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Visualizing the quality of dimensionality reduction
"... Abstract. Many different evaluation measures for dimensionality reduction can be summarized based on the coranking framework [6]. Here, we extend this framework in two ways: (i) we show that the current parameterization of the quality shows unpredictable behavior, even in simple settings, and we pr ..."
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Abstract. Many different evaluation measures for dimensionality reduction can be summarized based on the coranking framework [6]. Here, we extend this framework in two ways: (i) we show that the current parameterization of the quality shows unpredictable behavior, even in simple settings, and we propose a different parameterization which yields more intuitive results; (ii) we propose how to link the quality to pointwise quality measures which can directly be integrated into the visualization. 1
Data complexity measured by principal graphs
 COMPUTERS AND MATHEMATICS WITH APPLICATIONS
, 2013
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Nonlinear Quality of Life Index
, 2010
"... Abstract. We present details of the analysis of the nonlinear quality of life index for 171 countries. This index is based on four indicators: GDP per capita by Purchasing Power Parities, Life expectancy at birth, Infant mortality rate, and Tuberculosis incidence. We analyze the structure of the dat ..."
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Abstract. We present details of the analysis of the nonlinear quality of life index for 171 countries. This index is based on four indicators: GDP per capita by Purchasing Power Parities, Life expectancy at birth, Infant mortality rate, and Tuberculosis incidence. We analyze the structure of the data in order to find the optimal and independent on expert’s opinion way to map several numerical indicators from a multidimensional space onto the onedimensional space of the quality of life. In the 4D space we found a principal curve that goes “through the middle ” of the dataset and project the data points on this curve. The order along this principal curve gives us the ranking of countries. The measurement of the quality of life is very important for economic and social assessment and also for public policy, social legislation, and community programs. “There is a strong need for a systematic exploration of the content, reach, and relevance of the concept of the quality of life, and ways of making it concrete and usable ” [1]. Many of the existing indices of quality of life (for example, The Economist Intelligence Unit’s qualityoflife index and The Life Quality Index, LQI) are not free from certain problems and biases. For example, LQI uses a parameter K which cannot
A nonlinear principal component decomposition A nonlinear principal component decomposition *
"... Abstract The idea of summarizing the information contained in a large number of variables by a small number of "factors" or "principal components" has been widely adopted in economics and statistics. This paper introduces a generalization of the widely used principal component a ..."
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Abstract The idea of summarizing the information contained in a large number of variables by a small number of "factors" or "principal components" has been widely adopted in economics and statistics. This paper introduces a generalization of the widely used principal component analysis (PCA) to nonlinear settings, thus providing a new tool for dimension reduction and exploratory data analysis or representation. The distinguishing features of the method include (i) the ability to always deliver truly independent factors (as opposed to the merely uncorrelated factors of PCA); (ii) the reliance on the theory of optimal transport and Brenier maps to obtain a robust and efficient computational algorithm and (iii) the use of a new multivariate additive entropy decomposition to determine the principal nonlinear components that capture most of the information content of the data.
Geometrical Complexity of Data Approximators
"... Abstract. There are many methods developed to approximate a cloud of vectors embedded in highdimensional space by simpler objects: starting from principal points and linear manifolds to selforganizing maps, neural gas, elastic maps, various types of principal curves and principal trees, and so on. ..."
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Abstract. There are many methods developed to approximate a cloud of vectors embedded in highdimensional space by simpler objects: starting from principal points and linear manifolds to selforganizing maps, neural gas, elastic maps, various types of principal curves and principal trees, and so on. For each type of approximators the measure of the approximator complexity was developed too. These measures are necessary to find the balance between accuracy and complexity and to define the optimal approximations of a given type. We propose a measure of complexity (geometrical complexity) which is applicable to approximators of several types and which allows comparing data approximations of different types.
DistortionGuided StructureDriven Interactive Exploration of HighDimensional Data
"... Dimension reduction techniques are essential for feature selection and feature extraction of complex highdimensional data. These techniques, which construct lowdimensional representations of data, are typically geometrically motivated, computationally efficient and approximately preserve certain ..."
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Dimension reduction techniques are essential for feature selection and feature extraction of complex highdimensional data. These techniques, which construct lowdimensional representations of data, are typically geometrically motivated, computationally efficient and approximately preserve certain structural properties of the data. However, they are often used as black box solutions in data exploration and their results can be difficult to interpret. To assess the quality of these results, quality measures, such as coranking [LV09], have been proposed to quantify structural distortions that occur between highdimensional and lowdimensional data representations. Such measures could be evaluated and visualized pointwise to further highlight erroneous regions [MLGH13]. In this work, we provide an interactive visualization framework for exploring highdimensional data via its twodimensional embeddings obtained from dimension reduction, using a rich set of user interactions. We ask the following question: what new insights do we obtain regarding the structure of the data, with interactive manipulations of its embeddings in the visual space? We augment the twodimensional embeddings with structural abstractions obtained from hierarchical clusterings, to help users navigate and manipulate subsets of the data. We use pointwise distortion measures to highlight interesting regions in the domain, and further to guide our selection of the appropriate level of clusterings that are aligned with the regions of interest. Under the static setting, pointwise distortions indicate the level of structural uncertainty within the embeddings. Under the dynamic setting, onthefly updates of pointwise distortions due to data movement and data deletion reflect structural relations among different parts of the data, which may lead to new and valuable insights. 1.
Czerwinska et al. BMC Systems Biology (2015) 9:46 DOI 10.1186/s1291801501894 SOFTWARE Open Access
"... morphing datadriven and structuredriven network layouts Urszula Czerwinska1,2,3, Laurence Calzone1,2,3, Emmanuel Barillot1,2,3 and Andrei Zinovyev1,2,3* Background: Visualization and analysis of molecular profiling data together with biological networks are able to provide new mechanistic insights ..."
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morphing datadriven and structuredriven network layouts Urszula Czerwinska1,2,3, Laurence Calzone1,2,3, Emmanuel Barillot1,2,3 and Andrei Zinovyev1,2,3* Background: Visualization and analysis of molecular profiling data together with biological networks are able to provide new mechanistic insights into biological functions. Currently, it is possible to visualize highthroughput data on top of predefined network layouts, but they are not always adapted to a given data analysis task. A network layout based simultaneously on the network structure and the associated multidimensional data might be advantageous for data visualization and analysis in some cases. Results: We developed a Cytoscape app, which allows constructing biological network layouts based on the data from molecular profiles imported as values of node attributes. DeDaL is a Cytoscape 3 app, which uses linear and nonlinear algorithms of dimension reduction to produce datadriven network layouts based on multidimensional data (typically gene expression). DeDaL implements several data preprocessing and layout postprocessing steps such as continuous morphing between two arbitrary network layouts and aligning one network layout with respect to another one by rotating and mirroring. The combination of all these functionalities facilitates the creation of insightful network layouts representing both structural network features and correlation patterns in multivariate data. We demonstrate the added value of applying DeDaL in several practical applications, including an example of a large proteinprotein interaction network. Conclusions: DeDaL is a convenient tool for applying data dimensionality reduction methods and for designing insightful data displays based on datadriven layouts of biological networks, built within Cytoscape environment. DeDaL is freely available for downloading at