Search Results (7)

When the number of subjects, n, is large, paired comparisons are often sparse. Here, we study statistical inference in a class of paired comparison models parameterized by a set of merit parameters, under an Erdös–Rényi comparison graph, where the sparsity is measured by a probability $p_n$ tending to zero. We use the moment estimation base on the scores of subjects to infer the merit parameters. We establish a unified theoretical framework in which the uniform consistency and asymptotic normality of the moment estimator hold as the number of subjects goes to infinity. A key idea for the proof of the consistency is that we obtain the convergence rate of the Newton iterative sequence for solving the estimator. We use the Thurstone model to illustrate the unified theoretical results. Further extensions to a fixed sparse comparison graph are also provided. Numerical studies and real data analysis illustrate our theoretical findings. Full article

Visual attractiveness perception—an individual’s capacity to recognise and evaluate the visual appeal of urban scene safety—has direct implications for well-being, economic vitality, and social cohesion. However, most empirical studies rely on single-source metrics or algorithm-centric pipelines that under-represent human perception. Addressing this gap, we introduce a fully reproducible, multimodal framework that measures and models this domain-specific facet of human intelligence by coupling Generative Pre-trained Transformer 4o (GPT-4o) with 1000 Street View images. The pipeline first elicits pairwise aesthetic judgements from GPT-4o, converts them into a latent attractiveness scale via Thurstone’s law of comparative judgement, and then validates the scale against 1.17 M crowdsourced ratings from MIT’s Place Pulse 2.0 benchmark (Spearman ρ = 0.76, p < 0.001). Compared with a Siamese CNN baseline (ρ = 0.60), GPT-4o yields both higher criterion validity and an 88% reduction in inference time, underscoring its superior capacity to approximate human evaluative reasoning. In this study, we introduce a standardised and reproducible streetscape evaluation pipeline using GPT-4o. We then combine the resulting attractiveness scores with network-based accessibility modelling to generate a “aesthetic–accessibility map” of urban central districts in Chongqing, China. Cluster analysis reveals four statistically distinct street types—Iconic Core, Liveable Rings, Transit-Rich but Bland, and Peripheral Low-Appeal—providing actionable insights for landscape design, urban governance, and tourism planning. Full article

We study Thurstone-motivated paired comparison models from the perspective of data evaluability, focusing on cases where datasets cannot be directly evaluated. Despite this limitation, such datasets may still contain extractable information. Three main strategies are known in the literature: increasing the number of options, inserting artificial data through perturbation methods, and requesting new real comparisons. We propose a new approach closely related to the latter two. We analyze the structure of the data and introduce the concept of the optimal limit point, related to the supremum of the log-likelihood function. We prove a theorem for determining optimal limit points based on the data structure, which characterizes the information content of the available dataset. We also prove a theorem linking optimal limit points to the limiting behavior of evaluation results obtained via perturbations, thereby explaining why different perturbation methods may yield different outcomes. In addition, we propose a new perturbation method that adds the minimum possible amount of artificial data. Furthermore, the method identifies the most informative object pairs for new real comparisons, enabling a full evaluation of the dataset. Full article

In this paper, multi-option probabilistic paired comparison models are presented and applied for prediction. As these models operate on the basis of probabilities, they can estimate the likelihood of future outcomes and thus predict future events. The aim of the paper is to demonstrate that these models have strong predictive capabilities when the information embedded into the data is properly utilized. To this end, we incorporate the degree (e.g., large or small) of the differences between the compared objects. By refining the usual three-option model, we define a five-option model capable of leveraging information derived from the goal differences. To incorporate additional information, the model is further extended to account for potential advantages in the comparisons. As a further refinement, temporal weighting is also introduced. These models are applied to forecasting football match outcomes in the top five European leagues (Premier League, La Liga, Serie A, Bundesliga, and Ligue 1), and their predictive performance is evaluated using various metrics. Based on the most recent football seasons, this model consistently delivers better predictive metrics, on average, than those of the already strong benchmark model. The effect of a home-field advantage is statistically supported across all five leagues. The model fits are illustrated using confidence intervals, and, as an interesting insight, we also present the evolution of the team strengths for the top four English clubs during the 2023/24 season. Full article

This study had two purposes: (1) to develop a forced-choice personality inventory to assess student personality characteristics based on the five-factor (FFM) personality model and (2) to examine its factor structure via the Thurstonian Item Response Theory (TIRT) approach based on Thurstone’s law of comparative judgment. A total of 200 items were generated to represent the five dimensions, and through Principal Axis Factoring and the composite reliability index, a final pool of 75 items was selected. These items were then organized into 25 blocks, each containing three statements (triplets) designed to balance social desirability across the blocks. The study involved two samples: the first sample of 1484 students was used to refine the item pool, and the second sample of 823 university students was used to examine the factorial structure of the forced-choice inventory. After re-coding the responses into a binary format, the data were analyzed within a standard structural equation modeling (SEM) framework. Then, the TIRT model was applied to evaluate the factorial structure of the forced-choice inventory, with the results indicating an adequate fit. Further suggestions for future research with additional studies are provided to justify the scale’s reliability (e.g., test–retest) and validity (e.g., concurrent, convergent, and divergent). Full article

In this paper, paired comparison models with stochastic background are investigated. We focus on the models that allow three options for choice and the parameters are estimated by maximum likelihood method. The existence and uniqueness of the estimator are key issues of the evaluation. In the case of two options, a necessary and sufficient condition is given by Ford in the Bradley–Terry model. We generalize this statement for the set of strictly log-concave distribution. Although in the case of three options the necessary and sufficient condition is not known, there are two different sufficient conditions that are formulated in the literature. In this paper, we generalize them; moreover, we compare these conditions. Their capacities to indicate the existence of the maximum were analyzed using a large number of computer simulations. These simulations support that the new condition indicates the existence of the maximum much more frequently than the previously known ones. Full article

In this paper, we will describe a topological model for elementary particles based on 3-manifolds. Here, we will use Thurston’s geometrization theorem to get a simple picture: fermions as hyperbolic knot complements (a complement $C\left(K\right)=S^3\backslash (K\times D^2)$ of a knot K carrying a hyperbolic geometry) and bosons as torus bundles. In particular, hyperbolic 3-manifolds have a close connection to number theory (Bloch group, algebraic K-theory, quaternionic trace fields), which will be used in the description of fermions. Here, we choose the description of 3-manifolds by branched covers. Every 3-manifold can be described by a 3-fold branched cover of $S^3$ branched along a knot. In case of knot complements, one will obtain a 3-fold branched cover of the 3-disk $D^3$ branched along a 3-braid or 3-braids describing fermions. The whole approach will uncover new symmetries as induced by quantum and discrete groups. Using the Drinfeld–Turaev quantization, we will also construct a quantization so that quantum states correspond to knots. Particle properties like the electric charge must be expressed by topology, and we will obtain the right spectrum of possible values. Finally, we will get a connection to recent models of Furey, Stoica and Gresnigt using octonionic and quaternionic algebras with relations to 3-braids (Bilson–Thompson model). Full article