While clustering is one of the most popular methods for data mining, analysts lack adequate tools for quick, iterative clustering analysis, which is essential for hypothesis generation and data reasoning. We introduce Clustrophile, an interactive tool for iteratively computing discrete and continuous data clusters, rapidly exploring different choices of clustering parameters, and reasoning about clustering instances in relation to data dimensions. Clustrophile combines three basic visualizations -- a table of raw datasets, a scatter plot of planar projections, and a matrix diagram (heatmap) of discrete clusterings -- through interaction and intermediate visual encoding. Clustrophile also contributes two spatial interaction techniques, @math and @math , and a visualization method, @math , for reasoning about two-dimensional projections obtained through dimensionality reductions.
 Principle Component Analysis (PCA) is a widely used mathematical technique in many fields for factor and trend analysis, dimension reduction, etc. However, it is often considered to be a "black box" operation whose results are difficult to interpret and sometimes counter-intuitive to the user. In order to assist the user in better understanding and utilizing PCA, we have developed a system that visualizes the results of principal component analysis using multiple coordinated views and a rich set of user interactions. Our design philosophy is to support analysis of multivariate datasets through extensive interaction with the PCA output. To demonstrate the usefulness of our system, we performed a comparative user study with a known commercial system, SAS INSIGHT's Interactive Data Exploration. Participants in our study solved a number of high-level analysis tasks with each interface and rated the systems on ease of learning and usefulness. Based on the participants' accuracy, speed, and qualitative feedback, we observe that our system helps users to better understand relationships between the data and the calculated eigenspace, which allows the participants to more accurately analyze the data. User feedback suggests that the interactivity and transparency of our system are the key strengths of our approach. Current implementations of multidimensional scaling (MDS), an approach that attempts to best represent data point similarity in a low-dimensional representation, are not suited for many of today's large-scale datasets. We propose an extension to the spring model approach that allows the user to interactively explore datasets that are far beyond the scale of previous implementations of MDS. We present MDSteer, a steerable MDS computation engine and visualization tool that progressively computes an MDS layout and handles datasets of over one million points. Our technique employs hierarchical data structures and progressive layouts to allow the user to steer the computation of the algorithm to the interesting areas of the dataset. The algorithm iteratively alternates between a layout stage in which a subselection of points are added to the set of active points affected by the MDS iteration, and a binning stage which increases the depth of the bin hierarchy and organizes the currently unplaced points into separate spatial regions. This binning strategy allows the user to select onscreen regions of the layout to focus the MDS computation into the areas of the dataset that are assigned to the selected bins. We show both real and common synthetic benchmark datasets with dimensionalities ranging from 3 to 300 and cardinalities of over one million points This paper introduces an approach to exploration and discovery in high-dimensional data that incorporates a user's knowledge and questions to craft sets of projection functions meaningful to them. Unlike most prior work that defines projections based on their statistical properties, our approach creates projection functions that align with user-specified annotations. Therefore, the resulting derived dimensions represent concepts defined by the user's examples. These especially crafted projection functions, or explainers, can help find and explain relationships between the data variables and user-designated concepts. They can organize the data according to these concepts. Sets of explainers can provide multiple perspectives on the data. Our approach considers tradeoffs in choosing these projection functions, including their simplicity, expressive power, alignment with prior knowledge, and diversity. We provide techniques for creating collections of explainers. The methods, based on machine learning optimization frameworks, allow exploring the tradeoffs. We demonstrate our approach on model problems and applications in text analysis. In visual analytics, sensemaking is facilitated through interactive visual exploration of data. Throughout this dynamic process, users combine their domain knowledge with the dataset to create insight. Therefore, visual analytic tools exist that aid sensemaking by providing various interaction techniques that focus on allowing users to change the visual representation through adjusting parameters of the underlying statistical model. However, we postulate that the process of sensemaking is not focused on a series of parameter adjustments, but instead, a series of perceived connections and patterns within the data. Thus, how can models for visual analytic tools be designed, so that users can express their reasoning on observations (the data), instead of directly on the model or tunable parameters? Observation level (and thus “observation”) in this paper refers to the data points within a visualization. In this paper, we explore two possible observation-level interactions, namely exploratory and expressive, within the context of three statistical methods, Probabilistic Principal Component Analysis (PPCA), Multidimensional Scaling (MDS), and Generative Topographic Mapping (GTM). We discuss the importance of these two types of observation level interactions, in terms of how they occur within the sensemaking process. Further, we present use cases for GTM, MDS, and PPCA, illustrating how observation level interaction can be incorporated into visual analytic tools.  Visual analytics emphasizes sensemaking of large, complex datasets through interactively exploring visualizations generated by statistical models. For example, dimensionality reduction methods use various similarity metrics to visualize textual document collections in a spatial metaphor, where similarities between documents are approximately represented through their relative spatial distances to each other in a 2D layout. This metaphor is designed to mimic analysts' mental models of the document collection and support their analytic processes, such as clustering similar documents together. However, in current methods, users must interact with such visualizations using controls external to the visual metaphor, such as sliders, menus, or text fields, to directly control underlying model parameters that they do not understand and that do not relate to their analytic process occurring within the visual metaphor. In this paper, we present the opportunity for a new design space for visual analytic interaction, called semantic interaction, which seeks to enable analysts to spatially interact with such models directly within the visual metaphor using interactions that derive from their analytic process, such as searching, highlighting, annotating, and repositioning documents. Further, we demonstrate how semantic interactions can be implemented using machine learning techniques in a visual analytic tool, called ForceSPIRE, for interactive analysis of textual data within a spatial visualization. Analysts can express their expert domain knowledge about the documents by simply moving them, which guides the underlying model to improve the overall layout, taking the user's feedback into account. We discuss methodology for multidimensional scaling (MDS) and its implementation in two software systems, GGvis and XGvis. MDS is a visualization technique for proximity data, that is, data in the form of N × N dissimilarity matrices. MDS constructs maps (“configurations,” “embeddings”) in IRk by interpreting the dissimilarities as distances. Two frequent sources of dissimilarities are high-dimensional data and graphs. When the dissimilarities are distances between high-dimensional objects, MDS acts as a (often nonlinear) dimension-reduction technique. When the dissimilarities are shortest-path distances in a graph, MDS acts as a graph layout technique. MDS has found recent attention in machine learning motivated by image databases (“Isomap”). MDS is also of interest in view of the popularity of “kernelizing” approaches inspired by Support Vector Machines (SVMs; “kernel PCA”).This article discusses the following general topics: (1) the stability and multiplicity of MDS solutions; (2) the analysis of struc...