc</tt>, <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>125</mn> </mrow> </semantics></math>.</p> Full article ">Figure 11
<p>Seismic data from <a href="https://earthquake.usgs.gov/data/centennial/" target="_blank">https://earthquake.usgs.gov/data/centennial/</a>.</p> Full article ">Figure 12
<p>Daily commute data. Black dots represent collected locations <math display="inline"><semantics> <msub> <mi>x</mi> <mrow> <mn>1</mn> <mo>:</mo> <mi>t</mi> </mrow> </msub> </semantics></math>. The red point is the new observation <math display="inline"><semantics> <msub> <mi>x</mi> <mrow> <mi>t</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </semantics></math>. <tt>princurve</tt> (solid red) and <tt>slpc</tt> (solid green). (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math>, <tt>princurve</tt>. (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>127</mn> </mrow> </semantics></math>, <tt>princurve</tt>. (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math>, <tt>slpc</tt>. (<b>d</b>) <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>127</mn> </mrow> </semantics></math>, <tt>slpc</tt>.</p> Full article ">
14 pages, 1270 KiB  
Article
Weak Singularities of the Isothermal Entropy Change as the Smoking Gun Evidence of Phase Transitions of Mixed-Spin Ising Model on a Decorated Square Lattice in Transverse Field
by Jozef Strečka and Katarína Karl’ová
Entropy 2021, 23(11), 1533; https://doi.org/10.3390/e23111533 - 18 Nov 2021
Cited by 2 | Viewed by 1981
Abstract
The magnetocaloric response of the mixed spin-1/2 and spin-S (S>1/2) Ising model on a decorated square lattice is thoroughly examined in presence of the transverse magnetic field within the generalized decoration-iteration transformation, which provides an exact [...] Read more.
The magnetocaloric response of the mixed spin-1/2 and spin-S (S>1/2) Ising model on a decorated square lattice is thoroughly examined in presence of the transverse magnetic field within the generalized decoration-iteration transformation, which provides an exact mapping relation with an effective spin-1/2 Ising model on a square lattice in a zero magnetic field. Temperature dependencies of the entropy and isothermal entropy change exhibit an outstanding singular behavior in a close neighborhood of temperature-driven continuous phase transitions, which can be additionally tuned by the applied transverse magnetic field. While temperature variations of the entropy display in proximity of the critical temperature Tc a striking energy-type singularity (T−Tc)log|T−Tc|, two analogous weak singularities can be encountered in the temperature dependence of the isothermal entropy change. The basic magnetocaloric measurement of the isothermal entropy change may accordingly afford the smoking gun evidence of continuous phase transitions. It is shown that the investigated model predominantly displays the conventional magnetocaloric effect with exception of a small range of moderate temperatures, which contrarily promotes the inverse magnetocaloric effect. It turns out that the temperature range inherent to the inverse magnetocaloric effect is gradually suppressed upon increasing of the spin magnitude S. Full article
(This article belongs to the Section Statistical Physics)
►▼ Show Figures

Figure 1

Figure 1
<p>A cross-section from the decorated square lattice. The purple (dark) circles denote lattice positions of the nodal spin-1/2 magnetic ions and the green (light) circles schematically represent lattice positions of the decorating spin-<span class="html-italic">S</span> (<math display="inline"><semantics> <mrow> <mi>S</mi> <mo>&gt;</mo> <mn>1</mn> <mo>/</mo> <mn>2</mn> </mrow> </semantics></math>) magnetic ions. A rectangle delimits a three-spin cluster described by the bond Hamiltonian (<a href="#FD2-entropy-23-01533" class="html-disp-formula">2</a>).</p> Full article ">Figure 2
<p>Finite-temperature phase diagrams of the mixed spin-1/2 and spin-<span class="html-italic">S</span> Ising model on a decorated square lattice in the form of plots the critical temperature <math display="inline"><semantics> <mrow> <msub> <mi>k</mi> <mi mathvariant="normal">B</mi> </msub> <msub> <mi>T</mi> <mi>c</mi> </msub> <mo>/</mo> <mrow> <mo>|</mo> <mi>J</mi> <mo>|</mo> </mrow> </mrow> </semantics></math> versus the transverse magnetic field <math display="inline"><semantics> <mrow> <mo>Ω</mo> <mo>/</mo> <mo>|</mo> <mi>J</mi> <mo>|</mo> </mrow> </semantics></math> for four selected values of the spin magnitude <math display="inline"><semantics> <mrow> <mi>S</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mn>3</mn> <mo>/</mo> <mn>2</mn> </mrow> </semantics></math>, 2 and <math display="inline"><semantics> <mrow> <mn>5</mn> <mo>/</mo> <mn>2</mn> </mrow> </semantics></math>.</p> Full article ">Figure 3
<p>Temperature variations of the molar entropy of the mixed spin-1/2 and spin-<span class="html-italic">S</span> (<math display="inline"><semantics> <mrow> <mi>S</mi> <mo>&gt;</mo> <mn>1</mn> <mo>/</mo> <mn>2</mn> </mrow> </semantics></math>) Ising model on a decorated square lattice for a few different values of the transverse magnetic field and four selected values of the spin magnitude: (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>S</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>; (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>S</mi> <mo>=</mo> <mn>3</mn> <mo>/</mo> <mn>2</mn> </mrow> </semantics></math>; (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>S</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>; (<b>d</b>) <math display="inline"><semantics> <mrow> <mi>S</mi> <mo>=</mo> <mn>5</mn> <mo>/</mo> <mn>2</mn> </mrow> </semantics></math>. Filled symbols of different styles allocate singular points of the type <math display="inline"><semantics> <mrow> <mo>∝</mo> <mrow> <mo>(</mo> <mi>T</mi> <mo>−</mo> <msub> <mi>T</mi> <mi>c</mi> </msub> <mo>)</mo> </mrow> <mo form="prefix">log</mo> <mrow> <mo>|</mo> <mi>T</mi> <mo>−</mo> <msub> <mi>T</mi> <mi>c</mi> </msub> <mo>|</mo> </mrow> </mrow> </semantics></math>.</p> Full article ">Figure 4
<p>The molar entropy of the mixed spin-1/2 and spin-<span class="html-italic">S</span> (<math display="inline"><semantics> <mrow> <mi>S</mi> <mo>&gt;</mo> <mn>1</mn> <mo>/</mo> <mn>2</mn> </mrow> </semantics></math>) Ising model on a decorated square lattice as a function of the temperature deviation <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mi>T</mi> <mo>−</mo> <msub> <mi>T</mi> <mi>c</mi> </msub> <mo>)</mo> </mrow> <mo form="prefix">log</mo> <mrow> <mo>|</mo> <mi>T</mi> <mo>−</mo> <msub> <mi>T</mi> <mi>c</mi> </msub> <mo>|</mo> </mrow> </mrow> </semantics></math> from its critical value for two different values of the spin magnitude and transverse magnetic fields: (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>S</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>Ω</mo> <mo>/</mo> <mo>|</mo> <mi>J</mi> <mo>|</mo> <mo>=</mo> <mn>0.0</mn> </mrow> </semantics></math>; (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>S</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>Ω</mo> <mo>/</mo> <mo>|</mo> <mi>J</mi> <mo>|</mo> <mo>=</mo> <mn>1.0</mn> </mrow> </semantics></math>; (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>S</mi> <mo>=</mo> <mn>3</mn> <mo>/</mo> <mn>2</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>Ω</mo> <mo>/</mo> <mo>|</mo> <mi>J</mi> <mo>|</mo> <mo>=</mo> <mn>0.0</mn> </mrow> </semantics></math>; (<b>d</b>) <math display="inline"><semantics> <mrow> <mi>S</mi> <mo>=</mo> <mn>3</mn> <mo>/</mo> <mn>2</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>Ω</mo> <mo>/</mo> <mo>|</mo> <mi>J</mi> <mo>|</mo> <mo>=</mo> <mn>1.0</mn> </mrow> </semantics></math>. Blue broken lines are linear fits of the respective dependencies, which prove a singular character of the entropy that is in a close vicinity of the critical points (filled red circles) proportional to <math display="inline"><semantics> <mrow> <mo>∝</mo> <mrow> <mo>(</mo> <mi>T</mi> <mo>−</mo> <msub> <mi>T</mi> <mi>c</mi> </msub> <mo>)</mo> </mrow> <mo form="prefix">log</mo> <mrow> <mo>|</mo> <mi>T</mi> <mo>−</mo> <msub> <mi>T</mi> <mi>c</mi> </msub> <mo>|</mo> </mrow> </mrow> </semantics></math>.</p> Full article ">Figure 5
<p>Temperature dependencies of the isothermal entropy change for three different values of the transverse-field change <math display="inline"><semantics> <mrow> <mo>Δ</mo> <mo>Ω</mo> <mo>/</mo> <mo>|</mo> <mi>J</mi> <mo>|</mo> <mo>=</mo> <mn>0.5</mn> <mo>,</mo> <mn>1.0</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mn>2.0</mn> </mrow> </semantics></math> and four selected spin magnitude: (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>S</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>; (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>S</mi> <mo>=</mo> <mn>3</mn> <mo>/</mo> <mn>2</mn> </mrow> </semantics></math>; (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>S</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>; (<b>d</b>) <math display="inline"><semantics> <mrow> <mi>S</mi> <mo>=</mo> <mn>5</mn> <mo>/</mo> <mn>2</mn> </mrow> </semantics></math>. Thin dotted line at <math display="inline"><semantics> <mrow> <mo>−</mo> <mo>Δ</mo> <msub> <mi>S</mi> <mrow> <mi>i</mi> <mi>s</mi> <mi>o</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> is only guide for eyes, which enables to distinguish the conventional and inverse MCE. Open symbols denote weak singularities located at critical points of continuous phase transitions at the transverse magnetic fields <math display="inline"><semantics> <mrow> <mo>Ω</mo> <mo>/</mo> <mo>|</mo> <mi>J</mi> <mo>|</mo> <mo>=</mo> <mn>0.5</mn> <mo>,</mo> <mn>1.0</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mn>2.0</mn> </mrow> </semantics></math>, while filled symbols mark weak singularities of the zero-field entropy emergent at the critical temperature: (<b>a</b>) <math display="inline"><semantics> <mrow> <msub> <mi>k</mi> <mi mathvariant="normal">B</mi> </msub> <mi>T</mi> <mo>/</mo> <mrow> <mo>|</mo> <mi>J</mi> <mo>|</mo> </mrow> <mo>≈</mo> <mn>0.554</mn> </mrow> </semantics></math>; (<b>b</b>) <math display="inline"><semantics> <mrow> <msub> <mi>k</mi> <mi mathvariant="normal">B</mi> </msub> <mi>T</mi> <mo>/</mo> <mrow> <mo>|</mo> <mi>J</mi> <mo>|</mo> </mrow> <mo>≈</mo> <mn>0.767</mn> </mrow> </semantics></math>; (<b>c</b>) <math display="inline"><semantics> <mrow> <msub> <mi>k</mi> <mi mathvariant="normal">B</mi> </msub> <mi>T</mi> <mo>/</mo> <mrow> <mo>|</mo> <mi>J</mi> <mo>|</mo> </mrow> <mo>≈</mo> <mn>0.970</mn> </mrow> </semantics></math>; (<b>d</b>) <math display="inline"><semantics> <mrow> <msub> <mi>k</mi> <mi mathvariant="normal">B</mi> </msub> <mi>T</mi> <mo>/</mo> <mrow> <mo>|</mo> <mi>J</mi> <mo>|</mo> </mrow> <mo>≈</mo> <mn>1.180</mn> </mrow> </semantics></math>.</p> Full article ">Figure 6
<p>Density plots of the molar entropy of the mixed spin-1/2 and spin-<span class="html-italic">S</span> Ising model on a decorated square lattice in the plane transverse magnetic field versus temperature for four selected values of the decorated spins: (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>S</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>; (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>S</mi> <mo>=</mo> <mn>3</mn> <mo>/</mo> <mn>2</mn> </mrow> </semantics></math>; (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>S</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>; (<b>d</b>) <math display="inline"><semantics> <mrow> <mi>S</mi> <mo>=</mo> <mn>5</mn> <mo>/</mo> <mn>2</mn> </mrow> </semantics></math>. Broken lines show dependence of the critical temperature on the transverse magnetic field.</p> Full article ">
25 pages, 1324 KiB  
Article
How to Effectively Collect and Process Network Data for Intrusion Detection?
by Mikołaj Komisarek, Marek Pawlicki, Rafał Kozik, Witold Hołubowicz and Michał Choraś
Entropy 2021, 23(11), 1532; https://doi.org/10.3390/e23111532 - 18 Nov 2021
Cited by 15 | Viewed by 2958
Abstract
The number of security breaches in the cyberspace is on the rise. This threat is met with intensive work in the intrusion detection research community. To keep the defensive mechanisms up to date and relevant, realistic network traffic datasets are needed. The use [...] Read more.
The number of security breaches in the cyberspace is on the rise. This threat is met with intensive work in the intrusion detection research community. To keep the defensive mechanisms up to date and relevant, realistic network traffic datasets are needed. The use of flow-based data for machine-learning-based network intrusion detection is a promising direction for intrusion detection systems. However, many contemporary benchmark datasets do not contain features that are usable in the wild. The main contribution of this work is to cover the research gap related to identifying and investigating valuable features in the NetFlow schema that allow for effective, machine-learning-based network intrusion detection in the real world. To achieve this goal, several feature selection techniques have been applied on five flow-based network intrusion detection datasets, establishing an informative flow-based feature set. The authors’ experience with the deployment of this kind of system shows that to close the research-to-market gap, and to perform actual real-world application of machine-learning-based intrusion detection, a set of labeled data from the end-user has to be collected. This research aims at establishing the appropriate, minimal amount of data that is sufficient to effectively train machine learning algorithms in intrusion detection. The results show that a set of 10 features and a small amount of data is enough for the final model to perform very well. Full article
(This article belongs to the Special Issue Advances in Computer Recognition, Image Processing and Communications, Selected Papers from CORES 2021 and IP&C 2021)
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<p>The steps required to go from network traffic to publishing of a dataset suitable for ML methods. The red ellipse indicates the focus of this paper.</p> Full article ">Figure 2
<p>The correlations of features in the datasets: NF-UNSW-NB15, NF-BoT-IoT, NF-CSE-CIC-IDS2018, NF-UQ-NIDS, NF-ToN-IoT.</p> Full article ">Figure 3
<p>The top 10 best features determined by using three algorithms: LASSO L1, random forest Importance, and Chi2 for dataset nf-unsw-nb15.</p> Full article ">Figure 4
<p>The top 10 best features determined by using three algorithms: LASSO L1, random forest importance, and Chi2 for dataset nf-cse-cic-ids.</p> Full article ">Figure 5
<p>The top 10 best features determined by using three algorithms: LASSO L1, random forest importance, and Chi2 for dataset nf-ton-iot.</p> Full article ">Figure 6
<p>The top 10 best features determined by using three algorithms: LASSO L1, random forest importance, and Chi2 for dataset nf-uq-nids.</p> Full article ">Figure 7
<p>The top 10 best features determined by using three algorithms: LASSO L1, random forest importance, and Chi2 for dataset nf-bot-iot.</p> Full article ">Figure 8
<p>Comparison of the number of samples and machine learning effects—ROC curve plot for the random forest algorithm.</p> Full article ">Figure 9
<p>Comparison of the number of samples and machine learning effects—ROC curve plot for the AdaBoost algorithm.</p> Full article ">Figure 10
<p>Comparison of the number of samples and machine learning effects—ROC curve plot for the naïve Bayes classifier.</p> Full article ">Figure 11
<p>Comparison of the number of samples and machine learning effects—ROC curve plot for ANN.</p> Full article ">
19 pages, 1749 KiB  
Article
Katz Fractal Dimension of Geoelectric Field during Severe Geomagnetic Storms
by Agnieszka Gil, Vasile Glavan, Anna Wawrzaszek, Renata Modzelewska and Lukasz Tomasik
Entropy 2021, 23(11), 1531; https://doi.org/10.3390/e23111531 - 18 Nov 2021
Cited by 7 | Viewed by 3316
Abstract
We are concerned with the time series resulting from the computed local horizontal geoelectric field, obtained with the aid of a 1-D layered Earth model based on local geomagnetic field measurements, for the full solar magnetic cycle of 1996–2019, covering the two consecutive [...] Read more.
We are concerned with the time series resulting from the computed local horizontal geoelectric field, obtained with the aid of a 1-D layered Earth model based on local geomagnetic field measurements, for the full solar magnetic cycle of 1996–2019, covering the two consecutive solar activity cycles 23 and 24. To our best knowledge, for the first time, the roughness of severe geomagnetic storms is considered by using a monofractal time series analysis of the Earth electric field. We show that during severe geomagnetic storms the Katz fractal dimension of the geoelectric field grows rapidly. Full article
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<p>Temporal changes of the galactic cosmic ray (GCR) counts rate from Oulu neutron monitor: 27-day averages of daily data in the period April 1964–April 2020 (upper panel), half-day averages of 1-minute data in April 2008–August 2008 (middle panel), hourly data of GCR in 13–17 September 2008 (lower panel). Short periods marked by the brown dashed line in the upper panels are plotted with details in the lower panels displaying a self-similarity of the time series in various time-scales.</p> Full article ">Figure 2
<p>Geomagnetic field components, <math display="inline"><semantics> <mrow> <msub> <mi>B</mi> <mi>X</mi> </msub> <mo>,</mo> <msub> <mi>B</mi> <mi>Y</mi> </msub> </mrow> </semantics></math>, during the selected severe geomagnetic storms; the x-axis represents the day of year (Doy).</p> Full article ">Figure 3
<p>Geoelectric field, <span class="html-italic">E</span>, during the selected severe geomagnetic storms: (<b>a</b>) 6–7.04.2000, (<b>b</b>) 11–13.04.2001, (<b>c</b>) 29–30.05.2003, (<b>d</b>) 7–10.11.2004, (<b>e</b>) 14–15.12.2006 and (<b>f</b>) 7–8.09.2017, on the x–axis is marked the day of year (Doy).</p> Full article ">Figure 4
<p>Temporal changes of the geoelectric field during quiet days at the beginning of February 1999; the x-axis indicates the day of year (Doy).</p> Full article ">Figure 5
<p>Temporal changes of the geomagnetic field components, <math display="inline"><semantics> <mrow> <msub> <mi>B</mi> <mi>X</mi> </msub> <mo>,</mo> <msub> <mi>B</mi> <mi>Y</mi> </msub> </mrow> </semantics></math>, during quiet days at the beginning of February 1999; the x-axis indicates the day of year (Doy).</p> Full article ">Figure 6
<p>Fractal dimension: <math display="inline"><semantics> <mrow> <mi>f</mi> <msub> <mi>d</mi> <mi>K</mi> </msub> </mrow> </semantics></math> (blue straight line) and <math display="inline"><semantics> <mrow> <mi>f</mi> <msub> <mi>d</mi> <mn>2</mn> </msub> </mrow> </semantics></math> (red dashed line) of geoelectric field during the selected severe geomagnetic storms: (<b>a</b>) 6–7.04.2000, (<b>b</b>) 11–13.04.2001, (<b>c</b>) 29–30.05.2003, (<b>d</b>) 7–10.11.2004, (<b>e</b>) 14–15.12.2006 and (<b>f</b>) 7–8.09.2017, on the x–axis is marked the day of year (Doy).</p> Full article ">Figure 7
<p>Fractal dimension computed with the Katz approach, with moving windows of various lengths j = 60, 180, 300, 420 and 540 min, for the September 2017 Storm; the x-axis indicates the day of year (Doy).</p> Full article ">Figure 8
<p>Fractal dimension: <math display="inline"><semantics> <mrow> <mi>f</mi> <msub> <mi>d</mi> <mi>K</mi> </msub> </mrow> </semantics></math> (blue straight line) and <math display="inline"><semantics> <mrow> <mi>f</mi> <msub> <mi>d</mi> <mn>2</mn> </msub> </mrow> </semantics></math> (red dashed line) of geomagnetic <math display="inline"><semantics> <msub> <mi>B</mi> <mi>X</mi> </msub> </semantics></math> field component during the selected severe geomagnetic storms; the x-axis indicates the day of year (Doy).</p> Full article ">Figure 9
<p>Fractal dimension: <math display="inline"><semantics> <mrow> <mi>f</mi> <msub> <mi>d</mi> <mi>K</mi> </msub> </mrow> </semantics></math> (blue straight line) and <math display="inline"><semantics> <mrow> <mi>f</mi> <msub> <mi>d</mi> <mn>2</mn> </msub> </mrow> </semantics></math> (red dashed line) of geomagnetic <math display="inline"><semantics> <msub> <mi>B</mi> <mi>Y</mi> </msub> </semantics></math> field component during the selected severe geomagnetic storms; the x-axis indicates the day of year (Doy).</p> Full article ">Figure 10
<p>Temporal changes of the fractal dimension: <math display="inline"><semantics> <mrow> <mi>f</mi> <msub> <mi>d</mi> <mi>K</mi> </msub> </mrow> </semantics></math> (blue straight line) and <math display="inline"><semantics> <mrow> <mi>f</mi> <msub> <mi>d</mi> <mn>2</mn> </msub> </mrow> </semantics></math> (red dashed line) of the geoelectric field during quiet days at the beginning of February 1999; the x-axis indicates the day of year (Doy).</p> Full article ">Figure 11
<p>Temporal changes of the fractal dimension: <math display="inline"><semantics> <mrow> <mi>f</mi> <msub> <mi>d</mi> <mi>K</mi> </msub> </mrow> </semantics></math> (blue straight line) and <math display="inline"><semantics> <mrow> <mi>f</mi> <msub> <mi>d</mi> <mn>2</mn> </msub> </mrow> </semantics></math> (red dashed line) of the geomagnetic field components, <math display="inline"><semantics> <msub> <mi>B</mi> <mi>X</mi> </msub> </semantics></math> (left panel), and <math display="inline"><semantics> <msub> <mi>B</mi> <mi>Y</mi> </msub> </semantics></math> (right panel) during quiet days at the beginning of February 1999; the x-axis indicates the day of year (Doy).</p> Full article ">Figure 12
<p>Fractal dimension computed with the Katz <math display="inline"><semantics> <mrow> <mi>f</mi> <msub> <mi>d</mi> <mi>K</mi> </msub> </mrow> </semantics></math>, Sevcik <math display="inline"><semantics> <mrow> <mi>f</mi> <msub> <mi>d</mi> <mi>S</mi> </msub> </mrow> </semantics></math> and Higuchi <math display="inline"><semantics> <mrow> <mi>f</mi> <msub> <mi>d</mi> <mi>H</mi> </msub> </mrow> </semantics></math> (with <math display="inline"><semantics> <mrow> <msub> <mi>k</mi> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> </msub> <mo>=</mo> <mn>12</mn> </mrow> </semantics></math>) approach, for each 60 min, during the September 2017 geomagnetic storm; the x-axis indicates the day of year (Doy).</p> Full article ">
14 pages, 773 KiB  
Article
A Transnational and Transregional Study of the Impact and Effectiveness of Social Distancing for COVID-19 Mitigation
by Tarcísio M. Rocha Filho, Marcelo A. Moret and José F. F. Mendes
Entropy 2021, 23(11), 1530; https://doi.org/10.3390/e23111530 - 18 Nov 2021
Cited by 4 | Viewed by 2212
Abstract
We present an analysis of the relationship between SARS-CoV-2 infection rates and a social distancing metric from data for all the states and most populous cities in the United States and Brazil, all the 22 European Economic Community countries and the United Kingdom. [...] Read more.
We present an analysis of the relationship between SARS-CoV-2 infection rates and a social distancing metric from data for all the states and most populous cities in the United States and Brazil, all the 22 European Economic Community countries and the United Kingdom. We discuss why the infection rate, instead of the effective reproduction number or growth rate of cases, is a proper choice to perform this analysis when considering a wide span of time. We obtain a strong Spearman’s rank order correlation between the social distancing metric and the infection rate in each locality. We show that mask mandates increase the values of Spearman’s correlation in the United States, where a mandate was adopted. We also obtain an explicit numerical relation between the infection rate and the social distancing metric defined in the present work. Full article
(This article belongs to the Special Issue Statistical Methods for Medicine and Health Sciences)
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<p>Social distancing metric <span class="html-italic">M</span> in Equation (<a href="#FD5-entropy-23-01530" class="html-disp-formula">5</a>) for (<b>A</b>) Brazil states and (<b>B</b>) USA states.</p> Full article ">Figure 2
<p>(<b>A</b>) Time variation of <math display="inline"><semantics> <mrow> <mover> <mi>β</mi> <mo>¯</mo> </mover> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> and <math display="inline"><semantics> <msub> <mi>R</mi> <mi>t</mi> </msub> </semantics></math> for Los Angeles county. (<b>B</b>) Spearman’s rank-order correlation <math display="inline"><semantics> <mrow> <msub> <mi>r</mi> <mi>s</mi> </msub> <mrow> <mo>[</mo> <mi>M</mi> <mo>,</mo> <mover> <mi>β</mi> <mo>¯</mo> </mover> <mo>]</mo> </mrow> </mrow> </semantics></math> between the social distancing metric <span class="html-italic">M</span> and the infection rate <math display="inline"><semantics> <mover> <mi>β</mi> <mo>¯</mo> </mover> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>r</mi> <mi>s</mi> </msub> <mrow> <mo>[</mo> <mi>M</mi> <mo>,</mo> <msub> <mi>R</mi> <mi>t</mi> </msub> <mo>]</mo> </mrow> </mrow> </semantics></math> between <span class="html-italic">M</span> and the effective reproduction number <math display="inline"><semantics> <msub> <mi>R</mi> <mi>t</mi> </msub> </semantics></math> for Los Angeles county in the United States.</p> Full article ">Figure 3
<p>Spearman’s correlation index <math display="inline"><semantics> <msub> <mi>r</mi> <mi>s</mi> </msub> </semantics></math> between the social distancing metric <span class="html-italic">M</span> and the infection rate <math display="inline"><semantics> <mover> <mi>β</mi> <mo>¯</mo> </mover> </semantics></math> for: (<b>A</b>) Brazilian states; (<b>B</b>) Main Brazilian municipalities with population over 750 thousand; (<b>C</b>) 22 European countries; (D) US counties with at least one million inhabitants. (<b>D</b>) Main counties in the United States. Bar colors give the proportion of days with a mask mandate since the beginning of the pandemic in each location, up to 20 December 2020; (<b>E</b>) same as (<b>D</b>) but considering only the period with a mask mandate. States without a mask mandate in the period considered are marked in black. (<b>F</b>) Same as (<b>D</b>) but for all American states. (<b>G</b>) Same as (<b>E</b>) but for the American states.</p> Full article ">Figure 4
<p>Spearman’s correlation index <math display="inline"><semantics> <msub> <mi>r</mi> <mi>s</mi> </msub> </semantics></math> between each mobility variable and the infection rate <math display="inline"><semantics> <mover> <mi>β</mi> <mo>¯</mo> </mover> </semantics></math> for the main Brazilian municipalities, each Brazilian state and European countries.</p> Full article ">Figure 5
<p>Spearman’s correlation index <math display="inline"><semantics> <msub> <mi>r</mi> <mi>s</mi> </msub> </semantics></math> between changes in each mobility category and the infection rate <math display="inline"><semantics> <mover> <mi>β</mi> <mo>¯</mo> </mover> </semantics></math> for (<b>A</b>) counties with more than one million inhabitants and one thousand deaths for the period from the first COVID-19 case up to 20 December 2020; (<b>B</b>) same as (<b>A</b>) but for the period with a mask mandate, except those counties with no mask mandate in 2020 (marked in black in <a href="#entropy-23-01530-f003" class="html-fig">Figure 3</a>E), for which the whole period is considered; (<b>C</b>) same as (<b>A</b>) for all US states; (<b>D</b>) same as (<b>B</b>) for all US states.</p> Full article ">Figure 6
<p>Total number of deaths per 100 thousand inhabitants at the end of the considered period as a function of the average value of <math display="inline"><semantics> <mrow> <mover> <mi>β</mi> <mo>¯</mo> </mover> <mo>/</mo> <mi>γ</mi> </mrow> </semantics></math> during the same time span.</p> Full article ">Figure 7
<p>Coefficient <math display="inline"><semantics> <mrow> <mover> <mi>β</mi> <mo>¯</mo> </mover> <mo>/</mo> <mi>γ</mi> <mi>M</mi> <mo>=</mo> <mi>α</mi> <mo>/</mo> <mi>γ</mi> </mrow> </semantics></math> for (<b>A</b>) Brazilian states; (<b>B</b>) Brazilian municipalities; (<b>C</b>) US states; (<b>D</b>) US counties and (<b>E</b>) European countries. The normalized histogram (in red) and the log-normal distribution function (in black) for the values for <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>/</mo> <mi>γ</mi> </mrow> </semantics></math>: (<b>F</b>) Brazilian states; (<b>G</b>) Brazilian municipalities, (<b>H</b>) US states; (<b>I</b>) US counties and (<b>J</b>) European countries. The values for <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>/</mo> <mi>γ</mi> </mrow> </semantics></math> (CI <math display="inline"><semantics> <mrow> <mn>95</mn> <mo>%</mo> </mrow> </semantics></math>) are <math display="inline"><semantics> <mrow> <mn>0.015</mn> </mrow> </semantics></math> (<math display="inline"><semantics> <mrow> <mn>0.0096</mn> </mrow> </semantics></math>–<math display="inline"><semantics> <mrow> <mn>0.023</mn> </mrow> </semantics></math>), <math display="inline"><semantics> <mrow> <mn>0.019</mn> </mrow> </semantics></math> (<math display="inline"><semantics> <mrow> <mn>0.0081</mn> </mrow> </semantics></math>–<math display="inline"><semantics> <mrow> <mn>0.042</mn> </mrow> </semantics></math>), <math display="inline"><semantics> <mrow> <mn>0.014</mn> </mrow> </semantics></math> (<math display="inline"><semantics> <mrow> <mn>0.0089</mn> </mrow> </semantics></math>–<math display="inline"><semantics> <mrow> <mn>0.021</mn> </mrow> </semantics></math>), <math display="inline"><semantics> <mrow> <mn>0.015</mn> </mrow> </semantics></math> (<math display="inline"><semantics> <mrow> <mn>0.0091</mn> </mrow> </semantics></math>–<math display="inline"><semantics> <mrow> <mn>0.027</mn> </mrow> </semantics></math>) and <math display="inline"><semantics> <mrow> <mn>0.014</mn> </mrow> </semantics></math> (<math display="inline"><semantics> <mrow> <mn>0.0084</mn> </mrow> </semantics></math>–<math display="inline"><semantics> <mrow> <mn>0.024</mn> </mrow> </semantics></math>), respectively.</p> Full article ">
13 pages, 418 KiB  
Article
Still No Free Lunches: The Price to Pay for Tighter PAC-Bayes Bounds
by Benjamin Guedj and Louis Pujol
Entropy 2021, 23(11), 1529; https://doi.org/10.3390/e23111529 - 18 Nov 2021
Cited by 10 | Viewed by 2511
Abstract
“No free lunch” results state the impossibility of obtaining meaningful bounds on the error of a learning algorithm without prior assumptions and modelling, which is more or less realistic for a given problem. Some models are “expensive” (strong assumptions, such as sub-Gaussian tails), [...] Read more.
“No free lunch” results state the impossibility of obtaining meaningful bounds on the error of a learning algorithm without prior assumptions and modelling, which is more or less realistic for a given problem. Some models are “expensive” (strong assumptions, such as sub-Gaussian tails), others are “cheap” (simply finite variance). As it is well known, the more you pay, the more you get: in other words, the most expensive models yield the more interesting bounds. Recent advances in robust statistics have investigated procedures to obtain tight bounds while keeping the cost of assumptions minimal. The present paper explores and exhibits what the limits are for obtaining tight probably approximately correct (PAC)-Bayes bounds in a robust setting for cheap models. Full article
(This article belongs to the Special Issue Approximate Bayesian Inference)
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<p><math display="inline"><semantics> <mrow> <msqrt> <mn>2</mn> </msqrt> <mo>×</mo> <msqrt> <mrow> <mn>2</mn> <mo form="prefix">log</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>/</mo> <mi>δ</mi> <mo stretchy="false">)</mo> </mrow> </msqrt> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>/</mo> <msqrt> <mi>δ</mi> </msqrt> </mrow> </semantics></math> with respect to <math display="inline"><semantics> <mi>δ</mi> </semantics></math>.</p> Full article ">
19 pages, 27768 KiB  
Article
Constructal Optimization of Rectangular Microchannel Heat Sink with Porous Medium for Entropy Generation Minimization
by Wenlong Li, Zhihui Xie, Kun Xi, Shaojun Xia and Yanlin Ge
Entropy 2021, 23(11), 1528; https://doi.org/10.3390/e23111528 - 17 Nov 2021
Cited by 14 | Viewed by 2512
Abstract
A model of rectangular microchannel heat sink (MCHS) with porous medium (PM) is developed. Aspect ratio of heat sink (HS) cell and length-width ratio of HS are optimized by numerical simulation method for entropy generation minimization (EGM) according to constructal theory. The effects [...] Read more.
A model of rectangular microchannel heat sink (MCHS) with porous medium (PM) is developed. Aspect ratio of heat sink (HS) cell and length-width ratio of HS are optimized by numerical simulation method for entropy generation minimization (EGM) according to constructal theory. The effects of inlet Reynolds number (Re) of coolant, heat flux on bottom, porosity and volume proportion of PM on dimensionless entropy generation rate (DEGR) are analyzed. From the results, there are optimal aspect ratios to minimize DEGR. Given the initial condition, DEGR is 33.10% lower than its initial value after the aspect ratio is optimized. With the increase of Re, the optimal aspect ratio declines, and the minimum DEGR drops as well. DEGR gets larger and the optimal aspect ratio remains constant with the increasing of heat flux on bottom. For the different volume proportion of PM, the optimal aspect ratios are diverse, but the minimum DEGR almost stays unchanged. The twice minimized DEGR, which results from aspect ratio and length-width ratio optimized simultaneously, is 10.70% lower than the once minimized DEGR. For a rectangular bottom, a lower DEGR can be reached by choosing the proper direction of fluid flow. Full article
(This article belongs to the Special Issue Entropy in Computational Fluid Dynamics III)
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<p>Schematic diagram of MCHS with PM, (<b>a</b>) MCHS with PM, (<b>b</b>) MCHS cell with PM.</p> Full article ">Figure 2
<p>Variations of <span class="html-italic">T<sub>max</sub></span>, <span class="html-italic">P</span> versus <span class="html-italic">α</span>.</p> Full article ">Figure 3
<p>Variations of <math display="inline"><semantics> <mrow> <msub> <mover accent="true"> <mover accent="true"> <mi>S</mi> <mo>˙</mo> </mover> <mo>˜</mo> </mover> <mi>g</mi> </msub> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mover accent="true"> <mover accent="true"> <mi>S</mi> <mo>˙</mo> </mover> <mo>˜</mo> </mover> <mrow> <mi>g</mi> <mo>,</mo> <mi>s</mi> </mrow> </msub> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mover accent="true"> <mover accent="true"> <mi>S</mi> <mo>˙</mo> </mover> <mo>˜</mo> </mover> <mrow> <mi>g</mi> <mo>,</mo> <mi>p</mi> </mrow> </msub> </mrow> </semantics></math> versus <span class="html-italic">α</span>.</p> Full article ">Figure 4
<p>Variation of <span class="html-italic">Be</span> versus <span class="html-italic">α</span>.</p> Full article ">Figure 5
<p>Variations of <math display="inline"><semantics> <mrow> <msub> <mover accent="true"> <mover accent="true"> <mi>S</mi> <mo>˙</mo> </mover> <mo>˜</mo> </mover> <mi>g</mi> </msub> </mrow> </semantics></math> with <span class="html-italic">α</span> for different <span class="html-italic">Re</span>.</p> Full article ">Figure 6
<p>Temperature and temperature gradient profiles of optimal cell geometry for different <span class="html-italic">Re.</span> (<b>a</b>) Temperature profile, (<b>b</b>) Temperature gradient profile.</p> Full article ">Figure 6 Cont.
<p>Temperature and temperature gradient profiles of optimal cell geometry for different <span class="html-italic">Re.</span> (<b>a</b>) Temperature profile, (<b>b</b>) Temperature gradient profile.</p> Full article ">Figure 7
<p>Variations of <math display="inline"><semantics> <mrow> <msub> <mover accent="true"> <mover accent="true"> <mi>S</mi> <mo>˙</mo> </mover> <mo>˜</mo> </mover> <mi>g</mi> </msub> </mrow> </semantics></math> with <span class="html-italic">α</span> for different <math display="inline"><semantics> <msup> <mi>q</mi> <mo>″</mo> </msup> </semantics></math>.</p> Full article ">Figure 8
<p>Variations of <math display="inline"><semantics> <mrow> <msub> <mover accent="true"> <mover accent="true"> <mi>S</mi> <mo>˙</mo> </mover> <mo>˜</mo> </mover> <mi>g</mi> </msub> </mrow> </semantics></math> versus <span class="html-italic">α</span> and <math display="inline"><semantics> <mrow> <msub> <mi>ϕ</mi> <mi>p</mi> </msub> </mrow> </semantics></math>.</p> Full article ">Figure 9
<p>Temperature and temperature gradient profiles of optimal cell geometry for different <math display="inline"><semantics> <mrow> <msub> <mi>ϕ</mi> <mi>p</mi> </msub> </mrow> </semantics></math>. (<b>a</b>) Temperature profile, (<b>b</b>) Temperature gradient profile.</p> Full article ">Figure 10
<p>Variations of <math display="inline"><semantics> <mrow> <msub> <mover accent="true"> <mover accent="true"> <mi>S</mi> <mo>˙</mo> </mover> <mo>˜</mo> </mover> <mi>g</mi> </msub> </mrow> </semantics></math> with <span class="html-italic">α</span> for different <span class="html-italic">ε</span>.</p> Full article ">Figure 11
<p>Variations of <math display="inline"><semantics> <mrow> <msub> <mover accent="true"> <mover accent="true"> <mi>S</mi> <mo>˙</mo> </mover> <mo>˜</mo> </mover> <mi>g</mi> </msub> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mover accent="true"> <mover accent="true"> <mi>S</mi> <mo>˙</mo> </mover> <mo>˜</mo> </mover> <mrow> <mi>g</mi> <mo>,</mo> <mi>p</mi> </mrow> </msub> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mover accent="true"> <mover accent="true"> <mi>S</mi> <mo>˙</mo> </mover> <mo>˜</mo> </mover> <mrow> <mi>g</mi> <mo>,</mo> <mi>s</mi> </mrow> </msub> </mrow> </semantics></math> with <span class="html-italic">β</span> for different <span class="html-italic">Re.</span> (<b>a</b>) <span class="html-italic">Re</span> = 100, (<b>b</b>) <span class="html-italic">Re</span> = 200, (<b>c</b>) <span class="html-italic">Re</span> = 300.</p> Full article ">Figure 12
<p>Characteristics of <math display="inline"><semantics> <mrow> <msub> <mover accent="true"> <mover accent="true"> <mi>S</mi> <mo>˙</mo> </mover> <mo>˜</mo> </mover> <mi>g</mi> </msub> </mrow> </semantics></math> versus <span class="html-italic">α</span> and <span class="html-italic">β</span>.</p> Full article ">
14 pages, 796 KiB  
Article
Limits to Perception by Quantum Monitoring with Finite Efficiency
by Luis Pedro García-Pintos and Adolfo del Campo
Entropy 2021, 23(11), 1527; https://doi.org/10.3390/e23111527 - 17 Nov 2021
Cited by 5 | Viewed by 2186
Abstract
We formulate limits to perception under continuous quantum measurements by comparing the quantum states assigned by agents that have partial access to measurement outcomes. To this end, we provide bounds on the trace distance and the relative entropy between the assigned state and [...] Read more.
We formulate limits to perception under continuous quantum measurements by comparing the quantum states assigned by agents that have partial access to measurement outcomes. To this end, we provide bounds on the trace distance and the relative entropy between the assigned state and the actual state of the system. These bounds are expressed solely in terms of the purity and von Neumann entropy of the state assigned by the agent, and are shown to characterize how an agent’s perception of the system is altered by access to additional information. We apply our results to Gaussian states and to the dynamics of a system embedded in an environment illustrated on a quantum Ising chain. Full article
(This article belongs to the Special Issue Quantum Darwinism and Friends)
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<p><b>Illustration of the varying degrees of perception by different agents.</b> The amount of information that an agent possesses of a system can drastically alter its perception, as the expectations of outcomes for measurements performed on the system can differ. (<b>a</b>) The state <math display="inline"><semantics> <msubsup> <mi>ρ</mi> <mrow> <mi>t</mi> </mrow> <mi mathvariant="script">O</mi> </msubsup> </semantics></math> assigned by omniscient agent <math display="inline"><semantics> <mi mathvariant="script">O</mi> </semantics></math>, who has full access to the measurement outcomes, corresponds to a complete pure-state description of the system. <span class="html-italic">O</span> thus has the most accurate predictive power. (<b>b</b>) An agent <math display="inline"><semantics> <mi mathvariant="script">A</mi> </semantics></math> completely ignorant of measurement outcomes possesses the most incomplete description of the system. (<b>c</b>) A continuous transition between the two descriptions, corresponding to the worst and most complete perceptions of the system respectively, is obtained by considering an agent <math display="inline"><semantics> <mi mathvariant="script">B</mi> </semantics></math> with partial access to the measurement outcomes of the monitoring process.</p> Full article ">Figure 2
<p><b>Evolution of the average relative entropy.</b> Simulated evolution of the average <math display="inline"><semantics> <mrow> <mfenced separators="" open="&#x2329;" close="&#x232A;"> <mi>S</mi> <mfenced separators="" open="(" close=")"> <msubsup> <mi>ρ</mi> <mrow> <mi>t</mi> </mrow> <mi mathvariant="script">O</mi> </msubsup> <mrow> <mo>|</mo> <mo>|</mo> </mrow> <msubsup> <mi>ρ</mi> <mrow> <mi>t</mi> </mrow> <mi mathvariant="script">B</mi> </msubsup> </mfenced> </mfenced> <mo>=</mo> <mfenced separators="" open="&#x2329;" close="&#x232A;"> <mi>S</mi> <mfenced separators="" open="(" close=")"> <msubsup> <mi>ρ</mi> <mrow> <mi>t</mi> </mrow> <mi mathvariant="script">B</mi> </msubsup> </mfenced> </mfenced> </mrow> </semantics></math> of the relative entropy between complete and incomplete descriptions for a spin chain initially in a paramagnetic state on which individual spin components <math display="inline"><semantics> <msubsup> <mi>σ</mi> <mi>j</mi> <mi>z</mi> </msubsup> </semantics></math> are monitored. Here <math display="inline"><semantics> <mrow> <mo>〈</mo> <mo>·</mo> <mo>〉</mo> </mrow> </semantics></math> denotes an average over all measurement outcomes, and <math display="inline"><semantics> <mrow> <msubsup> <mi>ρ</mi> <mrow> <mi>t</mi> </mrow> <mi mathvariant="script">B</mi> </msubsup> <mo>=</mo> <msub> <mrow> <mo>〈</mo> <msubsup> <mi>ρ</mi> <mi>t</mi> <mi mathvariant="script">O</mi> </msubsup> <mo>〉</mo> </mrow> <mi mathvariant="script">B</mi> </msub> </mrow> </semantics></math> is the state assigned by agent <math display="inline"><semantics> <mi mathvariant="script">B</mi> </semantics></math> after discarding the outcomes unknown to him. The simulation corresponds to <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>6</mn> </mrow> </semantics></math> spins, with couplings <math display="inline"><semantics> <mrow> <mi>J</mi> <msub> <mi>τ</mi> <mi>m</mi> </msub> <mo>=</mo> <mi>h</mi> <msub> <mi>τ</mi> <mi>m</mi> </msub> <mo>=</mo> <mn>1</mn> <mo>/</mo> <mn>2</mn> </mrow> </semantics></math>. For <math display="inline"><semantics> <mrow> <mi>η</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> (black continuous curve), agent <math display="inline"><semantics> <mi mathvariant="script">A</mi> </semantics></math>, without any access to the measurement outcomes, has the most incomplete description of the system. For <math display="inline"><semantics> <mrow> <mi>η</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math> (red dashed curve), <math display="inline"><semantics> <mi mathvariant="script">B</mi> </semantics></math> gets closer to the complete description of the state of the system, after gaining access to partial measurement results. Finally, when <math display="inline"><semantics> <mrow> <mi>η</mi> <mo>=</mo> <mn>0.9</mn> </mrow> </semantics></math> (blue dotted curve), access to enough information provides <math display="inline"><semantics> <mi mathvariant="script">B</mi> </semantics></math> with an almost complete description of the state. Importantly, in all cases the agent can estimate how far the description possessed is from the complete one solely in terms of the entropy <math display="inline"><semantics> <mrow> <mi>S</mi> <mo>(</mo> <msubsup> <mi>ρ</mi> <mrow> <mi>t</mi> </mrow> <mi mathvariant="script">B</mi> </msubsup> <mo>)</mo> </mrow> </semantics></math>.</p> Full article ">Figure 3
<p><b>Transition between levels of perception.</b> Bounds on average trace distance (<b>left</b>) and average relative entropy (<b>right</b>) as function of measurement efficiency for a harmonic oscillator undergoing monitoring of its position. For such a system the purity of the state <math display="inline"><semantics> <msubsup> <mi>ρ</mi> <mrow> <mi>t</mi> </mrow> <mi mathvariant="script">B</mi> </msubsup> </semantics></math> depends solely on the measurement efficiency with which observer <math display="inline"><semantics> <mi mathvariant="script">B</mi> </semantics></math> monitors the system. This illustrates the transition from complete ignorance of the outcomes of measurements performed (<math display="inline"><semantics> <mrow> <mi>η</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>), to the most complete description as <math display="inline"><semantics> <mrow> <mi>η</mi> <mo>→</mo> <mn>1</mn> </mrow> </semantics></math>—the situation with the most accurate perception. Efficient use of information happens when a small fraction of the measurement output is incorporated at <math display="inline"><semantics> <mrow> <mi>η</mi> <mo>≪</mo> <mn>1</mn> </mrow> </semantics></math>, as then both <math display="inline"><semantics> <mrow> <mi mathvariant="script">D</mi> <mfenced separators="" open="(" close=")"> <msubsup> <mi>ρ</mi> <mrow> <mi>t</mi> </mrow> <mi mathvariant="script">B</mi> </msubsup> <mo>,</mo> <msubsup> <mi>ρ</mi> <mrow> <mi>t</mi> </mrow> <mi mathvariant="script">O</mi> </msubsup> </mfenced> </mrow> </semantics></math> and the relative entropy <math display="inline"><semantics> <mrow> <mi>S</mi> <mfenced separators="" open="(" close=")"> <msubsup> <mi>ρ</mi> <mrow> <mi>t</mi> </mrow> <mi mathvariant="script">O</mi> </msubsup> <mrow> <mo>|</mo> <mo>|</mo> </mrow> <msubsup> <mi>ρ</mi> <mrow> <mi>t</mi> </mrow> <mi mathvariant="script">B</mi> </msubsup> </mfenced> </mrow> </semantics></math> decay rapidly.</p> Full article ">Figure A1
<p><b>Evolution of the average trace distance and its bounds.</b> Simulated evolution of the average trace distance <math display="inline"><semantics> <mfenced separators="" open="&#x2329;" close="&#x232A;"> <mi mathvariant="script">D</mi> <mfenced separators="" open="(" close=")"> <msubsup> <mi>ρ</mi> <mrow> <mi>T</mi> </mrow> <mi mathvariant="script">O</mi> </msubsup> <mo>,</mo> <msubsup> <mi>ρ</mi> <mrow> <mi>T</mi> </mrow> <mi mathvariant="script">B</mi> </msubsup> </mfenced> </mfenced> </semantics></math> between complete and incomplete descriptions for a spin chain initially in a paramagnetic state on which individual spin components <math display="inline"><semantics> <msubsup> <mi>σ</mi> <mi>j</mi> <mi>z</mi> </msubsup> </semantics></math> are monitored. The simulation corresponds to <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>6</mn> </mrow> </semantics></math> spins, with couplings <math display="inline"><semantics> <mrow> <mi>J</mi> <msub> <mi>τ</mi> <mi>m</mi> </msub> <mo>=</mo> <mi>h</mi> <msub> <mi>τ</mi> <mi>m</mi> </msub> <mo>=</mo> <mn>1</mn> <mo>/</mo> <mn>2</mn> </mrow> </semantics></math>. The upper and lower bounds (16) on the average trace distance is depicted by dashed lines, while the shaded area represents the (one standard deviation) confidence region obtained from the upper bound (13) on the standard deviation in the main text, calculated with respect to the mean distance. For <math display="inline"><semantics> <mrow> <mi>η</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> (<b>left</b>), agent <math display="inline"><semantics> <mi mathvariant="script">A</mi> </semantics></math>, without any access to the measurement outcomes, has the most incomplete description of the system. After gaining access to partial measurement results, with <math display="inline"><semantics> <mrow> <mi>η</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math> (<b>center</b>) <math display="inline"><semantics> <mi mathvariant="script">B</mi> </semantics></math> gets closer to the complete description of the state of the system. Finally, when <math display="inline"><semantics> <mrow> <mi>η</mi> <mo>=</mo> <mn>0.9</mn> </mrow> </semantics></math> (<b>right</b>), access to enough information provides <math display="inline"><semantics> <mi mathvariant="script">B</mi> </semantics></math> with an almost complete description of the state. Importantly, in all cases the agent can bound how far the description possessed is from the complete one solely in terms solely of the purity <math display="inline"><semantics> <mrow> <mi mathvariant="script">P</mi> <mfenced separators="" open="(" close=")"> <msubsup> <mi>ρ</mi> <mrow> <mi>T</mi> </mrow> <mi mathvariant="script">B</mi> </msubsup> </mfenced> </mrow> </semantics></math>.</p> Full article ">Figure A2
<p><b>Evolution of the average relative entropy and its bounds.</b> Simulated evolution of the average relative entropy <math display="inline"><semantics> <mfenced separators="" open="&#x2329;" close="&#x232A;"> <mi>S</mi> <mfenced separators="" open="(" close=")"> <msubsup> <mi>ρ</mi> <mrow> <mi>T</mi> </mrow> <mi mathvariant="script">O</mi> </msubsup> <mrow> <mo>|</mo> <mo>|</mo> </mrow> <msubsup> <mi>ρ</mi> <mrow> <mi>T</mi> </mrow> <mi mathvariant="script">B</mi> </msubsup> </mfenced> </mfenced> </semantics></math> between complete and incomplete descriptions for a spin chain on which the <span class="html-italic">z</span> components of individual spins are monitored. The shaded area represents the (one standard deviation) confidence region obtained from the upper bound on the standard deviation of the relative entropy, Equation (14) in the main text. As in the case of the trace distance, access to more information leads to a more accurate state assigned by the agent.</p> Full article ">
13 pages, 291 KiB  
Article
The Solvability of the Discrete Boundary Value Problem on the Half-Line
by Magdalena Nockowska-Rosiak
Entropy 2021, 23(11), 1526; https://doi.org/10.3390/e23111526 - 17 Nov 2021
Viewed by 1747
Abstract
This paper provides conditions for the existence of a solution to the second-order nonlinear boundary value problem on the half-line of the form [...] Read more.
This paper provides conditions for the existence of a solution to the second-order nonlinear boundary value problem on the half-line of the form Δa(n)Δx(n)=f(n+1,x(n+1),Δx(n+1)),n∈N∪{0}, with αx(0)+βa(0)Δx(0)=0,x(∞)=d, where d,α,β∈R, α2+β2>0. To achieve our goal, we use Schauder’s fixed-point theorem and the perturbation technique for a Fredholm operator of index 0. Moreover, we construct the necessary condition for the existence of a solution to the considered problem. Full article
(This article belongs to the Section Complexity)
13 pages, 15264 KiB  
Article
Entropy-Based Combined Metric for Automatic Objective Quality Assessment of Stitched Panoramic Images
by Krzysztof Okarma, Wojciech Chlewicki, Mateusz Kopytek, Beata Marciniak and Vladimir Lukin
Entropy 2021, 23(11), 1525; https://doi.org/10.3390/e23111525 - 17 Nov 2021
Cited by 8 | Viewed by 2098
Abstract
Quality assessment of stitched images is an important element of many virtual reality and remote sensing applications where the panoramic images may be used as a background as well as for navigation purposes. The quality of stitched images may be decreased by several [...] Read more.
Quality assessment of stitched images is an important element of many virtual reality and remote sensing applications where the panoramic images may be used as a background as well as for navigation purposes. The quality of stitched images may be decreased by several factors, including geometric distortions, ghosting, blurring, and color distortions. Nevertheless, the specificity of such distortions is different than those typical for general-purpose image quality assessment. Therefore, the necessity of the development of new objective image quality metrics for such type of emerging applications becomes obvious. The method proposed in the paper is based on the combination of features used in some recently proposed metrics with the results of the local and global image entropy analysis. The results obtained applying the proposed combined metric have been verified using the ISIQA database, containing 264 stitched images of 26 scenes together with the respective subjective Mean Opinion Scores, leading to a significant increase of its correlation with subjective evaluation results. Full article
(This article belongs to the Special Issue Advances in Computer Recognition, Image Processing and Communications, Selected Papers from CORES 2021 and IP&C 2021)
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<p>Sample constituent and stitched images with various distortions from the ISIQA dataset.</p> Full article ">Figure 2
<p>Scatter plots for SIQE (red points), two variants of the EntSIQE metric (blue and black points), and the proposed two variants of the EntSIQE<math display="inline"><semantics> <msup> <mrow/> <mo>+</mo> </msup> </semantics></math> metric (green and violet points).</p> Full article ">
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