scholarly journals EXOTIC PARTICLES AND w∞-ALGEBRAS IN TWO- AND HIGH-DIMENSIONAL SPACES

2006 ◽  
Vol 21 (22) ◽  
pp. 4511-4518
Author(s):  
J. DOUARI
Keyword(s):  
Dimensional Space ◽  
High Dimensional ◽  
Two Dimensional ◽  
Exotic Particles ◽  
Translation Operators ◽  
Elementary Link

We construct a set of noncommuting translation operators in two- and high-dimensional lattices. The algebras they close are w∞-algebras. The construction is based on the introduction of noncommuting elementary link operators which link two neighborhood sites in the lattice. This type of operators preserve the braiding nature of exotic particles living basically in two-dimensional space.

scholarly journals Assessing relationships between chromatin interactions and regulatory genomic activities using the self-organizing map

2020 ◽  
Author(s):  
Timothy Kunz ◽  
Lila Rieber ◽  
Shaun Mahony
Keyword(s):  
Genome Organization ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Interaction Space ◽  
The Self ◽  
Chromatin Interaction ◽  
High Dimensional ◽  
Self Organizing Map ◽  
Two Dimensional ◽  
Chromatin Interactions

ABSTRACTFew existing methods enable the visualization of relationships between regulatory genomic activities and genome organization as captured by Hi-C experimental data. Genome-wide Hi-C datasets are often displayed using “heatmap” matrices, but it is difficult to intuit from these heatmaps which biochemical activities are compartmentalized together. High-dimensional Hi-C data vectors can alternatively be projected onto three-dimensional space using dimensionality reduction techniques. The resulting three-dimensional structures can serve as scaffolds for projecting other forms of genomic information, thereby enabling the exploration of relationships between genome organization and various genome annotations. However, while three-dimensional models are contextually appropriate for chromatin interaction data, some analyses and visualizations may be more intuitively and conveniently performed in two-dimensional space.We present a novel approach to the visualization and analysis of chromatin organization based on the Self-Organizing Map (SOM). The SOM algorithm provides a two-dimensional manifold which adapts to represent the high dimensional chromatin interaction space. The resulting data structure can then be used to assess the relationships between regulatory genomic activities and chromatin interactions. For example, given a set of genomic coordinates corresponding to a given biochemical activity, the degree to which this activity is segregated or compartmentalized in chromatin interaction space can be intuitively visualized on the 2D SOM grid and quantified using Lorenz curve analysis. We demonstrate our approach for exploratory analysis of genome compartmentalization in a high-resolution Hi-C dataset from the human GM12878 cell line. Our SOM-based approach provides an intuitive visualization of the large-scale structure of Hi-C data and serves as a platform for integrative analyses of the relationships between various genomic activities and genome organization.

scholarly journals Learning and Visualizing Chronic Latent Representations Using Electronic Health Records

2021 ◽  
Author(s):  
David Chushig-Muzo ◽  
Cristina Soguero-Ruiz ◽  
Pablo de Miguel Bohoyo ◽  
Inmaculada Mora-Jiménez
Keyword(s):  
Health Status ◽  
Electronic Health Records ◽  
Chronic Conditions ◽  
Dimensional Space ◽  
High Dimensional ◽  
Diabetic Patients ◽  
Two Dimensional ◽  
Health Records ◽  
Electronic Health ◽  
Latent Representations

Abstract Background: Nowadays, patients with chronic diseases such as diabetes and hypertension have reached alarming numbers worldwide. These diseases increase the risk of developing acute complications and involve a substantial economic burden and demand for health resources. The widespread adoption of Electronic Health Records (EHRs) is opening great opportunities for supporting decision-making. Nevertheless, data extracted from EHRs are complex (heterogeneous, high-dimensional and usually noisy), hampering the knowledge extraction with conventional approaches. Methods: We propose the use of the Denoising Autoencoder (DAE), a Machine Learning (ML) technique allowing to transform high-dimensional data into latent representations (LRs), thus addressing the main challenges with clinical data. We explore in this work how the combination of LRs with a visualization method can be used to map the patient data in a two-dimensional space, gaining knowledge about the distribution of patients with different chronic conditions. Furthermore, this representation can be also used to characterize the patient's health status evolution, which is of paramount importance in the clinical setting. Results: To obtain clinical LRs, we considered real-world data extracted from EHRs linked to the University Hospital of Fuenlabrada in Spain. Experimental results showed the great potential of DAEs to identify patients with clinical patterns linked to hypertension, diabetes and multimorbidity. The procedure allowed us to find patients with the same main chronic disease but different clinical characteristics. Thus, we identified two kinds of diabetic patients with differences in their drug therapy (insulin and non-insulin dependant), and also a group of women affected by hypertension and gestational diabetes. We also present a proof of concept for mapping the health status evolution of synthetic patients when considering the most significant diagnoses and drugs associated with chronic patients. Conclusions: Our results highlighted the value of ML techniques to extract clinical knowledge, supporting the identification of patients with certain chronic conditions. Furthermore, the patient's health status progression on the two-dimensional space might be used as a tool for clinicians aiming to characterize health conditions and identify their more relevant clinical codes.

scholarly journals Conditional t-SNE: more informative t-SNE embeddings

2020 ◽  
Author(s):  
Bo Kang ◽  
Darío García García ◽  
Jefrey Lijffijt ◽  
Raúl Santos-Rodríguez ◽  
Tijl De Bie
Keyword(s):  
Manifold Learning ◽  
Prior Information ◽  
Dimensional Space ◽  
Real Data ◽  
High Dimensional ◽  
Two Dimensional ◽  
Qualitative And Quantitative ◽  
Elegant Method ◽  
Effective Manner ◽  
Extra Parameter

AbstractDimensionality reduction and manifold learning methods such as t-distributed stochastic neighbor embedding (t-SNE) are frequently used to map high-dimensional data into a two-dimensional space to visualize and explore that data. Going beyond the specifics of t-SNE, there are two substantial limitations of any such approach: (1) not all information can be captured in a single two-dimensional embedding, and (2) to well-informed users, the salient structure of such an embedding is often already known, preventing that any real new insights can be obtained. Currently, it is not known how to extract the remaining information in a similarly effective manner. We introduce conditional t-SNE (ct-SNE), a generalization of t-SNE that discounts prior information in the form of labels. This enables obtaining more informative and more relevant embeddings. To achieve this, we propose a conditioned version of the t-SNE objective, obtaining an elegant method with a single integrated objective. We show how to efficiently optimize the objective and study the effects of the extra parameter that ct-SNE has over t-SNE. Qualitative and quantitative empirical results on synthetic and real data show ct-SNE is scalable, effective, and achieves its goal: it allows complementary structure to be captured in the embedding and provided new insights into real data.

On the Classification of Fractal and Non-Fractal Objects in Two-Dimensional Space

2020 ◽  
Vol 14 (6) ◽  
pp. 1232-1239
Author(s):  
P. M. Pustovoit ◽  
E. G. Yashina ◽  
K. A. Pshenichnyi ◽  
S. V. Grigoriev
Keyword(s):  
Dimensional Space ◽  
Two Dimensional

Political Cleavages and Political Parties

2018 ◽  
Author(s):  
Russell J. Dalton
Keyword(s):  
Political Parties ◽  
Dimensional Space ◽  
New Left ◽  
Party Systems ◽  
European Parliament ◽  
Two Dimensional ◽  
Far Right ◽  
Policy Positions ◽  
Cultural Policies ◽  
Policy Choices

This chapter uses the cleavage positions of Candidates to the European Parliament (CEPs) to as representative of their parties’ political positions. Three surveys of CEPs track the evolution of party supply in European party systems. In 1979 parties were primarily aligned along a Left–Right economic cleavage. Gradually new left and Green parties began to compete in elections and crystallized and represented liberal cultural policies. In recent decades new far-right parties arose to represent culturally conservative positions. The cross-cutting cultural cleavage has also prompted many of the established parties to alter their policy positions. In most multiparty systems, political parties now compete in a fully populated two-dimensional space. This increases the supply of policy choices for the voters. The analyses are based on the Candidates to the European Parliament Studies in 1979, 1994, and 2009.

scholarly journals Three-Dimensional Thematic Map Imaging of the Yacht Port on the Example of the Polish National Sailing Centre Marina in Gdańsk

2021 ◽  
Vol 11 (15) ◽  
pp. 7016
Author(s):  
Pawel S. Dabrowski ◽  
Cezary Specht ◽  
Mariusz Specht ◽  
Artur Makar
Keyword(s):  
Spatial Data ◽  
Convex Surface ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Future Research ◽  
Two Dimensional ◽  
Thematic Maps ◽  
Research Directions ◽  
Future Research Directions ◽  
The Many

The theory of cartographic projections is a tool which can present the convex surface of the Earth on the plane. Of the many types of maps, thematic maps perform an important function due to the wide possibilities of adapting their content to current needs. The limitation of classic maps is their two-dimensional nature. In the era of rapidly growing methods of mass acquisition of spatial data, the use of flat images is often not enough to reveal the level of complexity of certain objects. In this case, it is necessary to use visualization in three-dimensional space. The motivation to conduct the study was the use of cartographic projections methods, spatial transformations, and the possibilities offered by thematic maps to create thematic three-dimensional map imaging (T3DMI). The authors presented a practical verification of the adopted methodology to create a T3DMI visualization of the marina of the National Sailing Centre of the Gdańsk University of Physical Education and Sport (Poland). The profiled characteristics of the object were used to emphasize the key elements of its function. The results confirmed the increase in the interpretative capabilities of the T3DMI method, relative to classic two-dimensional maps. Additionally, the study suggested future research directions of the presented solution.

Neural networks trained with high-dimensional functions approximation data in high-dimensional space

2021 ◽  
pp. 1-12
Author(s):  
Jian Zheng ◽  
Jianfeng Wang ◽  
Yanping Chen ◽  
Shuping Chen ◽  
Jingjin Chen ◽  
...  
Keyword(s):  
Neural Networks ◽  
Dimensional Space ◽  
Data Distribution ◽  
High Dimensional ◽  
Sufficient Information ◽  
Sufficient Data ◽  
High Dimensional Space ◽  
Positive Effects ◽  
The Neural Networks ◽  
Using Data

Neural networks can approximate data because of owning many compact non-linear layers. In high-dimensional space, due to the curse of dimensionality, data distribution becomes sparse, causing that it is difficulty to provide sufficient information. Hence, the task becomes even harder if neural networks approximate data in high-dimensional space. To address this issue, according to the Lipschitz condition, the two deviations, i.e., the deviation of the neural networks trained using high-dimensional functions, and the deviation of high-dimensional functions approximation data, are derived. This purpose of doing this is to improve the ability of approximation high-dimensional space using neural networks. Experimental results show that the neural networks trained using high-dimensional functions outperforms that of using data in the capability of approximation data in high-dimensional space. We find that the neural networks trained using high-dimensional functions more suitable for high-dimensional space than that of using data, so that there is no need to retain sufficient data for neural networks training. Our findings suggests that in high-dimensional space, by tuning hidden layers of neural networks, this is hard to have substantial positive effects on improving precision of approximation data.

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