Universal, non-asymptotic confidence sets for circular means

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1 Universal, non-asymptotic confidence sets for circular means Institut für Mathematik Technische Universität Ilmenau 2nd Conference on Geometric Science of Information October 2015 École polytechnique, Palaiseau Institut für Mathematik, Technische Universität Ilmenau Page 1 / 20

2 Circular / directional data given: Z 1,..., Z n i.i.d. Z random variables on the circle S 1 examples: wind directions directions of a paleomagnetic field (ignoring vertical component) time of admittance to a hospital unit Problem: the circle is not a vector space how to define a mean for circular data? Introduction Institut für Mathematik, Technische Universität Ilmenau Page 2 / 20

3 Circular (or extrinsic) means embed S 1 in C: S 1 = {z C : z = 1} projection: π : C \ {0} S 1, π(z) = z z Euclidean sample mean: Z n := 1 n n i=1 Z i circular (a.k.a. extrinsic) sample mean: ˆµ n = argmin ζ S 1 Z n ζ 2 = π( Z n ) Euclidean population mean: E Z = C Z dp circular population mean: µ = argmin ζ S 1 E Z ζ 2 = π(e Z) unique iff Z n 0 or E Z 0, respectively see e.g. Mardia & Jupp: Directional Statistics (2000). Introduction Institut für Mathematik, Technische Universität Ilmenau Page 3 / 20

4 Aim Construct confidence sets for the circular population mean (set) µ on the circle S 1 Introduction Institut für Mathematik, Technische Universität Ilmenau Page 4 / 20

5 Asymptotics projection π : C \ {0} S 1 is differentiable, so by δ-method: Theorem (central limit theorem) If E Z 0, i.e. if µ is unique, then n Arg(µ 1ˆµ n ) D N (0, E( Im(µ 1 Z) 2) ) E Z 2 where Arg : C \ {0} ( π, π] R (and arbitrary at 0). note: asymptotic variance is small if E Z is large, and/or E ( Im(µ 1 Z) 2), i.e. the variance orthogonal to µ, is small. see e.g. Jammalamadaka & SenGupta: Topics in Circular Statistics (2001). Introduction Institut für Mathematik, Technische Universität Ilmenau Page 5 / 20

6 Asymptotics (cont.) consistent estimation of asympt. variance + Slutsky s lemma gives: Corollary (asymptotic confidence sets) If E Z 0 let δ A = q 1 α 2 n (Im(ˆµ 1 n Z k )) 2 n Z n k=1 where q 1 α denotes the (1 α 2 2 )-quantile of the standard normal distribution N (0, 1). Then C A = {ζ S 1 : Arg(ζ 1ˆµ n ) < δ A } is an asymptotic confidence set of level (1 α). Introduction Institut für Mathematik, Technische Universität Ilmenau Page 6 / 20

7 Why not to use asymptotics needs assumption E Z 0 should be tested, then needs post-test analysis coverage not guaranteed for finite sample sizes see example Introduction Institut für Mathematik, Technische Universität Ilmenau Page 7 / 20

8 Aim Construct confidence sets for the circular population mean (set) µ on the circle S 1 which are universal, i.e. no distributional assumptions (besides i.i.d.) non-asymptotic construction as acceptance region of a corresponding test Introduction Institut für Mathematik, Technische Universität Ilmenau Page 8 / 20

9 Bounding the deviation from the mean Theorem (Hoeffding s inequality) If U 1,..., U n are independent random variables taking values in the bounded interval [a, b] with < a < b < then Ū n = 1 n n k=1 U k with E Ū n = ν fulfills P(Ū n ν t) for any t (0, b ν). [ ( ν a ) ν a+t ( b ν ) ] n b ν t b a ν a+t b ν t = β [a,b] (t, ν) note: β [a,b] (t, ν) exp( 2nt 2 /(b a) 2 ) worse estimate (often called Hoeffding s inequality) β [a,b] (t, ν) = γ has unique solution t [a,b] (γ, ν) for γ (( ) ν a n b a, 1 ) which can easily be computed numerically First construction Institut für Mathematik, Technische Universität Ilmenau Page 9 / 20 see Hoeffding (1963).

10 First construction testing hypothesis E Z = 0: let s 0 = t [ 1,1] ( α 4, 0), exists if α 4 > 2 n 0 First construction Institut für Mathematik, Technische Universität Ilmenau Page 10 / 20

11 First construction testing hypothesis E Z = 0: let s 0 = t [ 1,1] ( α 4, 0), exists if α 4 > 2 n 0 s 0 P(Re Z n s 0 ) α 4 First construction Institut für Mathematik, Technische Universität Ilmenau Page 10 / 20

12 First construction testing hypothesis E Z = 0: let s 0 = t [ 1,1] ( α 4, 0), exists if α 4 > 2 n P(Re Z n s 0 ) α 4 0 s 0 P(Re Z n s 0 ) α 4 First construction Institut für Mathematik, Technische Universität Ilmenau Page 10 / 20

13 First construction testing hypothesis E Z = 0: let s 0 = t [ 1,1] ( α 4, 0), exists if α 4 > 2 n P(Im Z n s 0 ) α 4 s 0 P(Re Z n s 0 ) α 4 0 s 0 P(Re Z n s 0 ) α 4 First construction Institut für Mathematik, Technische Universität Ilmenau Page 10 / 20

14 First construction testing hypothesis E Z = 0: let s 0 = t [ 1,1] ( α 4, 0), exists if α 4 > 2 n P(Im Z n s 0 ) α 4 s 0 P(Re Z n s 0 ) α 4 0 s 0 P(Re Z n s 0 ) α 4 P(Im Z n s 0 ) α 4 First construction Institut für Mathematik, Technische Universität Ilmenau Page 10 / 20

15 First construction testing hypothesis E Z = 0: let s 0 = t [ 1,1] ( α 4, 0), exists if α 4 > 2 n P(Im Z n s 0 ) α 4 s 0 P(Re Z n s 0 ) α 4 0 s 0 P(Re Z n s 0 ) α 4 P(Im Z n s 0 ) α 4 P( Z n 2s 0 ) α First construction Institut für Mathematik, Technische Universität Ilmenau Page 10 / 20

16 First construction (cont.) testing hypothesis E Z = λζ, ζ S 1, λ > 0: ζ λζ 0 First construction Institut für Mathematik, Technische Universität Ilmenau Page 11 / 20

17 First construction (cont.) testing hypothesis E Z = λζ, ζ S 1, λ > 0: ζ λζ 0 P(Re ζ 1 Z n s 0 ) α 4 First construction Institut für Mathematik, Technische Universität Ilmenau Page 11 / 20

18 First construction (cont.) testing hypothesis E Z = λζ, ζ S 1, λ > 0: P(Im ζ 1 Z n s p) 3 8 α ζ s p λζ 0 P(Re ζ 1 Z n s 0 ) α 4 with s p = t [ 1,1] ( 3 8 α, 0) < s 0 First construction Institut für Mathematik, Technische Universität Ilmenau Page 11 / 20

19 First construction (cont.) testing hypothesis E Z = λζ, ζ S 1, λ > 0: P(Im ζ 1 Z n s p) 3 8 α ζ s p λζ 0 P(Re ζ 1 Z n s 0 ) α 4 P(Im ζ 1 Z n s p) 3 8 α with s p = t [ 1,1] ( 3 8 α, 0) < s 0 First construction Institut für Mathematik, Technische Universität Ilmenau Page 11 / 20

20 First construction (cont.) testing hypothesis E Z = λζ, ζ S 1, λ > 0: P(Im ζ 1 Z n s p) 3 8 α ζ s p 0 δ H λζ Z n P(Re ζ 1 Z n s 0 ) α 4 P(Im ζ 1 Z n s p) 3 8 α with s p = t [ 1,1] ( 3 8 α, 0) < s 0; critical angle δ H for Im ζ 1 Zn = s p First construction Institut für Mathematik, Technische Universität Ilmenau Page 11 / 20

21 Confidence set using Hoeffding s inequality Proposition (TH, FK & JW) C H = { S 1 if α 2 n+2 or Z n 2s 0, {ζ S 1 : Arg(ζ 1ˆµ n ) < δ H } otherwise, for δ H = arcsin(s p / Z n ) is a universal, non-asymptotic, level (1 α) confidence set for µ. note: if E Z 0 then P(C H = S 1 ) exp( nτ 2 /8) for some τ > 0 δ H = O P (n 1 2 ) δ H small for Z n large but variance perpendicular to µ not taken into account First construction Institut für Mathematik, Technische Universität Ilmenau Page 12 / 20

22 Estimating the variance varying the first construction: for Y k = Im ζ 1 Z k consider V n = 1 n n k=1 Y 2 k [0, 1] with E V n = Var Y k = σ 2 apply Hoeffding s inequality again to (numerically) get an upper bound on the variance fulfilling σ 2 = V n + t [0,1] ( α 4, 1 σ 2 ) such that P(σ 2 / [0, σ 2 ]) α 4 i.i.d. recall Hoeffding (1963) s Theorem 3: if W 1,..., W n W with W (, 1], E W = 0, Var W = ρ 2 then [ ( P( W n w) 1 + w ) ρ 2 w ( ) ] n w 1 1+ρ 2 ρ 2 1 w. for any w (0, 1) apply to Ȳ n using variance bound σ 2 sharper critical value s v = w( α 4, σ 2 ) perpendicular to µ Second construction Institut für Mathematik, Technische Universität Ilmenau Page 13 / 20

23 Confidence set taking variance into account Proposition (TH, FK & JW) C V = { S 1 if α 2 n+2 or Z n 2s 0, {ζ S 1 : Arg(ζ 1ˆµ n ) < δ V } otherwise, for δ V = arcsin(s v / Z n ) is a universal, non-asymptotic, level (1 α) confidence set for µ. Second construction Institut für Mathematik, Technische Universität Ilmenau Page 14 / 20

24 Simulation 1: two points of equal mass at ±10 50% 0 E Z µ 50% E Z highly concentrated distribution variance perpendicular to µ is maximal nominal coverage 1 α = 95% n = 400 independent draws in each experiment repetitions Simulations and applications Institut für Mathematik, Technische Universität Ilmenau Page 15 / 20

25 Simulation 1: two points of equal mass at ±10 Table: Results based on 10,000 repetitions with n = 400 observations each: average observed δ H, δ V, and δ A (with corresponding standard deviation), as well as frequency (with corresponding standard error) with which µ = 1 was covered by C H (first construction), C V (construction taking variance into account), and C A (asymptotic), respectively; the nominal coverage probability was 1 α = 95%. confidence set mean δ (± s.d.) coverage frequency (± s.e.) C H 8.2 (± ) 100.0% (±0.0%) C V 2.3 (± ) 100.0% (±0.0%) C A 1.0 (± ) 94.8% (±0.2%) C H, C V are conservative variance should be taken into account small standard deviation of δ Simulations and applications Institut für Mathematik, Technische Universität Ilmenau Page 16 / 20

26 Simulation 2: three points placed asymmetrically 1% 5% 0 E Z µ 94% E Z = 0.9 highly concentrated distribution nominal coverage 1 α = 90% n = 100 independent draws in each experiment repetitions Simulations and applications Institut für Mathematik, Technische Universität Ilmenau Page 17 / 20

27 Simulation 2: three points placed asymmetrically Table: Results based on 10,000 repetitions with n = 100 observations each: average observed δ H, δ V, and δ A (with corresponding standard deviation), as well as frequency (with corresponding standard error) with which µ = 1 was covered by C H (first construction), C V (construction taking variance into account), and C A (asymptotic), respectively; the nominal coverage probability was 1 α = 90%. confidence set mean δ (± s.d.) coverage frequency (± s.e.) C H 16.5 (± ) 100.0% (±0.0%) C V 5.0 (± ) 100.0% (±0.0%) C A 0.4 (± ) 62.8% (±0.5%) C H, C V still conservative coverage frequency of C A far from nominal level (90%) Simulations and applications Institut für Mathematik, Technische Universität Ilmenau Page 18 / 20

28 Summary constructed universal, non-asymptotic confidence sets for circular population means used mass concentration inequalities by Hoeffding variance perpendicular to mean can (and should) be taken into account asymptotic confidence sets are smaller but may not provide required coverage and presume uniqueness (requiring pretest?) Outlook: C V has practically useful size ( twice that of C A ) use sharper mass concentration inequalities spheres (etc.) intrinsic means (etc.) Summary Institut für Mathematik, Technische Universität Ilmenau Page 19 / 20

29 Thank you for your attention! Questions? Summary Institut für Mathematik, Technische Universität Ilmenau Page 20 / 20

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