Conflict analysis has been used as an important tool in economic, business, governmental and political dispute, games, management negotiations, military operations and etc. There are many mathematical formal models have been proposed to handle conflict situations and one of the most popular is rough set theory. With the ability to handle vagueness from the conflict data set, rough set theory has been successfully used. However, computational time is still an issue when determining the certainty, coverage, and strength of conflict situations. In this paper, we present an alternative approach to handle conflict situations, based on some ideas using soft set theory. The novelty of the proposed approach is that, unlike in rough set theory that uses decision rules, it is based on the concept of co-occurrence of parameters in soft set theory. We illustrate the proposed approach by means of a tutorial example of voting analysis in conflict situations. Furthermore, we elaborate the proposed approach on real world dataset of political conflict in Indonesian Parliament. We show that, the proposed approach achieves lower computational time as compared to rough set theory of up to 3.9%. Decision making is an important aspect when we consider the information about the result that we choose among two or more alternatives. There are many approaches in decision making problems which have ability to handle uncertainty, such as Fuzzy set theory , Rough set al. [et al. [et al. [et al. [et al. [et al. [Regarding conflict problems using rough sets, the model introduced by Deja  is an enet al. , 15 prop [et al.  proposed [et al.  presente [et al. . Skowron [et al.  proposed [et al.  proposed [et al.  proposed [et al.  applied  [et al.  presente [et al.  outlined [et al. .et al. [et al. [With the ability to handle uncertainty from the conflict data set, rough set theory has been successfully used. Another new approach in handling uncertainty is soft set theory . It is pet al.  proposedet al.  introduc [et al.  introduc [et al.  defined  [et al.  proposed [et al.  defined  [et al.  investiget al. [et al. [et al. [ivn-soft sets and implemented it on decision making problems. Maji et al. [There are also works on soft set theory implemented in decision making, Feng et al.  proposedet al.  discusse [et al.  proposed [et al.  proposed [et al.  extended [et al.  introduc [et al.  presente [et al.  combinedi et al.  proposedi et al.  presentei et al.  proposedet al. [et al. [t-norm and t-conorm of fuzzy parameterized soft sets (FP-soft sets) and investigated their properties. Çağman and Deli [et al. [In real life many problems are imprecise in nature, classical soft set theory is not fit of effectively dealing with such issues. Majumdar and Samanta  introducet al.  proposed [et al.  proposed [et al.  proposed [et al.  introduc [et al.  defined and Deli  defined and Deli  construcand Deli  proposed [et al.  proposedIn this paper, we present an alternative approach to handle conflict situations, based on some ideas using soft set theory. Our motivation is to improve the computational performance by the rough sets approach in handling conflict situations when determining the support, strength, certainty, and coverage of conflict situations.We propose conflict analysis based on soft set approach.The novelty of the proposed approach is that, unlike rough set theory that relies on decision rules, it is based on the concept of co-occurrence of parameters in soft set theory.We illustrate the proposed approach by means of a tutorial example of voting analysis in conflict situations.Comparative analysis of the propose approach and rough set-based approach in handling conflict of Indonesian political election is presented. Furthermore, we show the efficiency of our proposed approach in term of computational time to the rough set approach.In summary, the contribution of this work is described as follows:The reminder of this paper is organized as follows. In section 2, we review the basic concepts of soft set theory. Section 3 describes the analysis of rough set theory in handling conflict situations. In section 4, we present soft set approach for conflict analysis. In section 5, we present result and discussion. Finally, the conclusion of this work is given in section 6.U be an initial universe set and let E be set of parameters in relation to object in U. The set P(U) denote the power set of U. The definition of soft set is given as follows:In this section, we review some basic notion of soft set theory. To avoid difficulties, one must use an adequate parameterization. Let A pair  is called a soft set over U where F is a mapping given by F:E → P(U).(See ) A pair U. Every set F(e), for e ∈ E from this family may be considered as the set of e-elements of the soft set , or as the set of e-approximate elements of the soft set.In other words, the soft set is parameterized family of subsets of the set F,E) describe the “attractiveness of cars” that Mr. X is going to buy. Suppose that U = {c1,c2,c3,c4,c5,c6} and E = {e1,e2,e3,e4,e5}, where there are six cars in the universe U and E is a set of parameters, ei for i = 1,2,3,4,5 standing for the parameters “costly”, “safety”, “style”, “performance”, and “capacity” respectively. In this example, we consider a mapping F:E → P(U) which is given by “cars(.)”, where (.) is to be filled in by one of parameters e ∈ E. Suppose that, we have the following mapping values i.e.Let be a soft set (F(e3) means car with style characteristic, whose functional value is the set {c3,c4,c5}. Thus, we can view the soft set  as collection of approximation as follows:As we can refer to the example above, the mapping F,E) can be represented as a Boolean-valued information system   See , 53). Th. ThF,E) Soft set theory can also be used to handle multi-valued information systems and in the following section, we present the notion of multi soft sets representing multi-valued information systems.S =  to |A| number of Boolean-valued information systems which is based on decomposition of A = {a1,a2,…,aA||} into a single attribute {a1},{a2},…,{aA||}. In the following sub-section, we recall the construction of multi soft sets.Herawan and Deris proposed the idea of multi soft sets to for representing multi-valued information systems . The ideS =  be a multi-valued information system such that for every a ∈ A, f is a finite non-empty set and for every u ∈ U,|f| = 1. For every ai under ith-attribute consideration, ai ∈ A and v ∈ Va, we define the mapping In this section, we only consider for complete multi-valued information systems. Let Si = . The information systems Si = ,1≤i≤|A| is referred to as a decomposition of a multi-valued information system S =  into |A| number of Boolean-valued information systems, as depicted in The next step, we define a Boolean-valued information system as a quadruple Based on Proposition 3.1.If  is a soft set over the universe U, then  is a Boolean-valued information system S = .Si = , for 1≤i≤|A| in a ∈ A and for every object u ∈ U, the |f| is a total function. Hence, the structure of multi-valued information system and |A| number of Boolean-valued information systems give the same value of attribute related to objects.From Proposition 3.1, each Boolean-valued information system S =  be a multi-valued information system and Si = ,1≤i≤|A| be a |A| Boolean-valued information systems. Then we have the following multi-soft setsFrom sub-section 3.1, in this sub-section we present the notion of multi-soft set representing multi-valued information systems. Let F,A) = ,,…,) as a multi-soft sets over universe U representing a multi-valued information system S = .We further define the , where U and A are non-empty finite sets called universe, and the set of attributes, respectively. Pawlak models conflict analysis as follows:U is called objects (agents)Elements of A is set of attributes (issues).The In this section firstly we recall the concept of conflict analysis from the point of view of rough set theory. Pawlak , 11 inita ∈ A is a total function a:U → Va, where Va is the set of values of a, called the domain of a; elements of Va will be referred to as opinions, and a(x) is opinion of agent. The domain of each attribute is restricted to three values Va = {1,0,−1} representing opinion of agents, where 1 means alliance, 0 means neutrality and −1 means conflict. In the following sub-section, we introduce an alternative soft set approach for conflict analysis.Every attribute F,A) as presented in sub-section 3.2, let S =  be a multi-valued information system and Si = , for 1≤i≤|A| be a |A| decomposed Boolean-valued information systems. A conflict situation based on multi-soft sets is modelled as a  where quadruple Si = , where U and ai are non-empty finite sets, U is called universe, ai is called attributes, V = ∪a∈AVa, Va is the domain  of attribute a and element of Va is referred as opinions toward the issue, f:U × A → V is a total function such that f ∈ Va, for every  ∈ U × A.Based on the multi-soft sets , neutral, and conflict toward the issue, respectively. Therefore, here we have conflict model based on multi-soft sets as follows:On a conflict situation, we can say that is special type of multi-soft sets, because Si =  be a multi-valued information system, and Si = ,1≤i≤3, ai ∈ A, Va = {1,0,−1} be a conflict model Boolean-valued information systems with Va is restricted to three values Va = {1,0,−1} or Va = {+,0,−}. Therefore, here we haveLet In the following sub-section we discuss the binary relations on agents i.e. alliance , neutrality, and conflict.F,A) be a soft set representing a Boolean-valued information system Si = . Based on the idea from Pawlak [Let f = 1, between agents x and y have the same point of view or opinion about the issue f ;That means that, if b. Neutralityf = 0, at least there is one agent x or y has neutral view to conflict issue f ;In the case of neutrality, if c. Conflict (against)f = −1, both agents have different opinions about conflict issue f (agents x and y are conflict on f).Finally, in the case of conflict, if F,E) can be represented as a Boolean-valued information system , in the following definition we present the notion of similarity between two parameters in . We firstly define the notion of occurrence of parameters in soft set theory.From the fact that a standard soft set  be a soft set over the universe U representing  and an object u ∈ U. A parameter co-occurrence set of an object u can be defined as follows:coo(u) = {e∈E:f(e) = 1}. The following example illustrates the Definition 4.1.Obviously, Example 4.1. From soft set  in From Definition 4.1, we have the following definition of agent support.Definition 4.2.Let  be a soft set over the universe U and an agent u ∈ U. The support of an agent u is defined byFrom Definition 4.2, we have the following definition of rules strength.Definition 4.3.Let  be a soft set over the universe U representing . The strength of a rule A1 ⇒ A2for A1,A2 ⊂ A denoted by σx is defined byFrom Definition 4.3, we have the following definition of rules certainty.Definition 4.4.Let  be a soft set over the universe U representing . The certainty of a rule A1 ⇒ A2for A1,A2 ⊂ A denoted by cerx is defined byπ(A1(x)) = |A1(x)| / |U|.From Definition 4.4, we have the following definition of rules coverage.Definition 4.5.Let  be a soft set over the universe U representing . The coverage of a rule A1 ⇒ A2for A1,A2 ⊂ A denoted by covx is defined bywhere π(A2(x)) = |A2(x)| / |U|.x = πx(A1|A2)Similarly, covThe algorithm for handling conflict data using multi-soft sets is given in In the following section, we present a tutorial example of voting analysis in a conflict situation.U,A,V,f) where the domain agents (universe) and the voting function f are respectively defined byIn this section, we illustrate the proposed approach through an example of a conflict data set. Let a conflict situation given by a multi-valued information system  representations as in From the decomposition of a multi-valued information system  into mulF,E) in From multi-soft sets , neutrality, and against (conflict) among the agents. From steps 4, 5, and 6 in From The flow graph associated with A → 1 has the certainty factor 0.525 and coverage factor 0.206. The flow graph gives a clear insight into the voting structure of all parties. We can replace flow graph shown in From “approximate” flow graph depicted on A, B, C, and D form a coalition, which is in conflict with party E. This flow graph generated by using certainty factor greater than 0.5.The In the next section, we present an application of handling conflict in the problem of determining governor election model in Indonesia.In this section, we present a real world application of the proposed approach to solve the problem of determining governor election model in Indonesia i.e. direct or indirect methods. The dataset is taken from liputan6 online  and vivaFrom U,A,V,f) where the domain ag (universe U) of the voting function f is respectively defined byLet a conflict situation given by a multi-valued information system , abstention , and no (against/conflict), respectively.From F,E) representations as in From the decomposition of a multi-valued information system  into mulF,E) in From multi-soft sets . It is executed sequentially on a processor Intel Core i3 CPUs. The total main memory is 4GB and the operating system is Windows 10. The experimental results on all party supports are given in The strength, certainty, and coverage, of Indonesian parties voting result are given in The flow graph associated with From In the “approximate” flow graph depicted on The following graphs present the comparison of execution time between the proposed soft set based approach and rough set based approach on computing support, strength, certainty, and coverage.From From From From Conflict analysis has been used as an important tool in economic, business, governmental and political dispute, games, management negotiations, military operations and etc. In this paper we have presented an alternative approach for handling conflict situation involving uncertainty. It is based on multi-soft sets taking into account of co-occurrence of parameter related to object in universe. The novelty of the proposed approach is that, unlike in rough set theory that uses decision rules, it is based on the concept of co-occurrence of parameters in soft set theory. We have presented an illustrative example on how to handle conflict using multi soft sets. Furthermore, we elaborate the proposed approach of real world dataset of voting from political election data set from Indonesian parliament. However, we achieve lower computational time as compared to rough set approaches. In the future work, we will extend this proposed soft set-based method by refinement of the neutrosophic set , 58 and


Little is known about the effects of fear as a basic emotion on mental rotation (MR) performance. We expected that the emotional arousal evoked by fearful stimuli presented prior to each MR trial would enhance MR performance. Regarding the influence of anxiety, high anxious participants are supposed to show slower responses and higher error rates in this specific visuo-spatial ability. Furthermore, with respect to the embodied cognition viewpoint we wanted to investigate if the influence of fear on MR performance is the same for egocentric and object-based transformations. To investigate this, we presented either negative or neutral images prior to each MR trial. Participants were allocated to the specific emotion in a randomized order. Results show that fear enhances MR performance, expressed by a higher MR speed. Interestingly, this influence is dependent on the type of transformation: it is restricted to egocentric rotations. Both observation of emotional stimuli and egocentric strategies are associated with left hemisphere activation which could explain a stronger influence on this type of transformation during observation. Another possible notion is the conceptual link between visuo-spatial perspective taking and empathy based on the co-activation of parietal areas. Stronger responses in egocentric transformations could result from this specific link. Regarding the influence of anxiety, participants with high scores on the trait-anxiety scale showed poor results in both reaction time and MR speed. Findings of impoverished recruitment of prefrontal attentional control in patients with high scores in trait anxiety could be the explanation for this reduced performance. According to There is plentiful evidence that fear induced by presenting fearful stimuli affects visual perception in the ventral stream. For instance, By analyzing this special relationship, it raises the question whether affect and emotion also have an influence on mental imagery. This study tries to answer this question by investigating the influence of fear and anxiety on a certain visuo-spatial ability, specifically mental rotation (MR).MR involves the process of imagining how a two- or three-dimensional object would look if rotated away from its original upright position . In the In MR, there are two different strategies of mental transformations: object-based and egocentric transformations. In object-based transformations, the observer’s position remains fixed and moves the object in relation to the surrounding environment. In egocentric transformation tasks, participants are required to change their own perspective and thus imagine rotating their own body in order to make a decision . The useLittle is known about the effects of fear as basic emotion on visuo-spatial processing in the dorsal stream assessed by MR performance. The corresponding neural system that is activated when a stimulus evokes fear is the amygdala . AccordiRegarding the influence of anxiety on MR performance, Despite the fact that the role of emotion and affect has been investigated in the context of MR performance, it is still an open question if the influence of fear and anxiety is the same for egocentric and object-based transformations. The present article addresses this issue with regard to the embodied cognition approach. The key idea of this renewed viewpoint in cognitive neuroscience is that many cognitive processes that were formerly defined as purely “cognitive” are also deeply rooted in body-related experiences with the environment .There is plentiful evidence that motor processes are involved in both object-based  and egocWhat does this embodiment approach mean with regard to the influence of fear and anxiety on MR performance? Next to the postulation of “cognitions” being embodied, there is also a strict coupling between emotion and sensory-motor integration . For exaThe present investigation differs from previous research by investigating the influence of fear and anxiety on MR performance with focus on the differentiation of egocentric and object-based transformations. Based on the work of According to Concerning the role of fear, in line with the notion of an enhanced activation of the amygdala that in turn modulates activity in parietal areas through the presentation of aversive stimuli we expected that fear primes MR performance. It is still an open question if anxiety affects MR performance and if so, to what extent, but based on the negative influence of anxiety on cognitive processes cf.  it is renegative: 23.27, SD = 4.27; mean ageneutral: 23.27, SD = 4.27), t(84) = -0.69, n.s., and the anxiety trait score , t(84) = -0.46, n.s. Regarding intelligence, they showed comparable scores , t(84) = -0.26, n.s., see Table 1. Participants were recruited through advertisement at the university. All participants received either €10 for participation or credits for psychology courses. None of the participants have participated before on MR tests. All participants gave informed consent for participation.Eighty-six adults, 43 men  and 43 women  participated and were classified into the following two types of emotion: negative and neutral. The “negative emotion” group consisted of 22 men and 21 women, the “neutral emotion” group was composed of 21 men and 22 women with no significant difference in both age  served as between-subject factor. Since the IAPS consisted of 193 negative and 130 neutral pictures, which is not sufficient for the 336 MR trials in total, we had to choose 112 images for each block resulting in three repetitions of each image. Even if habituation and consequently a decrease of emotional response were risked, a less amount of trials would not have been arguable from the scientific viewpoint. The selection of the images was randomized. However, this selection was based on the fact that level of arousal and valence was comparable for both emotions. For this purpose, the IAPS provides a list with scores of valence and arousal. The primes were controlled for both levels. According to Before each trial pictures from the International Affective Picture System (IAPS) were presented, as illustrated in The individual test sessions lasted about 60 min and took place at a silent room at the University. At the beginning the participant completed the demographic questionnaire, the state-trait anxiety inventory and the ZVT.Later, the MR test with standardized task instruction was conducted. In the BFO and letter conditions, participants had to press the left mouse button (left-click) when the two stimuli were “same” and the right mouse button (right-click) when the two stimuli were “different.” In this case “same” means that the stimulus on the right side was identical to the comparison stimulus, “different” means that the stimulus on the right side was not identical to the comparison stimulus. In the BFE condition participants had to decide if the figure had the right (right mouse click) or the left arm (left mouse click) outstretched see .According to Each participant performed three blocks of 112 experimental trials, resulting in 336 trials: 3 transformation types (BFE vs. BFO vs. letters) * 2 trial types (same vs. different/left vs. right) * 7 angular disparities  * 4 repetitions of each combination * 2 stimuli per block . In each block, the order of the presentation of the stimuli was randomized.First, to exclude that the MR performance is influenced by possible IQ differences between gender and group an univariate analysis with the dependent measure IQ and the independent variables gender and group was conducted.Second, two repeated analyses of variance were conducted, with “reaction time” and “accuracy rate” as dependent variables, and with “angular disparity” , “transformation type” , “emotion”  as factors. The factors “angular disparity” and “transformation type” were the within-subject factors. Because preliminary analysis revealed no relevant effect with the factor gender only the factor “emotion”  served as between-subject factors. The variable “trait anxiety” was included as a co-variate. For reaction time, only the responses for “same” trials were analyzed because angular disparity is not clearly defined for mirror-reversed responses . For errThird, a repeated analysis of variance was calculated with “MR speed” as a dependent variable and “transformation type” as within-subject factor and “emotion” as between-subject factor. The variable “trait anxiety” was included as a co-variate. MR speed was calculated as the inverse of the slope of the regression line, calculated separately for each subject, relating RT to angular disparity and was expressed as degrees per second. A higher MR speed means that a larger angular disparity is rotated per second. According to the traditional theory of MR  claimingThe significance levels of the analyses of variance results were corrected according to the method of Greenhouse–Geisser to compensate for non-sphericity of the data if necessary.F = 0.38, n.s. nor a group effect, F = 0.008, n.s. nor an interaction between both factors, F = 1.52, n.s. concerning the transformed IQ values.There were neither gender differences, F = 10.13, p < 0.001, partial η2 = 0.11, and “angular disparity,” F = 54.68, p < 0.001, partial η2 = 0.39. The covariate also reached significance and could be expressed by a significant correlation between “trait anxiety” and the averaged reaction time for each transformation type . There were two positive significant correlations: (1) between “trait anxiety” and “BFO” , 2) between “trait anxiety” and “letters” . Regarding the significance of the factor “transformation type,” Bonferroni-corrected t-tests showed that the reaction was higher for the BFO condition  compared to the letter condition, , t = 12.84, p < 0.001, and to the BFE condition, , t = -12.61, p < 0.0001. Furthermore, there was a significant difference between the reaction time in the letters and BFE condition, t = 3.21, p < 0.01. Regarding the main effect of the factor “angular disparity,” post hoc pair-wise comparisons showed higher reaction times for each consecutive angular disparity (p = < 0.001) except the one at 30° which did not differ from that of 0°, t(85) = -2.34, n.s.Concerning reaction time, the analysis of variance showed three main effects for the factors “transformation type,” Furthermore, there were two interactions:(1)F = 1.84, p < 0.05, partial η2 = 0.02, and it is illustrated in Figure 3. Whereas reaction time in the BFO condition was overall increasing with angular disparity and higher for each consecutive angle (p = < 0.001), reaction times in the letters condition did not differ between angular disparities of 0 and 30°, t(85) = -2.06, n.s., and between angular disparities of 30 and 60°, t(85) = -1.01, n.s.. Increasing disparity in the BFE task only led to higher response times for disparities larger than 90° (p = < 0.001). All other effects did not reach significance at the 0.05 level. Furthermore, by trend reaction time in the egocentric transformation condition surprisingly decreased between the angular disparity of 0 and 60°. That is, whereas reaction times in the object-based condition roughly increased linearly with increasing disparity as expected, they showed a U-shaped pattern for the egocentric transformation condition. The “transformation type” × “angular disparity” interaction was significant, (2)r = -0.21, p < 0.01; 120°: r = -0.29, p < 0.01; 150°: r = -0.28, p < 0.01; 180°: r = -0.27, p < 0.01). All other effects failed to reach significance. The interaction between the covariate “trait anxiety” and “angular disparity” was significant, expressed by a correlation between “trait anxiety” and the averaged reaction time for each angular disparity. The correlation between “trait anxiety” and the angular disparities of 0, 120, 150, and 180° reached significance  = 8.932, p < 0.001, partial η2 = 0.09. Bonferroni-corrected t-tests revealed that from an angular disparity of 90°, there was a lower PR score for the following angular disparity compared to the preceding one . All other effects did not reach significance.The analysis of the PR score showed one main effect of the factor “angular disparity” F = 5.11, p < 0.01, and “angular disparity,” F = 136.28, p < 0.001. According to multiple post-hoc comparisons, participants showed higher accuracy rates in the BFE and the letters condition compared to that found in the BFO condition. Performance between the letters and BFE condition did not differ. Regarding the main effect of the factor “angular disparity,” the decrease of accuracy emerges for disparities larger than 90° (p = < 0.001). All other effects did not reach the 0.05 significance level.Regarding accuracy rate, results revealed two significant main effects of the factors “transformation type,” F = 5.00, p < 0.001. Whereas accuracy rate in both object-based conditions  was overall decreasing with angular disparity by trend and significant lesser for each consecutive angle from an angular disparity of 90° on (p = < 0.001), increasing disparity in the egocentric task only led to higher error rates for disparities larger than 120° (p < 0.001). Furthermore, by trend accuracy rate in the egocentric transformation condition surprisingly increased between the angular disparity of 0 and 60°, and between 90 and 120°. All other effects did not reach significance at the 0.05 level. Together, this small-angle advantage is more pronounced for the object-based transformations than for the BFE condition, as illustrated in Figure 4.Furthermore, the interaction between “transformation type” and “angular disparity” was significant, F = 3.91, p < 0.05, partial η2 = 0.05. Bonferroni-corrected t-tests showed that participants rotated stimuli in the BFE condition  significantly faster than those in the BFO condition , t = -5.67, p < 0.001, but not significantly faster than letters , t = -1.44, n.s.Due to negative rotation speed, two persons had to be excluded. The analysis of variance showed a significant main effect of “transformation type,” Furthermore, results showed two significant two-way interactions:(1)F = 5.37, p < 0.001, partial η2 = 0.07. Post-hoc comparisons showed that the rotation speed did not differ for participants who have either seen negative or neutral pictures before the MR task, both in the BFO condition, t = -0.18 n.s., and in the letters condition, t = -0.26 n.s., whereas MR speed differed between both groups in the BFE condition, t = 2.32, p < 0.05. The MR speed for the BFE condition was much higher if participants had seen negative pictures  compared to neutral ones , see Figure 5. All other effects were not significant. The interaction between “transformation type” and “emotion” reached significance at the 0.05 level, (2)F = 4.61, p < 0.05, partial η2 = 0.05 expressed by a negative correlation between “trait anxiety” and the rotation speed of “letters” . The interaction between “transformation type” and the co-variate “trait anxiety” was significant, The main goal of our study was to investigate if the influence of fear on MR performance is the same for egocentric and object-based transformations. Furthermore, we wanted to examine the effect of anxiety on MR performance. The main results were a facilitation effect of fear which is restricted to egocentric transformations: participants rotated stimuli in the BFE condition more quickly after seeing an aversive image compared to a neutral one. Concerning the influence of anxiety, individuals with high scores on the trait-anxiety scale of the STAI showed both partially higher reaction times and a partially slower MR speed.In line with previous findings , we coulLike mentioned above, both types of strategies differ in a crucial point: Whereas in object-based transformations participants are asked to mentally move/rotate the object in relation to the surrounding environment, in an egocentric chronometric MR tasks people are required to change their own perspective because they have to imagine themselves rotating in order to complete the task . TherefoAnother possible explanation for the enhanced effect of fearful stimuli on egocentric transformation is based on the idea that fearful primes may prepare the body to react. We tentatively propose that this motor pre-activation has a stronger impact on the BFE condition which is suggested to be embodied to a higher extent than the object-based conditions . The notion, that the processing of emotional stimuli activates bodily reactions has already been pointed out by Another attempt to explain the egocentric-specific influence of fear could involve the meaning of the working memory (WM). We tentatively propose that fear impairs functions of the WM which affects object-based rotations to a higher extent. This assumption stems from the fact that in egocentric transformations the visual buffer being the neuronal substrate for both imaginal and perceptual visuo-spatial transformations is not that highly loaded because there is no image interference in left–right judgments tasks . The widA further approach for this finding could be attributed to a conceptual link between visuo-spatial perspective taking and perspective-taking in the abstract sense, specifically empathy. Although purely speculative, it could be concluded that the presentation of aversive stimuli elicits a stronger activation of areas which are the neural correlate for egocentric transformations and therefore represent the social construct empathy in an abstract sense compared to those being activated during object-based transformations. In perspective transformations, subjects are required to transform themselves into the body of another person. There is evidence that this kind of self-other equivalence is a basic condition for empathy . AllportBased on the finding of the activation of the parietal cortex during both visuo-spatial processes and empathy , ThakkarIn contrast to their expectation, The positive correlation between “trait anxiety” and “reaction times” of letter and BFO condition and “MR speed” of letters in our study suggests that higher scores in the trait anxiety scale of the STAI are associated with higher reaction times of these conditions. These results are contradictory to the finding of previous research showing a facilitation effect of emotion: participants with high state-anxiety rotated objects more quickly after the presentation of fearful faces compared to neutral ones . HoweverThe findings of our study show that high anxiety scores interfere with MR performance which is in line with the negative influence of anxiety found in previous literature cf. . AccordiAccording to Interestingly, with respect to the simulation-based account the influence of anxiety is restricted to the two object-based conditions  where no motor simulation was required. This is in line with the specific effect of fearful stimuli being restricted to the egocentric transformation. Both results could provide further evidence for the importance of motor simulation in the assumed link between embodied cognitions and emotions.The investigation of the influence of fear by presenting aversive images is widespread. However, it still remains unclear to which extent this type of stimulus material elicits emotion. Furthermore, it raises the question which kind of emotion is triggered, whether it is rather disgust than fear. Even if the images were standardized regarding valence and arousal, the extent of the emotional response stays individually. This could be controlled by measuring the physiological response, specifically skin conductance response. However, this measurement still makes no statement about the quality of emotion. Conducting self-reported measurements could clarify the emotional state, but they can only be assessed after the MR task which may be a too long period after the presentation of the aversive stimuli.Furthermore, the direct comparison between egocentric and object-based transformations should be reconsidered in view of the fact that these types of transformations differ in some aspects: visual stimulation (2 stimuli vs. 1 stimulus, cf. Taken together, the explanation approaches mentioned above based on neuronal correlates still remain quite speculative at this point since no brain activity was measured in the present study. Further behavioral and neuroanatomical research is needed to clarify this specific link between emotion and cognition.Empathy seems to be associated with egocentric transformations based on the co-activation of parietal areas during visuo-spatial processes and this social construct . We hypoTo complete the emotional version of the MR task, the influence of positive images could be investigated in future. If the poor performance of anxious individuals is really stemming from an attentional deficit caused by threat-related stimuli, the adding of positive stimuli may clarify this assumption.The relationship between emotion and MR seems to be very close. The findings of our study suggest that there is a facilitation effect of fear which is restricted to egocentric transformations. To what extent empathy as social construct plays a role still remains unclear and demands a lot of future research. In contrast to The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.


There is an error in the sub-heading “Modulation of APP Processing by RC, RS, CP, and FG and S” under the “Results” section. The correct sub-heading for this section is:“Modulation of APP Processing by RC, RS, CP, and FG”Additionally, there is an error in the second paragraph of the sub-heading “Chromatographic Conditions” under the “Materials and Methods” section. There are no supplementary methods in the article, and Table S2 is incorrectly referenced as Table S1 in this paragraph. Therefore, the correct paragraph should read:Qualitative identification of 12 active compounds of HLJDT was also performed by ultra-high performance liquid chromatography with quadrupole time-of-flight mass spectrometry (UHPLC-Q-TOF-MS) analysis with positive mode ESI-MS. A database was prepared for the identification of phytoconstituents of HLJDT based upon their molecular masses, calculated m/z values and retention times (Rt) observed in the analysis (Table S2).


As well known, Boltzmann-Gibbs statistics is the correct way of thermostatistically approaching ergodic systems. On the other hand, nontrivial ergodicity breakdown and strong correlations typically drag the system into out-of-equilibrium states where Boltzmann-Gibbs statistics fails. For a wide class of such systems, it has been shown in recent years that the correct approach is to use Tsallis statistics instead. Here we show how the dynamics of the paradigmatic conservative (area-preserving) stan-dard map exhibits, in an exceptionally clear manner, the crossing from one statistics to the other. Our results unambiguously illustrate the domains of validity of both Boltzmann-Gibbs and Tsallis statistical distributions. Since various important physical systems from particle confinement in magnetic traps to autoionization of molecular Rydberg states, through particle dynamics in accelerators and comet dynamics, can be reduced to the standard map, our results are expected to enlighten and enable an improved interpretation of diverse experimental and observational results. These distributions are those that maximise the Boltzmann-Gibbs entropy and ensure the equilibrium state. The Maxwell distribution is an instance of the equilibrium distribution for the velocities of molecules in an ideal gas. The underlying mathematical reason for this is the existence of the standard Central Limit Theorem (CLT)q → 1. This feature permits to describe these special non-equilibrium states with the same formal framework of the equilibrium thermostatistics, known as Tsallis statisticsq attains a special limiting value (q → 1). In this case, the underlying mathematical mechanism is the generalized CLT4q-Gaussians. Therefore, the role of q-Gaussians in Tsallis statistics is basically the same as that of Gaussians in Boltzmann-Gibbs statistics. In this work we show, for the first time, that these two cases coexist in the classical standard map, and discuss the necessary conditions under which one case prevails over the other one. This neatly illustrates the respective domains of validity of Boltzmann-Gibbs and of Tsallis statistics. The results are important not only from the statistical mechanics theoretical viewpoint but also due to their potential of being applicable to diverse fields of physics. Indeed, many physical systems can, as a first approximation, be reduced to the standard map. This is so for particle confinement in magnetic trapswhere Boltzmann-Gibbs entropy is a special case as q-Gaussians, have been observed in nature in several experimental, observational and model systems1113q-Gaussians have been found as the chaos threshold is approached using the band splitting structure obeying the Huberman-Rudnick scaling law2021Non-Gaussian distributions, particularly q-Gaussians have also been recently observed in a conservative high dimensional modelα-XY model, i.e., a system of N classical localized planar rotators with two-body interactions and periodic boundary conditions, the potential is assumed to decay with distance as 1/rα, and α ≥ 0 is the parameter that controls the range of the interactions, short-range for α/d > 1, and long-range for 0 ≤ α/d ≤ 1 .α-XY model just mentioned. Ordinary Gaussians are observed when short-range interactions (α > 1) are present, and q-Gaussians are observed when long-range interactions (0 ≤ α ≤ 1) are presentλ asymptotically decreases as Nκ(α)−, in a rather similar way of that observed inα-XY model and the q-Gaussian distributions that emerge are characterized by the parameter q that depends on α.Recently a generalization of the conservative one-dimensional Fermi-Pasta-Ulam model, properly modified to account for linear and nonlinear long-range interactions, has been analyzed. The range of the interactions is controlled in the same way as for the λ and two regimes shall be distinguished in the thermodynamic limit (number of particles N → ∞). (i) Strongly chaotic regime corresponds to a large positive Lyapunov exponent, where the system is ergodic. The dynamics of the system evolves to an equilibrium state described by Boltzmann-Gibbs statistical mechanics, with exponential or Gaussian distributions ; (ii) Weakly chaotic regime corresponds to a very small positive Lyapunov exponent (λ ≈ 0), where the system behaves for a very long time as non-ergodic. Distributions of the dynamical variables are not exponential or Gaussians, and the Boltzmann-Gibbs framework is not suitable for this case. q-Gaussian distributions have been observed for this case by proper time and ensemble averagesSq227All these systems appear to share in common the following scenario: ergodicity in a region is characterized by the largest Lyapunov exponent B > 0 is the Lagrange parameter and the q-exponential is given bywhere A]+ ≡ max {0, A} and its inverse, q-logarithm, is defined by q → 1.where /[number of points with λ above λthreshold] to decrease; for fixed K, this ratio increases with increasing λthreshold, but remains almost constant for 5 × 10−5 < λthreshold < 10−2. The precise relation between α and this ratio remains to be studied but it is out of the scope of this manuscript and will be addressed elsewhere. It is easily seen that, as the stability islands dominate the whole phase space, the dominant distribution is the q-Gaussian with q = 1.935 (case K = 0.6), whereas the dominant one becomes the Gaussian if the chaotic sea invades more and more the whole phase space (case K = 2). It is in fact a very interesting result since, although for some portion of the phase space the system is ergodic, the signature of the q-Gaussian seems not to be erased even for very large T values. Therefore, apparently, for these two different portions of the phase space, the system behaves differently and each one is as robust as the other one. This mixture of Boltzmann-Gibbs statistical behaviour with Tsallis statistical behaviour is somewhat reminiscent of what was observed in a quite different system, namely one where overdamped motion is presentWe check this hypothesis using our two appropriate cases, namely, K, the phase space is almost entirely dominated by zero Lyapunov behaviour and the distributions  are q-Gaussians. As the value of K increases the measure of the zero Lyapunov regions decreases, and we see a continuous crossing, expressed by the parameter α in q-Gaussian distributions  and Gaussian ones (Boltzmann-Gibbs statistics) with βq → ∞ and β → π as K → ∞ . Remarkably enough, the distributions originated from initial conditions taken inside the region of islands, instead of over the entire phase space, yield one and the same value q = 1.935, independently on whether we consider one or many of these regions, and independently from K. Initial conditions taken within the chaotic sea always yield Gaussians with β = π. The variance of q-Gaussians with 5/3 < q < 3 diverges, though they have finite width. The N-fold convolution product of independent (or quasi-independent) q-Gaussians would asymptotically yield Lévy distributionsnot the case for the standard map: indeed, the actual time-averaging involves strong correlations. As previously discussed, since the standard map can be considered as a basic model for several physical systems , the present results are expected to be valid and useful for the statistical mechanical analysis of diverse phenomena, experimentally and observationally detected.The phase space of the standard map presents regions of positive Lyapunov exponents coexisting with regions of zero Lyapunov exponents. The positive Lyapunov regions present mixing and thus the system is ergodic in those regions. For sufficiently low values of the control parameter How to cite this article: Tirnakli, U. and Borges, E. P. The standard map: From Boltzmann-Gibbs statistics to Tsallis statistics. Sci. Rep.6, 23644; doi: 10.1038/srep23644 (2016).


Gb-metric space which is recently introduced by Aghajani et al. (Filomat 28(6):1087–1101, In this paper we prove the existence and uniqueness of couple fixed point theorems for three mappings satisfying some new rational contractive conditions. We prove our results in the frame work of  Due to this a flood of research work have been generated from this area. As a part of this study generalisation of metric space becomes one of the most interesting topic in which many researchers have devoted and continued working. Since the introduction of metric space by Frachet, there is a lot of generalisation of this space. Some of them which can be mentioned are 2-metric space, G-metric and b-metric and generated the new definition and named it as Gb-metric space. They also pointed out that the class of Gb-metric space is effectively larger than that of G-metric space and G-metric space becomes a particular case of Gb-metric space. They claimed that every Gb-metric space is topologically equivalent to a b-metric space. For more results on b-metric space one can study the research papers of Malhotra and Bansal  is called a G-metric space. is called a b-metric space with parameter s., Then G is called a generalized b-metric and pair  is called a generalized b-metric space or  be a G-metric space, and  be a r is be a complete symmetricmetric space with parameterand let the mappingssatisfyingfor allandwithThenS, TandRhave a unique common coupled fixed point inX.Let DefineTherefore,X. As X is complete By property  of DefinNow we will prove that Using inequality  we have\x, y) is a Common coupled fixed point of S, T and R.Similarly, we can prove that p, q) be the second common coupled fixed point of S, T and R.In order to prove the uniqueness of the coupled fixed point, if possible let   5) we haHence S, T, R have unique coupled common fixed point.Thus This completes the proof. Let  be a complete symmetricmetric space with parameterand let the mappingsatisfyingfor allandwithThenShas a unique coupled fixed point inX.Let  be a complete symmetricmetric space with parameterand let the mappingssatisfyfor allandare non-negative real numbers withThenS, T, Rhave unique coupled common fixed point.Let ThenAdding  and 8)  8) we haAlso, we can show thatBy property  of DefinSo, for X. Since X is complete Therefore Now, we will show that Therefore Which implies that Also, we can prove that x, y) is a common coupled fixed point of S, T and R.Hence,  be the second common copuled fixed point of S, T and R.In order to prove the uniqueness of the common coupled fixed point of Then by using inequality  we have9But Which implies that S, T, R have a unique common coupled fixed point.Thus This completes the proof. Let  be a complete symmetricmetric space with parameterand let the mappingsatisfyingfor allandare non-negative realnumbers withThenShas a unique coupled fixed point.Let S, T and R.Defineb-metric and G-metric spaces. These results may be extended to other spaces.We prove the existence and uniqueness of common coupled fixed point theorems for three mappings with a new rational contractive conditions in


Although a wide range of direction of arrival (DOA) estimation algorithms has been described for a diverse range of array configurations, no specific stochastic analysis framework has been established to assess the probability density function of the error on DOA estimates due to random errors in the array geometry. Therefore, we propose a stochastic collocation method that relies on a generalized polynomial chaos expansion to connect the statistical distribution of random position errors to the resulting distribution of the DOA estimates. We apply this technique to the conventional root-MUSIC and the Khatri-Rao-root-MUSIC methods. According to Monte-Carlo simulations, this novel approach yields a speedup by a factor of more than 100 in terms of CPU-time for a one-dimensional case and by a factor of 56 for a two-dimensional case. Recently, more advanced methods have been presented for DOA estimation, for example, the Khatri–Rao (KR) product approach. By assuming quasi-stationary sources, this concept can identify up to 2N − 1 sources using an N-element uniform linear array, without computing higher-order statistics , then vec(A) = T.Notations: We denote matrices and vectors by boldfaced capital letters and lower-case letters, respectively. SuperscriptThe remainder of the paper is organized as follows. In Section 2, we present the signal model and the KR-MUSIC algorithm . Section2.In this section, we give a short summary of the KR-MUSIC algorithm (see ), preced2.1.K sources impinging on a uniform linear sensor array, consisting of N omnidirectional sensors and neglecting mutual coupling. For a signal kS(t) emitted by the k-th source, let s(t) = T denote the K × 1 source signal vector. Let N × 1 steering vector corresponding to the direction θ, with d, the spacing between the N sensors and λ the signal wavelength. In this paper, we fix the element spacing to d/λ = 1/2. Our aim is to study the effect of random variations in the array geometry on algorithms estimating the directions of arrival kθ corresponding to the K source signals, in the presence of measurement noise. Therefore, A = [a(θ1) a(θ2) … a(kθ)] describes the N × K array manifold matrix and v(t) = T ∈ ℂN×1 represents the spatial and temporal white noise. For the observed signal nx(t) of the n-th sensor, let x(t) = T denote the N × 1 receiver signal vector. This vector equals:Consider v(t) is white Gaussian and zero-mean wide-sense stationary with covariance matrix C ≜ E{‖v(t)‖2}. It is statistically independent of the source signals.The noise kθ ≠ lθ for k ≠ l.All source DOAs are distinct: ks(t) are mutually uncorrected and have a zero mean.The source signals L, such that:m,kd the frame-dependent and sensor-dependent average normalized signal power. This assumption means that the second-order statistics of the source signals are time varying, but that they remain static over a short period of time. Under this quasi-stationarity assumption, we define a local covariance matrix:Each source signal is wide-sense quasi-stationary with frame length The DOA-estimation algorithms under study are the root-MUSC and KR-root-MUSIC algorithms. These algorithms rely on the following assumptions:ks(t) satisfying the above constraints, with sequence length T = 25,600 and an allowable range of the frame periods  = . Algorithm 1 is applied to generate the synthetic signals.In the remainder of the paper, we apply the DOA estimation algorithms to synthetic signals generated by the procedure proposed in  to constcurT = 0. curT is the start of a new frame of the signal;Set whilecurT < TdofL following a uniform distribution on ; Randomly generate sσ following a uniform distribution on ; Randomly generate fort = curTtocurT + fL− 1 do kS(t) = Rs(t) + Ijs(t), where Rs(t) and Is(t) are independent and identically Laplacian distributed with zero mean and variance   Randomly generate end curT := curT + fL; end  Algorithm 1: The algorithm to generate synthetic signals.L = 512. The number of frames is set to M = T/512 = 50.Observe that the frame intervals of the different impinging sources are not fixed and not synchronized. We apply the KR subspace methods by choosing a fixed frame period of 2.2.L, where L divides T, the KR-MUSIC algorithm proceeds as follows  ∈ ℂ(2N−1)×(2N−1−K)from U, we compute the DOA spectrum:By extracting the noise subspace matrix For a given receiver signal vector rb(z) ∊ ℂ1×(2N− 1) as:K largest local maxima of the DOA spectrum PKR-MUSIC(θ) are the DOA estimates.Define UnHW‖2. In the remainder of the paper, we call the resulting method the KR-root-MUSIC method.Instead of searching the peaks of the MUSIC spectrum, we can also apply the root-MUSIC method  to ‖UnHW3.N sensor elements, where the first sensor element is placed at the origin of the xy-plane and the other elements along the x-axis of this plane. The l-th sensor element is displaced by a vector In this section, we assume that one of the sensor elements is displaced due to an unintentional movement of an element. l-th sensor are given by r⃗ = . Define β = arg(r⃗) and r = ‖r⃗‖, such that u⃗ is the unit vector with the same direction of the position vector of the source, which is located in the array's far field. The l-th row of the array manifold matrix then becomes Denote 4.In order to investigate the influence of a stochastic displacement of one of the sensor elements, we need stochastic simulation tools to characterize the statistical distribution of the DOAs due to the distribution of the random displacement vectors.A Monte-Carlo simulation for such a task requires a large amount of realizations to accurately capture the statistics of the random process. Therefore, we apply the SCM proposed in  as a mor4.1.X  is given. We want to determine the variation of the output Y = f(X) , due to the statistical variation of X. The random variable X follows the cumulative distribution XP and probability density function XdP in the sample space Ω. To determine the statistics of Y, we rely on the Askey scheme . We apply the synthetic signal generation procedure of Section 2.1 to generate a random source signal and use this signal vector throughout the simulations. We furthermore assume that no noise has been added to the impinging signal. We assume that one of the sensor elements has moved along the x-axis. For d between two sensor elements) of the sensor element,Consider the case of In lgorithm  estimateK peaks. We also observe that in the KR-MUSIC case, the estimated curves are almost linear as a function of the shifts. Furthermore, we observe that DOAs estimated by using the KR-MUSIC algorithm are almost the same as those estimated by the KR-root-MUSIC algorithm.In 5.2.In this section, we start from the same assumptions as in Section 5.1, concerning the impinging signal and the sensor array.x-axis plσ = 0.08. As outlined in Section 4, we therefore apply Hermite polynomials in the GPoC expansion of Q quadrature points and a polynomial expansion of order P = Q − 1) to determine the expansion coefficients for the GPoC expansion. In the bottom figure of Let P = Q − 1 for plσ = 0.2. We observe that, in the 95% confidence interval  for In P = 4.In 5.3.v(t) = Rυ(t) + Iυ(t). Rυ(t) and Iυ(t) are both Gaussian distributed with zero mean and dispersion σ = 0.3 providing an SNR = 7.78 dB. Throughout the simulations, we use this source signal vector and noise vector. We also assume that the first sensor element is displaced along the x-axis and that the relative displacement plσ = 0.12. To obtain the cdfs, we use two different methods: Monte-Carlo (MC) simulation and SCM. We carry out an MC simulation  .In this section, we assume that the first sensor element is displaced by d,eyreld]. The relplσ = 0.12. We have carried out an MC simulation with 10,000 realizations. The simulation time is about 128 s for the root-MUSIC algorithm and 157 s for the KR-root-MUSIC algorithm. We also apply the two-dimensional GPoC expansion P in both variables to obtain the cdfs for the DOAs. We calculate the weight coefficients using Q1 = Q2 = P − 1 quadrature points in the x- and y-directions using Q = 44, Q = 99 and Q = 172, respectively, and with polynomial expansions of order P = 7, P = 10 and P = 15, respectively. The measured value of the KS test, which compares the samples of the MC method with those obtained with the cubature formulas, is also presented in α = 0.05). Using cubature formulas, only for the estimates of the third DOA, 172 quadrature points are insufficient to satisfy the null hypothesis. For the KR-root-MUSIC algorithm, four quadrature points in each direction are sufficient. If we use cubature formulas, Q = 44 quadrature points are sufficient to satisfy the null hypothesis. We also observe that, using the SCM theory, the simulation time is reduced by a factor of about 20, and using the cubature formulas, we can reduce the simulation time by a factor of 56.For the same impinging signal as in Section 5.3, we have plotted the cdfs of the different estimated DOAs, for P = Q − 1 = 13. We observe that both curves almost coincide.In pQ as the p-th percentile of the estimated DOA, meaning that p% of the estimates for the true DOA are below pQ. Define S around the true DOA. For a Gaussian distribution, the value S will correspond to the dispersion. In S as a function of plσ for a two-dimensional displacement of the first sensor element. Again, we observe that the SCM method provides a good approximation for the estimates, both for the root-MUSIC algorithm as for the KR-root-MUSIC algorithm. Observe that for values plσ less than 0.4, the root-MUSIC yields better estimates for the first and third DOA than the KR-root-MUSIC algorithm. However, the KR-root-MUSIC estimates the second DOA better. We furthermore observe that, for the KR-root-MUSIC algorithm, the S-values are straight lines as a function Of plσ.Define N,K) =  case with a displacement of the first sensor element and with the KR-root-MUSIC algorithm. The true DOAs to be estimated are . Observe that, with the same array configuration as in θ̂1 = −65° with a large error, even for small values of plσ. The estimation error for the remainder of the DOAs starts to increase dramatically for values of plσ larger than 0.2.In 6.In this paper, we have investigated the effect of a displacement of one of the sensor elements. We have applied the SCM formalism to efficiently predict the cdfs of the estimated DOAs, in the case that the relative displacement is Gaussian distributed. Comparison of the SCM method to the Monte-Carlo method demonstrates that SCM is an accurate method to predict the cdfs of the DOAs. For the one-dimensional case, the SCM method decreases the simulation time by a factor of more than 100. For the two-dimensional case and using cubature formulas, we can reduce the simulation time by a factor of 56. In the future, we will extend the applied method to array configurations in which more sensor elements are displaced. Moreover, the SCM will be applied to other DOA estimation algorithms and for other array configurations. The flexibility and non-intrusive character of the proposed method only requires that DOA-estimates need to be computed in the quadrature points, by use of the appropriate algorithm.


G′/G)-expansion and the generalized tanh-coth methods. These methods are developed for searching exact travelling wave solutions of nonlinear partial differential equations. It is shown that these methods, with the help of symbolic computation, provide a straightforward and powerful mathematical tool for solving nonlinear partial differential equations.We establish exact solutions including periodic wave and solitary wave solutions for the integrable sixth-order Drinfeld-Sokolov-Satsuma-Hirota system. We employ this system by using a generalized ( G′/G)-expansion method [G′/G)-expansion method to look for travelling wave solutions of NLEEs. Zhang et al. [G′/G)-expansion method and its applications. Authors of [G′/G)-expansion method. Also, Bekir [G′/G)-expansion method for nonlinear evolution equations. In this paper we explain the method which is called the (G′/G)-expansion method to look for travelling wave solutions of nonlinear evolution equations.The general form of the Drinfeld-Sokolov-Satsuma-Hirota system that is going to be studied in this paper is given by wt−6wwx+wloped in . System wt−6wwx+wwt−6wwx+wloped in . Howeverloped in . Wazwaz loped in , the sinloped in –10, the loped in , sine-coloped in , homotoploped in , homotoploped in , 15, varloped in , 17, Bäcloped in , 19, Exploped in , 21, -expansion method; we give the detailed description of the method which was first presented by Wang et al. [Another powerful analytical method is called  and dependent variable u, ξ = x + y − ct is wave variable. Also, c is a constant to be determined later.For a given NLPDE with independent variables G(ξ) satisfies the second order LODE in the following form: a0, ak , λ, and μ are constants which will be determined later, am = 0, but the degree of which is generally equal to or less than m − 1; the positive integer m can be determined by considering the homogeneous balance between the highest order derivatives and nonlinear terms appearing in /G(ξ))k , and then setting each coefficient to zero, we can get a set of overdetermined partial differential equations for a0, ai , λ, c, and μ with the aid of symbolic computation Maple.Substituting  and 5)  5) into ai,…, am, c and general solutions of (G(ξ) of ai,…, ations of  dependin(G(ξ) of .We now describe the generalized tanh-coth method for the given partial differential equations. We give the detailed description of the method which to use this method; we take the following steps.X =  and dependent variable u, we consider a general form of nonlinear equation ξ = x + y − ct is wave variable. Also, c is constant to be determined later.For a given NLPDE with independent variables ak , bk , p, r, and q are constants to be determined later. But, the positive integer m can be determined by considering the homogeneous balance between the highest order derivatives and nonlinear terms appearing in =S(Φ)m obtained in k , and then setting each coefficient to zero, we can get a set of overdetermined partial differential equations for a0, ai , bi  p, q, and r with the aid of symbolic computation Maple.Substituting  and 9)  9) into a0, a1, b1,…, am, bm, c in -expansion method, we have F = G′/G and then G′/G)-expansion are same as ones by the generalized tanh-coth methods. Hence we use only the generalized tanh-coth method.But from −18u′u′′′ in  we obtaiwt−6wwx+wions of (u(ξ)=a0+ae we get 2u(v)−18uituting =a0+a(I) The First Set for (ξ = x − (p2 − 4qr)t. Set for . By usin(II) The Second Set for (ξ = x − (p2 − 4qr)t. By using the w = ux, v = (1/6)(ut − 3(ux)2 + uxxx), can be used to get the solutions of the DSSH system (G′/G)-expansion and the generalized tanh-coth methods. Also, in this paper we can see correlation between (G′/G)-expansion method and tanh-coth methods. We have succeeded in identifying the equivalence of the two methods under special conditions [G′/G)-expansion method is exactly the same as the solution of tanh-coth method if the conditions are satisfied. In fact, we can see that the tanh-coth method is a special case of the (G′/G)-expansion method. Set for . By usinBy using u27=G′/G)-expansion and the generalized tanh-coth methods which are useful methods for finding travelling wave solutions of nonlinear evolution equations. These methods have been successfully applied to obtain some new generalized solitonary solutions to the DSSH equation. These exact solutions include three types: hyperbolic function solution, trigonometric function solution, and rational solution. The generalized (G′/G)-expansion method is more powerful in searching for exact solutions of NLPDEs. By comparing our results and Wazwaz's [In this paper we investigated the Drinfeld-Sokolov-Satsuma-Hirota system by using the generalized (Wazwaz's  results


Parkinson’s disease (PD) is a degenerative disorder of the central nervous system, in which a small naturally unfolded protein α-synuclein plays an essential role. α-Synuclein belongs to a synuclein family comprising three members: α, β, and γ-synucleins associated with neurodegenerative and neoplastic diseases and involved in development. Several studies revealed that α-synuclein is present not only in the brain, but also in the skin and other peripheral tissues. This finding open a new approach to PD diagnosis based on the assay of α-synuclein from a biological sample of a living patient. Furthermore, PD is associated with an increased risk of skin melanoma. An important posttranslational modification of α-synuclein is phosphorylation at serine-129, which may convert the protein into pathological species both in PD and melanoma. Thus, analysis of phosphorylated α-synuclein might be an important diagnostic test for both diseases providing additional information about the mechanism of pathology. Parkinson’s disease (PD) is a common severe neurodegenerative disorder that affects a significant proportion of the adult population. The risk of developing PD is age dependent, affecting 1%–2% of the population over 65 years . The undetc. . HoweveOne of the key proteins in PD pathogenesis is α-synuclein (reviewed in ), being The use of α-synuclein as a PD marker is relatively easy, because it is present not only in the brain, but also in peripheral organ and tissues .Currently, the study of α-synuclein phosphorylation mechanism related to the PD pathology has become a research hotspot, because after phosphorylation by G-protein-coupled kinase  or otherα-Synuclein is a naturally unfolded protein easily changing its conformation and only specific conformers possess toxic properties. Thus, generation of conformation-specific α-synuclein antibodies and/or antibodies distinguishing phosphorylated/nonphosphorylated α-synuclein is an important step for the improved specificity of PD diagnosis . Hopefuletc., have been made [PD diagnosis based on α-synuclein analysis may be simplified, because this protein is present not only in the brain, but also in peripheral tissues . Severaleen made ,18,19,20een made ).Initially testing of α-synuclein in CSF as early PD diagnostic tool gave promising results ,21, but in vivo biomarker for degenerative peripheral autonomic neuropathy. In a later study, the same research team applied similar method for analysis of native α-synuclein (n-syn) and misfolded phosphorylated (p-syn) α-synuclein in idiopathic Parkinson disease (IPD) attempting to define the importance of n-syn and p-syn for PD diagnosis [in vivo marker of IPD, and its analysis in skin shows that p-syn inclusions are different in PAF and IPD, pointing to dissimilar underlying pathogenic mechanisms [Recently, an important step forward in PD early diagnosis is described by Donadio and coauthors . These riagnosis . The autchanisms .in vivo, are promising because they are based on a direct, straightforward, low-priced, and minimally invasive technique with minor discomfort for the patient.Therefore, recent studies have shown that methods based on the use of n-syn/p-syn ratio analysis in skin biopsy to diagnose PD and other synucleinopathies Importantly, patients with PD have an increased risk of developing melanoma, which is the major cause of skin cancer death worldwide . It has Several putative players may be involved in this association, including tyrosine, tyrosinase, tyrosine hydroxylase, DJ-1 and melanocortin 1 receptor ,37,38,39An important ongoing international multicenter prospective study The Parkinson’s Progression Markers Initiative (PPMI)  is aimedetc. However, it is usually better with p-syn than with α-synuclein. For example, Wang and coauthors [The sensitivity and specificity of p-syn analysis in skin and other peripheral tissues in some studies is not high enough to use it as a routine method of PD diagnostic ,46, whicoauthors  concludeoauthors . Some mooauthors  proposedoauthors . The devoauthors  may be aThus, Ser129 phosphorylated α-synuclein may be a more reliable marker compared to unphosphorylated protein, and its analysis demonstrates higher specificity and better prognostic value. Collecting of skin samples is inexpensive and not invasive method. Therefore, the analysis of phosphorylated α-synuclein can be used as an important diagnostic test for both diseases and provide additional information about the mechanism of pathology.


Unlike the western medical approach where adrug is prescribed against specific symptoms of patients,traditional Chinese medicine (TCM) treatment has a uniquestep, which is called syndrome differentiation (SD). It is arguedthat SD is considered as patient classification because priorto the selection of the most appropriate formula from a setof relevant formulae for personalization, a practitioner hasto label a patient belonging to a particular class (syndrome)first. Hence, to detect the patterns between herbs and symptomsvia syndrome is a challenging problem; finding thesepatterns can help prepare a prescription that contributes tothe efficacy of a treatment. In order to highlight this uniquetriangular relationship of symptom, syndrome, and herb, wepropose a novel three-step mining approach. It first startswith the construction of a heterogeneous tripartite informationnetwork, which carries richer information. The second step isto systematically extract path-based topological features fromthis tripartite network. Finally, an unsupervised method is usedto learn the best parameters associated with different featuresin deciding the symptom-herb relationships. Experiments havebeen carried out on four real-world patient records  with comprehensivemeasurements. Interesting and insightful experimental resultsare noted and discussed. Traditional Chinese medicine (TCM) has a long history and has been accepted as one of the main medical approaches in China . Many ofIn recent years, interest in TCM has increased globally and the application of data mining to TCM –4 is alsIn general, a homogeneous information network can be derived from a heterogeneous information network, for example, an herb-herb network can be derived from a symptom-syndrome-herb network by a projection on herbs only. A heterogeneous information network is different from a homogeneous information network because it carries richer information than its corresponding projected homogeneous information networks. Therefore, it aimed to discover herb-symptom patterns, via syndromes, from a heterogeneous information network, which contains different types of attribute values associated with objects. To the best of our knowledge, this is the first attempt towards mining herb-symptom patterns in TCM utilizing heterogeneous information networks.In this research, we construct the heterogeneous information network leveraging the tripartite graph. Our heterogeneous information network contains multiple types of objects, such as herb, symptom, syndrome, and multiple types of links defining different relations among these objects, such as links existing between herbs and syndromes, between syndromes and symptoms, and between symptoms and herbs. Thus, the number of different types of objects there are in the network can be found out, as well as the identification of the possible links existing among objects. Furthermore, we can detect the patterns between herbs and symptoms.We construct the TCM heterogeneous information network utilizing the tripartite graph.We study the problem of the symptom-herb relationship prediction in TCM heterogeneous information network.We propose a novel three-step prediction approach based on the TCM heterogeneous information network to discover symptom-herb patterns.Experiments on real TCM patient records indicate that our proposed method can mine symptom-herb relationships with high accuracy.Treatments are proven to be more effective than a direct symptom-herb relationship; that is, classifying patients into different syndromes is a crucial step in TCM treatment.The major contributions of this paper are summarized.The remaining of the paper is organized as follows. We first introduce the background and preliminaries on TCM heterogeneous information networks and denote the task of symptom-herb pattern prediction in H = {h1, h2,…, hn}, a set of syndromes D = {d1, d2,…, dm}, and a set of symptoms P = {p1, p2,…, pq}. We assume that there are n herbs, m syndromes, and q symptoms. Here, symptoms refer to something that can be observed and measured, such as fever, nausea, coughing, and weight loss. Syndrome is a special phenomenon in TCM. A TCM doctor will base upon the patient's symptoms and classify them into one or two syndromes. After that, formulas will be prescribed according to the syndrome.In this work, we need to consider three types of entities: a set of herbs We first introduce the definitions of heterogeneous information network , 8, tripG =  with an entity type mapping function ϕ : V → 𝒜 and a link type mapping function ψ : E → ℛ, where each entity v⊆V belongs to one particular entity type ϕ(v)⊆𝒜, each link e⊆E belongs to a particular relation type ψ(e)⊆ℛ, and W : E → R+ is a weight mapping from an edge e⊆E to a real number w⊆R+. Notice that, when the types of entities |𝒜| > 1 and also the types of relations |ℛ| > 1, the network is called heterogeneous information network.A heterogeneous information network is denoted as a directed graph V1 ∪ V2 ∪ V3}, E〉 can be called as tripartite, if a set of graph nodes decomposed into three disjoint sets such that no two graph nodes within the same set are adjacent; that is, V1∩V2∩V3 = ∅.A graph TG = 〈{X, Y, and Z, where X = {x1, x2,…, xm}, Y = {y1, y2,…, yn}, and Z = {z1, z2,…, zq}, graph G = 〈V, E〉 is called a tritype information network on types X, Y, and Z, if V(G) = X ∪ Y ∪ Z and E(G) = {〈oi, oj〉}, where oi, oj ∈ X ∪ Y ∪ Z.Given three types of objects sets Wm+n)∗(m+n)( = {〈woioj〉} (or Wn+q)∗(n+q)( = {〈woioj〉} or Wm+q)∗(m+q)( = {〈woioj〉}) be the adjacency matrix of links, where 〈woioj〉 equals the weight of link 〈oi, oj〉, which is the observation number of the link, and we thus use G = 〈{X ∪ Y ∪ Z}, W〉 to define this tritype information network with weight. In the following, we use X, Y, and Z denoting the object set and their type name. For convenience, we decompose the link matrix into four blocks: WXX, WXY, WYX, and WYY , each denoting a subnetwork of objects between types of the subscripts. W can be denoted asLet H representing herbs, P representing symptoms, and D representing syndromes. Notations and similarity relations used in definitions as well as the rest part of the paper can be found in Notation section.This tritype information network, one of the heterogeneous information networks, denotes the rules of how entities exist and how links should be created. And, through analyzing this tritype information network, we can know how many types of objects there are in the network and where the possible links exist. In the following, we give an example of tritype information network, which is showed in G = 〈{H ∪ D ∪ P}, W〉, the target type P, and a set of herbs {Hj}, our goal is to find or predict the most reasonable herbs for each symptom Pi, that is, how to predict the target relationship E(G) = {〈Pi, Hj〉}, where Pi, Hj ∈ P ∪ H.Based on the previous definitions, our goal of this work can be summarized as follows: given a tritype network Different from symptom-syndrome patterns and syndrome-herb patterns, which are directed relationships , symptom-herb patterns are undirected relationships. Intuitively, the herb-symptom relationship detection is an implicit relationship mining, which is more difficult to detect than an explicit relationship mining. However, if new herb-symptom relationships can be discovered, they are beneficial for doctors configuring the prescriptions.In this work, our experiments were performed on four real TCM datasets: Insomnia, Infertility, Diabetes, Tourette. These four datasets were provided by Guang'anmen Hospital, China Academy of Chinese Medical Sciences. These four datasets include the symptoms, the syndromes, and prescription information of outpatients. Here, edges are formed among objects belonging to the same prescription. Properties of these four datasets are shown in In this section, we conduct following observations based on the four TCM datasets in order to get a better understanding on the symptom-syndrome-herb patterns and structural properties of TCM tripartite network.x-axis represents the 251 unique herbs, ordered by descending herb frequency. The y-axis refers to the herb frequency. As reported by other authors  to provide a range of tolerance to bad outcomes. In particular, given a relevance function R, the relevance threshold θ is used for creating the parameterized version of this relevance function, R, that is formalized asθ changes over different datasets. Ax, Ay ∈ X ∪ Y ∪ Z and ratio = Good_Outcome(Pattern)/(Good_Outcome(Pattern) + Bad_Outcome(Pattern)). Here, Good_Outcome(Pattern) refers to the total number of this pattern working effectively, and Bad_Outcome(Pattern) is the total number of this pattern having no effect on patients. In the next section, patterns of symptom-herb that are predicted above relevance threshold θ  = 1) are sorted according to predicted rating, while patterns of symptom-herb that are below θ  = 0) are ignored.In this work, relevance function is used to filter out the patterns with bad outcome. Here, the relevance function is parameterized with “relevance threshold” K nearest entities according to the similarity measures, simPH_Path or simPDH_Path; then, we predict rating for each potential entity pair; subsequently, we get Top-n predicted patterns by ranking prediction rating; lastly, we get Top-N list by filtering the patterns of bad outcome using relevance function. The pseudocode of Tri-TSPA is shown in Up to now, we have given a systematic way to extract and build the topological features in the tripartite networks. In this subsection, we will introduce our prediction algorithm (Tri-TSPA). Our prediction method is as follows: first, we discover F measurement strategy to calculate the rating. Actually, we can replace F with JC, AW, and TfIdf, respectively. In addition, PH_Path based on symptom-herb patterns mining is shown in Line 4–line 7, and PDH_Path based on symptom-herb patterns mining is shown in Line 8–Line 11.In In this section, we conduct many experiments to evaluate the effectiveness of the proposed algorithm. We show that our proposed three step prediction approach can mine a reasonable set for each symptom on the TCM networks.We first convert these datasets into heterogeneous tripartite information networks. We construct four TCM networks from TCM datasets, which consist of three types of objects: symptoms, syndromes, and herbs. Links exist between symptoms and syndromes, syndromes and herbs, and herbs and symptoms.PH_Path based strategy and PDH_Path based strategy. For each strategy, we apply four different measurement methods to set each term of each matrix related to this PH_Path (or PDH_Path). By combining these two kinds of strategies and four measurement methods together, we get total 8 different predicted methods. In the following section, a series of experiments will be carried on in order to find which predicted method can get the best performance.In order to effectively mine symptoms-herbs patterns, we adopt two kinds of strategies: F, JC, AM, and TfIdf). After collecting all associated features, a training model is then built to learn the best coefficients associated with different features in deciding the symptom-herb patterns by performing multiple experiments. In the test stage, we utilize the learned coefficients to predict the potential patterns between symptoms and herbs and record whether this pattern is to appear in the test dataset.In this work, we adopt twofold cross-validation  to evaluate the performance of the prediction for each TCM network. In the training stage, we first extract two kinds of paths, symptom-herb path and symptom-syndrome-herb path. In terms of these two paths, we further build five matrices in  accordinPDH_Path from Infertility and Diabetes datasets, such as p1-d1-h1, p2-d1-h1; meanwhile, we can obtain the existing patterns based on the PH_Path from Insomnia dataset, such as p1-h1, p2-h1. Second, we can further check whether the patterns based on the PH_Path from Insomnia dataset exist in the dataset Insomnia or Tourette. If they exist , we can assume a virtual syndrome d and construct the edge between d and p1 and the edge between d and h1 (or the edge between d and p2). Otherwise, we only assume a virtual syndrome d and produce the edges between d and other symptoms (or the edges between d and other herbs). Similarly, we can construct the tripartite graph based on the Tourette dataset.In addition, the Insomnia and Tourette dataset lacks the object of syndrome and symptom, respectively. In this case, we assume some virtual objects (representing syndromes or symptoms) which can be constructed according to the next method. Here, we take the Insomnia dataset as an example to explain how to construct the virtual objects, namely, syndromes. First, we can get the existing patterns based on the N highest ranked herbs as the predicted list for a target symptom. To evaluate the prediction accuracy, we focus on how many symptoms-herbs patterns previously removed in the preprocessing step reappear in the predicted results. Therefore, we apply two popular performance metrics, namely, Precision@N and Recall@N [Our proposed algorithm computes a ranking score for each candidate herb and returns the top-Recall@N –20, to cN is the ratio of recovered symptoms-herbs patterns to the N predicted symptoms-herbs patterns. Recall@N is the ratio of recovered symptoms-herbs patterns to the set of symptoms-herbs patterns deleted in preprocessing. We divide the symptoms-herbs patterns into two sets: the test set Th and the Top-N set Rh. Symptoms-herbs patterns that appear in both sets are members of the hit set. Precision and Recall are defined as follows:Precision@λ0, λ1, α, α0, α1, β, β0, β1, θ, and K. We have varied the neighborhood size from 10 to 50 by an interval of 10 and the other nine parameters from 0 to 1 by an interval of 0.1. Using the validation data (in Infertility dataset), we have found the best λ0 to be 0.8, λ1 to be 0.2, α to be 0.7, α0 to be 0.8, α1 to be 0.2, β to be 0.3, β0 to be 0.8, β1 to be 0.2, θ to be 0.5, and K to be 30. In addition, we have different values for these parameters in the other three datasets, but we get the similar experimental results. Here, we do not list all the values for these parameters because of the limitation of space.In our experiments, we divide each dataset into two parts: training set and test set. We further split the training data to validation data to optimize the parameters K as an example to explain how to install optimal value for each parameter. From N list the Precision changes over the neighborhood size K. We can further observe that when the neighborhood size K equals 30, our proposed method gets the best performance. Also, from K as 30.In PH_Path based strategy and PDH_Path based strategy by using the optimal measurement method.In this section, we first evaluate the performance of four different measurement methods for two kinds of paths. And then, we compare the performance of F, and JC, respectively. From F, and JC. Here, an interesting result is observed that JC gets the worst performance. Contrary to JC being known to be more useful to measure the similarity between two same type of objects, it may be due to the existence of different type of objects. Similarly, from It is worth noting that a comprehensive set of experiments was conducted using every measurement method in conjunction with every evaluation metric on every dataset, and the results are very consistent across all experiments. Because of the space limitations, we show the results based on the Infertility dataset in the Figures In this section, we will estimate the performance of our presented Tri-TSPA based on two kinds of paths.PDH_Path can find more symptom-herb relations than the one based on PH_Path, in general.First, we illustrate how our Tri-TSPA can serve as a powerful model for predicting potential symptom-herb relationships. The prediction processing performance results can be found in Figures PDH_Path improves Precision@30 by 10.8% compared with the one based on PH_Path. In addition, from PDH_Path improves Recall@30 by 11% when compared with PH_Path. Therefore, we can conclude that PDH_Path based prediction method gives a good performance overall. Here, we can see that when N reaches 30, the precision of both algorithms is optimal. Meanwhile, although Recall@50 of both algorithms reaches optimal value, the gap between Recall@30 of both algorithms and Recall@50 of both algorithms is very small. So we take N = 30 as an optimal value to achieve optimal prediction power for the Infertility dataset.From In addition to the Infertility dataset, we tested the proposed algorithm with other three datasets, and the same pattern is observed in the vast majority of cases.The symptoms in TCM are related to the body as a whole. A certain subset of symptoms belongs to a certain syndrome, and the typical treatment of a syndrome usually follows a therapeutic principle, which refers to the use of a certain combination of herbs .N list of herbs for each symptom  is a crucial step to a successful treatment in TCM. In the future, we intend to extend our work in the following three directions. Firstly, a new measure to estimate the performance in the proposed method should be explored. Secondly, another novel similarity measure method should be studied to capture the rich topological features. Thirdly, a new matching function to improve the predictive performance should be sought.


We prove the existence and uniqueness of fixed points of T-stability for an iteration on partial cone metric space of Zamfirescu contraction. As an application, we prove a theorem for integral equation. We also give illustrative examples to verify our results. Rhoades . Further, let T. In complete metric space, the Picard iteration process Banach’s contraction condition.Let us consider a self mapping entclass1pt{minimaWe shall state some of the iteration process generalising  as folloFor For Kannan  establisT, there exists For a self map atterjea  gave theT, there exists For a selfmap mfirescu  establisentclass1pt{minimamfirescu  and 6)..T, thereThen the mapping  Rhoades , 1991, L Rhoades , Yousefi Rhoades , Rhoades Rhoades , Qing an Rhoades , Asadi e Rhoades , Saadati Rhoades , Olatinw Rhoades  and refeHuang and Zhang  obtainedE be a real Banach space. A subset P of E is called a cone ifP is closed, non-empty and Let P by x ≤ y if and only if P).Given a cone There are two kinds of cone. They are normal cone and non-normal cone. A cone The aim of this paper is to show the existence and uniqueness of fixed points of T-stability for an iteration on partial cone metric space under Zamfirescu contraction. We give an application in integral equation. We also give illustrative examples that verifies our results.We have the following basic definitions:X be a nonempty set. Suppose the mapping  Then d is called a cone metric on X and  is called a cone metric space. such that X is non-empty set and p is a partial cone metric on X is called a partial cone metric space. We know that if p = 0, then x = y. But if x = y, then p may not be 0.Sonmez  A partiaA cone metric space is a partial cone metric space. But there are partial cone metric space which are not cone metric space. The following example verifies the statement.X, p) is a partial cone metric space which is not a cone metric space.Sonmez  Let \docX, p) be a partial cone metric space. Let X and x and x is called a limit of Sonmez  Let  be a partial cone metric space. Let X and N such that Sonmez  Let  is said to be complete if and only if every Cauchy sequence in X is convergent.Sonmez  A partiaX, d) be a complete metric space, X. Suppose that T. Let T which is defined by f is some function. Suppose p of T. Let T-stable or stable with respect to T if and only if Olatinwo  Let  be a partial cone metric space. Let X with X and posit that In this section we establish iteration procedure in partial cone metric spaces. This is to stretch out some recent results of T-stability. Let  be a complete partial cone metric space. Let P be a normal cone with normal constant K andwith. If there existssuch thatfor allandand in addition, wheneveris a sequence withas, then Picard iteration is T-stable.Let v be another fixed point of T, thenFor uniqueness: Let T is unique. Thus the fixed point of Let  be a complete partial cone metric space. Let P be a normal cone with normal constant K andwithIf there existssuch thatfor allandand in addition, wheneveris a sequence withasthen Picard iteration is T-stable.Let v be another fixed point of T, thenFor uniqueness: Let T is unique. Thus the fixed point of Let  be a complete partial cone metric space. Let P be a normal cone with normal constant K andwith. If there existssuch thatfor allandand in addition, wheneveris a sequence withas, then Picard iteration is T-stable.Let v be another fixed point of T, thenFor uniqueness: Let T is unique. We get p be the partial cone metric on X defined by Let mentclasspt{minimaLet X = C, withforand letdefined bywhereis a continuous function.The partial derivativeFwith respect to y exists andfor someFor every real numberone hasfor everyLetbe a normal cone and  the complete partial cone metric space definedwhere. Then Picard’s iteration is T-stable ifF satisfies of Theorem T has unique fixed point and Picard’s iteration is T-stable.Let We extend and prove the T-stability of Picard’s iteration satisfying Zamfirescu contraction in partial cone metric space. Our results are more general than that of the results of metric and cone metric spaces. This result can be extended to other spaces.


As a key regulatory mechanism of gene expression, DNA methylation patterns are widely altered in many complex genetic diseases, including cancer. DNA methylation is naturally quantified by bounded support data; therefore, it is non-Gaussian distributed. In order to capture such properties, we introduce some non-Gaussian statistical models to perform dimension reduction on DNA methylation data. Afterwards, non-Gaussian statistical model-based unsupervised clustering strategies are applied to cluster the data. Comparisons and analysis of different dimension reduction strategies and unsupervised clustering methods are presented. Experimental results show that the non-Gaussian statistical model-based methods are superior to the conventional Gaussian distribution-based method. They are meaningful tools for DNA methylation analysis. Moreover, among several non-Gaussian methods, the one that captures the bounded nature of DNA methylation data reveals the best clustering performance. Recentlβ, at a specific genomic locus provides an estimate of the fraction of cells that have that locus methylated. Although studies have considered using the logit-transform y = log2β/(1 − β) instead of β for subsequent statistical inferences   and canribution  and the ribution  are usuaIn the remaining part of this section, we will introduce some typical non-Gaussian distributions that can be applied in DNA methylation analysis.3.1.1.u and v. The pdf of the beta distribution is:The beta distribution is characterized by two positive shape parameters where Γ(·) is the gamma function. The beta distribution has a flexible shape, which is shown in 3.1.2.l2 norm equals one, i.e., ||x||2 = 1. The vMF distribution contains two parameters, namely the mean direction μ and the concentration parameter λ. The pdf of a K-dimensional vMF distribution can be expressed as:The vMF distribution is considered a popular distribution in the family of directional distributions ,51. The μ||2 = 1, λ ≥ 0 and K ≥ 2 [K(λ) is given by:where ||nd K ≥ 2 . The norν(·) represents the modified Bessel function of the first kind of order ν [where ℐ order ν . The pdf order ν ,54. It c order ν .3.1.3.x and −x are equivalent. In other words, that is ±x that are observed. Here, we need probability density functions for x on  which are axially symmetric, that is f(−x) = f(x). In such cases, the p-dimensional observation ±x can be regarded as being on the projective space ℙp−1, which is obtained by identifying opposite points on the sphere .Observations on the sphere might have an additional structure, such that the unit vectors One of the simplest distributions for axial data, with a rotational symmetry property, is the (Dimroth–Scheidegger–) Watson distribution. The Watson distribution is a special case of the Bingham distribution , which ix ∈ ℙp−1, or equivalently ±x ∈ , has the (p − 1)-dimensional Watson distribution pW , with the mean direction μ and the concentration parameter κ, if its probability density function is:A random vector κ ∈ ℝ, ||μ||2 = 1, and 1F1 is Kummer’s (confluent hypergeometric) function  reduces to the uniform density, and as κ → ∞, pW  tends to a point density. For κ < 0, as κ → − ∞, the density concentrates around the great circle orthogonal to the mean direction  is drawn according to the following generative model:With the above description, we assume that the matrix x; k, θ) is the gamma density with parameters k, θ defined as:where Gamma method  is appliwhich can be expressed in matrix form as:H, the reconstruction in where ⊘ means element-wise division. When placing sparsity constraints on the columns in W̄ is low-dimensional while retaining the bounded support constraint.Hence, the resulting pseudo-basis matrix 3.3.1.L are not corresponding to the convex region, the K-means algorithm cannot provide satisfactory clustering results. However, when mapping the data points to ℝK space via SC, they may form tight clusters [K-dimensional space, the reduced features, each of which is K-dimensional, are extracted by taking eigenvalue analysis of an intermediate matrix M. The reduced features will then be used for clustering, with conventional methods like K-means. The feature extraction procedure via the SC method [K space.Recently, spectral clustering (SC) has become one of the most popular clustering algorithms . It is aclusters . SC analC method  is summa3.4.K-dimensional space. These data are obviously non-Gaussian distributed. It has been shown in several studies that applying a suitable non-Gaussian distribution to model the non-Gaussian distributed data can improve the corresponding performance [The ultimate goal of dimension reduction is to benefit the clustering of DNA methylation data. The dimension reduction methods introduced above yield features with special properties. The BG-NMF method provides a basis matrix see , every eformance ,26,31. T3.4.1.For the beta-valued DNA methylation data, it is natural to consider the beta distribution as a candidate to model the underlying distribution. Since the DNA methylation data that show an obvious normal/cancer status are multi-modal, a beta mixture model (BMM) can be applied for modeling. One mixture component represents one cluster. In unsupervised clustering, selecting the optimal number of clusters is a big challenge. One popular method designed for such purpose is the recursive partitioning beta mixture model (RPBMM) . The RPB3.4.2.Another way of carrying out model selection is to employ the variational Bayesian estimation framework for BMM (VBBMM). Under this circumstance, the joint posterior distribution of the weighting factors is modeled by a sparse Dirichlet distribution, so that the component with a very small weight will be pruned from the mixture model. In , the ext3.4.3.l2 norm equaling one has a directional property. The von Mises-Fisher (vMF) distribution is suitable for such a type of data [The data with its  of data ,69. In o of data . This me3.4.4.i.e., l2 norm equals one) and its axial mirror by the Watson distribution. Similarly, when such data are multi-modally distributed, a Watson mixture model (WMM) can be applied. With a variational inference framework, Taghia et al. [The Watson distribution is a simple distribution for modeling axially symmetric data on the unit hypersphere ( (Chaptera et al.  proposed4.Cancer is characterized by alterations at the DNA methylation level. A Gaussian distribution, in general, cannot describe the DNA methylation data appropriately. Hence, the Gaussian distribution-based unsupervised clustering does not provide convincing performance.For the purpose of efficiently clustering DNA methylation data, we proposed several dimension reduction methods and consequent unsupervised learning methods, which are all based on non-Gaussian distributions. They all perform better than the Gaussian distribution-based method. In the dimension reduction step, both the BG-NMF and the SC methods can remove the redundant dimensions efficiently. In unsupervised clustering, the VBBMM method, the VBvMM method and the VBWMM method can all reveal the heterogeneity of the DNA methylation data appropriately. Clustering performance demonstrates that the proposed non-Gaussian distribution-based methods are meaningful tools for analyzing DNA methylation data. Experimental results also show that the BG-NMF + VBBMM method performs the best among all of the proposed methods and is faster than the benchmarked BG-NMF + RPBMM method. Furthermore, for the reduced features inferred from both the BG-NMF method and the SC method, the consequent unsupervised clustering method needs to be improved, so that better clustering accuracy can be obtained.Moreover, the methodology introduced in this paper can be easily extended to analyze other DNA methylation data sets. Some other non-Gaussian statistical models can also be applied for such purposes.