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PRINCIPAL MANIFOLDS AND GRAPHS IN PRACTICE: FROM MOLECULAR BIOLOGY TO DYNAMICAL SYSTEMS
"... We present several applications of nonlinear data modeling, using principal manifolds and principal graphs constructed using the metaphor of elasticity (elastic principal graph approach). These approaches are generalizations of the Kohonen’s selforganizing maps, a class of artificial neural networ ..."
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We present several applications of nonlinear data modeling, using principal manifolds and principal graphs constructed using the metaphor of elasticity (elastic principal graph approach). These approaches are generalizations of the Kohonen’s selforganizing maps, a class of artificial neural networks. On several examples we show advantages of using nonlinear objects for data approximation in comparison to the linear ones. We propose four numerical criteria for comparing linear and nonlinear mappings of datasets into the spaces of lower dimension. The examples are taken from comparative political science, from analysis of highthroughput data in molecular biology, from analysis of dynamical systems.
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
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.
ADAPTIVE GRAPHBASED ALGORITHMS FOR CONDITIONAL ANOMALY DETECTION AND SEMISUPERVISED LEARNING
, 2011
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Bull Math Biol DOI 10.1007/s1153801095971 ORIGINAL ARTICLE Law of the Minimum Paradoxes
, 2009
"... Abstract The “Law of the Minimum ” states that growth is controlled by the scarcest resource (limiting factor). This concept was originally applied to plant or crop growth ..."
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Abstract The “Law of the Minimum ” states that growth is controlled by the scarcest resource (limiting factor). This concept was originally applied to plant or crop growth
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