Modeling and replicating statistical topology, and evidence for CMB non-homogeneity

Under the banner of `Big Data', the detection and classification of structure in extremely large, high dimensional, data sets, is, one of the central statistical challenges of our times. Among the most intriguing approaches to this challenge is `TDA', or `Topological Data Analysis', one of the primary aims of which is providing non-metric, but topologically informative, pre-analyses of data sets which make later, more quantitative analyses feasible. While TDA rests on strong mathematical foundations from Topology, in applications it has faced challenges due to an inability to handle issues of statistical reliability and robustness and, most importantly, in an inability to make scientific claims with verifiable levels of statistical confidence. We propose a methodology for the parametric representation, estimation, and replication of persistence diagrams, the main diagnostic tool of TDA. The power of the methodology lies in the fact that even if only one persistence diagram is available for analysis -- the typical case for big data applications -- replications can be generated to allow for conventional statistical hypothesis testing. The methodology is conceptually simple and computationally practical, and provides a broadly effective statistical procedure for persistence diagram TDA analysis. We demonstrate the basic ideas on a toy example, and the power of the approach in a novel and revealing analysis of CMB non-homogeneity

Homology reveals significant anisotropy in the cosmic microwave background

We test the tenet of statistical isotropy of the standard cosmological model via a homology analysis of the cosmic microwave background temperature maps. Examining small sectors of the normalized maps, we find that the results exhibit a dependence on whether we compute the mean and variance locally from the masked patch, or from the full masked sky. Assigning local mean and variance for normalization, we find the maximum discrepancy between the data and model in the galactic northern hemisphere at more than $3.5$ s.d. for the PR4 dataset at degree-scale. For the PR3 dataset, the C-R and SMICA maps exhibit higher significance than the PR4 dataset at $\sim 4$ and $4.1$ s.d. respectively, however the NILC and SEVEM maps exhibit lower significance at $\sim 3.4$ s.d. The southern hemisphere exhibits high degree of consistency between the data and the model for both the PR4 and PR3 datasets. Assigning the mean and variance of the full masked sky decreases the significance for the northern hemisphere, the tails in particular. However the tails in the southern hemisphere are strongly discrepant at more than $4$ standard deviations at approximately $5$ degrees. The $p$-values obtained from the $\chi^2$-statistic exhibit commensurate significance in both the experiments. Examining the quadrants of the sphere, we find the first quadrant to be the major source of the discrepancy. Prima-facie, the results indicate a breakdown of statistical isotropy in the CMB maps, however more work is needed to ascertain the source of the anomaly. Regardless, these map characteristics may have serious consequences for downstream computations such as parameter estimation, and the related Hubble tension.Comment: 13 pages, 10 figures, 3 table

Response of a galactic disc to vertical perturbations:Strong dependence on density distribution

We study the self-consistent, linear response of a galactic disc to non-axisymmetric perturbations in the vertical direction as due to a tidal encounter, and show that the density distribution near the disc mid-plane has a strong impact on the radius beyond which distortions like warps develop. The self-gravity of the disc resists distortion in the inner parts. Applying this approach to a galactic disc with an exponential vertical profile, Saha & Jog showed that warps develop beyond 4-6 disc scalelengths, which could hence be only seen in HI. The real galactic discs, however, have less steep vertical density distributions that lie between a sech and an exponential profile. Here we calculate the disc response for such a general sech^(2/n) density distribution, and show that the warps develop from a smaller radius of 2-4 disc scalelengths. This naturally explains why most galaxies show stellar warps that start within the optical radius. Thus a qualitatively different picture of ubiquitous optical warps emerges for the observed less-steep density profiles. The surprisingly strong dependence on the density profile is due to the fact that the disc self-gravity depends crucially on its mass distribution close to the mid-plane. General results for the radius of onset of warps, obtained as a function of the disc scalelength and the vertical scaleheight, are presented as contour plots which can be applied to any galaxy.Comment: 11 pages, 7 figures. Accepted for publication in MNRAS

Cross-Geography Generalization of Machine Learning Methods for Classification of Flooded Regions in Aerial Images

Identification of regions affected by floods is a crucial piece of information required for better planning and management of post-disaster relief and rescue efforts. Traditionally, remote sensing images are analysed to identify the extent of damage caused by flooding. The data acquired from sensors onboard earth observation satellites are analyzed to detect the flooded regions, which can be affected by low spatial and temporal resolution. However, in recent years, the images acquired from Unmanned Aerial Vehicles (UAVs) have also been utilized to assess post-disaster damage. Indeed, a UAV based platform can be rapidly deployed with a customized flight plan and minimum dependence on the ground infrastructure. This work proposes two approaches for identifying flooded regions in UAV aerial images. The first approach utilizes texture-based unsupervised segmentation to detect flooded areas, while the second uses an artificial neural network on the texture features to classify images as flooded and non-flooded. Unlike the existing works where the models are trained and tested on images of the same geographical regions, this work studies the performance of the proposed model in identifying flooded regions across geographical regions. An F1-score of 0.89 is obtained using the proposed segmentation-based approach which is higher than existing classifiers. The robustness of the proposed approach demonstrates that it can be utilized to identify flooded regions of any region with minimum or no user intervention

Felix:A Topology Based Framework for Visual Exploration of Cosmic Filaments

The large-scale structure of the universe is comprised of virialized blob-like clusters, linear filaments, sheet-like walls and huge near empty three-dimensional voids. Characterizing the large scale universe is essential to our understanding of the formation and evolution of galaxies. The density range of clusters, walls and voids are relatively well separated, when compared to filaments, which span a relatively larger range. The large scale filamentary network thus forms an intricate part of the cosmic web. In this paper, we describe Felix, a topology based framework for visual exploration of filaments in the cosmic web. The filamentary structure is represented by the ascending manifold geometry of the 2-saddles in the Morse-Smale complex of the density field. We generate a hierarchy of Morse-Smale complexes and query for filaments based on the density ranges at the end points of the filaments. The query is processed efficiently over the entire hierarchical Morse-Smale complex, allowing for interactive visualization. We apply Felix to computer simulations based on the heuristic Voronoi kinematic model and the standard LCDM cosmology, and demonstrate its usefulness through two case studies. First, we extract cosmic filaments within and across cluster like regions in Voronoi kinematic simulation datasets. We demonstrate that we produce similar results to existing structure finders. Second, we extract different classes of filaments based on their density characteristics from the LCDM simulation datasets. Filaments that form the spine of the cosmic web, which exist in high density regions in the current epoch, are isolated using Felix. Also, filaments present in void-like regions are isolated and visualized. These filamentary structures are often over shadowed by higher density range filaments and are not easily characterizable and extractable using other filament extraction methodologies

Felix:A Topology Based Framework for Visual Exploration of Cosmic Filaments

The large-scale structure of the universe is comprised of virialized blob-like clusters, linear filaments, sheet-like walls and huge near empty three-dimensional voids. Characterizing the large scale universe is essential to our understanding of the formation and evolution of galaxies. The density range of clusters, walls and voids are relatively well separated, when compared to filaments, which span a relatively larger range. The large scale filamentary network thus forms an intricate part of the cosmic web. In this paper, we describe Felix, a topology based framework for visual exploration of filaments in the cosmic web. The filamentary structure is represented by the ascending manifold geometry of the 2-saddles in the Morse-Smale complex of the density field. We generate a hierarchy of Morse-Smale complexes and query for filaments based on the density ranges at the end points of the filaments. The query is processed efficiently over the entire hierarchical Morse-Smale complex, allowing for interactive visualization. We apply Felix to computer simulations based on the heuristic Voronoi kinematic model and the standard LCDM cosmology, and demonstrate its usefulness through two case studies. First, we extract cosmic filaments within and across cluster like regions in Voronoi kinematic simulation datasets. We demonstrate that we produce similar results to existing structure finders. Second, we extract different classes of filaments based on their density characteristics from the LCDM simulation datasets. Filaments that form the spine of the cosmic web, which exist in high density regions in the current epoch, are isolated using Felix. Also, filaments present in void-like regions are isolated and visualized. These filamentary structures are often over shadowed by higher density range filaments and are not easily characterizable and extractable using other filament extraction methodologies