Structural Analysis of Multivariate Point Clouds using Simplicial … · 2019. 2. 10. · Bastian...
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Structural Analysis of Multivariate Point Cloudsusing Simplicial Chains
Bastian Rieck Heike Leitte
Interdisciplinary Center for Scientific ComputingHeidelberg University
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Motivation
SPLOM
MDS Isomap
Common workflow:
Use e.g. SPLOM to make sense of data.Use dimensionality reduction methods to obtain visualizationfor exploratory data analysis.
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Motivation
SPLOM
MDS Isomap
Common workflow:Use e.g. SPLOM to make sense of data.
Use dimensionality reduction methods to obtain visualizationfor exploratory data analysis.
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![Page 4: Structural Analysis of Multivariate Point Clouds using Simplicial … · 2019. 2. 10. · Bastian RieckStructural Analysis of Multivariate Point Clouds1. Title: Structural Analysis](https://reader034.fdocuments.in/reader034/viewer/2022051910/5ffefb3c3c223548744d9472/html5/thumbnails/4.jpg)
Motivation
SPLOM
MDS Isomap
Common workflow:Use e.g. SPLOM to make sense of data.Use dimensionality reduction methods to obtain visualizationfor exploratory data analysis.
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![Page 5: Structural Analysis of Multivariate Point Clouds using Simplicial … · 2019. 2. 10. · Bastian RieckStructural Analysis of Multivariate Point Clouds1. Title: Structural Analysis](https://reader034.fdocuments.in/reader034/viewer/2022051910/5ffefb3c3c223548744d9472/html5/thumbnails/5.jpg)
Motivation
SPLOM MDS
Isomap
Common workflow:Use e.g. SPLOM to make sense of data.Use dimensionality reduction methods to obtain visualizationfor exploratory data analysis.
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![Page 6: Structural Analysis of Multivariate Point Clouds using Simplicial … · 2019. 2. 10. · Bastian RieckStructural Analysis of Multivariate Point Clouds1. Title: Structural Analysis](https://reader034.fdocuments.in/reader034/viewer/2022051910/5ffefb3c3c223548744d9472/html5/thumbnails/6.jpg)
Motivation
SPLOM MDS Isomap
Common workflow:Use e.g. SPLOM to make sense of data.Use dimensionality reduction methods to obtain visualizationfor exploratory data analysis.
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![Page 7: Structural Analysis of Multivariate Point Clouds using Simplicial … · 2019. 2. 10. · Bastian RieckStructural Analysis of Multivariate Point Clouds1. Title: Structural Analysis](https://reader034.fdocuments.in/reader034/viewer/2022051910/5ffefb3c3c223548744d9472/html5/thumbnails/7.jpg)
Beyond comparing projection scatterplots?
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Typical goalQuantify changes & differences
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Our approach
Combine geometrical (distance-preserving) and topological(structure-preserving) methods.Visualize internal connectivity of a data set as a graph
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Topology
b1 = 2, b2 = 1
Describe data sets by high-dimensional “holes”.Hole ≈ inhomogeneous region in the data.Detection using persistent homology.Description using simplicial chains
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The need for geometrical informationWithout geometrical information:
What we want
What we get
With geometrical information:
What we want What we get
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The need for geometrical informationWithout geometrical information:
What we want What we get
With geometrical information:
What we want What we get
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The need for geometrical informationWithout geometrical information:
What we want What we get
With geometrical information:
What we want
What we get
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The need for geometrical informationWithout geometrical information:
What we want What we get
With geometrical information:
What we want What we get
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Finding loops
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Localization of loops
Obtain unlocalized loops by traversal.Solve all-pairs shortest path problem.Obtain localized loops by traversal.
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The simplicial chain graphIdealized example
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The simplicial chain graphIn practice
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Data set: Voting data
Alice +1 +1 . . . +1Bob −1 −1 . . . −1Carol +1 −1 . . . 0Dave 0 0 . . . 0
Voting data from the United States House of Representatives.About 430 points of dimensions 600–900.Data sets from 1990–2011.
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Data set: Voting data
Democrats
Republicans
Republicans
Democrats
2008 2009
Boundaries between parties are defined by “dissenters” from theparty line.
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Data set: TAOTropical atmosphere ocean project
El Nino phenomenon.Continuous data stream from all buoy moorings.5-dimensional feature space: Wind velocities, humidity, airtemperature, and sea surface temperature.
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Data set: TAO
1993, 1996: No El Nino phenomenon in data set.1994–1995, 1997–1998: El Nino phenomenon.
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Lessons learned
Features obtained via persistent homology are suitable forcomparative analysis.Visualization of internal point cloud structure quickly becomestoo abstract.Thus: Aim to quantify differences in persistent homology usingwell-defined metrics.
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Summarize data even further
Data sets Persistence diagrams
Persistenthomology
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ApplicationQuantifying differences in models for multivariate data
bagged trees
regression trees
enet
enet tuned
knn
lm tuned
lm
m5
mars
plspls tuned
random forest
ridge
ridge tuned
rlm pca
rlm
rpart
svmsvm tuned
cubist
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ApplicationA framework for comparing dimensionality reduction methods
Define data descriptors on data set.Calculate their persistent homology.Compare their “topological fingerprints”.
Overall quality Worse
HLLE1.66
t-SNE2.26
Isomap2.26
PCA3.42
RP3.71
SPE10.67
Best
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Conclusion
Simplicial chain graphs as a visual metaphor for structures inmultivariate point clouds.Aspects: Comparison and quantifying differences graphically.Recent work concentrates on metric quantification usingpersistent homology.Open question: What “interesting” structures inhigh-dimensional space can we capture?
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