Mathematics

27 pages, 4452 KiB

Open AccessArticle

Self-Perceived Health, Life Satisfaction and Related Factors among Healthcare Professionals and the General Population: Analysis of an Online Survey, with Propensity Score Adjustment

by Ramón Ferri-García, María del Mar Rueda and Andrés Cabrera-León

Mathematics 2021, 9(7), 791; https://doi.org/10.3390/math9070791 - 6 Apr 2021

Cited by 3 | Viewed by 2496

Abstract

Healthcare professionals (HCPs) often suffer high levels of depression, stress, anxiety and burnout. Our main study aimswereto estimate the prevalences of poor self-perceived health, life dissatisfaction, chronic disease and unhealthy habits among HCPs and to explore the use of machine learning classification algorithms [...] Read more.

Healthcare professionals (HCPs) often suffer high levels of depression, stress, anxiety and burnout. Our main study aimswereto estimate the prevalences of poor self-perceived health, life dissatisfaction, chronic disease and unhealthy habits among HCPs and to explore the use of machine learning classification algorithms to remove selection bias. A sample of Spanish HCPs was asked to complete a web survey. Risk factors were identified by multivariate ordinal regression models. To counteract the absence of probabilistic sampling and representation, the sample was weighted by propensity score adjustment algorithms. The logistic regression algorithm was considered the most appropriate for dealing with misestimations. Male HCPs had significantly worse lifestyle habits than their female counterparts, together with a higher prevalence of chronic disease and of health problems. Members of the general population reported significantly poorer health and less satisfaction with life than the HCPs. Among HCPs, the prior existence of health problems was most strongly associated with worsening self-perceived health and decreased life satisfaction, while obesity had an important negative impact on female practitioners’ self-perception of health. Finally, the HCPs who worked as nurses had poorer self-perceptions of health than other HCPs, and the men who worked in primary care had less satisfaction with their lives than those who worked in other levels of healthcare. Full article

(This article belongs to the Special Issue Methods and Applications of Statistics in the Social and Health Sciences)

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15 pages, 3079 KiB

Open AccessArticle

Hierarchical Fractional Advection-Dispersion Equation (FADE) to Quantify Anomalous Transport in River Corridor over a Broad Spectrum of Scales: Theory and Applications

by Yong Zhang, Dongbao Zhou, Wei Wei, Jonathan M. Frame, Hongguang Sun, Alexander Y. Sun and Xingyuan Chen

Mathematics 2021, 9(7), 790; https://doi.org/10.3390/math9070790 - 6 Apr 2021

Cited by 4 | Viewed by 2354

Abstract

Fractional calculus-based differential equations were found by previous studies to be promising tools in simulating local-scale anomalous diffusion for pollutants transport in natural geological media (geomedia), but efficient models are still needed for simulating anomalous transport over a broad spectrum of scales. This [...] Read more.

Fractional calculus-based differential equations were found by previous studies to be promising tools in simulating local-scale anomalous diffusion for pollutants transport in natural geological media (geomedia), but efficient models are still needed for simulating anomalous transport over a broad spectrum of scales. This study proposed a hierarchical framework of fractional advection-dispersion equations (FADEs) for modeling pollutants moving in the river corridor at a full spectrum of scales. Applications showed that the fixed-index FADE could model bed sediment and manganese transport in streams at the geomorphologic unit scale, whereas the variable-index FADE well fitted bedload snapshots at the reach scale with spatially varying indices. Further analyses revealed that the selection of the FADEs depended on the scale, type of the geomedium (i.e., riverbed, aquifer, or soil), and the type of available observation dataset (i.e., the tracer snapshot or breakthrough curve (BTC)). When the pollutant BTC was used, a single-index FADE with scale-dependent parameters could fit the data by upscaling anomalous transport without mapping the sub-grid, intermediate multi-index anomalous diffusion. Pollutant transport in geomedia, therefore, may exhibit complex anomalous scaling in space (and/or time), and the identification of the FADE’s index for the reach-scale anomalous transport, which links the geomorphologic unit and watershed scales, is the core for reliable applications of fractional calculus in hydrology. Full article

(This article belongs to the Special Issue Fractional Calculus in Anomalous Transport Theory)

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Application 1: (a) The measured (symbols, from Martin et al. [<a href="#B46-mathematics-09-00790" class="html-bibr">46</a>]) versus the best-fit snapshots for bedload moving along a fixed gravel bed. (b) is the log-log plot of (a) to show the tail. Full article ">Figure 2
Application 1: Modeled (red lines, using the variable-index fractional advection-dispersion equation (FADE) (4)) versus measured (circles, by Sayre and Hubbell [<a href="#B47-mathematics-09-00790" class="html-bibr">47</a>]) snapshots of bed sediment at the North Loup River, Nebraska at ten sampling cycles (a–j). For comparison purposes, the results of the FADE (4) with a fixed index (dotted lines) and the original mode (without considering anomalous diffusion) from Sayre and Hubbell [<a href="#B47-mathematics-09-00790" class="html-bibr">47</a>] (dashed lines) are also shown. Full article ">Figure 3
Application 2: (a) three scales at the Pinal Creek Basin, Arizona (modified from Puckett et al. [<a href="#B1-mathematics-09-00790" class="html-bibr">1</a>]). (b–d): The measured (symbols) vs. modeled (lines) BTCs using the FADE (7) for the dissolved Mn transport at all three scales in the stream. Full article ">Figure 4
Finite elements for a laboratory-scale fractured medium, which has a dimension of 10 × 3 m (length × width). Full article ">Figure 5
Fracture: The best-fit (lines, using the FADE (7)) versus the Monte Carlo BTCs (symbols) of dissolved pollutants moving at different travel distances along the saturated fracture-matrix medium plotted in <a href="#mathematics-09-00790-f004" class="html-fig">Figure 4</a>. Full article ">Figure 6
Fracture: The change of the time index γ in the FADE (7a) with the travel distance for fitting the Breakthrough curves (BTCs) shown in <a href="#mathematics-09-00790-f005" class="html-fig">Figure 5</a>. The dashed line shows the asymptote. (b) is the semi-log plot of (a). Full article ">

17 pages, 365 KiB

Open AccessArticle

An Application of p-Fibonacci Error-Correcting Codes to Cryptography

by Emanuele Bellini, Chiara Marcolla and Nadir Murru

Mathematics 2021, 9(7), 789; https://doi.org/10.3390/math9070789 - 6 Apr 2021

Cited by 1 | Viewed by 2908

Abstract

In addition to their usefulness in proving one’s identity electronically, identification protocols based on zero-knowledge proofs allow designing secure cryptographic signature schemes by means of the Fiat–Shamir transform or other similar constructs. This approach has been followed by many cryptographers during the NIST [...] Read more.

In addition to their usefulness in proving one’s identity electronically, identification protocols based on zero-knowledge proofs allow designing secure cryptographic signature schemes by means of the Fiat–Shamir transform or other similar constructs. This approach has been followed by many cryptographers during the NIST (National Institute of Standards and Technology) standardization process for quantum-resistant signature schemes. NIST candidates include solutions in different settings, such as lattices and multivariate and multiparty computation. While error-correcting codes may also be used, they do not provide very practical parameters, with a few exceptions. In this manuscript, we explored the possibility of using the error-correcting codes proposed by Stakhov in 2006 to design an identification protocol based on zero-knowledge proofs. We showed that this type of code offers a valid alternative in the error-correcting code setting to build such protocols and, consequently, quantum-resistant signature schemes. Full article

(This article belongs to the Special Issue Algebra and Number Theory)

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20 pages, 2097 KiB

Open AccessEditor’s ChoiceArticle

An Exhaustive Power Comparison of Normality Tests

by Jurgita Arnastauskaitė, Tomas Ruzgas and Mindaugas Bražėnas

Mathematics 2021, 9(7), 788; https://doi.org/10.3390/math9070788 - 6 Apr 2021

Cited by 26 | Viewed by 5994

Abstract

A goodness-of-fit test is a frequently used modern statistics tool. However, it is still unclear what the most reliable approach is to check assumptions about data set normality. A particular data set (especially with a small number of observations) only partly describes the [...] Read more.

A goodness-of-fit test is a frequently used modern statistics tool. However, it is still unclear what the most reliable approach is to check assumptions about data set normality. A particular data set (especially with a small number of observations) only partly describes the process, which leaves many options for the interpretation of its true distribution. As a consequence, many goodness-of-fit statistical tests have been developed, the power of which depends on particular circumstances (i.e., sample size, outlets, etc.). With the aim of developing a more universal goodness-of-fit test, we propose an approach based on an N-metric with our chosen kernel function. To compare the power of 40 normality tests, the goodness-of-fit hypothesis was tested for 15 data distributions with 6 different sample sizes. Based on exhaustive comparative research results, we recommend the use of our test for samples of size

n \geq 118

. Full article

(This article belongs to the Special Issue Probability, Statistics and Their Applications 2021)

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Plot of out kernel function <math display="inline"><semantics> <mrow> <mi>K</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> with experimentally chosen optimal parameters <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>0.95</mn> <mo>,</mo> <mo> </mo> <mi>b</mi> <mo>=</mo> <mn>0.25</mn> <mo>,</mo> <mrow> <mtext> </mtext> <mi>and</mi> </mrow> <mo> </mo> <mi>c</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>. Full article ">Figure 2
Illustration of the power. Full article ">Figure 3
Significance levels of the statistic step function. Full article ">Figure 4
Average empirical power results, for the examined sample sizes, for the groups of symmetric, asymmetric, and modified normal distributions of five powerful goodness-of-fit tests. Full article ">Figure A1
Average empirical power results, for all sample sizes, for the groups of symmetric distributions of five powerful goodness-of-fit tests. Full article ">Figure A2
Average empirical power results for the examined sample sizes for the groups of asymmetric distributions of five powerful goodness-of-fit tests. Full article ">Figure A3
Average empirical power results for the examined sample sizes for the groups of the modified normal distributions of five powerful goodness-of-fit tests. Full article ">

19 pages, 832 KiB

Open AccessArticle

Supply Chain Coordination with a Risk-Averse Retailer and the Call Option Contract in the Presence of a Service Requirement

by Han Zhao, Hui Wang, Wei Liu, Shiji Song and Yu Liao

Mathematics 2021, 9(7), 787; https://doi.org/10.3390/math9070787 - 6 Apr 2021

Cited by 8 | Viewed by 2014

Abstract

This paper investigates a supply chain consisting of a single risk-neutral supplier and a single risk-averse retailer with the call option contract and a service requirement, where the retailer’s objective is to maximize the Conditional Value-at-Risk about profit. The optimal ordering quantity of [...] Read more.

This paper investigates a supply chain consisting of a single risk-neutral supplier and a single risk-averse retailer with the call option contract and a service requirement, where the retailer’s objective is to maximize the Conditional Value-at-Risk about profit. The optimal ordering quantity of the retailer and the optimal production quantity of the supplier are derived with the call option contract in the presence of a service requirement. Furthermore, by investigating the effect of the service level and the risk aversion on the supply chain, it is found that the retailer’s optimal Conditional Value-at-Risk is non-increasing in the service requirement and increasing in the risk aversion, while the supplier’s optimal expected profit is non-decreasing in the service and decreasing in the risk aversion. In addition, this paper demonstrates the impact of contract parameters on the service-constrained supply chain, and finds that the retailer’s optimal Conditional Value-at-Risk may be increasing, constant or decreasing in unit exercise price. Finally, with the call option contract, a distribution-free coordination condition is derived to achieve the Pareto improvement under Conditional Value-at-Risk criterion in the presence of a service requirement. Full article

24 pages, 2084 KiB

Open AccessArticle

A General Framework for Mixed and Incomplete Data Clustering Based on Swarm Intelligence Algorithms

by Yenny Villuendas-Rey, Eley Barroso-Cubas, Oscar Camacho-Nieto and Cornelio Yáñez-Márquez

Mathematics 2021, 9(7), 786; https://doi.org/10.3390/math9070786 - 6 Apr 2021

Cited by 2 | Viewed by 2015

Abstract

Swarm intelligence has appeared as an active field for solving numerous machine-learning tasks. In this paper, we address the problem of clustering data with missing values, where the patterns are described by mixed (or hybrid) features. We introduce a generic modification to three [...] Read more.

Swarm intelligence has appeared as an active field for solving numerous machine-learning tasks. In this paper, we address the problem of clustering data with missing values, where the patterns are described by mixed (or hybrid) features. We introduce a generic modification to three swarm intelligence algorithms (Artificial Bee Colony, Firefly Algorithm, and Novel Bat Algorithm). We experimentally obtain the adequate values of the parameters for these three modified algorithms, with the purpose of applying them in the clustering task. We also provide an unbiased comparison among several metaheuristics based clustering algorithms, concluding that the clusters obtained by our proposals are highly representative of the “natural structure” of data. Full article

(This article belongs to the Special Issue Swarm and Evolutionary Computation—Bridging Theory and Practice)

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Example of computation of a cluster center with 2D instances. (a) Instances in the cluster (b), Total dissimilarity among instances (using in this case the Euclidean distance) and (c) Instances in 2D; the cluster center (instance having minimum overall dissimilarity) is highlighted by a circle. Full article ">Figure 2
Dataset composed by 17 instances, described by <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math> features (x and y), forming <math display="inline"><semantics> <mrow> <mi>k</mi> <mo>=</mo> <mn>3</mn> </mrow> </semantics></math> clusters. The cluster centers are highlighted in gray. (a) The traditional <math display="inline"><semantics> <mrow> <mi>k</mi> <mo>×</mo> <mi>m</mi> </mrow> </semantics></math> representation; (b) Our proposed implementation. Note that our implementation has a constant size <math display="inline"><semantics> <mi>k</mi> </semantics></math>, despite the number of features in the dataset. Full article ">Figure 3
Modification strategy for updating individuals. (a) The instances in 2D, with cluster centers highlighted in gray, and the corresponding individual, (b) the same instances in 2D, with the updated centers after the modification of the individual. Full article ">Figure 4
Contingency table for the computation of the adjusted Rand index. Full article ">

10 pages, 1934 KiB

Open AccessArticle

Bivariate Infinite Series Solution of Kepler’s Equations

by Daniele Tommasini

Mathematics 2021, 9(7), 785; https://doi.org/10.3390/math9070785 - 6 Apr 2021

Cited by 3 | Viewed by 2231

Abstract

A class of bivariate infinite series solutions of the elliptic and hyperbolic Kepler equations is described, adding to the handful of 1-D series that have been found throughout the centuries. This result is based on an iterative procedure for the analytical computation of [...] Read more.

A class of bivariate infinite series solutions of the elliptic and hyperbolic Kepler equations is described, adding to the handful of 1-D series that have been found throughout the centuries. This result is based on an iterative procedure for the analytical computation of all the higher-order partial derivatives of the eccentric anomaly with respect to the eccentricity e and mean anomaly M in a given base point

(e_{c}, M_{c})

of the

(e, M)

plane. Explicit examples of such bivariate infinite series are provided, corresponding to different choices of

(e_{c}, M_{c})

, and their convergence is studied numerically. In particular, the polynomials that are obtained by truncating the infinite series up to the fifth degree reach high levels of accuracy in significantly large regions of the parameter space

(e, M)

. Besides their theoretical interest, these series can be used for designing 2-D spline numerical algorithms for efficiently solving Kepler’s equations for all values of the eccentricity and mean anomaly. Full article

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17 pages, 836 KiB

Open AccessArticle

Automatic Tempered Posterior Distributions for Bayesian Inversion Problems

by Luca Martino, Fernando Llorente, Ernesto Curbelo, Javier López-Santiago and Joaquín Míguez

Mathematics 2021, 9(7), 784; https://doi.org/10.3390/math9070784 - 6 Apr 2021

Cited by 10 | Viewed by 2292

Abstract

We propose a novel adaptive importance sampling scheme for Bayesian inversion problems where the inference of the variables of interest and the power of the data noise are carried out using distinct (but interacting) methods. More specifically, we consider a Bayesian analysis for [...] Read more.

We propose a novel adaptive importance sampling scheme for Bayesian inversion problems where the inference of the variables of interest and the power of the data noise are carried out using distinct (but interacting) methods. More specifically, we consider a Bayesian analysis for the variables of interest (i.e., the parameters of the model to invert), whereas we employ a maximum likelihood approach for the estimation of the noise power. The whole technique is implemented by means of an iterative procedure with alternating sampling and optimization steps. Moreover, the noise power is also used as a tempered parameter for the posterior distribution of the the variables of interest. Therefore, a sequence of tempered posterior densities is generated, where the tempered parameter is automatically selected according to the current estimate of the noise power. A complete Bayesian study over the model parameters and the scale parameter can also be performed. Numerical experiments show the benefits of the proposed approach. Full article

(This article belongs to the Special Issue Recent Advances in Data Science)

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16 pages, 1093 KiB

Open AccessArticle

Artificial Neural Network, Quantile and Semi-Log Regression Modelling of Mass Appraisal in Housing

by Jose Torres-Pruñonosa, Pablo García-Estévez and Camilo Prado-Román

Mathematics 2021, 9(7), 783; https://doi.org/10.3390/math9070783 - 6 Apr 2021

Cited by 16 | Viewed by 3179

Abstract

We used a large sample of 188,652 properties, which represented 4.88% of the total housing stock in Catalonia from 1994 to 2013, to make a comparison between different real estate valuation methods based on artificial neural networks (ANNs), quantile regressions (QRs) and semi-log [...] Read more.

We used a large sample of 188,652 properties, which represented 4.88% of the total housing stock in Catalonia from 1994 to 2013, to make a comparison between different real estate valuation methods based on artificial neural networks (ANNs), quantile regressions (QRs) and semi-log regressions (SLRs). A literature gap in regard to the comparison between ANN and QR modelling of hedonic prices in housing was identified, with this article being the first paper to include this comparison. Therefore, this study aimed to answer (1) whether QR valuation modelling of hedonic prices in the housing market is an alternative to ANNs, (2) whether it is confirmed that ANNs produce better results than SLRs when assessing housing in Catalonia, and (3) which of the three mass appraisal models should be used by Spanish banks to assess real estate. The results suggested that the ANNs and SLRs obtained similar and better performances than the QRs and that the SLRs performed better when the datasets were smaller. Therefore, (1) QRs were not found to be an alternative to ANNs, (2) it could not be confirmed whether ANNs performed better than SLRs when assessing properties in Catalonia and (3) whereas small and medium banks should use SLRs, large banks should use either SLRs or ANNs in real estate mass appraisal. Full article

(This article belongs to the Special Issue Statistical Data Modeling and Machine Learning with Applications)

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9 pages, 246 KiB

Open AccessArticle

Generalized Affine Connections Associated with the Space of Centered Planes

by Olga Belova

Mathematics 2021, 9(7), 782; https://doi.org/10.3390/math9070782 - 5 Apr 2021

Cited by 4 | Viewed by 1681

Abstract

Our purpose is to study a space

Π

of centered m-planes in n-projective space. Generalized fiberings (with semi-gluing) are investigated. Planar and normal affine connections associated with the space

Π

are set in the generalized fiberings. Fields of these affine connection [...] Read more.

Our purpose is to study a space

Π

of centered m-planes in n-projective space. Generalized fiberings (with semi-gluing) are investigated. Planar and normal affine connections associated with the space

Π

are set in the generalized fiberings. Fields of these affine connection objects define torsion and curvature tensors. The canonical cases of planar and normal generalized affine connections are considered. Full article

(This article belongs to the Special Issue Differential Geometry of Spaces with Special Structures)

14 pages, 635 KiB

Open AccessArticle

Tropical Balls and Its Applications to K Nearest Neighbor over the Space of Phylogenetic Trees

by Ruriko Yoshida

Mathematics 2021, 9(7), 779; https://doi.org/10.3390/math9070779 - 5 Apr 2021

Cited by 5 | Viewed by 2062

Abstract

A tropical ball is a ball defined by the tropical metric over the tropical projective torus. In this paper we show several properties of tropical balls over the tropical projective torus and also over the space of phylogenetic trees with a given set [...] Read more.

A tropical ball is a ball defined by the tropical metric over the tropical projective torus. In this paper we show several properties of tropical balls over the tropical projective torus and also over the space of phylogenetic trees with a given set of leaf labels. Then we discuss its application to the K nearest neighbors (KNN) algorithm, a supervised learning method used to classify a high-dimensional vector into given categories by looking at a ball centered at the vector, which contains K vectors in the space. Full article

(This article belongs to the Special Issue Mathematics in Biomedicine)

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20 pages, 7624 KiB

Open AccessArticle

Mexican Axolotl Optimization: A Novel Bioinspired Heuristic

by Yenny Villuendas-Rey, José L. Velázquez-Rodríguez, Mariana Dayanara Alanis-Tamez, Marco-Antonio Moreno-Ibarra and Cornelio Yáñez-Márquez

Mathematics 2021, 9(7), 781; https://doi.org/10.3390/math9070781 - 3 Apr 2021

Cited by 27 | Viewed by 5224

Abstract

When facing certain problems in science, engineering or technology, it is not enough to find a solution, but it is essential to seek and find the best possible solution through optimization. In many cases the exact optimization procedures are not applicable due to [...] Read more.

When facing certain problems in science, engineering or technology, it is not enough to find a solution, but it is essential to seek and find the best possible solution through optimization. In many cases the exact optimization procedures are not applicable due to the great computational complexity of the problems. As an alternative to exact optimization, there are approximate optimization algorithms, whose purpose is to reduce computational complexity by pruning some areas of the problem search space. To achieve this, researchers have been inspired by nature, because animals and plants tend to optimize many of their life processes. The purpose of this research is to design a novel bioinspired algorithm for numeric optimization: the Mexican Axolotl Optimization algorithm. The effectiveness of our proposal was compared against nine optimization algorithms (artificial bee colony, cuckoo search, dragonfly algorithm, differential evolution, firefly algorithm, fitness dependent optimizer, whale optimization algorithm, monarch butterfly optimization, and slime mould algorithm) when applied over four sets of benchmark functions (unimodal, multimodal, composite and competition functions). The statistical analysis shows the ability of Mexican Axolotl Optimization algorithm of obtained very good optimization results in all experiments, except for composite functions, where the Mexican Axolotl Optimization algorithm exhibits an average performance. Full article

(This article belongs to the Special Issue Bioinspired Computation: Recent Advances in Theory and Applications)

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17 pages, 991 KiB

Open AccessArticle

The Markovian Pattern of Social Deprivation for Mexicans with Diabetes

by José Carlos Ramírez, Francisco Ortiz-Arango and Leovardo Mata

Mathematics 2021, 9(7), 780; https://doi.org/10.3390/math9070780 - 3 Apr 2021

Viewed by 2304

Abstract

This paper aims to determine the Markovian pattern of the factors influencing social deprivation in Mexicans with Type 2 diabetes mellitus (DM2). To this end, we develop a methodology to meet the theoretical and practical considerations involved in applying a Hidden Markov Model [...] Read more.

This paper aims to determine the Markovian pattern of the factors influencing social deprivation in Mexicans with Type 2 diabetes mellitus (DM2). To this end, we develop a methodology to meet the theoretical and practical considerations involved in applying a Hidden Markov Model that uses non-panel data. After estimating the latent states and ergodic vectors for diabetic and non-diabetic populations, we find that the long-term state-dependent probabilities for people with DM2 show a darker perspective of impoverishment than the rest of the Mexican population. In the absence of extreme events that modify the present probability structure, the Markovian pattern confirms that people with DM2 will most likely become the poorest of Mexico’s poor. Full article

(This article belongs to the Special Issue Markov-Chain Modelling and Applications)

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18 pages, 2598 KiB

Open AccessArticle

The Impact of Rebate Distribution on Fairness Concerns in Supply Chains

by Xi Jiang and Jinsheng Zhou

Mathematics 2021, 9(7), 778; https://doi.org/10.3390/math9070778 - 2 Apr 2021

Cited by 4 | Viewed by 2424

Abstract

The reasonable distribution of supply chain profits among supply chain members is the core of the stability of a supply chain. Manufacturer rebates are a normal method to improve the performance of a supply chain and balance profit distribution. Based on consideration of [...] Read more.

The reasonable distribution of supply chain profits among supply chain members is the core of the stability of a supply chain. Manufacturer rebates are a normal method to improve the performance of a supply chain and balance profit distribution. Based on consideration of the behavior preferences of supply chain members, in this paper, we study the influence of rebate distribution on supply chain utility. We establish a supply chain utility model, including the proportion of distribution, fairness concern coefficient and effort level, and discuss three different situations of supply chain members. The results show that (i) a manufacturer’s rebate can more effectively improve the utility in a supply chain with fairness perception; (ii) with other conditions unchanged, the fairness perception of supply chain members will have a positive impact on their own utility; and (iii) at the same time, when the party who has more discourse power in the supply chain has a sense of fairness, this is conducive to realizing the stable development of the supply chain through changes in the proportion of rebate distribution. Full article

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17 pages, 1939 KiB

Open AccessArticle

Fuzzy Techniques Applied to the Analysis of the Causes and Effects of Tourism Competitiveness

by Martha B. Flores-Romero, Miriam E. Pérez-Romero, José Álvarez-García and María de la Cruz del Río-Rama

Mathematics 2021, 9(7), 777; https://doi.org/10.3390/math9070777 - 2 Apr 2021

Cited by 7 | Viewed by 2439

Abstract

The aim of this research is to identify and analyze the causes and effects of tourism competitiveness, as well as cause–effect relationships from the perspective of two groups of experts, which are decision makers versus academics/researchers, both from the tourism sector. The purpose [...] Read more.

The aim of this research is to identify and analyze the causes and effects of tourism competitiveness, as well as cause–effect relationships from the perspective of two groups of experts, which are decision makers versus academics/researchers, both from the tourism sector. The purpose is to respond to the question: do decision makers in the tourism sector share the same perspective as academics/researchers regarding the relationship between the causes and effects of tourism competitiveness? The methodology used is the theory of expertons, the theory of forgotten effects and the Hamming distance. It was found that in most cases, the groups of experts share perspective, since their differences are small or non-existent. However, in all the relationships analyzed (cause–effect, cause–cause, and effect–effect), academic experts reported the highest assessment. The greatest difference in opinion is identified in the evaluation of the “Environmental Commitment” and “Tourist Demand” relationship. Decision makers in the tourism sector are ignoring the growing inclination and sensitivity that tourists are adopting towards the environment. It is necessary for the tourism sector to develop and consolidate its commitment to caring for and preserving the environment, which is an element that contributes to a destination’s competitiveness and has two main effects: tourism demand and customer satisfaction. Full article

(This article belongs to the Special Issue Fuzzy Sets in Business Management, Finance, and Economics)

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28 pages, 13723 KiB

Open AccessArticle

A Method of Riemann–Hilbert Problem for Zhang’s Conjecture 1 in a Ferromagnetic 3D Ising Model: Trivialization of Topological Structure

by Osamu Suzuki and Zhidong Zhang

Mathematics 2021, 9(7), 776; https://doi.org/10.3390/math9070776 - 2 Apr 2021

Cited by 6 | Viewed by 2774

Abstract

A method of the Riemann–Hilbert problem is applied for Zhang’s conjecture 1 proposed in Philo. Mag. 87 (2007) 5309 for a ferromagnetic three-dimensional (3D) Ising model in the zero external field and the solution to the Zhang’s conjecture 1 is constructed by use [...] Read more.

A method of the Riemann–Hilbert problem is applied for Zhang’s conjecture 1 proposed in Philo. Mag. 87 (2007) 5309 for a ferromagnetic three-dimensional (3D) Ising model in the zero external field and the solution to the Zhang’s conjecture 1 is constructed by use of the monoidal transform. At first, the knot structure of the ferromagnetic 3D Ising model in the zero external field is determined and the non-local behavior of the ferromagnetic 3D Ising model can be described by the non-trivial knot structure. A representation from the knot space to the Clifford algebra of exponential type is constructed, and the partition function of the ferromagnetic 3D Ising model in the zero external field can be obtained by this representation (Theorem I). After a realization of the knots on a Riemann surface of hyperelliptic type, the monodromy representation is realized from the representation. The Riemann–Hilbert problem is formulated and the solution is obtained (Theorem II). Finally, the monoidal transformation is introduced for the solution and the trivialization of the representation is constructed (Theorem III). By this, we can obtain the desired solution to the Zhang’s conjecture 1 (Main Theorem). The present work not only proves the Zhang’s conjecture 1, but also shows that the 3D Ising model is a good platform for studying in deep the mathematical structure of a physical many-body interacting spin system and the connections between algebra, topology, and geometry. Full article

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28 pages, 476 KiB

Open AccessArticle

Chebyshev–Edgeworth-Type Approximations for Statistics Based on Samples with Random Sizes

by Gerd Christoph and Vladimir V. Ulyanov

Mathematics 2021, 9(7), 775; https://doi.org/10.3390/math9070775 - 2 Apr 2021

Cited by 2 | Viewed by 2544

Abstract

Second-order Chebyshev–Edgeworth expansions are derived for various statistics from samples with random sample sizes, where the asymptotic laws are scale mixtures of the standard normal or chi-square distributions with scale mixing gamma or inverse exponential distributions. A formal construction of asymptotic expansions is [...] Read more.

Second-order Chebyshev–Edgeworth expansions are derived for various statistics from samples with random sample sizes, where the asymptotic laws are scale mixtures of the standard normal or chi-square distributions with scale mixing gamma or inverse exponential distributions. A formal construction of asymptotic expansions is developed. Therefore, the results can be applied to a whole family of asymptotically normal or chi-square statistics. The random mean, the normalized Student t-distribution and the Student t-statistic under non-normality with the normal limit law are considered. With the chi-square limit distribution, Hotelling’s generalized

T_{0}^{2}

statistics and scale mixture of chi-square distributions are used. We present the first Chebyshev–Edgeworth expansions for asymptotically chi-square statistics based on samples with random sample sizes. The statistics allow non-random, random, and mixed normalization factors. Depending on the type of normalization, we can find three different limit distributions for each of the statistics considered. Limit laws are Student t-, standard normal, inverse Pareto, generalized gamma, Laplace and generalized Laplace as well as weighted sums of generalized gamma distributions. The paper continues the authors’ studies on the approximation of statistics for randomly sized samples. Full article

(This article belongs to the Special Issue Analytical Methods and Convergence in Probability with Applications)

25 pages, 435 KiB

Open AccessArticle

Analysis and Computation of Solutions for a Class of Nonlinear SBVPs Arising in Epitaxial Growth

by Amit K Verma, Biswajit Pandit and Ravi P. Agarwal

Mathematics 2021, 9(7), 774; https://doi.org/10.3390/math9070774 - 2 Apr 2021

Cited by 4 | Viewed by 1783

Abstract

In this work, the existence and nonexistence of stationary radial solutions to the elliptic partial differential equation arising in the molecular beam epitaxy are studied. Since we are interested in radial solutions, we focus on the fourth-order singular ordinary differential equation. It is [...] Read more.

In this work, the existence and nonexistence of stationary radial solutions to the elliptic partial differential equation arising in the molecular beam epitaxy are studied. Since we are interested in radial solutions, we focus on the fourth-order singular ordinary differential equation. It is non-self adjoint, it does not have exact solutions, and it admits multiple solutions. Here,

λ \in R

measures the intensity of the flux and G is stationary flux. The solution depends on the size of the parameter

λ

. We use a monotone iterative technique and integral equations along with upper and lower solutions to prove that solutions exist. We establish the qualitative properties of the solutions and provide bounds for the values of the parameter

λ

, which help us to separate existence from nonexistence. These results complement some existing results in the literature. To verify the analytical results, we also propose a new computational iterative technique and use it to verify the bounds on

λ

and the dependence of solutions for these computed bounds on

λ

. Full article

(This article belongs to the Special Issue New Trends on Boundary Value Problems)

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16 pages, 3727 KiB

Open AccessArticle

Identification of Inertial Parameters for Position and Force Control of Surgical Assistance Robots

by Pau Zamora-Ortiz, Javier Carral-Alvaro, Ángel Valera, José L. Pulloquinga, Rafael J. Escarabajal and Vicente Mata

Mathematics 2021, 9(7), 773; https://doi.org/10.3390/math9070773 - 2 Apr 2021

Cited by 3 | Viewed by 2526

Abstract

Surgeries or rehabilitation exercises are hazardous tasks for a mechanical system, as the device has to interact with parts of the human body without the hands-on experience that the surgeon or physiotherapist acquires over time. For various gynecological laparoscopic surgeries, such as laparoscopic [...] Read more.

Surgeries or rehabilitation exercises are hazardous tasks for a mechanical system, as the device has to interact with parts of the human body without the hands-on experience that the surgeon or physiotherapist acquires over time. For various gynecological laparoscopic surgeries, such as laparoscopic hysterectomy or laparoscopic pelvic endometriosis, Uterine Manipulators are used. These medical devices allow the uterus to be suitably mobilized. A gap needs to be filled in terms of the precise handling of this type of devices. In this sense, this manuscript first describes the mathematical procedure to identify the inertial parameters of uterine manipulators. These parameters are needed to establish an accurate position and force control for an electromechanical system to assist surgical operations. The method for identifying the mass and the center of mass of the manipulator is based on the solution of the equations for the static equilibrium of rigid solids. Based on the manipulator inertial parameter estimation, the paper shows how the force exerted by the manipulator can be obtained. For this purpose, it solves a matrix system composed of the torques and forces of the manipulator. Different manipulators have been used, and it has been verified that the mathematical procedures proposed in this work allow us to calculate in an accurate and efficient way the force exerted by these manipulators. Full article

(This article belongs to the Special Issue Mathematical Problems in Mechanical Engineering)

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17 pages, 805 KiB

Open AccessArticle

Graph Convolutional Network for Drug Response Prediction Using Gene Expression Data

by Seonghun Kim, Seockhun Bae, Yinhua Piao and Kyuri Jo

Mathematics 2021, 9(7), 772; https://doi.org/10.3390/math9070772 - 2 Apr 2021

Cited by 24 | Viewed by 8441

Abstract

Genomic profiles of cancer patients such as gene expression have become a major source to predict responses to drugs in the era of personalized medicine. As large-scale drug screening data with cancer cell lines are available, a number of computational methods have been [...] Read more.

Genomic profiles of cancer patients such as gene expression have become a major source to predict responses to drugs in the era of personalized medicine. As large-scale drug screening data with cancer cell lines are available, a number of computational methods have been developed for drug response prediction. However, few methods incorporate both gene expression data and the biological network, which can harbor essential information about the underlying process of the drug response. We proposed an analysis framework called DrugGCN for prediction of Drug response using a Graph Convolutional Network (GCN). DrugGCN first generates a gene graph by combining a Protein-Protein Interaction (PPI) network and gene expression data with feature selection of drug-related genes, and the GCN model detects the local features such as subnetworks of genes that contribute to the drug response by localized filtering. We demonstrated the effectiveness of DrugGCN using biological data showing its high prediction accuracy among the competing methods. Full article

(This article belongs to the Special Issue Fuzzy Sets and Soft Computing)

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Figure 1
Drug Graph Convolutional Network (DrugGCN) framework for drug response prediction. Full article ">Figure 2
Performance evaluation results of the six methods from L1000-IC50 dataset using the (a) RMSE, (b) PCC, (c) SCC, and (d) Normalized Discounted Cumulative Gain (NDCG). KRR, Kernelized Ridge Regression; BR, Bagging Regressor; KRL, Kernelized Ranked Learning; RWEN, Response-Weighted Elastic Net. Full article ">Figure 3
Box plot of ranks for each drug group predicted by DrugGCN using the L1000-IC50 dataset. Results from the RMSE, PCC, and SCC are combined. Full article ">Figure 4
Box plot of ranks for each drug group predicted by DrugGCN using the L1000-IC50 dataset. Ranks of drug groups are calculated from the (a) RMSE, (b) PCC, and (c) SCC. Full article ">Figure A1
Performance evaluation results of six methods from the L1000-AUC dataset using the (a) RMSE, (b) PCC, (c) SCC, and (d) NDCG. Full article ">Figure A2
Performance evaluation results of the six methods from the Var1000-IC50 dataset using the (a) RMSE, (b) PCC, (c) SCC, and (d) NDCG. Full article ">Figure A3
Performance evaluation results of the six methods from the Var1000-AUC dataset using the (a) RMSE, (b) PCC, (c) SCC, and (d) NDCG. Full article ">

15 pages, 1060 KiB

Open AccessArticle

Combining Classification and User-Based Collaborative Filtering for Matching Footwear Size

by Aleix Alcacer, Irene Epifanio, Jorge Valero and Alfredo Ballester

Mathematics 2021, 9(7), 771; https://doi.org/10.3390/math9070771 - 2 Apr 2021

Cited by 5 | Viewed by 2511

Abstract

Size mismatch is a serious problem in online footwear purchase because size mismatch implies an almost sure return. Not only foot measurements are important in selecting a size, but also user preference. This is the reason we propose several methodologies that combine the [...] Read more.

Size mismatch is a serious problem in online footwear purchase because size mismatch implies an almost sure return. Not only foot measurements are important in selecting a size, but also user preference. This is the reason we propose several methodologies that combine the information provided by a classifier with anthropometric measurements and user preference information through user-based collaborative filtering. As novelties: (1) the information sources are 3D foot measurements from a low-cost 3D foot digitizer, past purchases and self-reported size; (2) we propose to use an ordinal classifier after imputing missing data with different options based on the use of collaborative filtering; (3) we also propose an ensemble of ordinal classification and collaborative filtering results; and (4) several methodologies based on clustering and archetype analysis are introduced as user-based collaborative filtering for the first time. The hybrid methodologies were tested in a simulation study, and they were also applied to a dataset of Spanish footwear users. The results show that combining the information from both sources predicts the foot size better and the new proposals provide better accuracy than the classic alternatives considered. Full article

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19 pages, 335 KiB

Open AccessArticle

Extended Fuzzy Sets and Their Applications

by Bahram Farhadinia and Francisco Chiclana

Mathematics 2021, 9(7), 770; https://doi.org/10.3390/math9070770 - 2 Apr 2021

Cited by 3 | Viewed by 2322

Abstract

This contribution deals with introducing the innovative concept of extended fuzzy set (E-FS), in which the S-norm function of membership and non-membership grades is less than or equal to one. The proposed concept not only encompasses the concept of the fuzzy set (FS), [...] Read more.

This contribution deals with introducing the innovative concept of extended fuzzy set (E-FS), in which the S-norm function of membership and non-membership grades is less than or equal to one. The proposed concept not only encompasses the concept of the fuzzy set (FS), but it also includes the concepts of the intuitionistic fuzzy set (IFS), the Pythagorean fuzzy set (PFS) and the p-rung orthopair fuzzy set (p-ROFS). In order to explore the features of the E-FS concept, set and algebraic operations on E-FSs, average and geometric operations of E-FSs are studied and an E-FS score function is defined. The superiority of the E-FS concept is further confirmed with a score-based decision making technique in which the concepts of FS, IFS, PFS and p-ROFS do not make sense. Full article

(This article belongs to the Section Fuzzy Sets, Systems and Decision Making)

13 pages, 536 KiB

Open AccessArticle

Modelling an Industrial Robot and Its Impact on Productivity

by Carlos Llopis-Albert, Francisco Rubio and Francisco Valero

Mathematics 2021, 9(7), 769; https://doi.org/10.3390/math9070769 - 1 Apr 2021

Cited by 8 | Viewed by 2892

Abstract

This research aims to design an efficient algorithm leading to an improvement of productivity by posing a multi-objective optimization, in which both the time consumed to carry out scheduled tasks and the associated costs of the autonomous industrial system are minimized. The algorithm [...] Read more.

This research aims to design an efficient algorithm leading to an improvement of productivity by posing a multi-objective optimization, in which both the time consumed to carry out scheduled tasks and the associated costs of the autonomous industrial system are minimized. The algorithm proposed models the kinematics and dynamics of the industrial robot, provides collision-free trajectories, allows to constrain the energy consumed and meets the physical characteristics of the robot (i.e., restriction on torque, jerks and power in all driving motors). Additionally, the trajectory tracking accuracy is improved using an adaptive fuzzy sliding mode control (AFSMC), which allows compensating for parametric uncertainties, bounded external disturbances and constraint uncertainties. Therefore, the system stability and robustness are enhanced; thus, overcoming some of the limitations of the traditional proportional-integral-derivative (PID) controllers. The trade-offs among the economic issues related to the assembly line and the optimal time trajectory of the desired motion are analyzed using Pareto fronts. The technique is tested in different examples for a six-degrees-of-freedom (DOF) robot system. Results have proved how the use of this methodology enhances the performance and reliability of assembly lines. Full article

(This article belongs to the Special Issue Mathematical Problems in Mechanical Engineering)

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20 pages, 2066 KiB

Open AccessArticle

A Hybrid Metaheuristic for the Unrelated Parallel Machine Scheduling Problem

by Dung-Ying Lin and Tzu-Yun Huang

Mathematics 2021, 9(7), 768; https://doi.org/10.3390/math9070768 - 1 Apr 2021

Cited by 8 | Viewed by 3241

Abstract

The unrelated parallel machine scheduling problem aims to assign jobs to independent machines with sequence-dependent setup times so that the makespan is minimized. When many practical considerations are introduced, solving the resulting problem is challenging, especially when problems of realistic sizes are of [...] Read more.

The unrelated parallel machine scheduling problem aims to assign jobs to independent machines with sequence-dependent setup times so that the makespan is minimized. When many practical considerations are introduced, solving the resulting problem is challenging, especially when problems of realistic sizes are of interest. In this study, in addition to the conventional objective of minimizing the makespan, we further consider the burn-in (B/I) procedure that is required in practice; we need to ensure that the scheduling results satisfy the B/I ratio constrained by the equipment. To solve the resulting complicated problem, we propose a population-based simulated annealing algorithm embedded with a variable neighborhood descent technique. Empirical results show that the proposed solution strategy outperforms a commonly used commercial optimization package; it can obtain schedules that are better than the schedules used in practice, and it does so in a more efficient manner. Full article

(This article belongs to the Special Issue Theoretical and Computational Research in Various Scheduling Models)

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12 pages, 941 KiB

Open AccessArticle

A Concretization of an Approximation Method for Non-Affine Fractal Interpolation Functions

by Alexandra Băicoianu, Cristina Maria Păcurar and Marius Păun

Mathematics 2021, 9(7), 767; https://doi.org/10.3390/math9070767 - 1 Apr 2021

Cited by 1 | Viewed by 1877

Abstract

The present paper concretizes the models proposed by S. Ri and N. Secelean. S. Ri proposed the construction of the fractal interpolation function (FIF) considering finite systems consisting of Rakotch contractions, but produced no concretization of the model. N. Secelean considered countable systems [...] Read more.

The present paper concretizes the models proposed by S. Ri and N. Secelean. S. Ri proposed the construction of the fractal interpolation function (FIF) considering finite systems consisting of Rakotch contractions, but produced no concretization of the model. N. Secelean considered countable systems of Banach contractions to produce the fractal interpolation function. Based on the abovementioned results, in this paper, we propose two different algorithms to produce the fractal interpolation functions both in the affine and non-affine cases. The theoretical context we were working in suppose a countable set of starting points and a countable system of Rakotch contractions. Due to the computational restrictions, the algorithms constructed in the applications have the weakness that they use a finite set of starting points and a finite system of Rakotch contractions. In this respect, the attractor obtained is a two-step approximation. The large number of points used in the computations and the graphical results lead us to the conclusion that the attractor obtained is a good approximation of the fractal interpolation function in both cases, affine and non-affine FIFs. In this way, we also provide a concretization of the scheme presented by C.M. Păcurar. Full article

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16 pages, 1647 KiB

Open AccessArticle

Optimization of the ANNs Predictive Capability Using the Taguchi Approach: A Case Study

by Andrea Manni, Giovanna Saviano and Maria Grazia Bonelli

Mathematics 2021, 9(7), 766; https://doi.org/10.3390/math9070766 - 1 Apr 2021

Cited by 6 | Viewed by 2050

Abstract

Artificial neural networks (ANNs) are a valid alternative predictive method to the traditional statistical techniques currently used in many research fields where a massive amount of data is challenging to manage. In environmental analysis, ANNs can analyze pollution sources in large areas, estimating [...] Read more.

Artificial neural networks (ANNs) are a valid alternative predictive method to the traditional statistical techniques currently used in many research fields where a massive amount of data is challenging to manage. In environmental analysis, ANNs can analyze pollution sources in large areas, estimating difficult and expensive to detect contaminants from other easily measurable pollutants, especially for screening procedures. In this study, organic micropollutants have been predicted from heavy metals concentration using ANNs. Sampling was performed in an agricultural field where organic and inorganic contaminants concentrations are beyond the legal limits. A critical problem of a neural network design is to select its parametric topology, which can prejudice the reliability of the model. Therefore, it is very important to assess the performance of ANNs when applying different types of parameters of the net. In this work, based on Taguchi L₁₂ orthogonal array, turning experiments were conducted to identify the best parametric set of an ANNs design, considering different combinations of sample number, scaling, training rate, activation functions, number of hidden layers, and epochs. The composite desirability value for the multi-response variables has been obtained through the desirability function analysis (DFA). The parameters’ optimum levels have been identified using this methodology. Full article

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9 pages, 270 KiB

Open AccessArticle

Fuzzy Inner Product Space: Literature Review and a New Approach

by Lorena Popa and Lavinia Sida

Mathematics 2021, 9(7), 765; https://doi.org/10.3390/math9070765 - 1 Apr 2021

Cited by 5 | Viewed by 2655

Abstract

The aim of this paper is to provide a suitable definition for the concept of fuzzy inner product space. In order to achieve this, we firstly focused on various approaches from the already-existent literature. Due to the emergence of various studies on fuzzy [...] Read more.

The aim of this paper is to provide a suitable definition for the concept of fuzzy inner product space. In order to achieve this, we firstly focused on various approaches from the already-existent literature. Due to the emergence of various studies on fuzzy inner product spaces, it is necessary to make a comprehensive overview of the published papers on the aforementioned subject in order to facilitate subsequent research. Then we considered another approach to the notion of fuzzy inner product starting from P. Majundar and S.K. Samanta’s definition. In fact, we changed their definition and we proved some new properties of the fuzzy inner product function. We also proved that this fuzzy inner product generates a fuzzy norm of the type Nădăban-Dzitac. Finally, some challenges are given. Full article

(This article belongs to the Special Issue Fuzzy Logic and Soft Computing – Dedicated to the Centenary of the Birth of Lotfi A. Zadeh (1921-2017))

12 pages, 311 KiB

Open AccessArticle

The Size, Multipartite Ramsey Numbers for nK₂ Versus Path–Path and Cycle

by Yaser Rowshan, Mostafa Gholami and Stanford Shateyi

Mathematics 2021, 9(7), 764; https://doi.org/10.3390/math9070764 - 1 Apr 2021

Cited by 8 | Viewed by 2103

Abstract

For given graphs

G_{1}, G_{2}, \dots, G_{n}

and any integer j, the size of the multipartite Ramsey number

m_{j} (G_{1}, G_{2}, \dots, G_{n})

is the smallest positive [...] Read more.

For given graphs

G_{1}, G_{2}, \dots, G_{n}

and any integer j, the size of the multipartite Ramsey number

m_{j} (G_{1}, G_{2}, \dots, G_{n})

is the smallest positive integer t such that any n-coloring of the edges of

K_{j \times t}

contains a monochromatic copy of

G_{i}

in color i for some i,

1 \leq i \leq n

, where

K_{j \times t}

denotes the complete multipartite graph having j classes with t vertices per each class. In this paper, we computed the size of the multipartite Ramsey numbers

m_{j} (K_{1, 2}, P_{4}, n K_{2})

for any

j, n \geq 2

and

m_{j} (n K_{2}, C_{7})

, for any

j \leq 4

and

n \geq 2

. Full article

(This article belongs to the Special Issue Advances in Discrete Applied Mathematics and Graph Theory)

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14 pages, 1397 KiB

Open AccessArticle

Adaptive State-Quantized Control of Uncertain Lower-Triangular Nonlinear Systems with Input Delay

by Sung Jin Yoo

Mathematics 2021, 9(7), 763; https://doi.org/10.3390/math9070763 - 1 Apr 2021

Cited by 2 | Viewed by 1606

Abstract

In this paper, we investigate the adaptive state-quantized control problem of uncertain lower-triangular systems with input delay. It is assumed that all state variables are quantized for the feedback control design. The error transformation method using an auxiliary time-varying signal is presented to [...] Read more.

In this paper, we investigate the adaptive state-quantized control problem of uncertain lower-triangular systems with input delay. It is assumed that all state variables are quantized for the feedback control design. The error transformation method using an auxiliary time-varying signal is presented to deal with the compensation problem of input delay. Based on the error surfaces with the auxiliary variable, a neural-network-based adaptive state-quantized control scheme is constructed with the design of the input delay compensator. Different from existing results in the literature, the proposed method exhibits the following features: (i) compensating for the input delay effect by using quantized states; and (ii) establishing the stability of the adaptive quantized feedback control system in the presence of input delay. Furthermore, the boundedness of all the signals in the closed-loop and the convergence of the tracking error are analyzed. The effectiveness of the developed control strategy is demonstrated through the simulation on a hydraulic servo system. Full article

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21 pages, 658 KiB

Open AccessArticle

New Robust Cross-Variogram Estimators and Approximations of Their Distributions Based on Saddlepoint Techniques

by Alfonso García-Pérez

Mathematics 2021, 9(7), 762; https://doi.org/10.3390/math9070762 - 1 Apr 2021

Cited by 4 | Viewed by 2019

Abstract

Let

Z (s) = {(Z_{1} (s), \dots, Z_{p} (s))}^{t}

be an isotropic second-order stationary multivariate spatial process. We measure the statistical association between the p random components of

Z

with [...] Read more.

Let

Z (s) = {(Z_{1} (s), \dots, Z_{p} (s))}^{t}

be an isotropic second-order stationary multivariate spatial process. We measure the statistical association between the p random components of

Z

with the correlation coefficients and measure the spatial dependence with variograms. If two of the

Z

components are correlated, the spatial information provided by one of them can improve the information of the other. To capture this association, both within components of

Z (s)

and across

s

, we use a cross-variogram. Only two robust cross-variogram estimators have been proposed in the literature, both by Lark, and their sample distributions were not obtained. In this paper, we propose new robust cross-variogram estimators, following the location estimation method instead of the scale estimation one considered by Lark, thus extending the results obtained by García-Pérez to the multivariate case. We also obtain accurate approximations for their sample distributions using saddlepoint techniques and assuming a multivariate-scale contaminated normal model. The question of the independence of the transformed variables to avoid the usual dependence of spatial observations is also considered in the paper, linking it with the acceptance of linear variograms and cross-variograms. Full article

(This article belongs to the Special Issue Methodological and Applied Contributions on Stochastic Modelling and Forecasting)

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