102 research outputs found
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
Multiple factor analysis of distributional data
In the framework of Symbolic Data Analysis (SDA), distribution-variables are
a particular case of multi-valued variables: each unit is represented by a set
of distributions (e.g. histograms, density functions or quantile functions),
one for each variable. Factor analysis (FA) methods are primary exploratory
tools for dimension reduction and visualization. In the present work, we use
Multiple Factor Analysis (MFA) approach for the analysis of data described by
distributional variables. Each distributional variable induces a set new
numeric variable related to the quantiles of each distribution. We call these
new variables as \textit{quantile variables} and the set of quantile variables
related to a distributional one is a block in the MFA approach. Thus, MFA is
performed on juxtaposed tables of quantile variables. \\ We show that the
criterion decomposed in the analysis is an approximation of the variability
based on a suitable metrics between distributions: the squared
Wasserstein distance. \\ Applications on simulated and real distributional data
corroborate the method. The interpretation of the results on the factorial
planes is performed by new interpretative tools that are related to the several
characteristics of the distributions (location, scale and shape).Comment: Accepted from STATSTICA APPLICATA: Italian Journal of Applied
Statistics on 12/201
Combining unsupervised and supervised learning techniques for enhancing the performance of functional data classifiers
This paper offers a supervised classification strategy that combines functional data analysis with unsupervised and supervised classification methods. Specifically, a two-steps classification technique for high-dimensional time series treated as functional data is suggested. The first stage is based on extracting additional knowledge from the data using unsupervised classification employing suitable metrics. The second phase applies functional supervised classification of the new patterns learned via appropriate basis representations. The experiments on ECG data and comparison with the classical approaches show the effectiveness of the proposed technique and exciting refinement in terms of accuracy. A simulation study with six scenarios is also offered to demonstrate the efficacy of the suggested strategy. The results reveal that this line of investigation is compelling and worthy of further development
Pooling random forest and functional data analysis for biomedical signals supervised classification: theory and application to electrocardiogram data
Scientific progress has contributed to creating many devices to gather vast amounts of biomedical data over time. The goal of these devices is generally to monitor people's health conditions, diagnose, and prevent patients' diseases, for example, to discover cardiovascular disorders or predict epileptic seizures. A common way of investigating these data is classification, but these instruments generate signals often characterized by high dimensionality. Learning from these data is definitely a challenging task due to many issues, for example, the trade-off between complexity and accuracy and the course of dimensionality. This study proposes a supervised classification method based on the joint use of functional data analysis, classification trees, and random forest to deal with massive biomedical data recorded over time. For this purpose, this research suggests different original tools to extract features and train functional classifiers, interpret the classification rules, assess leaves' quality and composition, avoid the classical drawbacks due to the COD, and improve the accuracy of the functional classifiers. Focusing on ECG data as a possible example, the final purpose of this study is to offer an original approach to identify and classify patients at risk using different types of biomedical signals. The results confirm that this line of research is exciting; indeed, the interpretative tools show evidence to be very useful for understanding classification rules. Furthermore, the performance of the proposed functional classifier, in terms of accuracy, is excellent because the latter breaks the previous classification record regarding a well-known ECG dataset
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
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
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