4,461 research outputs found
Linear regression for numeric symbolic variables: an ordinary least squares approach based on Wasserstein Distance
In this paper we present a linear regression model for modal symbolic data.
The observed variables are histogram variables according to the definition
given in the framework of Symbolic Data Analysis and the parameters of the
model are estimated using the classic Least Squares method. An appropriate
metric is introduced in order to measure the error between the observed and the
predicted distributions. In particular, the Wasserstein distance is proposed.
Some properties of such metric are exploited to predict the response variable
as direct linear combination of other independent histogram variables. Measures
of goodness of fit are discussed. An application on real data corroborates the
proposed method
Basic statistics for probabilistic symbolic variables: a novel metric-based approach
In data mining, it is usually to describe a set of individuals using some
summaries (means, standard deviations, histograms, confidence intervals) that
generalize individual descriptions into a typology description. In this case,
data can be described by several values. In this paper, we propose an approach
for computing basic statics for such data, and, in particular, for data
described by numerical multi-valued variables (interval, histograms, discrete
multi-valued descriptions). We propose to treat all numerical multi-valued
variables as distributional data, i.e. as individuals described by
distributions. To obtain new basic statistics for measuring the variability and
the association between such variables, we extend the classic measure of
inertia, calculated with the Euclidean distance, using the squared Wasserstein
distance defined between probability measures. The distance is a generalization
of the Wasserstein distance, that is a distance between quantile functions of
two distributions. Some properties of such a distance are shown. Among them, we
prove the Huygens theorem of decomposition of the inertia. We show the use of
the Wasserstein distance and of the basic statistics presenting a k-means like
clustering algorithm, for the clustering of a set of data described by modal
numerical variables (distributional variables), on a real data set. Keywords:
Wasserstein distance, inertia, dependence, distributional data, modal
variables.Comment: 19 pages, 3 figure
Visualization of AE's Training on Credit Card Transactions with Persistent Homology
Auto-encoders are among the most popular neural network architecture for
dimension reduction. They are composed of two parts: the encoder which maps the
model distribution to a latent manifold and the decoder which maps the latent
manifold to a reconstructed distribution. However, auto-encoders are known to
provoke chaotically scattered data distribution in the latent manifold
resulting in an incomplete reconstructed distribution. Current distance
measures fail to detect this problem because they are not able to acknowledge
the shape of the data manifolds, i.e. their topological features, and the scale
at which the manifolds should be analyzed. We propose Persistent Homology for
Wasserstein Auto-Encoders, called PHom-WAE, a new methodology to assess and
measure the data distribution of a generative model. PHom-WAE minimizes the
Wasserstein distance between the true distribution and the reconstructed
distribution and uses persistent homology, the study of the topological
features of a space at different spatial resolutions, to compare the nature of
the latent manifold and the reconstructed distribution. Our experiments
underline the potential of persistent homology for Wasserstein Auto-Encoders in
comparison to Variational Auto-Encoders, another type of generative model. The
experiments are conducted on a real-world data set particularly challenging for
traditional distance measures and auto-encoders. PHom-WAE is the first
methodology to propose a topological distance measure, the bottleneck distance,
for Wasserstein Auto-Encoders used to compare decoded samples of high quality
in the context of credit card transactions.Comment: arXiv admin note: substantial text overlap with arXiv:1905.0989
PHom-GeM: Persistent Homology for Generative Models
Generative neural network models, including Generative Adversarial Network
(GAN) and Auto-Encoders (AE), are among the most popular neural network models
to generate adversarial data. The GAN model is composed of a generator that
produces synthetic data and of a discriminator that discriminates between the
generator's output and the true data. AE consist of an encoder which maps the
model distribution to a latent manifold and of a decoder which maps the latent
manifold to a reconstructed distribution. However, generative models are known
to provoke chaotically scattered reconstructed distribution during their
training, and consequently, incomplete generated adversarial distributions.
Current distance measures fail to address this problem because they are not
able to acknowledge the shape of the data manifold, i.e. its topological
features, and the scale at which the manifold should be analyzed. We propose
Persistent Homology for Generative Models, PHom-GeM, a new methodology to
assess and measure the distribution of a generative model. PHom-GeM minimizes
an objective function between the true and the reconstructed distributions and
uses persistent homology, the study of the topological features of a space at
different spatial resolutions, to compare the nature of the true and the
generated distributions. Our experiments underline the potential of persistent
homology for Wasserstein GAN in comparison to Wasserstein AE and Variational
AE. The experiments are conducted on a real-world data set particularly
challenging for traditional distance measures and generative neural network
models. PHom-GeM is the first methodology to propose a topological distance
measure, the bottleneck distance, for generative models used to compare
adversarial samples in the context of credit card transactions
Topological Signals of Singularities in Ricci Flow
We implement methods from computational homology to obtain a topological
signal of singularity formation in a selection of geometries evolved
numerically by Ricci flow. Our approach, based on persistent homology, produces
precise, quantitative measures describing the behavior of an entire collection
of data across a discrete sample of times. We analyze the topological signals
of geometric criticality obtained numerically from the application of
persistent homology to models manifesting singularities under Ricci flow. The
results we obtain for these numerical models suggest that the topological
signals distinguish global singularity formation (collapse to a round point)
from local singularity formation (neckpinch). Finally, we discuss the
interpretation and implication of these results and future applications.Comment: 24 pages, 14 figure
Dynamic Clustering of Histogram Data Based on Adaptive Squared Wasserstein Distances
This paper deals with clustering methods based on adaptive distances for
histogram data using a dynamic clustering algorithm. Histogram data describes
individuals in terms of empirical distributions. These kind of data can be
considered as complex descriptions of phenomena observed on complex objects:
images, groups of individuals, spatial or temporal variant data, results of
queries, environmental data, and so on. The Wasserstein distance is used to
compare two histograms. The Wasserstein distance between histograms is
constituted by two components: the first based on the means, and the second, to
internal dispersions (standard deviation, skewness, kurtosis, and so on) of the
histograms. To cluster sets of histogram data, we propose to use Dynamic
Clustering Algorithm, (based on adaptive squared Wasserstein distances) that is
a k-means-like algorithm for clustering a set of individuals into classes
that are apriori fixed.
The main aim of this research is to provide a tool for clustering histograms,
emphasizing the different contributions of the histogram variables, and their
components, to the definition of the clusters. We demonstrate that this can be
achieved using adaptive distances. Two kind of adaptive distances are
considered: the first takes into account the variability of each component of
each descriptor for the whole set of individuals; the second takes into account
the variability of each component of each descriptor in each cluster. We
furnish interpretative tools of the obtained partition based on an extension of
the classical measures (indexes) to the use of adaptive distances in the
clustering criterion function. Applications on synthetic and real-world data
corroborate the proposed procedure
Measure based metrics for aggregated data
Aggregated data arises commonly from surveys and censuses where groups of individuals are studied as coherent entities. The aggregated data can take many forms including sets, intervals, distributions and histograms. The data analyst needs to measure the similarity between such aggregated data items and a range of metrics are reported in the literature to achieve this (e.g. the Jaccard metric for sets and the Wasserstein metric for histograms). In this paper, a unifying theory based on measure theory is developed that establishes not only that known metrics are essentially similar but also suggests new metrics
Provenance of sedimentary rocks
Understanding the origins, or provenance, of a sedimentary deposit is an important aspect of geology. Sedimentary rocks are derived from the erosion of other rocks and thus provide important records of the geological environment at the time they were deposited. Some minerals found in sedimentary rocks, such as zircon particles, can be dated using uranium-lead techniques to trace the age of their parent rock thus providing useful information about the geological environment.
Statistical and mathematical analyses that can assist in the analysis of the distribution of ages of the zircon crystals are examined. Methods of defining a difference between the distributions of ages found in rock samples are proposed, and demonstrated in the division of multiple rock samples into clusters of similar types.
A test for the existence of a cluster is developed, and statistics for comparing different rock samples examined. Estimating an accurate age for the sedimentary deposit itself proves to be difficult unless prior distributions providing significant extra information are available
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