45,783 research outputs found
Introduction to Persistent Homology
This video presents an introduction to persistent homology, aimed at a viewer who has mathematical aptitude but not necessarily knowledge of algebraic topology. Persistent homology is an algebraic method of discerning the topological features of complex data, which in recent years has found applications in fields such as image processing and biological systems. Using smooth animations, the video conveys the intuition behind persistent homology, while giving a taste of its key properties, applications, and mathematical underpinnings
Introduction to the R package TDA
We present a short tutorial and introduction to using the R package TDA,
which provides some tools for Topological Data Analysis. In particular, it
includes implementations of functions that, given some data, provide
topological information about the underlying space, such as the distance
function, the distance to a measure, the kNN density estimator, the kernel
density estimator, and the kernel distance. The salient topological features of
the sublevel sets (or superlevel sets) of these functions can be quantified
with persistent homology. We provide an R interface for the efficient
algorithms of the C++ libraries GUDHI, Dionysus and PHAT, including a function
for the persistent homology of the Rips filtration, and one for the persistent
homology of sublevel sets (or superlevel sets) of arbitrary functions evaluated
over a grid of points. The significance of the features in the resulting
persistence diagrams can be analyzed with functions that implement recently
developed statistical methods. The R package TDA also includes the
implementation of an algorithm for density clustering, which allows us to
identify the spatial organization of the probability mass associated to a
density function and visualize it by means of a dendrogram, the cluster tree
On topological data analysis for structural dynamics: an introduction to persistent homology
Topological methods can provide a way of proposing new metrics and methods of
scrutinising data, that otherwise may be overlooked. In this work, a method of
quantifying the shape of data, via a topic called topological data analysis
will be introduced. The main tool within topological data analysis (TDA) is
persistent homology. Persistent homology is a method of quantifying the shape
of data over a range of length scales. The required background and a method of
computing persistent homology is briefly discussed in this work. Ideas from
topological data analysis are then used for nonlinear dynamics to analyse some
common attractors, by calculating their embedding dimension, and then to assess
their general topologies. A method will also be proposed, that uses topological
data analysis to determine the optimal delay for a time-delay embedding. TDA
will also be applied to a Z24 Bridge case study in structural health
monitoring, where it will be used to scrutinise different data partitions,
classified by the conditions at which the data were collected. A metric, from
topological data analysis, is used to compare data between the partitions. The
results presented demonstrate that the presence of damage alters the manifold
shape more significantly than the effects present from temperature
Introduction to Computational Topology Using Simplicial Persistent Homology
The human mind has a natural talent for finding patterns and shapes in nature where there are none, such as constellations among the stars. Persistent homology serves as a mathematical tool for accomplishing the same task in a more formal setting, taking in a cloud of individual points and assembling them into a coherent continuous image. We present an introduction to computational topology as well as persistent homology, and use them to analyze configurations of BuckyBalls®, small magnetic balls commonly used as desk toys
On topological data analysis for structural dynamics: an introduction to persistent homology
Topological methods can provide a way of proposing new metrics and methods of scrutinizing data, that otherwise may be overlooked. A method of quantifying the shape of data, via a topic called topological data analysis (TDA) will be introduced. The main tool of TDA is persistent homology. Persistent homology is a method of quantifying the shape of data over a range of length scales. The required background and a method of computing persistent homology are briefly discussed in this work. Ideas from topological data analysis are then used for nonlinear dynamics to analyze some common attractors, by calculating their embedding dimension, and then to assess their general topologies. A method will also be proposed, that uses topological data analysis to determine the optimal delay for a time-delay embedding. TDA will also be applied to a Z24 bridge case study in structural health monitoring, where it will be used to scrutinize different data partitions, classified by the conditions at which the data were collected. A metric, from topological data analysis, is used to compare data between the partitions. The results presented demonstrate that the presence of damage alters the manifold shape more significantly than the effects present from temperature
Towards Emotion Recognition: A Persistent Entropy Application
Emotion recognition and classification is a very active area of research. In
this paper, we present a first approach to emotion classification using
persistent entropy and support vector machines. A topology-based model is
applied to obtain a single real number from each raw signal. These data are
used as input of a support vector machine to classify signals into 8 different
emotions (calm, happy, sad, angry, fearful, disgust and surprised)
Towards Emotion Recognition: A Persistent Entropy Application
Emotion recognition and classification is a very active area of research. In this paper, we present
a first approach to emotion classification using persistent entropy and support vector machines. A
topology-based model is applied to obtain a single real number from each raw signal. These data are
used as input of a support vector machine to classify signals into 8 different emotions (calm, happy,
sad, angry, fearful, disgust and surprised)
Computing Persistent Homology within Coq/SSReflect
Persistent homology is one of the most active branches of Computational
Algebraic Topology with applications in several contexts such as optical
character recognition or analysis of point cloud data. In this paper, we report
on the formal development of certified programs to compute persistent Betti
numbers, an instrumental tool of persistent homology, using the Coq proof
assistant together with the SSReflect extension. To this aim it has been
necessary to formalize the underlying mathematical theory of these algorithms.
This is another example showing that interactive theorem provers have reached a
point where they are mature enough to tackle the formalization of nontrivial
mathematical theories
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