Dimensionality reduction (DR) is frequently used for analyzing and visualizing high-dimensional data as it provides a good first glance of the data. However, to interpret the DR result for gaining useful insights from the data, it would take additional analysis effort such as identifying clusters and understanding their characteristics. While there are many automatic methods (e.g., density-based clustering methods) to identify clusters, effective methods for understanding a cluster's characteristics are still lacking. A cluster can be mostly characterized by its distribution of feature values. Reviewing the original feature values is not a straightforward task when the number of features is large. To address this challenge, we present a visual analytics method that effectively highlights the essential features of a cluster in a DR result. To extract the essential features, we introduce an enhanced usage of contrastive principal component analysis (cPCA). Our method, called ccPCA (contrasting clusters in PCA), can calculate each feature's relative contribution to the contrast between one cluster and other clusters. With ccPCA, we have created an interactive system including a scalable visualization of clusters' feature contributions. We demonstrate the effectiveness of our method and system with case studies using several publicly available datasets.

Dimensionality reduction (DR) methods are commonly used for analyzing and visualizing multidimensional data. However, when data is a live streaming feed, conventional DR methods cannot be directly used because of their computational complexity and inability to preserve the projected data positions at previous time points. In addition, the problem becomes even more challenging when the dynamic data records have a varying number of dimensions as often found in real-world applications. This paper presents an incremental DR solution. We enhance an existing incremental PCA method in several ways to ensure its usability for visualizing streaming multidimensional data. First, we use geometric transformation and animation methods to help preserve a viewer's mental map when visualizing the incremental results. Second, to handle data dimension variants, we use an optimization method to estimate the projected data positions, and also convey the resulting uncertainty in the visualization. We demonstrate the effectiveness of our design with two case studies using real-world datasets.

The application fields of biotechnology are as diverse as healthcare, chemistry, material science, agriculture, and environmental protection. There exist nearly 1,300 biotechnology companies in the United States, providing employment to approximately 150,000 people. Today, most of these biotechnology firms in the United States are still relatively small, with approximately two-thirds employing fewer than 150 people and are located within a cluster of biotechnology companies, for example, in the areas of

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First, we have ensured that spherical nonrotating collisionless systems collapse with almost retaining spherical configurations during initial contraction phases even if they are allowed to collapse three-dimensionally. Next, on the assumption of spherical symmetry, we examine the evolution of velocity dispersions with collapse for the systems which have uniform or power-law density profiles with Maxwellian velocity distributions by integrating the collisionless Boltzmann equation directly. The results show that as far as the initial contraction phases are concerned, the radial velocity dispersion never grows faster than the tangential velocity dispersion except at small radii where the nearly isothermal nature remains, irrespective of the density profiles and virial ratios. This implies that velocity anisotropy as an initial condition should be a poor indicator for the radial orbit instability. The growing behavior of the velocity dispersions is briefly discussed from the viewpoint that phase space density is conserved in collisionless systems.