Comparison is a core task in visual analysis. Although there are numerous guidelines to help users design effective visualizations to aid known comparison tasks, there are few formalisms that define the semantics of comparison operations in a way that can serve as the basis for a grammar of comparison interactions. Recent work proposed a formalism called View Composition Algebra (VCA) that enables ad hoc comparisons between any combination of marks, trends, or charts in a visualization interface. However, VCA limits comparisons to visual representations of data that have an identical schema, or where the schemas form a strict subset relationship (e.g., comparing price per state with price, but not with price per county). In contrast, the majority of real-world data - temporal, geographical, organizational - are hierarchical. To bridge this gap, this paper presents an extension to VCA (called VCAH) that enables ad hoc comparisons between visualizations of hierarchical data. VCAH leverages known hierarchical relationships to enable ad hoc comparison of data at different hierarchical granularities. We illustrate applications to hierarchical and Tableau visualizations.

翻译：比较是视觉分析的一项核心任务。虽然有许多准则帮助用户设计有效的可视化来帮助已知的比较任务,但很少有形式主义来界定比较作业的语义,以便作为比较相互作用的语法基础。最近的工作提出了一种形式主义,称为View Constication Algebra(VCA),它使得能够对视觉化界面中任何标记、趋势或图表的组合进行临时比较。然而,VCA将比较限制在具有相同系统或系统形成严格分级关系的数据的直观表述上(例如,将每一州的价格与价格进行比较,而不是与每个州的价格进行比较)。相比之下,大多数真实世界数据――时间、地理、组织――是等级分级的。为了缩小这一差距,本文件将扩展至VCA(称为VCAHA),它使得对等级数据的各种可视化进行临时比较。VCAHA利用已知的等级关系,以便能够对不同等级粒子的数据进行临时比较。我们举例说明了对等级和表层可视化的应用。