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Keywords = Hölder divergences

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30 pages, 4794 KiB  
Article
Gathering and Cooking Seaweeds in Contemporary Ireland: Beyond Plant Foraging and Trendy Gastronomies
by Dauro M. Zocchi, Giulia Mattalia, Jeovana Santos Nascimento, Ryan Marley Grant, Jack Edwin Martin, Regina Sexton, Chiara Romano and Andrea Pieroni
Sustainability 2024, 16(8), 3337; https://doi.org/10.3390/su16083337 - 16 Apr 2024
Viewed by 1790
Abstract
Seaweed has historically been essential for coastal communities worldwide. Following a period of decline in the last century, Ireland has seen a recent resurgence in the appreciation and use of seaweed. This research explores the evolution in seaweed foraging practices, with a specific [...] Read more.
Seaweed has historically been essential for coastal communities worldwide. Following a period of decline in the last century, Ireland has seen a recent resurgence in the appreciation and use of seaweed. This research explores the evolution in seaweed foraging practices, with a specific focus on gastronomical uses in two Irish regions: the southwest and the west and midwest. It examines the diversity of seaweed and its present and past uses, comparing abandonment, continuation and revitalisation trajectories. Qualitative data were gathered through semi-structured interviews with 27 individuals who forage seaweed for commercial or personal use. We identified 22 seaweed species across the study areas, predominantly from the Fucaceae, Laminariaceae and Ulvaceae families. There was a fair divergence between the seaweed species used in the two study areas (16 seaweed species in the southwest region and 17 seaweed species in the west and midwest region), with 11 species mentioned in both areas. Different trajectories of resurgence were identified. In the west and midwest region, the revitalisation of local ecological and gastronomic knowledge related to seaweeds seems to be deeply entrenched in the territory’s historical legacy, showing a sort of continuation with the past and having followed a more commercially oriented path. Conversely, in the southwest region, the revival seems to be fostered by new knowledge holders with a contemporary interest in reconnecting with the marine landscape and promoting educational activities centred around seaweed. This research contributes to discussions on sustainable food systems and food heritage promotion, emphasising seaweed’s potential role in Irish coastal communities’ foodscapes. Full article
(This article belongs to the Special Issue Wild Food for Healthy, Sustainable, and Equitable Local Food Systems)
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<p>Map showing the study areas within Ireland (file credit: Creative Commons Attribution-Share Alike 3.0 licence).</p>
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<p>Sundried seaweeds collected on Squince Beach, West Cork, SW (Photo: Jeovana Santos Nascimento).</p>
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<p><span class="html-italic">Ascophyllum nodosum</span> (Photo: Jeovana Santos Nascimento).</p>
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<p><span class="html-italic">Porphyra</span> spp. cooked in seawater with potatoes in Quilty, County Claire (photo: Jack Edwin Martin).</p>
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<p>Potato bed fertilised with seaweed in Kinvarra, Galway Bay (photo: Jack Edwin Martin).</p>
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<p>Chord diagram showing the overlaps and differences in the species mentioned by the respondents in the two study areas. Key: SW (southwest), W (west and midwest).</p>
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25 pages, 398 KiB  
Article
Estimations of the Jensen Gap and Their Applications Based on 6-Convexity
by Muhammad Adil Khan, Asadullah Sohail, Hidayat Ullah and Tareq Saeed
Mathematics 2023, 11(8), 1957; https://doi.org/10.3390/math11081957 - 20 Apr 2023
Cited by 4 | Viewed by 1546
Abstract
The main purpose of this manuscript is to present some new estimations of the Jensen gap in a discrete sense along with their applications. The proposed estimations for the Jensen gap are provided with the help of the notion of 6-convex functions. Some [...] Read more.
The main purpose of this manuscript is to present some new estimations of the Jensen gap in a discrete sense along with their applications. The proposed estimations for the Jensen gap are provided with the help of the notion of 6-convex functions. Some numerical experiments are performed to determine the significance and correctness of the intended estimates. Several outcomes of the main results are discussed for the Hölder inequality and the power and quasi-arithmetic means. Furthermore, some applications are presented in information theory, which provide some bounds for the divergences, Bhattacharyya coefficient, Shannon entropy, and Zipf–Mandelbrot entropy. Full article
(This article belongs to the Special Issue Mathematical Inequalities, Models and Applications)
14 pages, 2577 KiB  
Article
A Qualitative Exploration of Conflicts in Human-Wildlife Interactions in Namibia’s Kunene Region
by Robert Luetkemeier, Ronja Kraus, Meed Mbidzo, Morgan Hauptfleisch, Stefan Liehr and Niels Blaum
Diversity 2023, 15(3), 440; https://doi.org/10.3390/d15030440 - 16 Mar 2023
Cited by 1 | Viewed by 2749
Abstract
Wildlife numbers are declining globally due to anthropogenic pressures. In Namibia, however, wildlife populations increased with policy instruments that allow private ownership and incentivize their sustainable use. Antithetically, this resulted in increased resource competition between humans and wildlife and triggered conflicts among various [...] Read more.
Wildlife numbers are declining globally due to anthropogenic pressures. In Namibia, however, wildlife populations increased with policy instruments that allow private ownership and incentivize their sustainable use. Antithetically, this resulted in increased resource competition between humans and wildlife and triggered conflicts among various stakeholder groups. This paper summarizes the results of a qualitative exploration of conflicts in wildlife management in Namibia’s Kunene Region, adjacent to Etosha National Park. We conducted a workshop and expert interviews with stakeholders from relevant sectors. Our qualitative research sheds light on societal conflicts over wildlife that originate from diverging interests, livelihood strategies, moral values, knowledge holders, personal relations and views on institutional procedures. We frame our insights into conflicting human–wildlife interactions with theoretical concepts of social-ecological systems, ecosystem services and ecosystem disservices and open the floor for quantitative assessments. Overall, our results may present a suitable way of understanding biodiversity conflicts in a theoretical way. Full article
(This article belongs to the Special Issue Human Wildlife Conflict across Landscapes)
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<p>Study area in the northwest of Namibia, south of the Etosha National Park. The map indicates the different land use types, key agglomerations and infrastructural features.</p>
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<p>Conceptualization of how conflicts between stakeholders emerge due to diverging views on elephants as either ES or ESD. Adapted from [<a href="#B27-diversity-15-00440" class="html-bibr">27</a>].</p>
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16 pages, 2270 KiB  
Article
Comparison of Nutri-Score and Health Star Rating Nutrient Profiling Models Using Large Branded Foods Composition Database and Sales Data
by Edvina Hafner and Igor Pravst
Int. J. Environ. Res. Public Health 2023, 20(5), 3980; https://doi.org/10.3390/ijerph20053980 - 23 Feb 2023
Cited by 8 | Viewed by 3229
Abstract
Front-of-package nutrition labelling (FOPNL) is known as an effective tool that can encourage healthier food choices and food reformulation. A very interesting type of FOPNL is grading schemes. Our objective was to compare two market-implemented grading schemes—European Nutri-Score (NS) and Australian Health Star [...] Read more.
Front-of-package nutrition labelling (FOPNL) is known as an effective tool that can encourage healthier food choices and food reformulation. A very interesting type of FOPNL is grading schemes. Our objective was to compare two market-implemented grading schemes—European Nutri-Score (NS) and Australian Health Star Rating (HSR), using large Slovenian branded foods database. NS and HSR were used for profiling 17,226 pre-packed foods and drinks, available in Slovenian food supply dataset (2020). Alignment between models was evaluated with agreement (% of agreement and Cohen’s Kappa) and correlation (Spearman rho). The 12-month nationwide sales-data were used for sale-weighing, to address market-share differences. Study results indicated that both models have good discriminatory ability between products based on their nutritional composition. NS and HSR ranked 22% and 33% of Slovenian food supply as healthy, respectively. Agreement between NS and HSR was strong (70%, κ = 0.62) with a very strong correlation (rho = 0.87). Observed profiling models were most aligned within food categories Beverages and Bread and bakery products, while less aligned for Dairy and imitates and Edible oils and emulsions. Notable disagreements were particularly observed in subcategories of Cheese and processed cheeses (8%, κ = 0.01, rho = 0.38) and Cooking oils (27%, κ = 0.11, rho = 0.40). Further analysis showed that the main differences in Cooking oils were due to olive oil and walnut oil, which are favoured by NS and grapeseed, flaxseed and sunflower oil that are favoured by HSR. For Cheeses and cheese products, we observed that HSR graded products across the whole scale, with majority (63%) being classified as healthy (≥3.5 *), while NS mostly graded lower scores. Sale-weighting analyses showed that offer in the food supply does not always reflect the sales. Sale-weighting increased overall agreement between profiles from 70% to 81%, with notable differences between food categories. In conclusion, NS and HSR were shown as highly compliant FOPNLs with few divergences in some subcategories. Even these models do not always grade products equally high, very similar ranking trends were observed. However, the observed differences highlight the challenges of FOPNL ranking schemes, which are tailored to address somewhat different public health priorities in different countries. International harmonization can support further development of grading type nutrient profiling models for the use in FOPNL, and make those acceptable for more stake-holders, which will be crucial for their successful regulatory implementation. Full article
(This article belongs to the Special Issue Food and Public Health: Food Supply, Marketing and Consumers)
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<p>Overall distribution of profiling scores according to Nutri-Score (FSAm-NPS (Food Standard Agency- Nutrient Profiling System)) and Health Star Rating algorithm in conjunction with their final grade. Colours represent different Nutri-Score and Health Star Rating grades across 2020 Slovenian food supply (n = 17,226). * refers to stars of Health Star Rating grades.</p>
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<p>Distribution of Nutri-Score and Health Star Rating grades across main categories. Colours represent different Nutri-Score and Health Star Rating grades. * refers to stars of Health Star Rating grades.</p>
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<p>Agreement (% of agreement and Cohen’s Kappa) and correlation (Spearman rho) between Nutri-Score and Health Star Rating for subcategories. The main categories of the analysed subcategories are presented on the right. Cut offs ranges for agreement and correlations were as follows (multiplied by 100% for % of agreement): 0–0.20 negligible (red); 0.21–0.40 weak (orange); 0.41–0.60 moderate (yellow); 0.61–0.80 strong (light green); 0.81–1 very strong (dark green); NA—not applicable.</p>
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<p>Average NS and HSR for different types of Cooking oils (<bold>A</bold>) and Cheeses (<bold>B</bold>). Special oils include less common oils (n &lt; 5), such as black cumin oil, apricot oil, and argan oil.</p>
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18 pages, 319 KiB  
Article
Refinement of Discrete Lah–Ribarič Inequality and Applications on Csiszár Divergence
by Đilda Pečarić, Josip Pečarić and Jurica Perić
Mathematics 2022, 10(5), 755; https://doi.org/10.3390/math10050755 - 26 Feb 2022
Viewed by 1207
Abstract
In this paper we give a new refinement of the Lah–Ribarič inequality and, using the same technique, we give a refinement of the Jensen inequality. Using these results, a refinement of the discrete Hölder inequality and a refinement of some inequalities for discrete [...] Read more.
In this paper we give a new refinement of the Lah–Ribarič inequality and, using the same technique, we give a refinement of the Jensen inequality. Using these results, a refinement of the discrete Hölder inequality and a refinement of some inequalities for discrete weighted power means and discrete weighted quasi-arithmetic means are obtained. We also give applications in the information theory; namely, we give some interesting estimations for the discrete Csiszár divergence and for its important special cases. Full article
(This article belongs to the Special Issue Mathematical Inequalities with Applications)
13 pages, 6456 KiB  
Article
Controlling Film Thickness Distribution by Magnetron Sputtering with Rotation and Revolution
by Handan Huang, Li Jiang, Yiyun Yao, Zhong Zhang, Zhanshan Wang and Runze Qi
Coatings 2021, 11(5), 599; https://doi.org/10.3390/coatings11050599 - 19 May 2021
Cited by 7 | Viewed by 4209
Abstract
The laterally graded multilayer collimator is a vital part of a high-precision diffractometer. It is applied as condensing reflectors to convert divergent X-rays from laboratory X-ray sources into a parallel beam. The thickness of the multilayer film varies with the angle of incidence [...] Read more.
The laterally graded multilayer collimator is a vital part of a high-precision diffractometer. It is applied as condensing reflectors to convert divergent X-rays from laboratory X-ray sources into a parallel beam. The thickness of the multilayer film varies with the angle of incidence to guarantee every position on the mirror satisfies the Bragg reflection. In principle, the accuracy of the parameters of the sputtering conditions is essential for achieving a reliable result. In this paper, we proposed a precise method for the fabrication of the laterally graded multilayer based on a planetary motion magnetron sputtering system for film thickness control. This method uses the fast and slow particle model to obtain the particle transport process, and then combines it with the planetary motion magnetron sputtering system to establish the film thickness distribution model. Moreover, the parameters of the sputtering conditions in the model are derived from experimental inversion to improve accuracy. The revolution and rotation of the substrate holder during the final deposition process are achieved by the speed curve calculated according to the model. Measurement results from the X-ray reflection test (XRR) show that the thickness error of the laterally graded multilayer film, coated on a parabolic cylinder Si substrate, is less than 1%, demonstrating the effectiveness of the optimized method for obtaining accurate film thickness distribution. Full article
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<p>Etched ring shape.</p>
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<p>The rectangular target etching ring orbit.</p>
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<p>Angular distribution of particle sputtering on the target surface.</p>
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<p>Planetary motion magnetron sputtering coating system.</p>
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<p>The trajectory of unit points at different rotation radii on the substrate holder.</p>
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<p>Comparison of periodic thickness gradient curves of calibration Samples A-1# and B.</p>
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<p>Comparison of the film thickness distribution calculated by the fitting model and experimental film thickness distribution.</p>
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<p>Film thickness deposition thickness distribution per unit area and unit revolution angle on the substrate holder at different rotation radii.</p>
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<p>Revolution speed distribution curve. (<b>a</b>) W target; (<b>b</b>) Si target.</p>
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<p>The placement of the finished product and the calibrated silicon wafer.</p>
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<p>Comparison chart of the d-spacing of calibration Samples 1# and 2#.</p>
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<p>Film thickness distribution curve (red line: target thickness curve; blue line: simulated thickness curve; brown line: actual coating thickness distribution curve).</p>
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<p>XRR test curve.</p>
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<p>The first-order peak of the XRR test curve (the straight line is the working grazing incident angle of the position).</p>
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10 pages, 482 KiB  
Article
Breast Milk for Term and Preterm Infants—Own Mother’s Milk or Donor Milk?
by Réka A. Vass, Gabriella Kiss, Edward F. Bell, Robert D. Roghair, Attila Miseta, József Bódis, Simone Funke and Tibor Ertl
Nutrients 2021, 13(2), 424; https://doi.org/10.3390/nu13020424 - 28 Jan 2021
Cited by 16 | Viewed by 3653
Abstract
Hormones are important biological regulators, controlling development and physiological processes throughout life. We investigated pituitary hormones such as follicle-stimulating hormone (FSH), luteinizing hormone (LH), prolactin (PRL) and total protein levels during the first 6 months of lactation. Breast milk samples were collected every [...] Read more.
Hormones are important biological regulators, controlling development and physiological processes throughout life. We investigated pituitary hormones such as follicle-stimulating hormone (FSH), luteinizing hormone (LH), prolactin (PRL) and total protein levels during the first 6 months of lactation. Breast milk samples were collected every fourth week of lactation from mothers who gave birth to preterm (n = 14) or term (n = 16) infants. Donor milk is suggested when own mother’s milk is not available; therefore, we collected breast milk samples before and after Holder pasteurization (HoP) from the Breast Milk Collection Center of Pécs, Hungary. Three infant formulas prepared in the Neonatal Intensive Care Unit of the University of Pécs were tested at three different time points. Our aim was to examine the hormone content of own mother’s milk and donor milk. There were no significant changes over time in the concentrations of any hormone. Preterm milk had higher PRL (28.2 ± 2.5 vs. 19.3 ± 2.3 ng/mL) and LH (36.3 ± 8.8 vs. 15.9 ± 4.1 mIU/L) concentrations than term milk during the first 6 months of lactation. Total protein and FSH concentrations did not differ between preterm and term breast milk. Holder pasteurization decreased the PRL concentration (30.4 ± 1.8 vs. 14.4 ± 0.6 ng/mL) and did not affect gonadotropin levels of donor milk. Infant formulas have higher total protein content than breast milk but do not contain detectable levels of pituitary hormones. Differences were detected in the content of pituitary hormones produced for preterm and term infants. Divergence between feeding options offers opportunities for improvement of nutritional guidelines for both hospital and home feeding practices. Full article
(This article belongs to the Special Issue Complementary Feeding in Preterm Newborns)
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<p>FSH:LH ratio in preterm, term, donor, and pasteurized donor milk. <sup>°°</sup> <span class="html-italic">p</span> &lt; 0.01 versus preterm milk; <sup>‡‡</sup> <span class="html-italic">p</span> &lt; 0.01 versus term milk; <sup>†</sup> <span class="html-italic">p</span> &lt; 0.05 versus donor milk.</p>
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15 pages, 3086 KiB  
Article
Optimizing Irradiation Geometry in LED-Based Photoacoustic Imaging with 3D Printed Flexible and Modular Light Delivery System
by Maju Kuriakose, Christopher D. Nguyen, Mithun Kuniyil Ajith Singh and Srivalleesha Mallidi
Sensors 2020, 20(13), 3789; https://doi.org/10.3390/s20133789 - 6 Jul 2020
Cited by 11 | Viewed by 4723
Abstract
Photoacoustic (PA) imaging–a technique combining the ability of optical imaging to probe functional properties of the tissue and deep structural imaging ability of ultrasound–has gained significant popularity in the past two decades for its utility in several biomedical applications. More recently, light-emitting diodes [...] Read more.
Photoacoustic (PA) imaging–a technique combining the ability of optical imaging to probe functional properties of the tissue and deep structural imaging ability of ultrasound–has gained significant popularity in the past two decades for its utility in several biomedical applications. More recently, light-emitting diodes (LED) are being explored as an alternative to bulky and expensive laser systems used in PA imaging for their portability and low-cost. Due to the large beam divergence of LEDs compared to traditional laser beams, it is imperative to quantify the angular dependence of LED-based illumination and optimize its performance for imaging superficial or deep-seated lesions. A custom-built modular 3-D printed hinge system and tissue-mimicking phantoms with various absorption and scattering properties were used in this study to quantify the angular dependence of LED-based illumination. We also experimentally calculated the source divergence of the pulsed-LED arrays to be 58° ± 8°. Our results from point sources (pencil lead phantom) in non-scattering medium obey the cotangential relationship between the angle of irradiation and maximum PA intensity obtained at various imaging depths, as expected. Strong dependence on the angle of illumination at superficial depths (−5°/mm at 10 mm) was observed that becomes weaker at intermediate depths (−2.5°/mm at 20 mm) and negligible at deeper locations (−1.1°/mm at 30 mm). The results from the tissue-mimicking phantom in scattering media indicate that angles between 30–75° could be used for imaging lesions at various depths (12 mm–28 mm) where lower LED illumination angles (closer to being parallel to the imaging plane) are preferable for deep tissue imaging and superficial lesion imaging is possible with higher LED illumination angles (closer to being perpendicular to the imaging plane). Our results can serve as a priori knowledge for the future LED-based PA system designs employed for both preclinical and clinical applications. Full article
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<p>(<b>a</b>–<b>c</b>) Schematics of light emitting diode (LED) illumination cross-sectional view of photoacoustic (PA) setup at representative angles: 15°, 45°, and 75°, respectively, orthogonal to the imaging plane <span class="html-italic">XZ</span>, that contains a hypothetical absorber, p. M<sub>0</sub> is the medium that facilitates acoustic coupling between the transducer and the phantom material M<sub>1</sub>; (<b>d</b>–<b>f</b>) Photographs of LED source pivoted at representative angles, <span class="html-italic">θ</span> = 15°, 45°, and 75°, respectively, using 3D printed modular hinge system.</p>
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<p>(<b>a</b>) Modular hinge system holding the LED arrays and attached to the US transducer. (<b>b</b>) The photographs of the individual pieces are shown in the top panel. The 3D renderings of the hinge pieces are shown on the bottom panel. Part A fits around the transducer and extends the horizontal reach of the holder to allow the LEDs to be placed at angles approaching 90°. Part B in conjunction with Part A allows precise horizontal and vertical height adjustment. The holder consists of two-part B pieces, and schematic of only one piece is shown in the panel. Part C holds the LEDs using the heat sinks and provides flexibility for any final adjustments on the LED illumination angle. Scale bar = 10 mm.</p>
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<p>(<b>a</b>) Photograph of Pencil lead matrix; (<b>b</b>) photograph of an experimental arrangement using intralipid medium; (<b>c</b>–<b>h</b>) PA image acquired at representative angles 15° (<b>c</b>,<b>f</b>), 45° (<b>d</b>,<b>g</b>), and 75° (<b>e</b>,<b>h</b>) in water (<b>c</b>–<b>e</b>) and 1% intralipid (<b>f</b>–<b>h</b>); (<b>i</b>,<b>k</b>) Mean PA signal intensities and their standard deviations (of 5 lateral positions at each depth) plotted as a function of LED angles in water (<b>j</b>) and intralipid (<b>k</b>); and (<b>j</b>,<b>l</b>) corresponding contrast to noise ratios (CNRs) obtained as a function of LED angles in water (<b>j</b>) and intralipid (<b>l</b>). Different depths from the transducer are indicated by 12 mm (line with black circles), 18 mm (line with red squares), 24 mm (line with blue diamonds), and 30 mm (line with green downward triangles). The noise background levels corresponding to 12, 18, 24, and 30 mm are represented by black, red, blue, and green dash-dotted lines, respectively, in (<b>i</b>) in water and (<b>j</b>) in intralipid. The backgrounds were obtained right below from each signal regions, e.g., as indicated by the yellow rectangles in (<b>c</b>,<b>f</b>). Images for all the angles can be found in <a href="#app1-sensors-20-03789" class="html-app">Figure S2</a> (for water) and <a href="#app1-sensors-20-03789" class="html-app">Figure S3</a> (for intralipid).</p>
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<p>(<b>a</b>) Illustration of the source wavefront profile, while assuming the LED array as a line source with Gaussian profile having divergence described by Equation (4); (<b>b</b>) Source profile after pivoting LED arrays at three different angles in the imaging plane–green at 0°, red at 45°, and blue at 90°; (<b>c</b>) Normalized PA intensity obtained from 18 mm target vs. LED illumination angle, <span class="html-italic">θ</span>, is indicated in blue squares. The red solid curve shows the Gaussian fit using Equation (5), with an R-squared value of 0.95; (<b>d</b>) Blue squares with error bar show the peak PA intensity with standard deviation for each depth plotted as a function of <span class="html-italic">θ</span> for pencil lead phantom data in water. The black solid curve displays the best fit using a cotangent function (R<sup>2</sup> = 0.95).</p>
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<p>(<b>a</b>) Schematic of the tissue-mimicking phantom: Absorbing lesion (cylindrical in shape, 2 mm in diameter) placed on chicken tissue and arranged obliquely in the <span class="html-italic">XZ</span> plane. (<b>b</b>) Photograph of the experiment. The dotted line indicated by arrowhead shows the projection of lesion to the <span class="html-italic">x</span>-axis; (<b>c</b>,<b>d</b>) US image of the sample showing water top layer (<b>c</b>) or chicken layer (<b>d</b>) lesion and bottom chicken tissue in both cases; (<b>e</b>–<b>j</b>) PA intensity images captured using water as the top layer (<b>e</b>–<b>g</b>) or chicken tissue as the top layer (<b>h</b>–<b>j</b>), by choosing LED array directions: 15° (<b>e</b>,<b>h</b>), 45° (<b>f</b>,<b>i</b>), and 75° (<b>g</b>,<b>j</b>) with respect to the imaging plane.</p>
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<p>(<b>a</b>) PA image of a sample containing absorbing lesion (2 mm diameter) with the top water layer. White parallelograms indicate the regions selected for PA intensity and yellow for the background; (<b>b</b>,<b>d</b>) PA signal intensity plotted as a function of LED angles for selected depths in mm as labeled by numbers with the corresponding color, right next to each plot. Black circle with line corresponds to 12 mm, red squares with line corresponds to 20 mm and blue diamond with line indicates 28 mm. The black dotted lines, red dash-dotted lines, and blue dashed lines represent the background levels for 12, 20, and 28 mm depths (from the transducer) respectively; a gap of 10.5 mm exists between the transducer and the phantom surface. (<b>c</b>) CNR of the lesion under water; (<b>e</b>) CNR of the lesion under chicken breast. The error bars represent the standard deviation.</p>
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16 pages, 982 KiB  
Article
Interpreting the Sustainable Development of Human Capital and the Sheepskin Effects in Returns to Higher Education: Empirical Evidence from Pakistan
by Zhimin Liu, Aftab Ahmed Memon, Woubshet Negussie and Haile Ketema
Sustainability 2020, 12(6), 2393; https://doi.org/10.3390/su12062393 - 19 Mar 2020
Cited by 4 | Viewed by 3195
Abstract
According to poststructuralists, workers with higher level of education and possession of potential experience are supposed to have higher wages. Yet, there are plausible questions that arise as to what levels of education or work history are needed for the enhancement of wage [...] Read more.
According to poststructuralists, workers with higher level of education and possession of potential experience are supposed to have higher wages. Yet, there are plausible questions that arise as to what levels of education or work history are needed for the enhancement of wage discrimination. Additionally, the outcomes arising from rehashing years of schooling are worth considering. We used a several methods, employing the administrative Household Integrated Economic Survey (HIES) data from Pakistan without ignoring environmental effects. Our estimated results support the conventional assumptions of linearity of log-wage. First, we found substantial returns for postgraduate diploma holders in both public and private sectors, even after controlling the individual’s heterogeneity. Second, we did notice a significant divergence in return to low-level education (LLE) and job history. Third, rehashing years of education may create suspiciousness regarding the lack of competence. Our results suggest that continuous investment in human capital toward postgraduate diploma may result in higher premiums. Full article
(This article belongs to the Special Issue Economics of Education and Sustainable Development)
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<p>Average earnings for each level of education. Source: Pakistan Bureau of Statistics 2011 sample from public sector employees; author’s calculation.</p>
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<p>Capital investment toward the outcomes. Author-designed schematic diagram.</p>
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<p>This figure depicts the monthly log earnings of public sector employees with 16 years (BS(honors)) education and 18 years (MS(honors)) education from the sample period.</p>
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<p>Estimated mean of actual years of degree attainment according to MOE system. Source: Pakistan Bureau of Statistics 2011; author’s calculation.</p>
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27 pages, 612 KiB  
Article
Geometric Estimation of Multivariate Dependency
by Salimeh Yasaei Sekeh and Alfred O. Hero
Entropy 2019, 21(8), 787; https://doi.org/10.3390/e21080787 - 12 Aug 2019
Cited by 6 | Viewed by 3687
Abstract
This paper proposes a geometric estimator of dependency between a pair of multivariate random variables. The proposed estimator of dependency is based on a randomly permuted geometric graph (the minimal spanning tree) over the two multivariate samples. This estimator converges to a quantity [...] Read more.
This paper proposes a geometric estimator of dependency between a pair of multivariate random variables. The proposed estimator of dependency is based on a randomly permuted geometric graph (the minimal spanning tree) over the two multivariate samples. This estimator converges to a quantity that we call the geometric mutual information (GMI), which is equivalent to the Henze–Penrose divergence. between the joint distribution of the multivariate samples and the product of the marginals. The GMI has many of the same properties as standard MI but can be estimated from empirical data without density estimation; making it scalable to large datasets. The proposed empirical estimator of GMI is simple to implement, involving the construction of an minimal spanning tree (MST) spanning over both the original data and a randomly permuted version of this data. We establish asymptotic convergence of the estimator and convergence rates of the bias and variance for smooth multivariate density functions belonging to a Hölder class. We demonstrate the advantages of our proposed geometric dependency estimator in a series of experiments. Full article
(This article belongs to the Section Information Theory, Probability and Statistics)
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<p>The MST and FR statistic of spanning the merged set of normal points when <math display="inline"><semantics> <mi mathvariant="bold">X</mi> </semantics></math> and <math display="inline"><semantics> <mi mathvariant="bold">Y</mi> </semantics></math> are independent (denoted in blue points) and when <math display="inline"><semantics> <mi mathvariant="bold">X</mi> </semantics></math> and <math display="inline"><semantics> <mi mathvariant="bold">Y</mi> </semantics></math> are highly dependent (denoted in red points). The FR test statistic is the number of edges in the MST that connect samples from different color nodes (denoted in green) and it is used to estimate the GMI <math display="inline"><semantics> <msub> <mi>I</mi> <mi>p</mi> </msub> </semantics></math>.</p>
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<p>(<b>left</b>) Log–log plot of theoretical and experimental MSE of the proposed MST-based GMI estimator as a function of sample size <span class="html-italic">n</span> for <math display="inline"><semantics> <mrow> <mi>d</mi> <mo>=</mo> <mn>6</mn> <mo>,</mo> <mn>10</mn> <mo>,</mo> <mn>12</mn> </mrow> </semantics></math> and fixed smoothness parameter <math display="inline"><semantics> <mi>η</mi> </semantics></math>. (<b>right</b>) The GMI estimator was implemented using two approaches, Algorithm 1 and KDE method where the KDE-GMI used KDE density estimators in the formula (<a href="#FD2-entropy-21-00787" class="html-disp-formula">2</a>). In this experiment, samples are generated from the two-dimensional normal distribution with zero mean and covariance matrix (<a href="#FD21-entropy-21-00787" class="html-disp-formula">21</a>) for various value of <math display="inline"><semantics> <mrow> <mi>ρ</mi> <mo>∈</mo> <mo>[</mo> <mn>0.1</mn> <mo>,</mo> <mn>0.9</mn> <mo>]</mo> </mrow> </semantics></math>.</p>
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<p>MSE log–log plots as a function of sample size <span class="html-italic">n</span> (<b>left</b>) for the proposed MST-GMI estimator (“Estimated GMI”) and the standard KDE-GMI plug-in estimator of GMI. The right column of plots correspond to the GMI estimated for dimension <math display="inline"><semantics> <mrow> <mi>d</mi> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math> (<b>top</b>) and <math display="inline"><semantics> <mrow> <mi>d</mi> <mo>=</mo> <mn>12</mn> </mrow> </semantics></math> (<b>bottom</b>). In both cases the proportionality parameter <math display="inline"><semantics> <mi>α</mi> </semantics></math> is <math display="inline"><semantics> <mrow> <mn>0.5</mn> </mrow> </semantics></math>. The MST-GMI estimator in both plots for sample size <span class="html-italic">n</span> in <math display="inline"><semantics> <mrow> <mo>[</mo> <mn>700</mn> <mo>,</mo> <mn>1600</mn> <mo>]</mo> </mrow> </semantics></math> outperforms the KDE-GMI estimator, especially for larger dimensions.</p>
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<p>MSE log–log plots as a function of sample size <span class="html-italic">n</span> for the proposed FR estimator. We compare the MSE of our proposed FR estimator for various dimensions <math display="inline"><semantics> <mrow> <mi>d</mi> <mo>=</mo> <mn>15</mn> <mo>,</mo> <mn>20</mn> <mo>,</mo> <mn>50</mn> </mrow> </semantics></math> (<b>left</b>). As <span class="html-italic">d</span> increases, the blue curve takes larger values than green and orange curves i.e., MSE increases as <span class="html-italic">d</span> grows. However, this is more evidential for large sample size <span class="html-italic">n</span>. The second experiment (<b>right</b>) focuses on optimal proportion <math display="inline"><semantics> <mi>α</mi> </semantics></math> for <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>500</mn> <mo>,</mo> <msup> <mn>10</mn> <mn>4</mn> </msup> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>ρ</mi> <mo>=</mo> <mn>0.7</mn> <mo>,</mo> <mn>0.5</mn> </mrow> </semantics></math>. <math display="inline"><semantics> <mover accent="true"> <mi>α</mi> <mo stretchy="false">˜</mo> </mover> </semantics></math> is the optimal <math display="inline"><semantics> <mi>α</mi> </semantics></math> for <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>∈</mo> <mo>[</mo> <msubsup> <mi>α</mi> <mn>0</mn> <mi>L</mi> </msubsup> <mo>,</mo> <msubsup> <mi>α</mi> <mn>0</mn> <mi>U</mi> </msubsup> <mo>]</mo> </mrow> </semantics></math>.</p>
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<p>Runtime of KDE approach and proposed MST-based estimator of GMI vs sample size. The proposed GMI estimator achieves significant speedup, while for small sample size, the KDE method becomes overly fast. Please note that in this experiment the sample is generated from the Gaussian distribution in dimension <math display="inline"><semantics> <mrow> <mi>d</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>.</p>
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44 pages, 550 KiB  
Article
Chattel or Child: The Liminal Status of Companion Animals in Society and Law
by Nicole R. Pallotta
Soc. Sci. 2019, 8(5), 158; https://doi.org/10.3390/socsci8050158 - 23 May 2019
Cited by 13 | Viewed by 12969
Abstract
Companion animals in the U.S. are increasingly regarded as members of the family with whom one may share a strong emotional bond. However, despite an evolving social construction that has elevated their status in the dominant culture, companion animals lack meaningful legal rights, [...] Read more.
Companion animals in the U.S. are increasingly regarded as members of the family with whom one may share a strong emotional bond. However, despite an evolving social construction that has elevated their status in the dominant culture, companion animals lack meaningful legal rights, and “family member” is a provisional status that can be dissolved at will based on the discretion of the sole rights-holder in the relationship: the human owner. Because they are still defined within the U.S. legal system as property, it is a common lament within the animal protection movement that the law has not kept pace with the emergent cultural perception of companion animals as family or best friends who may occupy a significant place in one’s constellation of interpersonal relationships. But how divergent are the laws that govern our treatment of companion animals from prevailing social norms? This article examines current trends in animal law and society to shed light on this question. I find that while a new family member cultural status is emerging for companion animals in the U.S., their legal status as property is a countervailing force, enabling contradictory practices and beliefs that construct animals as expendable. The fact that their cultural status is in flux in turn reinforces their status under the law. I conclude with proposed policy reforms that will facilitate the integration of companion animals into society as true rather than rhetorical family members. Full article
(This article belongs to the Special Issue We Are Best Friends: Animals in Society)
14 pages, 5219 KiB  
Article
Convex Optimization via Symmetrical Hölder Divergence for a WLAN Indoor Positioning System
by Osamah Abdullah
Entropy 2018, 20(9), 639; https://doi.org/10.3390/e20090639 - 25 Aug 2018
Cited by 7 | Viewed by 3258
Abstract
Modern indoor positioning system services are important technologies that play vital roles in modern life, providing many services such as recruiting emergency healthcare providers and for security purposes. Several large companies, such as Microsoft, Apple, Nokia, and Google, have researched location-based services. Wireless [...] Read more.
Modern indoor positioning system services are important technologies that play vital roles in modern life, providing many services such as recruiting emergency healthcare providers and for security purposes. Several large companies, such as Microsoft, Apple, Nokia, and Google, have researched location-based services. Wireless indoor localization is key for pervasive computing applications and network optimization. Different approaches have been developed for this technique using WiFi signals. WiFi fingerprinting-based indoor localization has been widely used due to its simplicity, and algorithms that fingerprint WiFi signals at separate locations can achieve accuracy within a few meters. However, a major drawback of WiFi fingerprinting is the variance in received signal strength (RSS), as it fluctuates with time and changing environment. As the signal changes, so does the fingerprint database, which can change the distribution of the RSS (multimodal distribution). Thus, in this paper, we propose that symmetrical Hölder divergence, which is a statistical model of entropy that encapsulates both the skew Bhattacharyya divergence and Cauchy–Schwarz divergence that are closed-form formulas that can be used to measure the statistical dissimilarities between the same exponential family for the signals that have multivariate distributions. The Hölder divergence is asymmetric, so we used both left-sided and right-sided data so the centroid can be symmetrized to obtain the minimizer of the proposed algorithm. The experimental results showed that the symmetrized Hölder divergence consistently outperformed the traditional k nearest neighbor and probability neural network. In addition, with the proposed algorithm, the position error accuracy was about 1 m in buildings. Full article
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Figure 1

Figure 1
<p>Signal-to-noise ratio of received strength signal indicator variations over time.</p>
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<p>The Bregman divergence represents the vertical distance between the potential function and hyperplane at <span class="html-italic">q</span>.</p>
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<p>Interpreting the Jensen-Bregman divergence.</p>
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<p>Hölder divergence encompasses the skew Bhattacharyya divergence and the Cauchy-Schwarz divergence.</p>
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<p>The offline and online stages of location WiFi-based fingerprinting architecture.</p>
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<p>The layout used in the experimental work in the College of Engineering and Applied.</p>
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<p>The implementation results of different number of clusters with respect to the average of the localization distance.</p>
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<p>The implementation result of the average localization error under different AP selection schemes.</p>
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<p>Experiment results: The Cumulative distribution function (CDF) of localization error when using 50 nearest neighbors.</p>
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6948 KiB  
Article
On Hölder Projective Divergences
by Frank Nielsen, Ke Sun and Stéphane Marchand-Maillet
Entropy 2017, 19(3), 122; https://doi.org/10.3390/e19030122 - 16 Mar 2017
Cited by 15 | Viewed by 6155
Abstract
We describe a framework to build distances by measuring the tightness of inequalities and introduce the notion of proper statistical divergences and improper pseudo-divergences. We then consider the Hölder ordinary and reverse inequalities and present two novel classes of Hölder divergences and pseudo-divergences [...] Read more.
We describe a framework to build distances by measuring the tightness of inequalities and introduce the notion of proper statistical divergences and improper pseudo-divergences. We then consider the Hölder ordinary and reverse inequalities and present two novel classes of Hölder divergences and pseudo-divergences that both encapsulate the special case of the Cauchy–Schwarz divergence. We report closed-form formulas for those statistical dissimilarities when considering distributions belonging to the same exponential family provided that the natural parameter space is a cone (e.g., multivariate Gaussians) or affine (e.g., categorical distributions). Those new classes of Hölder distances are invariant to rescaling and thus do not require distributions to be normalized. Finally, we show how to compute statistical Hölder centroids with respect to those divergences and carry out center-based clustering toy experiments on a set of Gaussian distributions which demonstrate empirically that symmetrized Hölder divergences outperform the symmetric Cauchy–Schwarz divergence. Full article
(This article belongs to the Special Issue Information Geometry II)
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Graphical abstract

Graphical abstract
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<p>Hölder proper divergence (bi-parametric) and Hölder improper pseudo-divergence (tri-parametric) encompass Cauchy–Schwarz divergence and skew Bhattacharyya divergence.</p>
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<p>First row: the Hölder pseudo divergence (HPD) <math display="inline"> <semantics> <mrow> <msubsup> <mi>D</mi> <mrow> <mi>α</mi> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> <mi mathvariant="monospace">H</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mi>r</mi> </msub> <mo>:</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics> </math> for <math display="inline"> <semantics> <mrow> <mi>α</mi> <mo>∈</mo> <mo>{</mo> <mn>4</mn> <mo>/</mo> <mn>3</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>4</mn> <mo>}</mo> </mrow> </semantics> </math>, KL divergence and reverse KL divergence. Remaining rows: the HD <math display="inline"> <semantics> <mrow> <msubsup> <mi>D</mi> <mrow> <mi>α</mi> <mo>,</mo> <mi>γ</mi> </mrow> <mi mathvariant="monospace">H</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mi>r</mi> </msub> <mo>:</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics> </math> for <math display="inline"> <semantics> <mrow> <mi>α</mi> <mo>∈</mo> <mo>{</mo> <mn>4</mn> <mo>/</mo> <mn>3</mn> <mo>,</mo> <mn>1.5</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>10</mn> <mo>}</mo> </mrow> </semantics> </math> (from top to bottom) and <math display="inline"> <semantics> <mrow> <mi>γ</mi> <mo>∈</mo> <mo>{</mo> <mn>0.5</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>5</mn> <mo>,</mo> <mn>10</mn> <mo>}</mo> </mrow> </semantics> </math> (from left to right). The reference distribution <math display="inline"> <semantics> <msub> <mi>p</mi> <mi>r</mi> </msub> </semantics> </math> is presented as “★”. The minimizer of <math display="inline"> <semantics> <mrow> <msubsup> <mi>D</mi> <mrow> <mi>α</mi> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> <mi mathvariant="monospace">H</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mi>r</mi> </msub> <mo>:</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics> </math>, if different from <math display="inline"> <semantics> <msub> <mi>p</mi> <mi>r</mi> </msub> </semantics> </math>, is presented as “•”. Notice that <math display="inline"> <semantics> <mrow> <msubsup> <mi>D</mi> <mrow> <mn>2</mn> <mo>,</mo> <mn>2</mn> </mrow> <mi mathvariant="monospace">H</mi> </msubsup> <mo>=</mo> <msubsup> <mi>D</mi> <mrow> <mn>2</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> <mi mathvariant="monospace">H</mi> </msubsup> </mrow> </semantics> </math>. (<b>a</b>) Reference categorical distribution <math display="inline"> <semantics> <mrow> <msub> <mi>p</mi> <mi>r</mi> </msub> <mo>=</mo> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>/</mo> <mn>3</mn> <mo>,</mo> <mn>1</mn> <mo>/</mo> <mn>3</mn> <mo>,</mo> <mn>1</mn> <mo>/</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics> </math>; (<b>b</b>) reference categorical distribution <math display="inline"> <semantics> <mrow> <msub> <mi>p</mi> <mi>r</mi> </msub> <mo>=</mo> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>/</mo> <mn>2</mn> <mo>,</mo> <mn>1</mn> <mo>/</mo> <mn>3</mn> <mo>,</mo> <mn>1</mn> <mo>/</mo> <mn>6</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics> </math>.</p>
Full article ">Figure 2 Cont.
<p>First row: the Hölder pseudo divergence (HPD) <math display="inline"> <semantics> <mrow> <msubsup> <mi>D</mi> <mrow> <mi>α</mi> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> <mi mathvariant="monospace">H</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mi>r</mi> </msub> <mo>:</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics> </math> for <math display="inline"> <semantics> <mrow> <mi>α</mi> <mo>∈</mo> <mo>{</mo> <mn>4</mn> <mo>/</mo> <mn>3</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>4</mn> <mo>}</mo> </mrow> </semantics> </math>, KL divergence and reverse KL divergence. Remaining rows: the HD <math display="inline"> <semantics> <mrow> <msubsup> <mi>D</mi> <mrow> <mi>α</mi> <mo>,</mo> <mi>γ</mi> </mrow> <mi mathvariant="monospace">H</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mi>r</mi> </msub> <mo>:</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics> </math> for <math display="inline"> <semantics> <mrow> <mi>α</mi> <mo>∈</mo> <mo>{</mo> <mn>4</mn> <mo>/</mo> <mn>3</mn> <mo>,</mo> <mn>1.5</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>10</mn> <mo>}</mo> </mrow> </semantics> </math> (from top to bottom) and <math display="inline"> <semantics> <mrow> <mi>γ</mi> <mo>∈</mo> <mo>{</mo> <mn>0.5</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>5</mn> <mo>,</mo> <mn>10</mn> <mo>}</mo> </mrow> </semantics> </math> (from left to right). The reference distribution <math display="inline"> <semantics> <msub> <mi>p</mi> <mi>r</mi> </msub> </semantics> </math> is presented as “★”. The minimizer of <math display="inline"> <semantics> <mrow> <msubsup> <mi>D</mi> <mrow> <mi>α</mi> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> <mi mathvariant="monospace">H</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mi>r</mi> </msub> <mo>:</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics> </math>, if different from <math display="inline"> <semantics> <msub> <mi>p</mi> <mi>r</mi> </msub> </semantics> </math>, is presented as “•”. Notice that <math display="inline"> <semantics> <mrow> <msubsup> <mi>D</mi> <mrow> <mn>2</mn> <mo>,</mo> <mn>2</mn> </mrow> <mi mathvariant="monospace">H</mi> </msubsup> <mo>=</mo> <msubsup> <mi>D</mi> <mrow> <mn>2</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> <mi mathvariant="monospace">H</mi> </msubsup> </mrow> </semantics> </math>. (<b>a</b>) Reference categorical distribution <math display="inline"> <semantics> <mrow> <msub> <mi>p</mi> <mi>r</mi> </msub> <mo>=</mo> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>/</mo> <mn>3</mn> <mo>,</mo> <mn>1</mn> <mo>/</mo> <mn>3</mn> <mo>,</mo> <mn>1</mn> <mo>/</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics> </math>; (<b>b</b>) reference categorical distribution <math display="inline"> <semantics> <mrow> <msub> <mi>p</mi> <mi>r</mi> </msub> <mo>=</mo> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>/</mo> <mn>2</mn> <mo>,</mo> <mn>1</mn> <mo>/</mo> <mn>3</mn> <mo>,</mo> <mn>1</mn> <mo>/</mo> <mn>6</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics> </math>.</p>
Full article ">Figure 3
<p>First row: <math display="inline"> <semantics> <mrow> <msubsup> <mi>D</mi> <mrow> <mi>α</mi> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> <mi mathvariant="monospace">H</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mi>r</mi> </msub> <mo>:</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics> </math>, where <math display="inline"> <semantics> <msub> <mi>p</mi> <mi>r</mi> </msub> </semantics> </math> is the standard Gaussian distribution and <math display="inline"> <semantics> <mrow> <mi>α</mi> <mo>∈</mo> <mo>{</mo> <mn>4</mn> <mo>/</mo> <mn>3</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>4</mn> <mo>}</mo> </mrow> </semantics> </math> compared to the KL divergence. The rest of the rows: <math display="inline"> <semantics> <mrow> <msubsup> <mi>D</mi> <mrow> <mi>α</mi> <mo>,</mo> <mi>γ</mi> </mrow> <mi mathvariant="monospace">H</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mi>r</mi> </msub> <mo>:</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics> </math> for <math display="inline"> <semantics> <mrow> <mi>α</mi> <mo>∈</mo> <mo>{</mo> <mn>4</mn> <mo>/</mo> <mn>3</mn> <mo>,</mo> <mn>1.5</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>10</mn> <mo>}</mo> </mrow> </semantics> </math> (from top to bottom) and <math display="inline"> <semantics> <mrow> <mi>γ</mi> <mo>∈</mo> <mo>{</mo> <mn>0.5</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>5</mn> <mo>,</mo> <mn>10</mn> <mo>}</mo> </mrow> </semantics> </math> (from left to right). Notice that <math display="inline"> <semantics> <mrow> <msubsup> <mi>D</mi> <mrow> <mn>2</mn> <mo>,</mo> <mn>2</mn> </mrow> <mi mathvariant="monospace">H</mi> </msubsup> <mo>=</mo> <msubsup> <mi>D</mi> <mrow> <mn>2</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> <mi mathvariant="monospace">H</mi> </msubsup> </mrow> </semantics> </math>. The coordinate system is formed by <span class="html-italic">μ</span> (mean) and <span class="html-italic">σ</span> (standard deviation).</p>
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<p>Variational <span class="html-italic">k</span>-means clustering results on a toy dataset consisting of a set of 2D Gaussians organized into two or three clusters. The cluster centroids are represented by contour plots using the same density levels. (<b>a</b>) <math display="inline"> <semantics> <mrow> <mi>α</mi> <mo>=</mo> <mi>γ</mi> <mo>=</mo> <mn>1.1</mn> </mrow> </semantics> </math> (Hölder clustering); (<b>b</b>) <math display="inline"> <semantics> <mrow> <mi>α</mi> <mo>=</mo> <mi>γ</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics> </math> (Cauchy–Schwarz clustering); (<b>c</b>) <math display="inline"> <semantics> <mrow> <mi>α</mi> <mo>=</mo> <mi>γ</mi> <mo>=</mo> <mn>1.1</mn> </mrow> </semantics> </math> (Hölder clustering); (<b>d</b>) <math display="inline"> <semantics> <mrow> <mi>α</mi> <mo>=</mo> <mi>γ</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics> </math> (Cauchy–Schwarz clustering).</p>
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214 KiB  
Article
Scale-Invariant Divergences for Density Functions
by Takafumi Kanamori
Entropy 2014, 16(5), 2611-2628; https://doi.org/10.3390/e16052611 - 13 May 2014
Cited by 10 | Viewed by 5115
Abstract
Divergence is a discrepancy measure between two objects, such as functions, vectors, matrices, and so forth. In particular, divergences defined on probability distributions are widely employed in probabilistic forecasting. As the dissimilarity measure, the divergence should satisfy some conditions. In this paper, we [...] Read more.
Divergence is a discrepancy measure between two objects, such as functions, vectors, matrices, and so forth. In particular, divergences defined on probability distributions are widely employed in probabilistic forecasting. As the dissimilarity measure, the divergence should satisfy some conditions. In this paper, we consider two conditions: The first one is the scale-invariance property and the second is that the divergence is approximated by the sample mean of a loss function. The first requirement is an important feature for dissimilarity measures. The divergence will depend on which system of measurements we used to measure the objects. Scale-invariant divergence is transformed in a consistent way when the system of measurements is changed to the other one. The second requirement is formalized such that the divergence is expressed by using the so-called composite score. We study the relation between composite scores and scale-invariant divergences, and we propose a new class of divergences called H¨older divergence that satisfies two conditions above. We present some theoretical properties of H¨older divergence. We show that H¨older divergence unifies existing divergences from the viewpoint of scale-invariance. Full article
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