The Intersection of Statistics and Topology:

Transcription

1 The Intersection of Statistics and Topology: Confidence Sets for Persistence Diagrams Brittany Terese Fasy joint work with S. Balakrishnan, F. Lecci, A. Rinaldo, A. Singh, L. Wasserman 3 June 2014 [FLRWBS] Statistical Inference for Persistent Homology: Confidence Sets for Persistence Diagrams. ArXiv Tentatively accepted, Annals of Statistics. B. Fasy (Tulane) Statistics and Topology 3 June / 48

2 The Intersection of Statistics and Topology: Stochastic Convergence of Persistence Landscapes and Silhouettes Brittany Terese Fasy joint work with F. Chazal, F. Lecci, B. Michel A. Rinaldo, L. Wasserman 3 June 2014 [CFLRW] Stochastic Convergence of Persistence Landscapes and Silhouettes. ArXiv. SoCG 2014 Proceedings, Kyoto. Journal version pending. B. Fasy (Tulane) Statistics and Topology 3 June / 48

3 The Intersection of Statistics and Topology: Confidence Sets for Persistence Diagrams Brittany Terese Fasy joint work with S. Balakrishnan, F. Lecci, A. Rinaldo, A. Singh, L. Wasserman 3 June 2014 [FLRWBS] Statistical Inference for Persistent Homology: Confidence Sets for Persistence Diagrams. ArXiv Tentatively accepted, Annals of Statistics. B. Fasy (Tulane) Statistics and Topology 3 June / 48

4 Introduction Motivation Extracting Structure from Point Cloud Data B. Fasy (Tulane) Statistics and Topology 3 June / 48

5 Introduction Motivation Extracting Structure from Point Cloud Data B. Fasy (Tulane) Statistics and Topology 3 June / 48

6 Introduction Motivation Extracting Structure from Point Cloud Data B. Fasy (Tulane) Statistics and Topology 3 June / 48

7 Introduction Motivation Extracting Structure from Point Cloud Data B. Fasy (Tulane) Statistics and Topology 3 June / 48

8 Introduction Motivation Extracting Structure from Point Cloud Data B. Fasy (Tulane) Statistics and Topology 3 June / 48

9 Introduction Motivation Extracting Structure from Point Cloud Data B. Fasy (Tulane) Statistics and Topology 3 June / 48

10 Introduction Motivation Extracting Structure from Point Cloud Data B. Fasy (Tulane) Statistics and Topology 3 June / 48

11 Introduction Motivation Extracting Structure from Point Cloud Data B. Fasy (Tulane) Statistics and Topology 3 June / 48

12 Introduction Motivation Extracting Structure from Point Cloud Data B. Fasy (Tulane) Statistics and Topology 3 June / 48

13 Introduction Homeomorphisms An Informal Discussion A homeomorphism is a continuous bending, stretching, or shrinking (but not tearing or glueing) of one object into another object. B. Fasy (Tulane) Statistics and Topology 3 June / 48

14 Introduction Homeomorphisms An Informal Discussion A homeomorphism is a continuous bending, stretching, or shrinking (but not tearing or glueing) of one object into another object. Circle = Square B. Fasy (Tulane) Statistics and Topology 3 June / 48

15 Introduction Homeomorphisms An Informal Discussion A homeomorphism is a continuous bending, stretching, or shrinking (but not tearing or glueing) of one object into another object. Circle = Square Donut = Coffee Cup B. Fasy (Tulane) Statistics and Topology 3 June / 48

16 Introduction Homeomorphisms An Informal Discussion A homeomorphism is a continuous bending, stretching, or shrinking (but not tearing or glueing) of one object into another object. Circle = Square Donut Sugar Bowl B. Fasy (Tulane) Statistics and Topology 3 June / 48

17 Topological Spaces Nerve Complexes Confidence Sets for Persistence Diagrams B. Fasy (Tulane) Statistics and Topology 3 June / 48

18 Topological Spaces Nerve Complexes Confidence Sets for Persistence Diagrams B. Fasy (Tulane) Statistics and Topology 3 June / 48

19 Topological Spaces Nerve Complexes Confidence Sets for Persistence Diagrams B. Fasy (Tulane) Statistics and Topology 3 June / 48

20 Topological Spaces Nerve Complexes Confidence Sets for Persistence Diagrams Nerve Lemma If X is a set of closed convex sets, then Nrv(X ) is topologically equivalent to X (up to homotopy type). B. Fasy (Tulane) Statistics and Topology 3 June / 48

21 Homology β 0 = n Simplicial Homology X p is the power set of all p-simplices. C p = (X p, + 2 ). p : C p C p 1 is the boundary map. H p = Ker( p )/Im( p+1 ) β p = rank(h p ). β 1 = 0 B. Fasy (Tulane) Statistics and Topology 3 June / 48

22 Homology Confidence Sets for Persistence Diagrams β 0 = n β 1 = 0 β 0 = 5 β 1 = 1 B. Fasy (Tulane) Statistics and Topology 3 June / 48

23 Homology Confidence Sets for Persistence Diagrams β 0 = n β 1 = 0 β 0 = 5 β 1 = 1 β 0 = 1 β 1 = 1 B. Fasy (Tulane) Statistics and Topology 3 June / 48

24 Homology Confidence Sets for Persistence Diagrams β 0 = n β 0 = 5 β 0 = 1 β 0 = 1 β 1 = 0 β 1 = 1 β 1 = 1 β 1 = 0 B. Fasy (Tulane) Statistics and Topology 3 June / 48

25 Homology Confidence Sets for Persistence Diagrams d 1 (0) d 1 ([0, r 1 ]) d 1 ([0, r 2 ]) d 1 ([0, r 4 ]) β 0 = n β 0 = 5 β 0 = 1 β 0 = 1 β 1 = 0 β 1 = 1 β 1 = 1 β 1 = 0 B. Fasy (Tulane) Statistics and Topology 3 June / 48

26 Persistent Homology Lower-Level Set Filtration Let M be a compact topological space. Let f : M R be continuous. We consider all sets of the form M t := f 1 ((, t]). B. Fasy (Tulane) Statistics and Topology 3 June / 48

27 Persistent Homology Lower-Level Set Filtration Let M be a compact topological space. Let f : M R be continuous. We consider all sets of the form M t := f 1 ((, t]). B. Fasy (Tulane) Statistics and Topology 3 June / 48

28 Persistent Homology Lower-Level Set Filtration Let M be a compact topological space. Let f : M R be continuous. We consider all sets of the form M t := f 1 ((, t]). B. Fasy (Tulane) Statistics and Topology 3 June / 48

29 Persistent Homology Lower-Level Set Filtration Let M be a compact topological space. Let f : M R be continuous. We consider all sets of the form M t := f 1 ((, t]). B. Fasy (Tulane) Statistics and Topology 3 June / 48

30 Persistent Homology Lower-Level Set Filtration Let M be a compact topological space. Let f : M R be continuous. We consider all sets of the form M t := f 1 ((, t]). B. Fasy (Tulane) Statistics and Topology 3 June / 48

31 Persistent Homology Lower-Level Set Filtration Let M be a compact topological space. Let f : M R be continuous. We consider all sets of the form M t := f 1 ((, t]). B. Fasy (Tulane) Statistics and Topology 3 June / 48

32 Persistent Homology Lower-Level Set Filtration Let M be a compact topological space. Let f : M R be continuous. We consider all sets of the form M t := f 1 ((, t]). B. Fasy (Tulane) Statistics and Topology 3 June / 48

33 Persistent Homology Lower-Level Set Filtration Let M be a compact topological space. Let f : M R be continuous. We consider all sets of the form M t := f 1 ((, t]). B. Fasy (Tulane) Statistics and Topology 3 June / 48

34 Persistent Homology Lower-Level Set Filtration Let M be a compact topological space. Let f : M R be continuous. We consider all sets of the form M t := f 1 ((, t]). B. Fasy (Tulane) Statistics and Topology 3 June / 48

35 Persistent Homology Upper-Level Set Filtration Let M be a compact topological space. Let f : M R be continuous. We consider all sets of the form M t := f 1 ([t, )). B. Fasy (Tulane) Statistics and Topology 3 June / 48

36 Distance and Stability A Metric on Persistence Diagrams B. Fasy (Tulane) Statistics and Topology 3 June / 48

37 Distance and Stability A Metric on Persistence Diagrams B. Fasy (Tulane) Statistics and Topology 3 June / 48

38 Bottleneck Distance Given two persistence diagrams, find the best perfect matching M between the point sets. Minimize Cost We wish to find W = min M { max (p,q) M p q }. B. Fasy (Tulane) Statistics and Topology 3 June / 48

39 Bottleneck Distance Given two persistence diagrams, find the best perfect matching M between the point sets. Minimize Cost We wish to find W = min M { max (p,q) M p q }. B. Fasy (Tulane) Statistics and Topology 3 June / 48

40 Stability of Matchings Bottleneck Stability Theorem [CDGO] f 1 f 2 W (D 1, D 2 ) B. Fasy (Tulane) Statistics and Topology 3 June / 48

41 Topological Inference B. Fasy (Tulane) Statistics and Topology 3 June / 48

42 Objective Let D T denote the set of all T -bounded persistence diagrams. Confidence Sets Given α (0, 1) and unknown diagram D, we want C α D T such that P(D C α ) 1 α. Question If D is an estimate of D, how close is D to D? B. Fasy (Tulane) Statistics and Topology 3 June / 48

43 Confidence Sets for Persistent Diagrams C α = {D D T : W (D, ˆD) t(α)} B. Fasy (Tulane) Statistics and Topology 3 June / 48

44 Confidence Sets for Persistent Diagrams C α = {D D T : W (D, ˆD) t(α)} B. Fasy (Tulane) Statistics and Topology 3 June / 48

45 Statistical Model Confidence Sets for Persistence Diagrams M is a manifold. P is a probability distribution supported on M. Observe data X 1, X 2,..., X n P. Compute D n = Dgm (X 1,..., X n ) B. Fasy (Tulane) Statistics and Topology 3 June / 48

46 Statistical Model Confidence Sets for Persistence Diagrams M is a manifold. P is a probability distribution supported on M. Observe data X 1, X 2,..., X n P. Compute D n = Dgm (X 1,..., X n ) Question How does D n compare to E( D n ) Dgm (M)? B. Fasy (Tulane) Statistics and Topology 3 June / 48

47 Statistical Model Confidence Sets for Persistence Diagrams M is a manifold. P is a probability distribution supported on M. Observe data X 1, X 2,..., X n P. Compute D n = Dgm (X 1,..., X n ) Question How does D n compare to E( D n ) Dgm (M)? Answer Find C α such that P(E( D n ) C α ) 1 α. B. Fasy (Tulane) Statistics and Topology 3 June / 48

48 Statistical Model Confidence Sets for Persistence Diagrams M is a manifold. P is a probability distribution supported on M. Observe data X 1, X 2,..., X n P. Compute D n = Dgm (X 1,..., X n ) Question How does D n compare to E( D n ) Dgm (M)? Answer Find C α such that P(E( D n ) C α ) 1 α. How? B. Fasy (Tulane) Statistics and Topology 3 June / 48

49 Computing a Confidence Interval With Infinite Resources Repeatedly sample n data points, obtaining: Confidence Intervals P(Θ n (M) [0, q α ]) 1 α. B. Fasy (Tulane) Statistics and Topology 3 June / 48

50 Computing a Confidence Interval With Infinite Resources Repeatedly sample n data points, obtaining: ˆΘ n,1,..., ˆΘ n,n Confidence Intervals P(Θ n (M) [0, q α ]) 1 α. B. Fasy (Tulane) Statistics and Topology 3 June / 48

51 Computing a Confidence Interval With Infinite Resources Repeatedly sample n data points, obtaining: ˆΘ n,1,..., ˆΘ n,n via simulation. Confidence Intervals P(Θ n (M) [0, q α ]) 1 α. B. Fasy (Tulane) Statistics and Topology 3 June / 48

52 Bootstrapping When We Can Only Take One Sample We have one sample: S n = {X 1,..., X n } B. Fasy (Tulane) Statistics and Topology 3 June / 48

53 Bootstrapping When We Can Only Take One Sample We have one sample: S n = {X 1,..., X n } Subsample (with replacement), obtaining: {X 1,..., X n } B. Fasy (Tulane) Statistics and Topology 3 June / 48

54 Bootstrapping When We Can Only Take One Sample We have one sample: S n = {X 1,..., X n } Subsample (with replacement), obtaining: {X 1,..., X n } Compute ˆΘ n = Θ(X 1,..., X n ). B. Fasy (Tulane) Statistics and Topology 3 June / 48

55 Bootstrapping When We Can Only Take One Sample We have one sample: S n = {X 1,..., X n } Subsample (with replacement), obtaining: {X 1,..., X n } Compute ˆΘ n = Θ(X 1,..., X n ). Repeat N times, obtaining: ˆΘ n,1,..., ˆΘ n,n. B. Fasy (Tulane) Statistics and Topology 3 June / 48

56 Bootstrapping When We Can Only Take One Sample We have one sample: S n = {X 1,..., X n } Subsample (with replacement), obtaining: {X 1,..., X n } Compute ˆΘ n = Θ(X 1,..., X n ). Repeat N times, obtaining: ˆΘ n,1,..., ˆΘ n,n. B. Fasy (Tulane) Statistics and Topology 3 June / 48

57 Distance to a Subset Distance Function d M (a) = inf x M x a D = Dgm p (d M ) B. Fasy (Tulane) Statistics and Topology 3 June / 48

58 Distance to a Subset Distance Function d M (a) = inf x M x a D = Dgm p (d M ) P has continuous density p. support(p) = M. S n = {X 1,..., X n } P D = Dgm p (d Sn ) B. Fasy (Tulane) Statistics and Topology 3 June / 48

59 Distance Function Subsampling A Variant of the Bootstrap Let Sb i be a subsample of size b < n, for i = 1,..., B. L b is the CDF of H(Sb i, S n). B. Fasy (Tulane) Statistics and Topology 3 June / 48

60 Distance Function Subsampling A Variant of the Bootstrap Let Sb i be a subsample of size b < n, for i = 1,..., B. L b is the CDF of H(Sb i, S n). Confidence Sets from Subsampling [FLRWBS] Assume that p(x) is bounded away from zero. Then, almost surely, for all large n, P ( d M d Sn > 2L 1 b (α)) α + O ( 1/n ) B. Fasy (Tulane) Statistics and Topology 3 June / 48

61 Distance Function Subsampling A Variant of the Bootstrap Let Sb i be a subsample of size b < n, for i = 1,..., B. L b is the CDF of H(Sb i, S n). Confidence Sets from Subsampling [FLRWBS] Assume that p(x) is bounded away from zero. Then, almost surely, for all large n, P ( d M d Sn > 2L 1 b (α)) α + O ( 1/n ) [RS-2002] On the uniform asymptotic validity of subsampling and the bootstrap. Annals of Statistics. B. Fasy (Tulane) Statistics and Topology 3 June / 48

62 Putting It All Together Distance Function Subsampling Theorem P ( d M d Sn > 2L 1 b (α)) α + O ( 1/n ) B. Fasy (Tulane) Statistics and Topology 3 June / 48

63 Putting It All Together Distance Function Subsampling Theorem P ( d M d Sn > 2L 1 b (α)) α + O Bottleneck Stability Theorem d M d Sn W (D, D n ) ( 1/n ) B. Fasy (Tulane) Statistics and Topology 3 June / 48

64 Putting It All Together Distance Function Subsampling Theorem P ( d M d Sn > 2L 1 b (α)) α + O Bottleneck Stability Theorem d M d Sn W (D, D n ) ( 1/n ) Confidence Sets for Persistence Diagrams ( P W (D, D ) ( ) n ) > 2L 1 b (α) α + O 1/n B. Fasy (Tulane) Statistics and Topology 3 June / 48

65 Putting It All Together Distance Function Subsampling Theorem P ( d M d Sn > 2L 1 b (α)) α + O Bottleneck Stability Theorem d M d Sn W (D, D n ) ( 1/n ) Confidence Sets for Persistence Diagrams ( P W (D, D ) ( ) n ) > 2L 1 b (α) α + O 1/n Asymptotic Confidence Sets for Persistence Diagrams ( lim P W (D, D ) n ) 2L 1 n b (α) 1 α B. Fasy (Tulane) Statistics and Topology 3 June / 48

66 Varying α Confidence Sets for Persistence Diagrams Distance Function B. Fasy (Tulane) Statistics and Topology 3 June / 48

67 Varying α Confidence Sets for Persistence Diagrams Distance Function α = 0.001, 0.05, 0.25 B. Fasy (Tulane) Statistics and Topology 3 June / 48

68 Two More Methods Distance Function S n = S 1,n S2,n. Theorem (Concentration of Measure) There exists ˆt cm = ˆt cm (α, d, n, S 1,n ) such that ( P (W (D, D ) (log ) ) n 1/d+2 n ) > ˆt cm α + O. n Theorem (Method of Shells) There exists ˆt s = ˆt s (α, d, n, K, S 1,n ) such that ( P (W (D, D ) (log ) ) n 1/d+2 n ) > ˆt s α + O. n B. Fasy (Tulane) Statistics and Topology 3 June / 48

69 Distance Function These Methods are Different Concentration of Measure ˆt cm is found by solving the following for t: 2 d+1 ( ) t d exp ntd ˆρ 1,n = α. ˆρ 1,n 2 Shells ˆt s is found by solving the following for t: 2 d+1 t d ˆρ n ( ) ĝ(v) v exp nvtd ˆρ 1,n dv = α. 2 B. Fasy (Tulane) Statistics and Topology 3 June / 48

70 Distance Function Examples Uniform Distribution on Unit Circle Distance Function B. Fasy (Tulane) Statistics and Topology 3 June / 48

71 Distance Function Examples Uniform Distribution on Cassini Curve Distance Function B. Fasy (Tulane) Statistics and Topology 3 June / 48

72 Distance Function Examples Cassini Curve with Outliers Distance Function B. Fasy (Tulane) Statistics and Topology 3 June / 48

73 Distance Function Examples Normal Distribution on Unit Circle Distance Function B. Fasy (Tulane) Statistics and Topology 3 June / 48

74 Distance Function The Intersection of Statistics and Topology: Stochastic Convergence of Persistence Landscapes and Silhouettes Brittany Terese Fasy joint work with F. Chazal, F. Lecci, B. Michel A. Rinaldo, L. Wasserman 3 June 2014 [CFLRW] Stochastic Convergence of Persistence Landscapes and Silhouettes. ArXiv. SoCG 2014 Proceedings, Kyoto. Journal version pending. B. Fasy (Tulane) Statistics and Topology 3 June / 48

75 Persistence Landscapes Distance Function B. Fasy (Tulane) Statistics and Topology 3 June / 48

76 Persistence Landscapes Distance Function /2 ( ( - ( + ( /2 B. Fasy (Tulane) Statistics and Topology 3 June / 48

77 Persistence Landscapes Distance Function /2 ( ( - ( + ( /2 B. Fasy (Tulane) Statistics and Topology 3 June / 48

78 Persistence Landscapes Distance Function /2 ( ( - ( + ( /2 [B-2012] Statistical Topology Using Persistent Homology. ArXiv B. Fasy (Tulane) Statistics and Topology 3 June / 48

79 Weak Convergence of Landscapes Distance Function Let λ 1,..., λ n L T. B. Fasy (Tulane) Statistics and Topology 3 June / 48

80 Weak Convergence of Landscapes Distance Function Let λ 1,..., λ n L T. λ : true (unknown) landscape µ = E(λ i ) B. Fasy (Tulane) Statistics and Topology 3 June / 48

81 Weak Convergence of Landscapes Distance Function n Let λ 1,..., λ n L T. λ : true (unknown) landscape µ = E(λ i ) λ n : average landscape B. Fasy (Tulane) Statistics and Topology 3 June / 48

82 Distance Function Weak Convergence of Landscapes n Let λ 1,..., λ n L T. λ : true (unknown) landscape µ = E(λ i ) λ n : average landscape t Pointwise Convergence [B-2012]. λ n converges pointwise to µ. B. Fasy (Tulane) Statistics and Topology 3 June / 48

83 Distance Function Weak Convergence of Landscapes n Let λ 1,..., λ n L T. λ : true (unknown) landscape µ = E(λ i ) λ n : average landscape t Pointwise Convergence [B-2012]. λ n converges pointwise to µ. Gaussian Process on [0, T ] For t [0, T ], we define G n (f t ) = G n (t) := 1 n ( λ n (t) µ(t)). B. Fasy (Tulane) Statistics and Topology 3 June / 48

84 Uniform Convergence Distance Function Weak Convergence G n converges weakly to the Brownian bridge G with covariance function κ(f, g) = f (u)g(y)dp(u) ( f (u)dp(u))( g(u)dp(u)). B. Fasy (Tulane) Statistics and Topology 3 June / 48

85 Uniform Convergence Distance Function Weak Convergence G n converges weakly to the Brownian bridge G with covariance function κ(f, g) = f (u)g(y)dp(u) ( f (u)dp(u))( g(u)dp(u)). Uniform CLT There exists a random variable W d = sup t [t,t ] G(f t ) such that ( sup P z R sup t [t,t ] ) G n (t) z ( (log n) P (W z) = O n ). B. Fasy (Tulane) Statistics and Topology 3 June / 48

86 Uniform Convergence Distance Function Weak Convergence G n converges weakly to the Brownian bridge G with covariance function κ(f, g) = f (u)g(y)dp(u) ( f (u)dp(u))( g(u)dp(u)). Uniform CLT There exists a random variable W d = sup t [t,t ] G(f t ) such that ( sup P z R sup t [t,t ] ) G n (t) z ( (log n) P (W z) = O Proofs rely on [CCK-2013]: Anti-concentration and honest adaptive confidence bands. ArXiv B. Fasy (Tulane) Statistics and Topology 3 June / 48 n ).

87 Distance Function Confidence Bands E λ λ n variance {}}{ E µ λ n + E λ µ }{{} bais B. Fasy (Tulane) Statistics and Topology 3 June / 48

88 Confidence Bands Distance Function 5 (death-birth)/ E λ λ n variance {}}{ E µ λ n (birth+death)/ Confidence Band Find c such that l n = λ n c and u n = λ n + c such that P(l n (t) µ(t) u n (t) for all t) 1 α. B. Fasy (Tulane) Statistics and Topology 3 June / 48

89 Asymptotic Confidence Bands Distance Function E λ λ n variance {}}{ E µ λ n Asymptotic Confidence Band Find c such that l n = λ n c and u n = λ n + c such that lim P(l n(t) µ(t) u n (t) for all t) 1 α. n B. Fasy (Tulane) Statistics and Topology 3 June / 48

90 Confidence Bands for Landscapes Distance Function ξ 1,..., ξ n N(0, 1) α-quantile G n (f t ) := 1 n Z α is the unique value such that ( ) P sup G n (f t ) > Z α {λ i } t n ξ i (λ i (t) λ n (t)) i=1 = α B. Fasy (Tulane) Statistics and Topology 3 June / 48

91 Confidence Bands for Landscapes Distance Function ξ 1,..., ξ n N(0, 1) α-quantile G n (f t ) := 1 n Z α is the unique value such that ( ) P sup G n (f t ) > Z α {λ i } t *Approx. Z α by MC simulation n ξ i (λ i (t) λ n (t)) i=1 = α B. Fasy (Tulane) Statistics and Topology 3 June / 48

92 Distance Function Confidence Bands for Landscapes The Multiplier Bootstrap l n = λ n (t) Z(α) n u n = λ n (t) + Z(α) n Uniform Band ( 7 (log n) ) 8 P (l n (t) µ(t) u n (t) for all t) 1 α. n 1 8 B. Fasy (Tulane) Statistics and Topology 3 June / 48

93 Variable Width Confidence Bands Distance Function H n (f t ) := G n (t)/σ(t) = 1 n n i=1 λ i (t) µ(t) σ(t) Ĥ n (f t ) := 1 n n i=1 ξ i λ i (t) λ n (t) ˆσ n (t) Q α is the unique value such that ( P sup H n (f t ) > Q α {λ i } t ) = α B. Fasy (Tulane) Statistics and Topology 3 June / 48

94 Distance Function Confidence Bands for Landscapes The Multiplier Bootstrap l n = λ n (t) ˆQ(α)ˆσ n (t) n u n = λ n (t) + ˆQ(α)ˆσ n (t) n Variable Band ( 7 (log n) ) 8 P (l n (t) µ(t) u n (t) for all t) 1 α. n 1 8 B. Fasy (Tulane) Statistics and Topology 3 June / 48

95 Distance Function The Intersection of Statistics and Topology:... And More! Brittany Terese Fasy joint work with S. Balakrishnan, F. Lecci, A. Rinaldo, A. Singh, L. Wasserman 3 June 2014 B. Fasy (Tulane) Statistics and Topology 3 June / 48

96 Persistence Silhouettes Definitions Distance Function /2 ( ( - ( + ( /2 B. Fasy (Tulane) Statistics and Topology 3 June / 48

97 Persistence Silhouettes Definitions Distance Function /2 ( ( - ( + ( /2 B. Fasy (Tulane) Statistics and Topology 3 June / 48

98 Persistence Silhouettes Definitions Distance Function B. Fasy (Tulane) Statistics and Topology 3 June / 48

99 Persistence Silhouettes Definitions Distance Function Weighted Silhouette n i=1 φ(t) = w iλ j (t) m j=1 wj Power-Weighted Silhouette w i = d i b i p B. Fasy (Tulane) Statistics and Topology 3 June / 48

100 Power-Weighted Silhouettes Two Examples Distance Function Persistence Diagram Silhouette (p=0.1) Silhouette (p=1) (Death Birth)/ φ(t) φ(t) (Birth+Death)/ t t Persistence Diagram Silhouette (p=0.1) Silhouette (p=3) (Death Birth)/ φ(t) φ(t) (Birth+Death)/ t t B. Fasy (Tulane) Statistics and Topology 3 June / 48

101 Persistence Silhouettes Results Distance Function Since φ is one-lipschitz for non-negative weights w j... Convergence of Empirical Process ( n ) 1 φ i (t) E[φ(t)] n i=1 converges weakly to a Brownian bridge, with known rate of convergence. Confidence Bands We can use the multiplier bootstrap to create a uniform (or a variable width) confidence band defined by l sil n and un sil such that ( ) lim P l sil n n (t) µ(t) un sil (t) for all t = 1 α. B. Fasy (Tulane) Statistics and Topology 3 June / 48

102 Confidence Sets for Persistence Diagrams Distance Function Example I A Toy Example B. Fasy (Tulane) Statistics and Topology 3 June / 48

103 Example II Earthquake Epicenters Confidence Sets for Persistence Diagrams Distance Function B. Fasy (Tulane) Statistics and Topology 3 June / 48

104 Summary Conclusion B. Fasy (Tulane) Statistics and Topology 3 June / 48

105 Collaborator Collage Conclusion B. Fasy (Tulane) Statistics and Topology 3 June / 48

106 Thank you! Conclusion Brittany Terese Fasy B. Fasy (Tulane) Statistics and Topology 3 June / 48

107 References More References [CDGO] The Structure and Stability of Persistence Modules. ArXiv [CFLRSW] On the Bootstrap for Persistence Diagrams and Landscapes. Modeling and Analysis of Information Systems, 20:6 (Dec. 2013), B. Fasy (Tulane) Statistics and Topology 3 June / 48

108 Other Stuff B. Fasy (Tulane) Statistics and Topology 3 June / 48