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Geometrical properties of a generalized cone and its 2D image 广义锥及其二维图像的几何性质
IF 1.8 4区 心理学 Q2 MATHEMATICS, INTERDISCIPLINARY APPLICATIONS Pub Date : 2023-06-01 DOI: 10.1016/j.jmp.2023.102765
Tadamasa Sawada , Zygmunt Pizlo

The generalized cone is a simple 3D shape that is produced by sweeping a planar cross-section along a curve. Many complex and articulated 3D objects can be represented by combining generalized cones. It has been shown that generalized cones play an important role in our visual system for perceiving the shapes of these objects and recognizing them. In this study, we analyzed the geometrical properties of generalized cones and their 2D images and found that there are invariant features in the images of the generalized cones under both 2D orthographic and perspective projections that facilitate the recovery of the 3D shapes of the cones from the images. We found that the 3D translational symmetry of generalized cones can be analyzed using tools designed for 3D mirror-symmetry.

广义圆锥体是一种简单的三维形状,它是通过沿曲线扫掠平面横截面而产生的。许多复杂和铰接的三维对象可以通过组合广义锥来表示。研究表明,广义锥在我们的视觉系统中扮演着重要的角色,可以感知这些物体的形状并识别它们。在本研究中,我们分析了广义锥及其二维图像的几何性质,发现在二维正投影和透视投影下,广义锥的图像中都存在不变特征,有助于从图像中恢复锥的三维形状。我们发现,广义锥的三维平移对称性可以使用为三维镜像对称设计的工具进行分析。
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引用次数: 0
Adjacencies on random ordering polytopes and flow polytopes 随机排序多面体和流动多面体上的邻接关系
IF 1.8 4区 心理学 Q2 MATHEMATICS, INTERDISCIPLINARY APPLICATIONS Pub Date : 2023-06-01 DOI: 10.1016/j.jmp.2023.102768
Jean-Paul Doignon , Kota Saito

The Multiple Choice Polytope (MCP) is the prediction range of a random utility model due to Block and Marschak(1960). Fishburn(1998) offers a nice survey of the findings on random utility models at the time. A complete characterization of the MCP is a remarkable achievement of Falmagne (1978). To derive a more enlightening proof of Falmagne Theorem, Fiorini(2004) assimilates the MCP with the flow polytope of some acyclic network. However, apart from a recognition of the facets by Suck(2002), the geometric structure of the MCP was apparently not much investigated. We characterize the adjacency of vertices and the adjacency of facets. Our characterization of the edges of the MCP helps understand recent findings in economics papers such as Chang, Narita and Saito(2022) and Turansick(2022). Moreover, our results on adjacencies also hold for the flow polytope of any acyclic network. In particular, they apply not only to the MCP, but also to three polytopes which Davis-Stober, Doignon, Fiorini, Glineur and Regenwetter (2018) introduced as extended formulations of the weak order polytope, interval order polytope and semiorder polytope (the prediction ranges of other models, see for instance Fishburn and Falmagne, 1989, and Marley and Regenwetter, 2017).

多选多点(MCP)是Block和Marschak(1960)提出的随机效用模型的预测范围。Fishburn(1998)对当时的随机实用新型的发现进行了很好的调查。对MCP的完整表征是Falmagne(1978)的一项非凡成就。为了得到Falmagne定理的更具启发性的证明,Fiorini(2004)将MCP与一些非循环网络的流多面体同化。然而,除了Suck(2002)对小平面的认识外,MCP的几何结构显然没有得到太多研究。我们刻画了顶点的邻接性和小平面的邻接性。我们对MCP边缘的描述有助于理解Chang、Narita和Saito(2022)以及Turansick(2022)等经济学论文中的最新发现。此外,我们关于邻接的结果也适用于任何非循环网络的流多面体。特别是,它们不仅适用于MCP,还适用于Davis Stober、Doignon、Fiorini、Glineur和Regenwetter(2018)引入的三个多面体,作为弱阶多面体、区间阶多面体和半阶多面体的扩展公式(其他模型的预测范围,见Fishburn和Falmagne,1989,以及Marley和Regenwitter,2017)。
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引用次数: 4
Stochastic choice with bounded processing capacity 具有有限处理能力的随机选择
IF 1.8 4区 心理学 Q2 MATHEMATICS, INTERDISCIPLINARY APPLICATIONS Pub Date : 2023-06-01 DOI: 10.1016/j.jmp.2023.102771
Thierry Marchant , Arunava Sen

We propose and characterize a class of stochastic decision functions for a decision-maker who has a capacity for processing at most k-alternatives at a time. When faced with a menu containing more than k alternatives, she randomly chooses a sub-menu of size k with uniform probability and selects the best alternative according to a strict ordering . For smaller menus, she chooses the best alternative according to .

我们为一个决策者提出并刻画了一类随机决策函数,该决策者一次处理最多k个备选方案。当面对包含k个以上备选方案的菜单时,她以均匀的概率随机选择一个大小为k的子菜单,并根据严格的排序选择最佳备选方案。对于较小的菜单,她根据≻选择最佳选择。
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引用次数: 2
Comparison of type I error and statistical power between state trace analysis and analysis of variance 状态跟踪分析与方差分析的I型误差和统计能力比较
IF 1.8 4区 心理学 Q2 MATHEMATICS, INTERDISCIPLINARY APPLICATIONS Pub Date : 2023-06-01 DOI: 10.1016/j.jmp.2023.102767
Wei Liu , Yu-Xue Jia

State-Trace Analysis (STA) is a methodology for investigating the number of latent variables. Recently, a quantitative STA technique based on conjoint monotonic regression and double bootstrap method (STA-CMR) has been proposed. More discussion is needed on the type I error and the statistical power of this technique, as it adopts null hypothesis significance testing (NHST) to draw statistical inference. Because the results of STA are comparable with analysis of variance (ANOVA) in a three-factor experiment with linearity assumption, it is necessary to compare STA-CMR with ANOVA accordingly. This study investigated the type I error and the statistical power of STA-CMR and ANOVA in specific linear and nonlinear models using simulated data. Results demonstrated that both techniques were effective in the linear models, where ANOVA had a greater statistical power and STA-CMR had a more rigorous control of type I error. In the nonlinear models, although STA-CMR worked just as well as in the linear models, ANOVA completely lost its effectiveness. Besides, we found that the estimated type I error rate of STA-CMR was always smaller than the preset significance level in both linear and non-linear models. We suggest that the suppressed type I error rate may be caused by the bootstrap procedure, but the exact causes need further investigation. In conclusion, despite the suppressed type I error rate, STA-CMR can be a useful tool for determining the number of latent variables, particularly in non-linear models.

状态跟踪分析(STA)是一种研究潜在变量数量的方法。最近,提出了一种基于联合单调回归和双自举方法的定量STA技术(STA-CMR)。由于该技术采用零假设显著性检验(NHST)进行统计推断,因此需要对I型误差和该技术的统计能力进行更多的讨论。由于STA的结果与线性假设的三因素实验中的方差分析(ANOVA)具有可比性,因此有必要相应地将STA-CMR与ANOVA进行比较。本研究使用模拟数据研究了特定线性和非线性模型中STA-CMR和ANOVA的I型误差和统计功效。结果表明,这两种技术在线性模型中都是有效的,其中ANOVA具有更大的统计能力,STA-CMR对I型误差有更严格的控制。在非线性模型中,尽管STA-CMR与线性模型一样有效,但ANOVA完全失去了有效性。此外,我们发现在线性和非线性模型中,STA-CMR的估计I型错误率总是小于预设的显著性水平。我们认为,被抑制的I型错误率可能是由引导程序引起的,但确切的原因需要进一步调查。总之,尽管抑制了I型错误率,但STA-CMR可以是确定潜在变量数量的有用工具,特别是在非线性模型中。
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引用次数: 1
The best-worst-choice polytope on four alternatives 四种选择上的最佳最差选择多晶体
IF 1.8 4区 心理学 Q2 MATHEMATICS, INTERDISCIPLINARY APPLICATIONS Pub Date : 2023-06-01 DOI: 10.1016/j.jmp.2023.102769
Jean-Paul Doignon

Several papers co-authored by A.A.J. Marley helped in popularizing the best-worst-choice paradigm due to Finn and Louviere (1992). Inspired by Block and Marschak (1960), Marley conceived a random utility model for the choice frequencies of the best and worst alternatives in any proposed set of alternatives (Marley and Louviere, 2005). He then asked for a characterization of the prediction range of the model. The range being a convex polytope, an affine description of this polytope would provide a solution to Marley problem. For four alternatives, we show that a minimal such description consists in 26 affine equalities and 144 affine inequalities. The result derives from the Gale transform of the set of polytope vertices: the transform being a family of 24 vectors in a one-dimensional vector space, it plainly reveals the affine structure of the polytope. As far as we know, Marley problem is still open when the number of alternatives exceeds 4.

A.A.J.Marley合著的几篇论文帮助推广了Finn和Louviere(1992)提出的最佳-最差选择范式。受Block和Marschak(1960)的启发,Marley构思了一个随机实用模型,用于在任何拟议的备选方案集中选择最佳和最差备选方案的频率(Marley和Louviere,2005)。然后,他要求对模型的预测范围进行表征。范围是一个凸多面体,对这个多面体的仿射描述将为Marley问题提供一个解决方案。对于四个备选方案,我们证明了一个最小的这样的描述由26个仿射不等式和144个仿射不等式组成。结果来自多面体顶点集的Gale变换:该变换是一维向量空间中的24个向量族,它清楚地揭示了多面体的仿射结构。据我们所知,当备选方案的数量超过4时,马利问题仍然存在。
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引用次数: 0
Length of the state trace: A method for partitioning model complexity 状态跟踪长度:一种划分模型复杂性的方法
IF 1.8 4区 心理学 Q2 MATHEMATICS, INTERDISCIPLINARY APPLICATIONS Pub Date : 2023-04-01 DOI: 10.1016/j.jmp.2023.102755
F. Gregory Ashby

A novel and easy-to-compute measure is proposed that compares the relative contribution of each parameter of a mathematical model to the model’s mathematical flexibility or complexity, with respect to accounting for the results of some specific experiment. When the data space is a two-dimensional plot of the type used in standard state-trace analysis, then the model complexity contributed by a single parameter equals the length of the state trace (LOST) that results when that parameter is varied and all other parameters are held constant. For the normal, equal-variance, signal-detection model, the average LOST when the response-criterion parameter XC is varied is about four times greater than the average LOST when the sensitivity parameter d is varied. As a result, applying the signal-detection model to random data almost always leads to the conclusion that all the points share the same value of d but were generated under different values of XC. Parameters that have non-monotonic effects on performance, such as the attention-weight parameter that is used in popular exemplar and prototype models of categorization, tend to have large LOSTs, and therefore contribute to model flexibility more than parameters that have monotonic effects on performance. Comparing LOSTs for exemplar and prototype models also leads to some deep new insights into the structure of both models.

本文提出了一种新颖且易于计算的度量方法,比较数学模型的每个参数对模型的数学灵活性或复杂性的相对贡献,从而考虑到某些特定实验的结果。如果数据空间是标准状态跟踪分析中使用的那种类型的二维图,那么单个参数所贡献的模型复杂性等于当该参数变化而所有其他参数保持不变时所产生的状态跟踪(LOST)的长度。对于正态、等方差的信号检测模型,响应准则参数XC变化时的平均loss约为灵敏度参数d '变化时的平均loss的4倍。因此,将信号检测模型应用于随机数据时,几乎总是得出所有的点都具有相同的d '值,但在不同的XC值下生成。对性能具有非单调影响的参数,例如在流行的分类范例和原型模型中使用的注意力权重参数,往往具有较大的loss,因此比对性能具有单调影响的参数更有助于模型的灵活性。比较范例模型和原型模型的loss还可以对这两种模型的结构产生一些深刻的新见解。
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引用次数: 0
Stochastic additive differences 随机加性差异
IF 1.8 4区 心理学 Q2 MATHEMATICS, INTERDISCIPLINARY APPLICATIONS Pub Date : 2023-04-01 DOI: 10.1016/j.jmp.2022.102744
Yutaka Nakamura

Properties of a binary choice probability function p defined on multiattributed outcomes are studied to represent p as a transformation of additive difference evaluations of chosen and unchosen outcomes into the unit interval. We use an algebraic assumption to obtain an additive difference representation, but allow for restricting strict increasingness of the transformation to the subset of the domain on which transformed values are strictly between 0 and 1. We also apply a topological assumption to axiomatize the cases of homogeneous product sets in the context of finite-state decision making under uncertainty.

研究了在多属性结果上定义的二元选择概率函数p的性质,以将p表示为选择和未选择结果的加性差异评估到单位区间的转换。我们使用代数假设来获得加性差分表示,但允许将变换的严格增量限制在域的子集上,在该子集上,变换值严格在0和1之间。我们还应用拓扑假设来公理化在不确定性下的有限状态决策中齐次乘积集的情况。
{"title":"Stochastic additive differences","authors":"Yutaka Nakamura","doi":"10.1016/j.jmp.2022.102744","DOIUrl":"https://doi.org/10.1016/j.jmp.2022.102744","url":null,"abstract":"<div><p><span>Properties of a binary choice probability function </span><span><math><mi>p</mi></math></span> defined on multiattributed outcomes are studied to represent <span><math><mi>p</mi></math></span><span> as a transformation of additive difference evaluations of chosen and unchosen outcomes into the unit interval. We use an algebraic assumption to obtain an additive difference representation, but allow for restricting strict increasingness of the transformation to the subset of the domain on which transformed values are strictly between 0 and 1. We also apply a topological assumption to axiomatize the cases of homogeneous product sets in the context of finite-state decision making under uncertainty.</span></p></div>","PeriodicalId":50140,"journal":{"name":"Journal of Mathematical Psychology","volume":"113 ","pages":"Article 102744"},"PeriodicalIF":1.8,"publicationDate":"2023-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49868432","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"心理学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On the correspondence between granular polytomous spaces and polytomous surmising functions 关于粒度多胞空间与多胞猜测函数的对应关系
IF 1.8 4区 心理学 Q2 MATHEMATICS, INTERDISCIPLINARY APPLICATIONS Pub Date : 2023-04-01 DOI: 10.1016/j.jmp.2022.102743
Xun Ge

By modifying the concept of polytomous surmise functions, this paper introduces polytomous surmising functions. Then, it is shown that there is a one-to-one correspondence f between granular polytomous spaces and polytomous surmising functions where polytomous surmising functions cannot be replaced with polytomous surmise functions. This result gives a correction for a correspondence between granular polytomous spaces and polytomous surmise functions. As an application of the correspondence f, this paper demonstrates that the pair (f,f1) of mappings forms a Galois connection where all granular polytomous spaces and all polytomous surmising functions are closed elements of this Galois connection.

通过对多棱猜测函数概念的修正,引入了多棱猜测函数。然后,证明了颗粒多边形空间与多边形猜测函数之间存在一一对应关系,其中多边形猜测函数不能被多边形猜测函数所取代。这一结果给出了颗粒多边形空间和多边形猜测函数之间对应关系的修正。作为对应关系f的一个应用,证明了映射对(f,f−1)形成了一个伽罗瓦连接,其中所有颗粒多边形空间和所有多边形推测函数都是该伽罗瓦连接的闭元。
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引用次数: 0
Semiorders and continuous Scott–Suppes representations. Debreu’s Open Gap Lemma with a threshold 半序和连续scott - supes表示。带阈值的德布鲁开隙引理
IF 1.8 4区 心理学 Q2 MATHEMATICS, INTERDISCIPLINARY APPLICATIONS Pub Date : 2023-04-01 DOI: 10.1016/j.jmp.2023.102754
A. Estevan

The problem of finding a utility function for a semiorder has been studied since 1956, when the notion of semiorder was introduced by Luce. But few results on continuity and no result like Debreu’s Open Gap Lemma, but for semiorders, was found. In the present paper, we characterize semiorders that accept a continuous representation (in the sense of Scott–Suppes). Two weaker theorems are also proved, which provide a programmable approach to Open Gap Lemma, yield a Debreu’s Lemma for semiorders, and enable us to remove the open-closed and closed-open gaps of a set of reals while keeping the threshold.

自1956年Luce提出半阶的概念以来,就一直在研究半阶的效用函数问题。但对于连续性的结果很少,也没有像Debreu的开间隙引理那样的结果,但对于半阶,却没有发现。在本文中,我们刻画了接受连续表示的半阶(在Scott–Suppes的意义上)。还证明了两个较弱的定理,它们为开间隙引理提供了一种可编程的方法,产生了半阶的Debreu引理,并使我们能够在保持阈值的同时去除一组实数的开闭和闭开间隙。
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引用次数: 0
A test of attribute normalization via a double decoy effect 通过双重诱饵效应的属性归一化检验
IF 1.8 4区 心理学 Q2 MATHEMATICS, INTERDISCIPLINARY APPLICATIONS Pub Date : 2023-04-01 DOI: 10.1016/j.jmp.2022.102741
Remi Daviet , Ryan Webb

We report a “Double Decoy” experiment designed to separate two competing accounts of the asymmetric dominance effect. The experiment places an additional decoy alternative within the range of existing alternatives, which should leave choice behaviour unaltered if attributes are weighted by their range. Instead, we observe a decrease in the relative proportion of targets chosen, particularly for subjects who exhibited an initial decoy effect. We also observe considerably more variation in individual behaviour than expected. We therefore consider an alternative theory in which attributes values are compared with diminishing sensitivity (via divisive normalization) and assess its performance in an additional discrete choice experiment previously used in the discrete choice literature. We find that divisive normalization captures behaviour better than range normalization and the linear additive Logit model typically used in applied settings. We therefore propose divisive normalization as both a neuro-computational explanation for context effects and a useful empirical tool for applied researchers.

我们报告了一个“双重诱饵”实验,旨在分离不对称优势效应的两个竞争账户。该实验在现有选择的范围内放置了一个额外的诱饵选择,如果属性按其范围加权,则应该保持选择行为不变。相反,我们观察到选择目标的相对比例下降,特别是对于那些表现出最初诱饵效应的受试者。我们还观察到,个体行为的差异比预期的要大得多。因此,我们考虑一种替代理论,其中属性值与递减灵敏度(通过分裂归一化)进行比较,并在先前在离散选择文献中使用的额外离散选择实验中评估其性能。我们发现分裂归一化比范围归一化和线性加性Logit模型更好地捕获行为,通常用于应用设置。因此,我们提出分裂归一化既可以作为上下文效应的神经计算解释,也可以作为应用研究人员的有用经验工具。
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引用次数: 2
期刊
Journal of Mathematical Psychology
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