15,544 research outputs found
Wholeness as a Hierarchical Graph to Capture the Nature of Space
According to Christopher Alexander's theory of centers, a whole comprises
numerous, recursively defined centers for things or spaces surrounding us.
Wholeness is a type of global structure or life-giving order emerging from the
whole as a field of the centers. The wholeness is an essential part of any
complex system and exists, to some degree or other, in spaces. This paper
defines wholeness as a hierarchical graph, in which individual centers are
represented as the nodes and their relationships as the directed links. The
hierarchical graph gets its name from the inherent scaling hierarchy revealed
by the head/tail breaks, which is a classification scheme and visualization
tool for data with a heavy-tailed distribution. We suggest that (1) the degrees
of wholeness for individual centers should be measured by PageRank (PR) scores
based on the notion that high-degree-of-life centers are those to which many
high-degree-of-life centers point, and (2) that the hierarchical levels, or the
ht-index of the PR scores induced by the head/tail breaks can characterize the
degree of wholeness for the whole: the higher the ht-index, the more life or
wholeness in the whole. Three case studies applied to the Alhambra building
complex and the street networks of Manhattan and Sweden illustrate that the
defined wholeness captures fairly well human intuitions on the degree of life
for the geographic spaces. We further suggest that the mathematical model of
wholeness be an important model of geographic representation, because it is
topological oriented that enables us to see the underlying scaling structure.
The model can guide geodesign, which should be considered as the
wholeness-extending transformations that are essentially like the unfolding
processes of seeds or embryos, for creating beautiful built and natural
environments or with a high degree of wholeness.Comment: 14 pages, 7 figures, 2 table
One-class classifiers based on entropic spanning graphs
One-class classifiers offer valuable tools to assess the presence of outliers
in data. In this paper, we propose a design methodology for one-class
classifiers based on entropic spanning graphs. Our approach takes into account
the possibility to process also non-numeric data by means of an embedding
procedure. The spanning graph is learned on the embedded input data and the
outcoming partition of vertices defines the classifier. The final partition is
derived by exploiting a criterion based on mutual information minimization.
Here, we compute the mutual information by using a convenient formulation
provided in terms of the -Jensen difference. Once training is
completed, in order to associate a confidence level with the classifier
decision, a graph-based fuzzy model is constructed. The fuzzification process
is based only on topological information of the vertices of the entropic
spanning graph. As such, the proposed one-class classifier is suitable also for
data characterized by complex geometric structures. We provide experiments on
well-known benchmarks containing both feature vectors and labeled graphs. In
addition, we apply the method to the protein solubility recognition problem by
considering several representations for the input samples. Experimental results
demonstrate the effectiveness and versatility of the proposed method with
respect to other state-of-the-art approaches.Comment: Extended and revised version of the paper "One-Class Classification
Through Mutual Information Minimization" presented at the 2016 IEEE IJCNN,
Vancouver, Canad
Multimapper: Data Density Sensitive Topological Visualization
Mapper is an algorithm that summarizes the topological information contained
in a dataset and provides an insightful visualization. It takes as input a
point cloud which is possibly high-dimensional, a filter function on it and an
open cover on the range of the function. It returns the nerve simplicial
complex of the pullback of the cover. Mapper can be considered a discrete
approximation of the topological construct called Reeb space, as analysed in
the -dimensional case by [Carriere et al.,2018]. Despite its success in
obtaining insights in various fields such as in [Kamruzzaman et al., 2016],
Mapper is an ad hoc technique requiring lots of parameter tuning. There is also
no measure to quantify goodness of the resulting visualization, which often
deviates from the Reeb space in practice. In this paper, we introduce a new
cover selection scheme for data that reduces the obscuration of topological
information at both the computation and visualisation steps. To achieve this,
we replace global scale selection of cover with a scale selection scheme
sensitive to local density of data points. We also propose a method to detect
some deviations in Mapper from Reeb space via computation of persistence
features on the Mapper graph.Comment: Accepted at ICDM
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