Introduction to Registration Problem for Functional Data

Transcription

1 Introducton to Regstraton Problem for Functonal Data Tan Wang Functonal Data Regstraton Mar / 24

2 Functonal Data Introducton to Regstraton Data avalable n R package fda Functonal Data Regstraton Mar / 24

3 Introducton to Regstraton Unallgned v.s. Allgned Functons Functonal Data Regstraton Mar / 24

4 Warpng Functon Introducton to Regstraton Functonal Data Regstraton Mar / 24

5 Warpng Functon Introducton to Regstraton Functonal Data Regstraton Mar / 24

6 Warpng Functon Introducton to Regstraton Functonal Data Regstraton Mar / 24

7 Warpng Functon Introducton to Regstraton Functonal Data Regstraton Mar / 24

8 Introducton to Regstraton Requrements on Warpng Functons h(0) = 0, h(1) = 1. Functonal Data Regstraton Mar / 24

9 Introducton to Regstraton Requrements on Warpng Functons h(t) s strctly monotone ncreasng. Functonal Data Regstraton Mar / 24

10 Introducton to Regstraton Landmarks Functonal Data Regstraton Mar / 24

11 Introducton to Regstraton Landmark Regstraton Lnear Warpng Functons Regstered Functons h(t) x(h(t)) t t Nonlnear Warpng Functons Regstered Functons h(t) x(h(t)) t t Functonal Data Regstraton Mar / 24

12 Introducton to Regstraton Regstered Functon Space Functonal Data Regstraton Mar / 24

13 Introducton to Regstraton Regstered Functon Space Goal: Regstered functons form a convex subspace. Functonal Data Regstraton Mar / 24

14 Decomposton of Total Varaton Phase Varaton and Ampltude Varaton Functonal Data Regstraton Mar / 24

15 Decomposton of Total Varaton Quantfyng Phase and Ampltude Varaton Regstered Functon: y(t) = x(h(t)) Cross-sectonal Mean: µ(t) = E[x(t)] Structural Mean: ν(t) = E[y(t)] Total Varance: MS total = E [x(t) µ(t)] 2 dt Constant C (Knep and Ramsay, 2008): C := E [x(t)] 2 dt cov[h E [y(t)] 2 dt = 1 + (t), y 2 (t)]dt E [y(t)] 2 dt Functonal Data Regstraton Mar / 24

16 Decomposton of Total Varaton Total Varance Decomposton MS total = E [x(t) µ(t)] 2 dt = (C E [y(t) ν(t)] 2 dt) + (C ν 2 (t)dt µ 2 (t)dt) Total Ampltude Varaton: MS amp = C E [y(t) ν(t)] 2 dt Total Phase Varaton: MS phase = C ν 2 (t)dt µ 2 (t)dt Functonal Data Regstraton Mar / 24

17 Regstraton Methods Combnng Regstraton and Functonal PCA #zeros(x ): the number of ponts τ [0, 1]wth x (τ) = 0. #feat(x): the number of strct local mnma/maxma. Concluson (Knep and Ramsay, 2008): Assume that for some 0 < k <, Pr (#feat(x ) k, #zeros(x ) k)=1. Then there exsts a consstent regstraton procedure such that for some p k + 2, the regstered functons y can be wrtten as x [h (t)] = y (t) = p θ j ξ j (t), t [0, 1] j=1 for some random coeffcents θ j R. Functonal Data Regstraton Mar / 24

18 Regstraton Methods Combnng Regstraton and Functonal PCA Algorthm (Knep and Ramsay, 2008): Iteraton 0: y (0) = x h (0) = I Iteraton l > 0: ν (l) = n 1 n =1 y (l 1) ŷ (l) = ν (l) + PCAAPPROX(y (l 1) ν (l) ) ŷ (l) y (l 1) h (l) nc (REGIST ) y (l) = y (l 1) h (l) nc Stoppng Rule: h (l) nc, (t) h d(t). Functonal Data Regstraton Mar / 24

19 Regstraton Methods Combnng Regstraton and Functonal PCA More Detals about REGIST : Deformaton Functon: d (l) (t) = h (l) nc, (t) t Frst-order Approxmaton at t: y (l) or alternatvely, (t) = y (l 1) ŷ (l) y (l 1) = y (l 1) [h (l) nc, (t)] (t) + y (l 1) (t)[h (l) (t) t] (t) + y (l 1) (t) y (l 1) (t) y (l 1) nc, (t)d (l) (t) (t)d (l) (t) Functonal Data Regstraton Mar / 24

20 Regstraton Methods Combnng Regstraton and Functonal PCA Representaton of d (l) (t): d (l) (t) = L s=1 d (l) s ψ s(t) = (d (l) ) T ψ(t) Goal: Mnmze the Integrated Squared Error SSE(d (l) ) = 1 0 PENSSE(d (l) [ŷ (l) (t) y (l 1) (t) y (l 1) λ) = SSE(d (l) ) + λ 1 0 (t)ψ T (t)d (l) ] 2 dt [ψ T (t)d (l) ] 2 dt Functonal Data Regstraton Mar / 24

21 Regstraton by Fsher-Rao Metrc Regstraton Usng Fsher-Rao Remannan Metrc Fsher-Rao Remannan Metrc (Srvastava, et al., 2011): v 1, v 2 T x (X), where T x (X) s the tangent space to X at x, then the Fsher-Rao Remannan metrc s defned as the nner product: v 1, v 2 x = v 1 (t)v 2 (t) 1 x (t) dt Property1: d FR (x 1, x 2 ) = d FR (x 1 h, x 2 h) for any h. Functonal Data Regstraton Mar / 24

22 Regstraton by Fsher-Rao Metrc Regstraton Usng Fsher-Rao Remannan Metrc Fsher-Rao Remannan Metrc (Srvastava, et al., 2011): v 1, v 2 T x (X), where T x (X) s the tangent space to X at x, then the Fsher-Rao Remannan metrc s defned as the nner product: v 1, v 2 x = v 1 (t)v 2 (t) 1 x (t) dt Property1: d FR (x 1, x 2 ) = d FR (x 1 h, x 2 h) for any h. Square-root Velocty Functon (SRVF): q(t) = sgn(x(t)) x (t) Property2: d FR (x 1, x 2 ) = q 1 q 2 L 2. Functonal Data Regstraton Mar / 24

23 Regstraton by Fsher-Rao Metrc Regstraton Usng Fsher-Rao Remannan Metrc SRVF of x h : (q, h) = (q h) h Elastc Dstance: d([q 1 ], [q 2 ]) = nf h q 1 (q 2, h) L 2 Functonal Data Regstraton Mar / 24

24 Regstraton by Fsher-Rao Metrc Regstraton Usng Fsher-Rao Remannan Metrc SRVF of x h : (q, h) = (q h) h Elastc Dstance: d([q 1 ], [q 2 ]) = nf h q 1 (q 2, h) L 2 Algorthm (Srvastava, et al., 2011): Step1: Compute the Karcher Mean [µ] n of [q 1 ], [q 2 ],..., [q n ]. Step2: Fnd the center µ n of [µ] n wth respect to {q }. Step3: For each, fnd h = argmn h µ n (q, h) L 2. Functonal Data Regstraton Mar / 24

25 Thank you! Functonal Data Regstraton Mar / 24

26 Reference Reference Knep A, Ramsay JO. Combnng regstraton and fttng for functonal model. Journal of the Amercan Statstcal Assocaton, 2008; 103: Marron JS, Ramsay JO, Sangall LM, Srvastava A. Functon data analyss of ampltude and phase varaton, 2015; arxv: Ramsay JO, Slverman BO. Functonal data analyss, Second ed. Sprnger, Srvastava A, Wu W, Kurtek S, Klassen E, Marron JS. Regstraton of functonal data usng Fsher-Rao metrc, 2011a; arxv: v2. Tucker JD, Wu W, Srvastava A. Generatve models for functonal data usng phase and ampltude separaton. Computatonal Statstcs and Data Analyss, 2013; 61: Pcture Source: Functonal Data Regstraton Mar / 24