Dimensionality reduction is a common method for analyzing and visualizing high-dimensional data across domains. Dimensionality-reduction algorithms involve complex optimizations and the reduced dimensions computed by these algorithms generally lack clear relation to the initial data dimensions. Therefore, interpreting and reasoning about dimensionality reductions can be difficult. In this work, we introduce two interaction techniques, and , for reasoning dynamically about scatter plots of dimensionally reduced data. We also contribute two related visualization techniques, and to facilitate and enrich the effective use of the proposed interactions, which we integrate in a new tool called . To evaluate our techniques, we first analyze their time and accuracy performance across varying sample and dimension sizes. We then conduct a user study in which twelve data scientists use so as to assess the usefulness of the techniques in performing exploratory data analysis tasks. Results suggest that our visual interactions are intuitive and effective for exploring dimensionality reductions and generating hypotheses about the underlying data.
Prior work introduces various visualizations in planar scatter plots of DRs, in order to improve the user experience by communicating projection errors @cite_33 @cite_6 @cite_45 @cite_40 , change in dimensionality projection positions @cite_18 , data properties and clustering results @cite_6 @cite_35 , and contributions of original data dimensions in reduced dimensions @cite_12 . Low-dimensional projections are generally lossy representations of the original data relations: therefore, it is useful to convey both overall and per-point dimensionality-reduction errors to users when desired. Researchers visualized errors in DR scatter plots using Voronoi diagrams @cite_45 @cite_40 and corrected (undistorted) the errors by adjusting the projection layout with respect to the examined point @cite_33 @cite_6 .
@cite_35 @cite_18 @cite_33 @cite_6 @cite_40 @cite_45 @cite_12
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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. We introduce a set of integrated interaction techniques to interpret and interrogate dimensionality-reduced data. Projection techniques generally aim to make a high-dimensional information space visible in form of a planar layout. However, the meaning of the resulting data projections can be hard to grasp. It is seldom clear why elements are placed far apart or close together and the inevitable approximation errors of any projection technique are not exposed to the viewer. Previous research on dimensionality reduction focuses on the efficient generation of data projections, interactive customisation of the model, and comparison of different projection techniques. There has been only little research on how the visualization resulting from data projection is interacted with. We contribute the concept of probing as an integrated approach to interpreting the meaning and quality of visualizations and propose a set of interactive methods to examine dimensionality-reduced data as well as the projection itself. The methods let viewers see approximation errors, question the positioning of elements, compare them to each other, and visualize the influence of data dimensions on the projection space. We created a web-based system implementing these methods, and report on findings from an evaluation with data analysts using the prototype to examine multidimensional datasets. Multidimensional scaling is a must-have tool for visual data miners, projecting multidimensional data onto a two-dimensional plane. However, what we see is not necessarily what we think about. In many cases, end-users do not take care of scaling the projection space with respect to the multidimensional space. Anyway, when using non-linear mappings, scaling is not even possible. Yet, without scaling geometrical structures which might appear do not make more sense than considering a random map. Without scaling, we shall not make inference from the display back to the multidimensional space. No clusters, no trends, no outliers, there is nothing to infer without first quantifying the mapping quality. Several methods to qualify mappings have been devised. Here, we propose CheckViz, a new method belonging to the framework of Verity Visualization. We define a two-dimensional perceptually uniform colour coding which allows visualizing tears and false neighbourhoods, the two elementary and complementary types of geometrical mapping distortions, straight onto the map at the location where they occur. As examples shall demonstrate, this visualization method is essential to help users make sense out of the mappings and to prevent them from over interpretations. It could be applied to check other mappings as well. The visualization of continuous multi-dimensional data based on their projection to a 2-dimensional space is a way to detect visually interesting patterns, as far as the projection provides a faithful image of the original data. In order to evaluate this faithfulness, we propose to visualize any measure associated to a projected datum or to a pair of projected data, by coloring the corresponding Voronoi cell in the projection space. We also define specific measures and show how they allow estimating visually whether some part of the projection is or is not a reliable image of the original manifolds. It also helps to figure out what the original topology of the data is, telling where the high-dimensional manifolds have been torn or glued during the projection. We experiment these techniques with the principal component analysis and the curvilinear component analysis applied to artificial and real databases. SUMMARY Any matrix of rank two can be displayed as a biplot which consists of a vector for each row and a vector for each column, chosen so that any element of the matrix is exactly the inner product of the vectors corresponding to its row and to its column. If a matrix is of higher rank, one may display it approximately by a biplot of a matrix of rank two which approximates the original matrix. The biplot provides a useful tool of data analysis and allows the visual appraisal of the structure of large data matrices. It is especially revealing in principal component analysis, where the biplot can show inter-unit distances and indicate clustering of units as well as display variances and correlations of the variables. Any matrix may be represented by a vector for each row and another vector for each column, so chosen that the elements of the matrix are the inner products of the vectors representing the corresponding rows and columns. This is conceptually helpful in understanding properties of matrices. When the matrix is of rank 2 or 3, or can be closely approximated by a matrix of such rank, the vectors may be plotted or modelled and the matrix representation inspected physically. This is of obvious practical interest for the analysis of large matrices. Any n x m matrix Y of rank r can be factorized as