An Introduction to Topological Data Analysis
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1 An Introduction to Topological Data Analysis Elizabeth Munch Duke University :: Dept of Mathematics Tuesday, June 11, 2013 Elizabeth Munch (Duke) CompTop-SUNYIT Tuesday, June 11, / 43
2 There is no branch of mathematics, however abstract, which may not some day be applied to phenomena of the real world. -Nikolai Lobachevsky Elizabeth Munch (Duke) CompTop-SUNYIT Tuesday, June 11, / 43
3 Finding Structure in Data Sets Elizabeth Munch (Duke) CompTop-SUNYIT Tuesday, June 11, / 43
4 Outline 1 Homology & Persistent Homology 2 Statistics 3 Behavioral Clustering and Future Work Elizabeth Munch (Duke) CompTop-SUNYIT Tuesday, June 11, / 43
5 Outline 1 Homology & Persistent Homology 2 Statistics 3 Behavioral Clustering and Future Work Elizabeth Munch (Duke) CompTop-SUNYIT Tuesday, June 11, / 43
6 Homology & Persistent Homology What is Homology? A topological invariant which assigns a sequence of vector spaces, H k (X ), to a given topological space X. Elizabeth Munch (Duke) CompTop-SUNYIT Tuesday, June 11, / 43
7 Homology & Persistent Homology Elizabeth Munch (Duke) CompTop-SUNYIT Tuesday, June 11, / 43
8 Homology & Persistent Homology Elizabeth Munch (Duke) CompTop-SUNYIT Tuesday, June 11, / 43
9 Homology & Persistent Homology Elizabeth Munch (Duke) CompTop-SUNYIT Tuesday, June 11, / 43
10 Homology & Persistent Homology Elizabeth Munch (Duke) CompTop-SUNYIT Tuesday, June 11, / 43
11 Homology & Persistent Homology Elizabeth Munch (Duke) CompTop-SUNYIT Tuesday, June 11, / 43
12 Homology & Persistent Homology Elizabeth Munch (Duke) CompTop-SUNYIT Tuesday, June 11, / 43
13 Homology & Persistent Homology What is Persistent Homology? A way to watch how the homology of a filtration (sequence) of topological spaces changes so that we can understand something about the space. Elizabeth Munch (Duke) CompTop-SUNYIT Tuesday, June 11, / 43
14 Homology & Persistent Homology What is Persistent Homology? A way to watch how the homology of a filtration (sequence) of topological spaces changes so that we can understand something about the space. Elizabeth Munch (Duke) CompTop-SUNYIT Tuesday, June 11, / 43
15 Homology & Persistent Homology What is Persistent Homology? A way to watch how the homology of a filtration (sequence) of topological spaces changes so that we can understand something about the space. Elizabeth Munch (Duke) CompTop-SUNYIT Tuesday, June 11, / 43
16 Homology & Persistent Homology What is Persistent Homology? A way to watch how the homology of a filtration (sequence) of topological spaces changes so that we can understand something about the space. Elizabeth Munch (Duke) CompTop-SUNYIT Tuesday, June 11, / 43
17 Homology & Persistent Homology What is Persistent Homology? A way to watch how the homology of a filtration (sequence) of topological spaces changes so that we can understand something about the space. Elizabeth Munch (Duke) CompTop-SUNYIT Tuesday, June 11, / 43
18 Homology & Persistent Homology What is Persistent Homology? A way to watch how the homology of a filtration (sequence) of topological spaces changes so that we can understand something about the space. Elizabeth Munch (Duke) CompTop-SUNYIT Tuesday, June 11, / 43
19 Homology & Persistent Homology What is Persistent Homology? A way to watch how the homology of a filtration (sequence) of topological spaces changes so that we can understand something about the space. Elizabeth Munch (Duke) CompTop-SUNYIT Tuesday, June 11, / 43
20 Homology & Persistent Homology What is Persistent Homology? A way to watch how the homology of a filtration (sequence) of topological spaces changes so that we can understand something about the space. Elizabeth Munch (Duke) CompTop-SUNYIT Tuesday, June 11, / 43
21 Filtration Given topological space K and filtration K 0 K 1 K 2 K n gives a sequence of maps on homology H 2 (K 0 ) H 2 (K 1 ) H 2 (K 2 ) H 2 (K n ) Elizabeth Munch (Duke) CompTop-SUNYIT Tuesday, June 11, / 43
22 Birth and Death Birth A class γ H p (K i ) is born at K i if it is not in the image of H p (K i 1 ) H p (K i ) Death A class γ dies entering K j if merges with an older class. Elizabeth Munch (Duke) CompTop-SUNYIT Tuesday, June 11, / 43
23 Simplicial Complex Simplices Simplicial Complex Elizabeth Munch (Duke) CompTop-SUNYIT Tuesday, June 11, / 43
24 Boundary Matrix Data Structure Elizabeth Munch (Duke) CompTop-SUNYIT Tuesday, June 11, / 43
25 Large Data Sets Elizabeth Munch (Duke) CompTop-SUNYIT Tuesday, June 11, / 43
26 Large Data Sets Elizabeth Munch (Duke) CompTop-SUNYIT Tuesday, June 11, / 43
27 Large Data Sets Elizabeth Munch (Duke) CompTop-SUNYIT Tuesday, June 11, / 43
28 Large Data Sets Elizabeth Munch (Duke) CompTop-SUNYIT Tuesday, June 11, / 43
29 Rips complex vs Čech complex Čech Complex Set of points χ R n Čech complex C σ C Br (V i ) Rips Complex Set of points χ R n Rips complex R σ R v i v j r for all v i, v j σ Elizabeth Munch (Duke) CompTop-SUNYIT Tuesday, June 11, / 43
30 Persistence Algorithm Elizabeth Munch (Duke) CompTop-SUNYIT Tuesday, June 11, / 43
31 Persistence Algorithm R = D V = I for j = 1 to m: while j 0 < j with low(j 0 ) = low(j): add column j 0 to column j Elizabeth Munch (Duke) CompTop-SUNYIT Tuesday, June 11, / 43
32 Visualizing Birth and Death Elizabeth Munch (Duke) CompTop-SUNYIT Tuesday, June 11, / 43
33 Large Data Sets Elizabeth Munch (Duke) CompTop-SUNYIT Tuesday, June 11, / 43
34 Wasserstein Distance on D p c y d b z x a p-wasserstein distance for diagrams Given diagrams X and Y, the distance between them is ( W p [L q ](X, Y ) = inf ϕ:x Y x X ( x ϕ(x) q ) p ) 1/p. Elizabeth Munch (Duke) CompTop-SUNYIT Tuesday, June 11, / 43
35 Stability Theorem (Cohen-Steiner, Edelsbrunner, Harer, Mileyko) Let X be a triangulable metric space that implies bounded degree k total persistence, for k 1, and let f, g : X R be two tame Lipschitz functions. Then W p (f, g) C 1/p f g 1 k/p for all p k, where C = C X max{lip(f ) k, Lip(g) k }. Elizabeth Munch (Duke) CompTop-SUNYIT Tuesday, June 11, / 43
36 Properties of D p Mileyko, Mukherjee, Harer Complete Separable Characterization of Compact Sets Turner, Mileyko, Mukherjee, Harer Non-negatively curved Alexandrov space Geodesics Elizabeth Munch (Duke) CompTop-SUNYIT Tuesday, June 11, / 43
37 1 Homology & Persistent Homology 2 Statistics 3 Behavioral Clustering and Future Work Elizabeth Munch (Duke) CompTop-SUNYIT Tuesday, June 11, / 43
38 Statistics Elizabeth Munch (Duke) CompTop-SUNYIT Tuesday, June 11, / 43
39 Statistics Elizabeth Munch (Duke) CompTop-SUNYIT Tuesday, June 11, / 43
40 Statistics How do we give a summary of the data? Will it play nicely with time varying persistence diagrams? Elizabeth Munch (Duke) CompTop-SUNYIT Tuesday, June 11, / 43
41 Fréchet mean Fréchet mean Given a probability space (D p, B(D p ), P), the quantity [ ] Var P = inf F P (X ) = W p (X, Y ) 2 dp(y ) < X D p D p is the Fréchet variance of P. The set at which the value is obtained E(P) = {X F P (X ) = VarP} is the Fréchet expectation, also called Fréchet mean. Elizabeth Munch (Duke) CompTop-SUNYIT Tuesday, June 11, / 43
42 Properties of the Fréchet mean on D p Theorem (Mileyko et al.) Let P be a probability measure on (D p, B(D p )) with a finite second moment. If P has compact support, then E(P). Theorem (Mileyko et al.) Let P be a tight probability measure on (D p, B(D p )) with the rate of decay at infinity q > max{2, p}. Then E(P). Elizabeth Munch (Duke) CompTop-SUNYIT Tuesday, June 11, / 43
43 Goal Given diagrams X 1,, X N, find a diagram Y which minimizes the Fréchet function. Elizabeth Munch (Duke) CompTop-SUNYIT Tuesday, June 11, / 43
44 Example b x f z c y g h a Elizabeth Munch (Duke) CompTop-SUNYIT Tuesday, June 11, / 43
45 Example Elizabeth Munch (Duke) CompTop-SUNYIT Tuesday, June 11, / 43
46 Algorithm for Computation - Outline Algorithm - Kate Turner et al. 1 Persistence Diagrams X 1,, X N 2 Pick one at random to start, b x f z c Y = X i 3 Repeat: Find best matching for Wasserstein distance W p (Y, X i ) Create matching G with a with a selection for each point y j Y with the points x i X i for each i paired a h y g with y j Replace Y with mean X (G) Elizabeth Munch (Duke) CompTop-SUNYIT Tuesday, June 11, / 43
47 Algorithm for Computation - Outline b x f z c Theorem (Turner et al.) y g The algorithm terminates at a local minimum of the Fréchet function. h a Elizabeth Munch (Duke) CompTop-SUNYIT Tuesday, June 11, / 43
48 The Problem with Pointwise Fréchet means 1 b a 2 Elizabeth Munch (Duke) CompTop-SUNYIT Tuesday, June 11, / 43
49 The Problem with Pointwise Fréchet means 1 u a x y b v 2 Elizabeth Munch (Duke) CompTop-SUNYIT Tuesday, June 11, / 43
50 The Solution Instead of a single diagram, return a distribution on diagrams! Elizabeth Munch (Duke) CompTop-SUNYIT Tuesday, June 11, / 43
51 The Solution Instead of a single diagram, return a distribution on diagrams! Main Idea: 1 Perturb the diagrams and compute the matching. Elizabeth Munch (Duke) CompTop-SUNYIT Tuesday, June 11, / 43
52 The Solution Instead of a single diagram, return a distribution on diagrams! Main Idea: 1 Perturb the diagrams and compute the matching. 2 Do this repeatedly to get a distribution on matchings. Elizabeth Munch (Duke) CompTop-SUNYIT Tuesday, June 11, / 43
53 The Solution Instead of a single diagram, return a distribution on diagrams! Main Idea: 1 Perturb the diagrams and compute the matching. 2 Do this repeatedly to get a distribution on matchings. 3 Associate the probability of getting a particular matching to the mean of the matching. Elizabeth Munch (Duke) CompTop-SUNYIT Tuesday, June 11, / 43
54 The Solution Instead of a single diagram, return a distribution on diagrams! Main Idea: 1 Perturb the diagrams and compute the matching. 2 Do this repeatedly to get a distribution on matchings. 3 Associate the probability of getting a particular matching to the mean of the matching. Limit to S M,K D p : inside triangle of height M with at most K off-diagonal points. Elizabeth Munch (Duke) CompTop-SUNYIT Tuesday, June 11, / 43
55 The Drawing Procedure - Make X i Pick α Pick distribution η x with mean at x and support contained in B α (x). For each x X i, make X i by: 1 Draw point from η x 2 If contained in B x (x), add it to X i. Elizabeth Munch (Duke) CompTop-SUNYIT Tuesday, June 11, / 43
56 The Drawing Procedure - Determine Matching b x f z c y g h a Elizabeth Munch (Duke) CompTop-SUNYIT Tuesday, June 11, / 43
57 The Drawing Procedure - Determine Matching b' f ' x' z' h' y' g' Elizabeth Munch (Duke) CompTop-SUNYIT Tuesday, June 11, / 43
58 The Drawing Procedure - Determine Matching b' f ' x' z' d d d 1 b x f 2 y g 3 h. 4 z y' g' h' Elizabeth Munch (Duke) CompTop-SUNYIT Tuesday, June 11, / 43
59 The Drawing Procedure - Determine Matching b x f z c d d d 1 b x f 2 y g 3 h. 4 z y g a h d d d 1 b x f 2 y g 3 h 4 z. 5 a 6 c Elizabeth Munch (Duke) CompTop-SUNYIT Tuesday, June 11, / 43
60 The Drawing Procedure - Create Distribution 1 b Definition a 2 µ X = G P(G) δ meanx (G) Elizabeth Munch (Duke) CompTop-SUNYIT Tuesday, June 11, / 43
61 The Drawing Procedure - Create Distribution 1 x b u v Definition a y 2 µ X = G P(G) δ meanx (G) Elizabeth Munch (Duke) CompTop-SUNYIT Tuesday, June 11, / 43
62 Continuity Theorem (Munch et al) The map is continuous. S M,K S M,K P(S M,K ) X 1,, X N µ X Elizabeth Munch (Duke) CompTop-SUNYIT Tuesday, June 11, / 43
63 Examples Elizabeth Munch (Duke) CompTop-SUNYIT Tuesday, June 11, / 43
64 1 Homology & Persistent Homology 2 Statistics 3 Behavioral Clustering and Future Work Elizabeth Munch (Duke) CompTop-SUNYIT Tuesday, June 11, / 43
65 How do we automate behavioral analysis? Image: Wikipedia Elizabeth Munch (Duke) CompTop-SUNYIT Tuesday, June 11, / 43
66 Behavior Vectors Main Idea: Quantify a particular behavior as a vector in R D and cluster these points. Elizabeth Munch (Duke) CompTop-SUNYIT Tuesday, June 11, / 43
67 Behavior Vectors Main Idea: Quantify a particular behavior as a vector in R D and cluster these points. Example: Loop time vector: Put an entry into a list for each loop in a track, sort, and compare. Elizabeth Munch (Duke) CompTop-SUNYIT Tuesday, June 11, / 43
68 Behavior Vectors Main Idea: Quantify a particular behavior as a vector in R D and cluster these points. Example: Loop time vector: Put an entry into a list for each loop in a track, sort, and compare. Black: (6.52, 0, 0, 0) Purple: (6.63, 0, 0, 0) Orange: (5.01, 4.85, 0, 0) Red: (8.20, 6.46, 0, 0) Blue: (21.17, 17.60, 13.26, 7.54) Green: (15.40, 11.78, 8.28, 7.12) Elizabeth Munch (Duke) CompTop-SUNYIT Tuesday, June 11, / 43
69 Dendrograms Elizabeth Munch (Duke) CompTop-SUNYIT Tuesday, June 11, / 43
70 Experimental Results Experiment Tracks generated using open-source SUMO software. Additional tracks forced to make small circles, larger circles, and figure eights. Loops computed by: 1 checking for crossings, looking at time needed to complete loop. 2 computing persistence, using lifetime of class as score. Elizabeth Munch (Duke) CompTop-SUNYIT Tuesday, June 11, / 43
71 Experimental Results - NonPersistence Random Small Loop Large Loop Figure 8 Elizabeth Munch (Duke) CompTop-SUNYIT Tuesday, June 11, / 43
72 Experimental Results - Persistence Random Small Loop Large Loop Figure 8 Elizabeth Munch (Duke) CompTop-SUNYIT Tuesday, June 11, / 43
73 Group Formation Image: Wikipedia Elizabeth Munch (Duke) CompTop-SUNYIT Tuesday, June 11, / 43
74 Acknowledgements Funding: DDR&E Honors: Jo Rae Wright Fellowship for Outstanding Women in Science, Collaborators: John Harer Paul Bendich Kate Turner (UChicago) Sayan Mukherjee Jonathan Mattingly Ingrid Daubachies Robert Calderbank Peter Sang Chin (APL/JHU) Elizabeth Munch (Duke) CompTop-SUNYIT Tuesday, June 11, / 43
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