Symmetry, Volume 12, Issue 2 (February 2020) – 140 articles

The fusion of light detection and ranging (LiDAR) and camera data in real-time is known to be a crucial process in many applications, such as in autonomous driving, industrial automation, and robotics. Especially in the case of autonomous vehicles, the efficient fusion of data from these two types of sensors is important to enabling the depth of objects as well as the detection of objects at short and long distances. As both the sensors are capable of capturing the different attributes of the environment simultaneously, the integration of those attributes with an efficient fusion approach greatly benefits the reliable and consistent perception of the environment. This paper presents a method to estimate the distance (depth) between a self-driving car and other vehicles, objects, and signboards on its path using the accurate fusion approach. Based on the geometrical transformation and projection, low-level sensor fusion was performed between a camera and LiDAR using a 3D marker. Further, the fusion information is utilized to estimate the distance of objects detected by the RefineDet detector. Finally, the accuracy and performance of the sensor fusion and distance estimation approach were evaluated in terms of quantitative and qualitative analysis by considering real road and simulation environment scenarios. Thus the proposed low-level sensor fusion, based on the computational geometric transformation and projection for object distance estimation proves to be a promising solution for enabling reliable and consistent environment perception ability for autonomous vehicles. Full article

Wellbores are largely constructed during coal mining, shale gas production, and geothermal exploration. Studying the shape and size of the disturbed zone in surrounding rock is of great significance for wellbore stability control. In this paper, a theoretical model for elastic-plastic-damage analysis around a deep circular wellbore under non-uniform compression is proposed. Based on the elastoplastic softening constitutive model and Mohr-Coulomb strength criterion, the analytical expressions of stresses in the elastic, plastic and damaged zones around a circle wellbore are derived. Further, the boundary line equations among the three zones are obtained according to the conditions of stress continuity. Then, the influence rules of non-uniform in-situ stress and mechanical parameters on the stress distribution and plastic zone size in surrounding rock mass are analyzed. The plastic and the damaged zones are both approximately elliptical in shape. When the lateral stress coefficient of the in-situ stress field takes the value 1, the model degenerates into the Yuan Wenbo’s Solution. If the brittleness coefficient of the surrounding rock is 0, the model degenerates into the Kastner’s Equation. Finally, the results are compared with those under two special cases (in the elastoplastic softening rock under a uniform stress field, in the ideal elastoplastic rock under a non-uniform stress field) and a common approximation method (the perturbation method). Full article

This paper addresses the problem of the optimal location and sizing of photovoltaic (PV) sources in direct current (DC) electrical networks considering time-varying load and renewable generation curves. To represent this problem, a mixed-integer nonlinear programming (MINLP) model is developed. The main idea of including PV sources in the DC grid is minimizing the total greenhouse emissions produced by diesel generators in isolated areas. An artificial neural network is employed for short-term forecasting to deal with uncertainties in the PV power generation. The general algebraic modeling system (GAMS) package is employed to solve the MINLP model by using the CONOPT solver that works with mixed and integer variables. Numerical results demonstrate important reductions of harmful gas emissions to the atmosphere when PV sources are optimally integrated (size and location) to the DC grid. Full article

When analyzing the dynamic behavior of multi-body elastic systems, a commonly used method is the finite element method conjunctively with Lagrange’s equations. The central problem when approaching such a system is determining the equations of motion for a single finite element. The paper presents an alternative method of calculation theses using the Gibbs–Appell (GA) formulation, which requires a smaller number of calculations and, as a result, is easier to apply in practice. For this purpose, the energy of the accelerations for one single finite element is calculated, which will be used then in the GA equations. This method can have advantages in applying to the study of multi-body systems with elastic elements and in the case of robots and manipulators that have in their composition some elastic elements. The number of differentiation required when using the Gibbs–Appell method is smaller than if the Lagrange method is used which leads to a smaller number of operations to obtain the equations of motion. Full article

The modern state of the Pauli exclusion principle is shortly discussed. We describe the discovery by Pauli, his principle for electrons, and how it was generalized for all elementary particles in the framework of quantum mechanics. The motivations and theoretical conceptions that induced the experiments for verification of the Pauli exclusion principle are analyzed. The results and methodology of two different types of experiments are discussed: (1) the search of unusual atoms and nuclei in the stable non-Pauli states, and (2) the experiments in which the emitted radiation of non-Pauli transitions is measured. In conclusion, the comments on the discussed experiments that follow from the general quantum mechanical conceptions and group theory are formulated. Full article

Several facial expression recognition methods cluster facial elements according to similarity and weight them considering the importance of each element in classification. However, these methods are limited by the pre-definitions of units restricting modification of the structure during optimization. This study proposes a modified support vector machine classifier called Grid Map, which is combined with reinforcement learning to improve the classification accuracy. To optimize training, the input image size is normalized according to the cascade rules of a pre-processing detector, and the regional weights are assigned by an adaptive cell size that divides each region of the image using bounding grids. Reducing the size of the bounding grid reduces the area used for feature extraction, allowing more detailed weighted features to be extracted. Error-correcting output codes with a histogram of gradient is selected as the classification method via an experiment to determine the optimal feature and classifier selection. The proposed method is formulated into a decision process and solved via Q-learning. To classify seven emotions, the proposed method exhibits accuracies of 96.36% and 98.47% for four databases and Extended Cohn-–Kanade Dataset (CK+), respectively. Compared to the basic method exhibiting a similar accuracy, the proposed method requires 68.81% fewer features and only 66.33% of the processing time. Full article

In this paper, we aim to study the impact of the shift in herding tendency on the diffusion of internet investment products in modular social networks. The epidemic spreading mechanism is applied and numerical analyses are conducted. The results suggest that the increase in herding tendency slows down the diffusion process and postpones the outbreak time of the diffusion, but such negative effects can be compromised when the independent acceptance willingness is high. When independent acceptance willingness is low, the limited extent of the herding tendency increases the diffusion scope. In addition, the expansion of the propagation lifetime or the increase of the clustering coefficient increases the threshold so that the herding tendency has an effect on outbreak size. Further, the growth of the herding propensity tends to magnify the positive influence of the clustering coefficient and the negative effect of the modularity. Full article

We discuss the tight-binding models of solid state physics with the $Z_2$ sublattice symmetry in the presence of elastic deformations in an important particular case—the tight binding model of graphene. In order to describe the dynamics of electronic quasiparticles, the Wigner–Weyl formalism is explored. It allows the calculation of the two-point Green’s function in the presence of two slowly varying external electromagnetic fields and the inhomogeneous modification of the hopping parameters that result from elastic deformations. The developed formalism allows us to consider the influence of elastic deformations and the variations of magnetic field on the quantum Hall effect. Full article

Recently, Tahmasebi and Eskandarzadeh introduced a new extended cumulative entropy (ECE). In this paper, we present results on shift-dependent measure of ECE and its dynamic past version. These results contain stochastic order, upper and lower bounds, the symmetry property and some relationships with other reliability functions. We also discuss some properties of conditional weighted ECE under some assumptions. Finally, we propose a nonparametric estimator of this new measure and study its practical results in blind image quality assessment. Full article

Cyclic associativity can be regarded as a kind of variation symmetry, and cyclic associative groupoid (CA-groupoid) is a generalization of commutative semigroup. In this paper, the various cancellation properties of CA-groupoids, including cancellation, quasi-cancellation and power cancellation, are studied. The relationships among cancellative CA-groupoids, quasi-cancellative CA-groupoids and power cancellative CA-groupoids are found out. Moreover, the concept of variant CA-groupoid is proposed firstly, some examples are presented. It is shown that the structure of variant CA-groupoid is very interesting, and the construction methods and decomposition theorem of variant CA-groupoids are established. Full article

The maldistribution of fluid flow through multi-channels is a critical issue encountered in many areas, such as multi-channel heat exchangers, electronic device cooling, refrigeration and cryogenic devices, air separation and the petrochemical industry. In this paper, the uniformity of flow distribution in a printed circuit heat exchanger (PCHE) is investigated. The flow distribution and resistance characteristics of a PCHE plate are studied with numerical models under different flow distribution cases. The results show that the sudden change in the angle of the fluid at the inlet of the channel can be greatly reduced by using a spreader plate with an equal inner and outer radius. The flow separation of the fluid at the inlet of the channel can also be weakened and the imbalance of flow distribution in the channel can be reduced. Therefore, the flow uniformity can be improved and the pressure loss between the inlet and outlet of PCHEs can be reduced. The flow maldistribution in each PCHE channel can be reduced to ± 0.2%, and the average flow maldistribution in all PCHE channels can be reduced to less than 5% when the number of manifolds reaches nine. The numerical simulation of fluid flow distribution can provide guidance for the subsequent research and the design and development of multi-channel heat exchangers. In summary, the symmetry of the fluid flow in multi-channels for PCHE was analyzed in this work. This work presents the frequently encountered problem of maldistribution of fluid flow in engineering, and the performance promotion leads to symmetrical aspects in both the structure and the physical process. Full article

We show that in a relativistically covariant formulation of the two-body problem, the bound state spectrum is in agreement, up to relativistic corrections, with the non-relativistic bound-state spectrum. This solution is achieved by solving the problem with support of the wave functions in an $O(2,1)$ invariant submanifold of the Minkowski spacetime. The $O(3,1)$ invariance of the differential equation requires, however, that the solutions provide a representation of $O(3,1)$ . Such solutions are obtained by means of the method of induced representations, providing a basic insight into the subject of the symmetries of relativistic dynamics. Full article

In this paper, we consider the eigenproblems for Latin squares in a bipartite min-max-plus system. The focus is upon developing a new algorithm to compute the eigenvalue and eigenvectors (trivial and non-trivial) for Latin squares in a bipartite min-max-plus system. We illustrate the algorithm using some examples. The proposed algorithm is implemented in MATLAB, using max-plus algebra toolbox. Computationally speaking, our algorithm has a clear advantage over the power algorithm presented by Subiono and van der Woude. Because our algorithm takes $0.088783$ sec to solve the eigenvalue problem for Latin square presented in Example 2, while the compared one takes $1.718662$ sec for the same problem. Furthermore, a time complexity comparison is presented, which reveals that the proposed algorithm is less time consuming when compared with some of the existing algorithms. Full article

Encountering a problem or error in the final stages of providing products or services increases costs and delays scheduling. The key task is to ensure quality and reliability in the early stages of the production process and prevent errors from occurring from the beginning. Failure mode and effect analysis (FMEA) is one of the tools for identifying potential problems and their impact on products and services. The conventional FMEA technique has been criticized extensively due to its disadvantages. In this study, the concepts of uncertainty and reliability are considered simultaneously. The processes of weighting risk factors, prioritizing failures by using the stepwise weight assessment ratio analysis (SWARA)–gray relational analysis (GRA) integrated method based on Ζ-number theory and complete prioritization of failures are implemented. Crucial management indices, such as cost and time, are considered in addition to severity, occurrence and detection factors along with assigning symmetric form of the weights to them. This, in turn, increases the interpretability of results and reduces the decision-maker’s subjectivity in risk prioritization. The developed model is implemented on solar panel data with 19 failure modes determined by the FMEA team. Results show that the proposed approach provides a more complete and realistic prioritization of failures than conventional FMEA and fuzzy GRA methods do. Full article

In recent decades, an interest has been developed towards the thermal consequences of nanofluid because of utilization of nano-materials to improve the thermal conductivity of traditional liquid and subsequently enhance the heat transportation phenomenon. Following this primarily concept, this current work investigates the thermal developed flow of third-grade nanofluid configured by a stretched surface with additional features of activation energy, viscous dissipation and second-order slip. Buongiorno’s nanofluid model is used to explore the thermophoresis and Brownian motion features based on symmetry fundamentals. It is further assumed that the nanoparticles contain gyrotactic microorganisms, which are associated with the most fascination bioconvection phenomenon. The flow problem owing to the partial differential equations is renovated into dimensional form, which is numerically simulated with the help of bvp4c, by using MATLAB software. The aspects of various physical parameters associated to the current analysis are graphically examined against nanoparticles’ velocity, temperature, concentration and gyrotactic microorganisms’ density distributions. Further, the objective of local Nusselt number, local Sherwood number and motile density number are achieved numerically with variation of various parameters. The results presented here may find valuable engineering applications, like cooling liquid metals, solar systems, power production, solar energy, thermal extrusion systems cooling of machine equipment, transformer oil and microelectronics. Further, flow of nanoparticles containing gyrotactic microorganisms has interesting applications in microbial fuel cells, microfluidic devices, bio-technology and enzyme biosensors. Full article

In this paper, the newly proposed concept of Raina’s function and quantum calculus are utilized to anticipate the quantum behavior of two variable Ostrowski-type inequalities. This new technique is the convolution of special functions with hypergeometric and Mittag–Leffler functions, respectively. This new concept will have the option to reduce self-similitudes in the quantum attractors under investigation. We discuss the implications and other consequences of the quantum Ostrowski-type inequalities by deriving an auxiliary result for a $q_1{q}_2$ -differentiable function by inserting Raina’s functions. Meanwhile, we present a numerical scheme that can be used to derive variants for Ostrowski-type inequalities in the sense of coordinated generalized $\mathsf{\Phi }$ -convex functions with the quantum approach. This new scheme of study for varying values of parameters with the involvement of Raina’s function yields extremely intriguing outcomes with an illustrative example. It is supposed that this investigation will provide new directions for the capricious nature of quantum theory. Full article

Face recognition (FR) has received considerable attention in the field of security, especially in the use of closed-circuit television (CCTV) cameras in security monitoring. Although significant advances in the field of computer vision are made, advanced face recognition systems provide satisfactory performance only in controlled conditions. They deteriorate significantly in the face of real-world scenarios such as lighting conditions, motion blur, camera resolution, etc. This article shows how we design, implement, and conduct the empirical comparisons of machine learning open libraries in building attendance taking (AT) support systems using indoor security cameras called ATSS. Our trial system was deployed to record the appearances of 120 students in five classes who study on the third floor of FPT Polytechnic College building. Our design allows for flexible system scaling, and it is not only usable for a school but a generic attendance system with CCTV. The measurement results show that the accuracy is suitable for many different environments. Full article

Fifth-generation (5G) communications system is commercially introduced by several mobile operators where sub-6 GHz bands are the backbone of the 5G networks. A large-scale multiple-input multiple-output (MIMO), or massive MIMO (mMIMO), technology has a major impact to secure high data rate, high spectral efficiency, and quality of service (QoS). It could also have a major role in the beyond-5G systems. A massive number of antennas seek advanced signal processing to detect and equalize the signal. However, optimal detectors, such as the maximum likelihood (ML) and maximum posterior (MAP), are not desirable in implementation due to extremely high complexity. Therefore, sub-optimum solutions have been introduced to obtain and guarantee enough balance between the performance and the computational complexity. In this paper, a robust and joint low complexity detection algorithm is proposed based on the Jacobi (JA) and Gauss–Seidel (GS) methods. In such iterative methods, the performance, complexity, and convergence rate are highly dependent on the initial vector. In this paper, initial solution is proposed by exploiting the benefits of a stair matrix to obtain a fast convergence rate, high performance, and low complexity. Numerical results show that proposed algorithm achieves high accuracy and relieve the computational complexity even when the BS-to-user-antenna ratio (BUAR) is small. Full article

Sustainable products and their marketing have played a crucial role in developing more sustainable consumption patterns and solutions for socio-ecological problems. They have been demonstrated to significantly decrease social consumption problems. Neuromarketing has recently gained considerable popularity and helped companies generate deeper insights into consumer behavior. It has provided new ways of conceptualizing consumer behavior and decision making. Thus, this research aims to investigate the factors influencing managers’ decisions to adopt neuromarketing techniques in sustainable product marketing using the fuzzy analytic hierarchy process (AHP) approach. Symmetric triangular fuzzy numbers were used to indicate the relative strength of the elements in the hierarchy. Data were collected from the marketing managers of several companies who have experience with sustainable product marketing through online shopping platforms. The results revealed that the accuracy and bias of neuromarketing techniques have been the main critical factors for managers to select neuromarketing in their business for advertising and branding purposes. This research provides important results on the use of neuromarketing techniques for sustainable product marketing, as well as their limitations and implications, and it also presents useful information on the factors impacting business managers’ decision making in adopting neuroscience techniques for sustainable product development and marketing. Full article

The weighted independent set problem on $P_5$ -free graphs has numerous applications, including data mining and dispatching in railways. The recognition of $P_5$ -free graphs is executed in polynomial time. Many problems, such as chromatic number and dominating set, are NP-hard in the class of $P_5$ -free graphs. The size of a minimum independent feedback vertex set that belongs to a $P_5$ -free graph with n vertices can be computed in $O(n^{16})$ time. The unweighted problems, clique and clique cover, are NP-complete and the independent set is polynomial. In this work, the $P_5$ -free graphs using the weak decomposition are characterized, as is the dominating clique, and they are given an $O(n(n+m))$ recognition algorithm. Additionally, we calculate directly the clique number and the chromatic number; determine in $O(n)$ time, the size of a minimum independent feedback vertex set; and determine in $O(n+m)$ time the number of stability, the dominating number and the minimum clique cover. Full article

There are currently different types of commercial optical disdrometers to measure the rainfall intensity, of which many are commonly used for monitoring road conditions. Having information about the amount of rain, the composition of the precipitation particles and visibility are essential to avoid accidents, which requires intelligent systems that warn drivers and redirect traffic. However, few studies related to Intelligent Transport Systems (ITS) have been performed regarding why these devices are not optimized for this type of applications. Therefore, this paper analyzes and evaluates the operating mode of these equipment through their theoretical model, which will allow the design of prototypes of disdrometers with different characteristics. In addition, this model will be implemented in a simulation program, through which an exhaustive study analyzing how the type of precipitation and its intensity affect the measures provided by the model will be conducted. In this way, the results will help optimize its operation to be thus used in urban settings, which will allow obtaining more accurate real-time information, better traffic management, and a reduction in the number of accidents. Full article

With the increasing pressure from global competition, manufacturers have realized that sustainable production is significant in supply chain management. Sustainable supplier selection and order allocation (SSS&OA) play a distinct and critical role for organizations to achieve sustainable development and build competitive advantage. In this paper, we aim to develop a novel SSS&OA model for selecting the most suitable sustainable suppliers and determining the optimal order sizes among them. First, double hierarchy hesitant linguistic term sets (DHHLTSs) are adopted to deal with uncertainty in evaluating the sustainable performance of alternative suppliers. Then, an extended decision field theory is proposed to choose efficient sustainable suppliers dynamically. Considering quantity discount, a multi-objective linear programming (MOLP) model is established to allocate reasonable order quantities among the selected suppliers. Finally, the applicability and effectiveness of the developed model are illustrated through its application in the electronic industry and through a comparative analysis with other methods. Full article

Network traffic classification based on machine learning is an important branch of pattern recognition in computer science. It is a key technology for dynamic intelligent network management and enhanced network controllability. However, the traffic classification methods still facing severe challenges: The optimal set of features is difficult to determine. The classification method is highly dependent on the effective characteristic combination. Meanwhile, it is also important to balance the experience risk and generalization ability of the classifier. In this paper, an improved network traffic classification model based on a support vector machine is proposed. First, a filter-wrapper hybrid feature selection method is proposed to solve the false deletion of combined features caused by a traditional feature selection method. Second, to balance the empirical risk and generalization ability of support vector machine (SVM) traffic classification model, an improved parameter optimization algorithm is proposed. The algorithm can dynamically adjust the quadratic search area, reduce the density of quadratic mesh generation, improve the search efficiency of the algorithm, and prevent the over-fitting while optimizing the parameters. The experiments show that the improved traffic classification model achieves higher classification accuracy, lower dimension and shorter elapsed time and performs significantly better than traditional SVM and the other three typical supervised ML algorithms. Full article

Thermal Leptogenesis and the gravitino problem are reviewed in the framework of non-standard cosmologies. We consider in particular the $f\left(\mathcal{T}\right)$ cosmology, where $\mathcal{T}$ is the torsion field. We constrain the parameters space of these cosmological models consistently with thermal Leptogenesis scenario (with degenerate mass spectrum of light neutrinos), and we show that they allow to solve the gravitino problem as well. Owing to the similar characteristics to $f\left(\mathcal{T}\right)$ cosmology, we shortly discuss also the case of the shear dominated Universe. Full article

Incremental feature extraction algorithms are designed to analyze large-scale data streams. Many of them suffer from high computational cost, time complexity, and data dependency, which adversely affects the processing of the data stream. With this motivation, this paper presents a novel incremental feature extraction approach based on the Discrete Cosine Transform (DCT) for the data stream. The proposed approach is separated into initial and sequential phases, and each phase uses a fixed-size windowing technique for processing the current samples. The initial phase is performed only on the first window to construct the initial model as a baseline. In this phase, normalization and DCT are applied to each sample in the window. Subsequently, the efficient feature subset is determined by a particle swarm optimization-based method. With the construction of the initial model, the sequential phase begins. The normalization and DCT processes are likewise applied to each sample. Afterward, the feature subset is selected according to the initial model. Finally, the k-nearest neighbor classifier is employed for classification. The approach is tested on the well-known streaming data sets and compared with state-of-the-art incremental feature extraction algorithms. The experimental studies demonstrate the proposed approach’s success in terms of recognition accuracy and learning time. Full article

In recent decades, there has been a massive growth towards the prime interest of the hydrogen energy industry in automobile transportation fuel. Hydrogen is the most plentiful component and a perfect carrier of energy. Generally, evaluating a suitable hydrogen power plant site is a complex selection of multi-criteria decision-making (MCDM) problem concerning proper location assessment based on numerous essential criteria, the decision-makers expert opinion, and other qualitative/quantitative aspects. This paper presents the novel single-valued neutrosophic (SVN) multi-attribute decision-making method to help decision-makers choose the optimal hydrogen power plant site. At first, novel operating laws based on sine trigonometric function for single-valued neutrosophic sets (SVNSs) are introduced. The well-known sine trigonometry function preserves the periodicity and symmetric in nature about the origin, and therefore it satisfies the decision-maker preferences over the multi-time phase parameters. In conjunction with these properties and laws, we define several new aggregation operators (AOs), called SVN weighted averaging and geometric operators, to aggregate SVNSs. Subsequently, on the basis of the proposed AOs, we introduce decision-making technique for addressing multi-attribute decision-making (MADM) problems and provide a numerical illustration of the hydrogen power plant selection problem for validation. A detailed comparative analysis, including a sensitivity analysis, was carried out to improve the understanding and clarity of the proposed methodologies in view of the existing literature on MADM problems. Full article

The neighborhood rough set theory was adopted for attributes reduction and the weight distribution of condition attributes based on the concept of importance level. Smart elderly care coverage rate is low in China. A decisive role in the adoption of smart elderly care is still a problem that needs to be addressed. This study contributes to the adoption of smart elderly care was selected as the decision attribute. The remaining attributes are used as conditional attributes and the multi-level symmetric attribute set for assessing acceptance of smart elderly care. Prior studies are not included smart elderly care adoption attributes in multi-levels; hence, this problem needs to be addressed. The results of this study indicate that the condition attribute of gender has the greatest influence on the decision attribute. The condition attribute of living expenses for smart elderly care has the second largest impact on decision attribute. Children’s support for the elderly decency of the novel elderly care system and the acceptance of non-traditional elderly care methods belong to the primary condition attribute of traditional concept. The result indicates traditional concepts have a certain impact on the adoption of smart elderly care and a condition attribute of residence also has a slight influence on the symmetric decision attribute. The sensitivity analysis shows the insights for uncertainties and provides as a basis for the analysis of the attributes in the smart elderly care service adoption. Full article

This paper studies a simple skewness measure to detect symmetry and asymmetry in samples. The statistic can be obviously applied with only three short central tendencies; i.e., the first and ninth deciles, and the median. The strength of the statistic to find symmetry and asymmetry is studied by employing numerous Monte Carlo simulations and is compared with some alternative measures by applying some simulation studies. The results show that the performance of this statistic is generally good in the simulation. Full article

This paper studies in depth the sensitivity of Spanish companies’ returns to changes in several risk factors between January 2000 and December 2018 using the quantile regression approach. Concretely, this research applies extensions of the Fama and French three- and five-factor models (1993 and 2015), according to González and Jareño (2019), adding relevant explanatory factors, such as nominal interest rates, the Carhart (1997) risk factor for momentum and for momentum reversal and the Pastor and Stambaugh (2003) traded liquidity factor. Additionally, for robustness, this paper splits the entire sample period into three sub-sample periods (pre-crisis, crisis and post-crisis) to analyse the results according to the economic cycle. The main conclusions of this paper are fourfold: First, these two models have the greatest explanatory power in the extreme quantiles of the return distribution (0.1 and 0.9) and more specifically in the lowest quantile 0.1. Second, the second model, based on the Fama and French five-factor model, shows the highest explanatory power not only in the full period but also in the three sub-periods. Third, the bank BBVA is the company that shows the highest sensitivity to changes in the explanatory factors in most periods because its adjusted R2 is the highest. Fourth, the stage of the economy with the highest explanatory power is the crisis subperiod. Thus, the final conclusion of this paper is that the second model explains best variations in Spanish companies’ returns in crisis stages and low quantiles. Full article