new dpt. of computational mathematics, science & engineering … · 2017. 3. 20. · thanks!!...
TRANSCRIPT
The shape of data
Jose PereaDpt. of Computational Mathematics, Science & Engineering (CMSE)
Dpt. of Mathematics
Model
Model Error
Model Error
Least-squaresproblem
Let’s (try to) do it again
• 7 × 7 grey-scale images depicting a single straight line
• 7 × 7 grey-scale images depicting a single straight line
• intensity-centered
• 7 × 7 grey-scale images depicting a single straight line
• intensity-centered
• subset of , via column concatenation
The shape of data
In Topology
Objects are equal up to continuous deformations:
The circle The torus
The projectiveplane
The Klein bottle
Homology of simplicial complexes
Homology of simplicial complexes
Fix a field ( e.g., , , , )
Homology of simplicial complexes
Fix a field ( e.g., , , , )
Let and a simplicial complex
Homology of simplicial complexes
Fix a field ( e.g., , , , )
Let and a simplicial complex
Homology of simplicial complexes
- th homology group of with coefficients in
Fix a field ( e.g., , , , )
Let and a simplicial complex
Homology of simplicial complexes
- th homology group of with coefficients in
Vector space over
Fix a field ( e.g., , , , )
Let and a simplicial complex
Homology of simplicial complexes
- th homology group of with coefficients in
Vector space over
Number of -dimensional -holes
Homology of simplicial complexes
- th homology group of with coefficients in
Vector space over Number of -dimensional -holes
Homology of simplicial complexes
- th homology group of with coefficients in
Vector space over Number of -dimensional -holes
# of connected components
Homology of simplicial complexes
- th homology group of with coefficients in
Vector space over Number of -dimensional -holes
# of connected components
# of 1-dimensional holes
The torus
The torus
The torus
The torusNumber of -dimensional -holes
The torusNumber of -dimensional -holes
The torusNumber of -dimensional -holes
The torusNumber of -dimensional -holes
The Klein bottle
The Klein bottle
The Klein bottle
The Klein bottleNumber of -dimensional -holes
The Klein bottleNumber of -dimensional -holes
The Klein bottleNumber of -dimensional -holes
The Klein bottleNumber of -dimensional -holes
The projective plane
Number of -dimensional -holes The projective plane
Number of -dimensional -holes The projective plane
Number of -dimensional -holes The projective plane
The Persistent Homology of Data
Barcodes: The Persistent Topology of Data, R. Ghrist, 2008
The Persistent Homology of Data:
Barcodes: The Persistent Topology of Data, R. Ghrist, 2008
The Persistent Homology of Data:
Barcodes: The Persistent Topology of Data, R. Ghrist, 2008
The Persistent Homology of Data:
Barcodes: The Persistent Topology of Data, R. Ghrist, 2008
The Persistent Homology of Data:
The Persistent Homology of Data:
Back to
The Projective Plane
Main idea
Main idea
First cohomology
group
Homotopyclasses of maps
Main idea
Lines in
Main idea
Lines in
Main idea
Main idea
Main idea
Main idea
Main idea
How we do this in practice
The Nerve Complex
• Open cover for
The Nerve Complex
• Open cover for
•
The Nerve Complex
• Open cover for
•
The Nerve Complex
• Open cover for
•
Theorem (P.)
Let be a metric space and let
be so that
If and then
is well defined and classifies the - vector bundle with transition functions
Theorem (P.)
Let be a metric space and let
be so that
If and then
is well defined and classifies the - vector bundle with transition functions
Theorem (P.)
Let be a metric space and let
be so that
If and then
is well defined and classifies the - vector bundle with transition functions
Theorem (P.)
Let be a metric space and let
be so that
If and then
is well defined and classifies the - vector bundle with transition functions
Projective coordinates
Projective coordinates for the analysis of data, J. Perea, Preprint, 2016
Projective coordinates
Steps:
1. Compute salient cohomologicalfeature of the data
2. Use bundle theory (and ….) to construct classifying map
3. Do “PCA” in projective space
Projective coordinates for the analysis of data, J. Perea, Preprint, 2016
Projective coordinates
Steps:
1. Compute salient cohomologicalfeature of the data
2. Use bundle theory (and ….) to construct classifying map
3. Do “PCA” in projective space
Projective coordinates for the analysis of data, J. Perea, Preprint, 2016
Projective coordinates
Steps:
1. Compute salient cohomologicalfeature of the data
2. Use bundle theory (and ….) to construct classifying map
3. Do “PCA” in projective space
Projective coordinates for the analysis of data, J. Perea, Preprint, 2016
Projective coordinates
Projective coordinates for the analysis of data, J. Perea, Preprint, 2016
Projective coordinates
Projective coordinates for the analysis of data, J. Perea, Preprint, 2016
Data
Suggested Model and map
Projective coordinates for the analysis of data, J. Perea, Preprint, 2016
Data
More examples
The torus
Projective coordinates for the analysis of data, J. Perea, Preprint, 2016
The torus
Projective coordinates for the analysis of data, J. Perea, Preprint, 2016
The torus
The torus
The torus
The torus
The Klein bottle
Projective coordinates for the analysis of data, J. Perea, Preprint, 2016
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The Klein bottle
Projective coordinates for the analysis of data, J. Perea, Preprint, 2016
The Klein bottle
Projective coordinates for the analysis of data, J. Perea, Preprint, 2016
Quantifying recurrence in time-evolving systems(no equations allowed)
Recording
Dynamic networkVideo dataEvolving solution (RB conv.)
4 Data modalities – 1 Theme
Problem:
Given time series data
• Determine
• Quantify
• Leverage
the global geometric/topological structure of attractors
Problem:
Given time series data
• Determine
• Quantify
• Leverage
the global geometric/topological structure of attractors
Problem:
Given time series data
• Determine
• Quantify
• Leverage
the global geometric/topological structure of attractors
Problem:
Given time series data
• Determine – generative models
• Quantify
• Leverage
the global geometric/topological structure of attractors
Problem:
Given time series data
• Determine – generative models
• Quantify – dynamic regimes
• Leverage
the global geometric/topological structure of attractors
Problem:
Given time series data
• Determine – generative models
• Quantify – dynamic regimes
• Leverage – for prediction and control
the global geometric/topological structure of attractors
Topological Time Series Analysis
Set up:
Metric space Time series
Topological Time Series Analysis
Set up:
Metric space Time series
• Real/complex numbers • Images• Functions• Graphs
Examples ( ):
Topological Time Series Analysis
Set up:
Metric space Time series
• Real/complex numbers • Images• Functions• Graphs
Examples ( ):
Topological Time Series Analysis
Set up:
Metric space Time series
Sliding window embedding:
Topological Time Series Analysis
Set up:
Metric space Time series
Sliding window embedding:
parameters
Topological Time Series Analysis
Set up:
Metric space Time series
Sliding window embedding:
parameters
Topological Time Series Analysis
Set up:
Metric space Time series
Sliding window embedding:
parameters
Topological Time Series Analysis
Set up:
Metric space Time series
Sliding window embedding:
parameters
Time Series
Time Series
Commensurate
Non-Commensurate
Sliding Window EmbeddingTime Series
Commensurate
Non-Commensurate
Sliding Window EmbeddingTime Series
Commensurate
Non-Commensurate
(joint with Chris Tralie)
(joint with Chris Tralie)
PCA of sliding window embedding
Eigenvalues
Applications
Results: Humans (Amazon turk) vs Computers
Kendall’s Tau SWCutlerDavis
FreqCutlerDavis
LatticeHumans
SW 1 -0.315 0.221 0.663
CutlerDavisFreq
1 -0.0842 -0.294
CutlerDavisLattice
1 0.347
Humans 1
Correlation of rakings (from most periodic to least periodic)
across 20 videos
(joint with Chris Tralie)
Physiology of speech Disorders
normal Mucus irregular AP-BiphonationClinical asymmetry
(joint with Chris Tralie)
Video Name Periodicity Score Quasiperiodicity Score
HerbstPeriodic 0.794 0.005
NormalPeriodic 0.561 0.010
APBiphonation 0.296 0.298
APBiphonation2 0.486 0.133
ClinicalAsymmetry 0.072 0.423
MucusPerturbedPeriodic 0.019 0.005
HerbstIrregular 0.019 0.037
(joint with Chris Tralie)
Thanks!!J. Perea and J. Harer, Sliding Windows and Persistence: An Application of Topological Methods to Signal Analysis, Foundations of Computational Mathematics, 2014.
J. Perea, A. Deckard, S. Haase and J. Harer, SW1PerS: Sliding Windows and 1-Persistence Socring; Discovering Periodicity in Time Series Data, BMC Bioinformatics, 2015.
J. Perea, Multi-Scale Projective Coordinates via Persistent Cohomology of Sparse Filtrations, arXiv, 2016.
J. Perea, Persistent Homology of Toroidal Sliding Window Embeddings, ICASSP, 2016.