Strong Consistency of Set-Valued Frechet Sample Mean in Metric Spaces

Transcription

1 Strong Consistency of Set-Valued Frechet Sample Mean in Metric Spaces Cedric E. Ginestet Department of Mathematics and Statistics Boston University JSM 2013

2 The Frechet Mean Barycentre as Average Given a set of points x 1,..., x n R k ; And a set of masses w 1,..., w n R; The barycentre is x := 1 n w i x i. (1) When the distribution of mass is uniform, this is the centroid.

3 The Frechet Mean Barycentre as Average Given a set of points x 1,..., x n R k ; And a set of masses w 1,..., w n R; The barycentre is x := 1 n w i x i. (1) When the distribution of mass is uniform, this is the centroid. Barycentre as Minimizer Take the L 2 -metric on R k ; The barycentre is a minimizer, 1 x = argmin x R n k w i x i x 2 2. (2)

4 The Frechet Mean The Most Central Element Given a metric space (, d), Let an abstract-valued r.v. : (Ω, F, P) (, B), Fréchet (1948) defined the mean value, for every r 1, Θ r := arginf d(x, x ) r dµ(x). x

5 The Frechet Mean The Most Central Element Given a metric space (, d), Let an abstract-valued r.v. : (Ω, F, P) (, B), Fréchet (1948) defined the mean value, for every r 1, Θ r := arginf d(x, x ) r dµ(x). x Frechet Sample Mean Given a family of abstract-valued r.v.s i s, i = 1,..., n, Θ r n := arginf x 1 n d( i, x ) r. In general, Θ r, Θ r n will not be unique: i.e. Θ r, Θ r n.

6 Motivating Example G 1 G 2 Figure: A Sample of Two Simple Graphs, G with V (G) = 4.

7 Motivating Example G 1 G 2 Figure: A Sample of Two Simple Graphs, G with V (G) = 4. Hamming Distance d(g, G ) := i<j I{e ij e ij}.

8 Non-unique Means G 1 G 2 Figure: A Sample of Two Simple Graphs, G with V (G) = 4. Θ 1 Θ 2 Θ 3 Θ 4 Figure: Set of Frechet means (Θ) is a superset of the sampled graphs.

9 Questions and Assumptions Convergence of Frechet Sample Mean Θ r n := arginf x 1 n d( i, x ) r arginf x d(x, x ) r dµ(x) =: Θ r, Ziezold (1977) in separable finite metric spaces, for r = 2; Sverdrup-Thygeson (1981) in compact metric spaces, for r 1.

10 Questions and Assumptions Convergence of Frechet Sample Mean Θ r n := arginf x 1 n d( i, x ) r arginf x d(x, x ) r dµ(x) =: Θ r, Ziezold (1977) in separable finite metric spaces, for r = 2; Sverdrup-Thygeson (1981) in compact metric spaces, for r 1. Topological and Measure-theoretic Assumptions Separable bounded (, d). Possibly empty Θ r n, Θ r. Possibly non-unique Θ r n, Θ r.

11 Set-valued Outer and Inner Limits Figure: Outer and inner limits, defined wrt set inclusion on. limsup S n liminf S n limsup S n := n n=1 m=n S n, and liminf n S n := n=1 m=n S n. Outer Limit: The x s that belong to infinitely many S n. Inner Limit: The x s that belong to all but finitely many S n.

12 Kuratowski Upper Limit Equivalent Definitions Given a sequence of subsets A n : Set of cluster points of sequences a n A n ; The Kuratowski upper limit is { } Limsup A n := x : liminf d(x, A n) = 0. n n where liminf and Limsup are taken with respect to real numbers and subsets of, respectively. Lemma Given a metric space (, d), for any sequence of sets A n, Limsup A n = n n=1 m=n A m.

13 Almost Sure Consistency of Frechet Sample Mean Main Result In a separable bounded metric space (, d), For any 1,..., n be an iid sequence, Convergence of the infima: σ n r 1 := inf x n d( i, x ) r inf d(x, x ) r dx =: σ r x a.s., Convergence of the arginf s: Limsup n Θ r n Θ r a.s.. For every r > 0.

14 Almost Sure Uniform Weak Convergence Glivenko-Cantelli Lemma (Rao, 1962) i. Let F( ) be a class of real-valued functions on separable, ii. Let a sequence of finite measures µ n, and µ, iii. F( ) is dominated by a continuous integrable function on, iv. F( ) is equicontinuous, and v. µ n µ, a.s.; then we obtain uniform a.s. weak convergence, lim fdµ n fdµ = 0, a.s.. sup n f F

15 Point Functions Definition For some z, the z-point function is d z (x) := d(z, x), for every x. The class of point functions on (, d) is then denoted by D r ( ) := {d r z : z }. for every r 1. Lemma If (, d) is a bounded metric space, then for every r 1, D r ( ) is i. Uniformly bounded; ii. Uniformly equicontinuous.

16 Almost Sure Consistency of Frechet Sample Mean Strengthening of a.s. Weak Convergence Using the Glivenko-Cantelli lemma, we obtain for every r 1, lim sup n d r z dµ n d r z dµ z D r = 0 a.s.. where µ n := 1 n n δ i.

17 Almost Sure Consistency of Frechet Sample Mean Strengthening of a.s. Weak Convergence Using the Glivenko-Cantelli lemma, we obtain for every r 1, lim sup n d r z dµ n d r z dµ z D r = 0 a.s.. where µ n := 1 n n δ i. Almost Sure Convergence of σ n r In order to show that σ n r σ r a.s., We also need the following sandwich relationship, Tn r σ n r σ r Tn. r

18 Almost Sure Consistency of Frechet Sample Mean Sandwich Argument I By the minimality of θ Θ 2, T n (ˆθ n ) := 1 n 1 n d 2ˆθn ( i ) d 2ˆθn (x)dµ(x) d 2ˆθn ( i ) d 2 θ(x)dµ(x) =: Tn(ˆθ n ).

19 Almost Sure Consistency of Frechet Sample Mean Sandwich Argument I By the minimality of θ Θ 2, T n (ˆθ n ) := 1 n 1 n d 2ˆθn ( i ) d 2ˆθn (x)dµ(x) d 2ˆθn ( i ) d 2 θ(x)dµ(x) =: Tn(ˆθ n ). Sandwich Argument II By the minimality of ˆθ n Θ 2 n, T n(ˆθ n ) := 1 n 1 n d 2ˆθn ( i ) d 2 θ(x)dµ(x) d 2 θ( i ) d 2 θ(x)dµ(x) =: T n (θ).

20 Almost Sure Consistency of Frechet Sample Mean Main Result In a separable bounded metric space (, d), For any 1,..., n be an iid sequence, Convergence of the infima: σ n r 1 := inf x n d( i, x ) r inf d(x, x ) r dx =: σ r x a.s., Convergence of the arginf s: Limsup n Θ r n Θ r a.s.. For every r > 0.

21 Conclusion and Extensions Consistency of Frechet Sample Mean Under separable and bounded d. For non-unique means. For every r 1. Possible Extensions Distance functions without the triangle inequality. Non-iid random variables (See Kendall et al., 2012). Rate of convergence. Statistical inference.

22 Publications & Funding Agencies Papers Ginestet, C.E., Simmons, A. and Kolaczyk, E.D. (2012). Probability and Statistics Letters, 82(10), Fournel, A.P., Reynaud, E., Brammer, M.J., Simmons, A. and Ginestet, C.E. (2013). Neuroimage, 76, Ginestet, C.E. (Submitted). Strong Consistency of Fréchet Sample Mean Sets for Graph-Valued Random Variables. Ariv. Acknowledgements Eric D. Kolaczyk (Boston University) Tom Nichols and Wilfrid S. Kendall (Warwick University) Funding Agency Air Force Office for Scientific Research (AFOSR).