Optimal Base Station Network Based on Topological Data Analysis

The decision of which base stations need to be removed due to the cost is always a difficult problem, because the influence on the cover rate of the network caused by the removal should be kept to a minimum. However, the common methods to solve this problem such as K-means Clustering show a low accuracy. Barcode, which belongs to TDA, has the possibility to show the result by identifying the Persistent Homology of base station network. This essay mainly illustrates the specific problem of optimal base station network, which applies the TDA(Topological Data Analysis) methods to find which base stations need removing due to the cost K-means Clustering and Topological Data Analysis methods were mainly used. With the simulated distribution of telecommunication users, K-means Clustering algorithm was used to locate 30 best base stations. By comparing the minimum distance between the results (K=25 and K=30), K-means Clustering was used again to decide base station points to be removed. Then TDA was used to select which 5 base stations should be removed through observing barcode. By repeating above steps five times, Finally the average and variance of cover area in original network, K-means Clustering and TDA were compared. The experiment showed that the average cover rate of original network was 81.20% while the result of TDA and K-means Clustering were 92.13% and 89.87%. It was proved by simulation that it is more efficient to use TDA methods to construct the optimal base station network.

Topological models for stable motivic invariants of regular number rings

For an infinity of number rings we express stable motivic invariants in terms of topological data determined by the complex numbers, the real numbers and finite fields. We use this to extend Morel’s identification of the endomorphism ring of the motivic sphere with the Grothendieck–Witt ring of quadratic forms to deeper base schemes.

Topological data analysis identifies distinct biomarker phenotypes during the inflammatory phase of COVID-19

OBJECTIVES: The relationships between baseline clinical phenotypes and the cytokine milieu of the peak inflammatory phase of coronavirus 2019 (COVID-19) are not yet well understood. We used Topological Data Analysis (TDA), a dimensionality reduction technique to identify patterns of inflammation associated with COVID-19 severity and clinical characteristics. DESIGN: Exploratory analysis from a multi-center prospective cohort study. SETTING: Eight military hospitals across the United States between April 2020 and January 2021. PATIENTS: Adult (≥18 years of age) SARS-CoV-2 positive inpatient and outpatient participants were enrolled with plasma samples selected from the putative inflammatory phase of COVID-19, defined as 15-28 days post symptom onset. INTERVENTIONS: None. MEASUREMENTS AND MAIN RESULTS: Concentrations of 12 inflammatory protein biomarkers were measured using a broad dynamic range immunoassay. TDA identified 3 distinct inflammatory protein expression clusters. Peak severity (outpatient, hospitalized, ICU admission or death), Charlson Comorbidity Index (CCI), and body mass index (BMI) were evaluated with logistic regression for associations with each cluster. The study population (n=129, 33.3% female, median 41.3 years of age) included 77 outpatient, 31 inpatient, 16 ICU-level, and 5 fatal cases. Three distinct clusters were found that differed by peak disease severity (p <0.001), age (p <0.001), BMI (p<0.001), and CCI (p=0.001). CONCLUSIONS: Exploratory clustering methods can stratify heterogeneous patient populations and identify distinct inflammation patterns associated with comorbid disease, obesity, and severe illness due to COVID-19.

The shape of cancer relapse: Topological data analysis predicts recurrence in paediatric acute lymphoblastic leukaemia

Acute Lymphoblastic Leukaemia (ALL) is the most frequent paediatric cancer. Modern therapies have improved survival rates, but approximately 15-20 % of patients relapse. At present, patients' risk of relapse are assessed by projecting high-dimensional flow cytometry data onto a subset of biomarkers and manually estimating the shape of this reduced data. Here, we apply methods from topological data analysis (TDA), which quantify shape in data via features such as connected components and loops, to pre-treatment ALL datasets with known outcomes. We combine these fully unsupervised analyses with machine learning to identify features in the pre-treatment data that are prognostic for risk of relapse. We find significant topological differences between relapsing and non-relapsing patients and confirm the predictive power of CD10, CD20, CD38, and CD45. Further, we are able to use the TDA descriptors to predict patients who relapsed. We propose three prognostic pipelines that readily extend to other haematological malignancies.

3D TOPOLOGICAL SUPPORT IN SPATIAL DATABASES: AN OVERVIEW

The storage of spatial data that consists of spatial and non-spatial properties requires a database management system that possesses spatial functions that can cater to the spatial characteristics of data. These characteristics include the geometrical shape, topological and positional information. Parallel to how geometries describe the shape of an object, topological information is also an important spatial property which describes how the geometries in a space are related to each other. This information describes the connectivity, containment and adjacencies of spatial objects which are the foundation for more complex analysis such as navigation, data reconstruction, spatial queries and others. However, the topological support provided by spatial databases varies. This paper provided an overview on the current implementations of topological support in spatial databases such as ArcGIS, QGIS, PostgreSQL and others. The native topology in most spatial databases was found to be 2D topology maintained by 2D topology rules with limited representation of 3D topological relationships. Consequently, 3D objects represented by 2D topology had to be decomposed into objects of lower dimensions. Approaches to implement additional topological support for spatial databases included the use of topological data models, data structures, operators, and rules. 3D applications such as 3D cadastre required more detailed representations of topological information which required a more comprehensive 3D topological data model. Nonetheless, comprehensive preservation of topological information also mandates voluminous storage and higher computational efficiency. Thus, the appropriate 3D topological support should be provided in spatial databases to accurately represent 3D objects and meet 3D analysis requirements.