Fuzzy clusters, where ambiguous samples belong to multiple clusters, are common in real-world applications. Analyzing such ambiguous samples in large-scale datasets is crucial for practical applications, such as diagnosing machine learning models. A promising method to support such analysis is through hierarchical cluster-aware grid visualizations, which offer high space efficiency and clear cluster perception. However, existing cluster-aware grid layout methods cannot clarify ambiguity among fuzzy clusters, which limits their effectiveness in fuzzy cluster analysis. To tackle this issue, we introduce a hierarchical fuzzy-cluster-aware grid layout method that supports hierarchical exploration of large-scale datasets. Throughout the hierarchical exploration, it is crucial to facilitate fuzzy cluster analysis while maintaining visual continuity for users. To achieve this, we propose a two-step optimization strategy for enhancing cluster perception, clarifying ambiguity, and preserving stability during the exploration. The first step is to create cluster-aware partitions, where each partition corresponds to a cluster. This step focuses on enhancing cluster perception and maintaining the previous shapes and positions of clusters to preserve stability at the cluster level. The second step is to generate a grid layout for each partition. In addition to placing similar samples together, this step also places ambiguous samples near the boundaries to clarify ambiguity and reveal the root causes of their occurrences and maintains the relative positions of the samples in the same cluster to preserve stability at the sample level. Several quantitative experiments and a use case are conducted to demonstrate the effectiveness and usefulness of our method in analyzing large-scale datasets, especially in fuzzy cluster analysis.

Matrix reordering permutes the rows and columns of a matrix to reveal meaningful visual patterns, such as blocks that represent clusters. A comprehensive collection of matrices, along with a scoring method for measuring the quality of visual patterns in these matrices, contributes to building a benchmark. This benchmark is essential for selecting or designing suitable reordering algorithms for revealing specific patterns. In this paper, we build a matrix-reordering benchmark, ReorderBench, with the goal of evaluating and improving matrix-reordering techniques. This is achieved by generating a large set of representative and diverse matrices and scoring these matrices with a convolution- and entropy-based method. Our benchmark contains 2,835,000 binary matrices and 5,670,000 continuous matrices, each generated to exhibit one of four visual patterns: block, off-diagonal block, star, or band, along with 450 real-world matrices featuring hybrid visual patterns. We demonstrate the usefulness of ReorderBench through three main applications in matrix reordering: 1) evaluating different reordering algorithms, 2) creating a unified scoring model to measure the visual patterns in any matrix, and 3) developing a deep learning model for matrix reordering.

Grid visualizations are widely used in many applications to visually explain a set of data and their proximity relationships. However, existing layout methods face difficulties when dealing with the inherent cluster structures within the data. To address this issue, we propose a cluster-aware grid layout method that aims to better preserve cluster structures by simultaneously considering proximity, compactness, and convexity in the optimization process. Our method utilizes a hybrid optimization strategy that consists of two phases. The global phase aims to balance proximity and compactness within each cluster, while the local phase ensures the convexity of cluster shapes. We evaluate the proposed grid layout method through a series of quantitative experiments and two use cases, demonstrating its effectiveness in preserving cluster structures and facilitating analysis tasks.

This paper investigates structural variation of agricultural productivity growth in China using a multiple-frontier approach. Agricultural productivity gain in Sichuan province is decomposed into technological progress, technical, mix and scale efficiency changes in terms of economic and environmental performance. The results show that agricultural Luenberger productivity indicator in Sichuan presents a significant increase trend with annual growth rate of 2.5% over the period of 1997-2017. The agricultural green growth in Sichuan is most driven by technological progress. Of the decline in the OI changes, over 90% results from the decline of environmental inefficiency. The technological progress is the main driven force of the economic growth while the TEC (technical inefficiency) incumbers its growth. The structural effects (mix and scale components) incumber environmental growth. The results imply that policymakers should pay more attention to new agricultural technological development and resource misallocation to improve agricultural green growth in Sichuan.

The Belt and Road Initiative (BRI) initiated by Chinese government could be regarded as a systematic framework for promoting economic cooperation and development among the countries along the Belt and Road and China. This paper attempts to analyze economic and environmental performance in 61 developing countries along Belt and Road. An additive total factor productivity growth measure allows aggregating contributions of individual countries along the BRI to construct a reasonable measure. Both desirable and undesirable outputs are considered. The growth in the total factor productivity is decomposed with respect to the economic and environmental contributions. The annual average growth rate of green productivity is 3.1% and the disparity of economic and environmental performance could be observed among countries. Some countries show robust economic growths while environmental performance slows down green growth. This indicates that developing economies should pay attention to environmental impacts and promote sustainable development by sharing emission reduction technologies.

The integration of economies always attracts much attention from policymakers and researchers. This paper introduces a novel approach to evaluate potential economic and environmental gains from integrating economies. Based on aggregate production technology and directional distance functions, we regard all decision-making units as a whole, allowing free resource reallocation among units. The level of resource misallocation is identified by a structural measure, which is obtained by the difference between overall potential improvement and individual technical inefficiency. Taking China as an empirical example, possible economic output expansions are estimated at 43.2% and 10.1% under convex and nonconvex production technologies, respectively; potential pollution reductions are around 28.4% and 5.1% under convex and nonconvex production technologies, respectively. A significant disparity of structural inefficiencies is detected, indicating a high level of resource misallocation in China. Economic cooperation is vital to promote potential green gains for all provinces in China.