Statistical topological data analysis using persistence landscapes - PowerPoint PPT Presentation

Application Theory Analysis Statistical topological data analysis using persistence landscapes applied to brain arteries CANSSI–SAMSI Workshop: Geometric Topological and Graphical Model Methods in Statistics Peter Bubenik Department of Mathematics Cleveland State University p.bubenik@csuohio.edu http://academic.csuohio.edu/bubenik_p/ May 23, 2014 funded by AFOSR 1/20 Peter Bubenik Persistence landscapes

Application Theory Analysis Statistical topological data analysis The plan: Geometric Topological Statistical Data object summary analysis 2/20 Peter Bubenik Persistence landscapes

Application Theory Analysis Brain arteries Geometry Topology Brain arteries Joint work with Ezra Miller (Duke/SAMSI), J.S. Marron (UNC-CH), Paul Bendich (Duke) and Sean Skwerer (UNC-CH). 3/20 Peter Bubenik Persistence landscapes

Application Theory Analysis Brain arteries Geometry Topology Brain arteries Goal: Analyze the shape of brain arteries in order to understand normal changes with respect to age detect and locate pathology (tumors) predict stroke risk 3/20 Peter Bubenik Persistence landscapes

Application Theory Analysis Brain arteries Geometry Topology The data Bullitt and Aylward (2002) MRA → Tubes 4/20 Peter Bubenik Persistence landscapes

Application Theory Analysis Brain arteries Geometry Topology Filling the arteries – increasing sublevel sets 5/20 Peter Bubenik Persistence landscapes

Application Theory Analysis Brain arteries Geometry Topology Mathematical viewpoint Let X be a graph representing the brain arteries of one subject: vertices with ( x , y , z , r ) coordinates edges connecting adjacent vertices 6/20 Peter Bubenik Persistence landscapes

Application Theory Analysis Brain arteries Geometry Topology Mathematical viewpoint Let X be a graph representing the brain arteries of one subject: vertices with ( x , y , z , r ) coordinates edges connecting adjacent vertices Let X t denotes the full subgraph on the vertices with z coordinate at most t . ∅ = X 0 ⊆ X 1 ⊆ X 2 ⊆ · · · ⊆ X N = X Take homology in degree 0. H 0 ( X 0 ) → H 0 ( X 1 ) → H 0 ( X 2 ) → · · · → H 0 ( H N ) 6/20 Peter Bubenik Persistence landscapes

Application Theory Analysis Setup Persistence landscape Mean Banach space More general setup For each t , have a simplicial complex X t a vector space H ( X t ) an inclusion X t ⊆ X t ′ For t ≤ t ′ , have a linear map H ( X t ) → H ( X t ′ ) Persistent homology is the image of this map. This set of vector spaces and linear maps is called a persistence module. We want a summary of the persistence module that is amenable to statistical analysis. 7/20 Peter Bubenik Persistence landscapes

Application Theory Analysis Setup Persistence landscape Mean Banach space Persistence landscape Recall that the persistence module consisted of linear maps H ( X t ) → H ( X t ′ ) , for t ≤ t ′ . For k = 1 , 2 , 3 , . . . , define λ k : R → R by λ k ( t ) = max( h | rank( H ( X t − h ) → H ( X t + h ) ≥ k ) We can combine these to get one function λ : N × R → R , where λ ( k , t ) = λ k ( t ). 8/20 Peter Bubenik Persistence landscapes

Application Theory Analysis Setup Persistence landscape Mean Banach space Persistence landscape examples 9/20 Peter Bubenik Persistence landscapes

Application Theory Analysis Setup Persistence landscape Mean Banach space Mean landscapes n Persistence landscapes, λ (1) , . . . , λ ( n ) , have mean, λ = 1 � λ ( i ) . n i =1 That is, n λ k ( t ) = 1 λ ( i ) � k ( t ) n i =1 10/20 Peter Bubenik Persistence landscapes

Application Theory Analysis Setup Persistence landscape Mean Banach space Mean landscape for brain arteries 12 10 8 6 4 2 0 1020304050 90 100 50 60 70 80 10 20 30 40 11/20 Peter Bubenik Persistence landscapes

Application Theory Analysis Setup Persistence landscape Mean Banach space Summary space � 1 �� p � λ kp Let 1 ≤ p < ∞ . Then � λ � p = . k We assume � λ � := � λ � p < ∞ . That is, λ ∈ L p ( N × R ). So λ is a random variable with values in a Banach space. 12/20 Peter Bubenik Persistence landscapes

Application Theory Analysis Setup Persistence landscape Mean Banach space Asymptotics λ ∈ L p ( N × R ), � λ � is a real random variable. If E � λ � < ∞ then there exists E ( λ ) ∈ L p ( N × R ) such that E ( f ( λ )) = f ( E ( λ )) for all continuous linear functionals f . Theorem (Strong Law of Large Numbers (SLLN)) ( n ) → E ( λ ) almost surely if and only if E � λ � < ∞ . λ Theorem (Central Limit Theorem (CLT)) Assume p ≥ 2 . If E � λ � < ∞ and E ( � λ � 2 ) < ∞ then √ n [ λ ( n ) − E ( λ )] converges weakly to a Gaussian random variable with the same covariance structure as λ . 13/20 Peter Bubenik Persistence landscapes

Application Theory Analysis Setup Persistence landscape Mean Banach space Weighted norms � 1 �� p � λ kp Recall that � λ � p = . k � 1 j � p � � λ kp Fix i ≤ j . Define � λ � p , i , j = . k = i The previous SLLN and CLT also apply to this weighted norm. 14/20 Peter Bubenik Persistence landscapes

Statistical topological data analysis using persistence landscapes - PowerPoint PPT Presentation

Application Theory Analysis Statistical topological data analysis using persistence landscapes applied to brain arteries CANSSISAMSI Workshop: Geometric Topological and Graphical Model Methods in Statistics Peter Bubenik Department of

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