Universal, non-asymptotic confidence sets for circular means

28/10/2015
Publication GSI2015
OAI : oai:www.see.asso.fr:11784:14334

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Based on Hoeffding’s mass concentration inequalities, nonasymptotic confidence sets for circular means are constructed which are universal in the sense that they require no distributional assumptions. These are then compared with asymptotic confidence sets in simulations and for a real data set.

Universal, non-asymptotic confidence sets for circular means

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Based on Hoeffding’s mass concentration inequalities, nonasymptotic confidence sets for circular means are constructed which are universal in the sense that they require no distributional assumptions. These are then compared with asymptotic confidence sets in simulations and for a real data set.

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Universal, non-asymptotic confidence sets for circular means Thomas Hotz∗, Florian Kelma & Johannes Wieditz Institut f¨ur Mathematik Technische Universit¨at Ilmenau 2nd Conference on Geometric Science of Information 28–30 October 2015 ´Ecole polytechnique, Palaiseau Universal, non-asymptotic confidence sets for circular means Thomas Hotz∗ , Florian Kelma & Johannes Wieditz Institut f¨ur Mathematik, Technische Universit¨at Ilmenau Page 1 / 20 Circular / directional data given: Z1, . . . , Zn i.i.d. ∼ Z random variables on the circle S1 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? Universal, non-asymptotic confidence sets for circular means Introduction Thomas Hotz∗ , Florian Kelma & Johannes Wieditz Institut f¨ur Mathematik, Technische Universit¨at Ilmenau Page 2 / 20 Circular (or extrinsic) means embed S1 in C: S1 = {z ∈ C : |z| = 1} projection: π : C \ {0} → S1, π(z) = z |z| Euclidean sample mean: ¯Zn := 1 n n i=1 Zi circular (a.k.a. extrinsic) sample mean: ˆµn = argmin ζ∈S1 ¯Zn − ζ 2 = π(¯Zn) Euclidean population mean: E Z = C Z dP circular population mean: µ = argmin ζ∈S1 E Z − ζ 2 = π(E Z) unique iff ¯Zn = 0 or E Z = 0, respectively see e.g. Mardia & Jupp: Directional Statistics (2000). Universal, non-asymptotic confidence sets for circular means Introduction Thomas Hotz∗ , Florian Kelma & Johannes Wieditz Institut f¨ur Mathematik, Technische Universit¨at Ilmenau Page 3 / 20 Aim Construct confidence sets for the circular population mean (set) µ on the circle S1 Universal, non-asymptotic confidence sets for circular means Introduction Thomas Hotz∗ , Florian Kelma & Johannes Wieditz Institut f¨ur Mathematik, Technische Universit¨at Ilmenau Page 4 / 20 Asymptotics projection π : C \ {0} → S1 is differentiable, so by δ-method: Theorem (central limit theorem) If E Z = 0, i.e. if µ is unique, then √ n Arg(µ−1 ˆµn) D → N 0, E Im(µ−1Z)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−1Z)2 , i.e. the variance orthogonal to µ, is small. see e.g. Jammalamadaka & SenGupta: Topics in Circular Statistics (2001). Universal, non-asymptotic confidence sets for circular means Introduction Thomas Hotz∗ , Florian Kelma & Johannes Wieditz Institut f¨ur Mathematik, Technische Universit¨at Ilmenau Page 5 / 20 Asymptotics (cont.) consistent estimation of asympt. variance + Slutsky’s lemma gives: Corollary (asymptotic confidence sets) If E Z = 0 let δA = q1−α 2 n|¯Zn| n k=1 (Im(ˆµ−1 n Zk))2 where q1−α 2 denotes the (1 − α 2 )-quantile of the standard normal distribution N(0, 1). Then CA = {ζ ∈ S1 : | Arg(ζ−1 ˆµn)| < δA} is an asymptotic confidence set of level (1 − α). Universal, non-asymptotic confidence sets for circular means Introduction Thomas Hotz∗ , Florian Kelma & Johannes Wieditz Institut f¨ur Mathematik, Technische Universit¨at Ilmenau Page 6 / 20 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 Universal, non-asymptotic confidence sets for circular means Introduction Thomas Hotz∗ , Florian Kelma & Johannes Wieditz Institut f¨ur Mathematik, Technische Universit¨at Ilmenau Page 7 / 20 Aim Construct confidence sets for the circular population mean (set) µ on the circle S1 which are universal, i.e. no distributional assumptions (besides i.i.d.) non-asymptotic construction as acceptance region of a corresponding test Universal, non-asymptotic confidence sets for circular means Introduction Thomas Hotz∗ , Florian Kelma & Johannes Wieditz Institut f¨ur Mathematik, Technische Universit¨at Ilmenau Page 8 / 20 Bounding the deviation from the mean Theorem (Hoeffding’s inequality) If U1, . . . , Un are independent random variables taking values in the bounded interval [a, b] with −∞ < a < b < ∞ then ¯Un = 1 n n k=1 Uk with E ¯Un = ν fulfills P(¯Un − ν ≥ t) ≤ ν−a ν−a+t ν−a+t b−ν b−ν−t b−ν−t n b−a = β[a,b](t, ν) for any t ∈ (0, b − ν). note: β[a,b](t, ν) ≤ exp(−2nt2/(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 b−a n, 1 which can easily be computed numerically see Hoeffding (1963). Universal, non-asymptotic confidence sets for circular means First construction Thomas Hotz∗ , Florian Kelma & Johannes Wieditz Institut f¨ur Mathematik, Technische Universit¨at Ilmenau Page 9 / 20 First construction testing hypothesis E Z = 0: let s0 = t[−1,1](α 4 , 0), exists if α 4 > 2−n 0 Universal, non-asymptotic confidence sets for circular means First construction Thomas Hotz∗ , Florian Kelma & Johannes Wieditz Institut f¨ur Mathematik, Technische Universit¨at Ilmenau Page 10 / 20 First construction testing hypothesis E Z = 0: let s0 = t[−1,1](α 4 , 0), exists if α 4 > 2−n 0 P(Re ¯Zn ≥ s0) ≤ α 4 s0 Universal, non-asymptotic confidence sets for circular means First construction Thomas Hotz∗ , Florian Kelma & Johannes Wieditz Institut f¨ur Mathematik, Technische Universit¨at Ilmenau Page 10 / 20 First construction testing hypothesis E Z = 0: let s0 = t[−1,1](α 4 , 0), exists if α 4 > 2−n 0 P(Re ¯Zn ≥ s0) ≤ α 4 s0 P(Re ¯Zn ≤ −s0) ≤ α 4 Universal, non-asymptotic confidence sets for circular means First construction Thomas Hotz∗ , Florian Kelma & Johannes Wieditz Institut f¨ur Mathematik, Technische Universit¨at Ilmenau Page 10 / 20 First construction testing hypothesis E Z = 0: let s0 = t[−1,1](α 4 , 0), exists if α 4 > 2−n 0 P(Re ¯Zn ≥ s0) ≤ α 4 s0 P(Re ¯Zn ≤ −s0) ≤ α 4 P(Im ¯Zn ≥ s0) ≤ α 4 s0 Universal, non-asymptotic confidence sets for circular means First construction Thomas Hotz∗ , Florian Kelma & Johannes Wieditz Institut f¨ur Mathematik, Technische Universit¨at Ilmenau Page 10 / 20 First construction testing hypothesis E Z = 0: let s0 = t[−1,1](α 4 , 0), exists if α 4 > 2−n 0 P(Re ¯Zn ≥ s0) ≤ α 4 s0 P(Re ¯Zn ≤ −s0) ≤ α 4 P(Im ¯Zn ≥ s0) ≤ α 4 s0 P(Im ¯Zn ≤ −s0) ≤ α 4 Universal, non-asymptotic confidence sets for circular means First construction Thomas Hotz∗ , Florian Kelma & Johannes Wieditz Institut f¨ur Mathematik, Technische Universit¨at Ilmenau Page 10 / 20 First construction testing hypothesis E Z = 0: let s0 = t[−1,1](α 4 , 0), exists if α 4 > 2−n 0 P(Re ¯Zn ≥ s0) ≤ α 4 s0 P(Re ¯Zn ≤ −s0) ≤ α 4 P(Im ¯Zn ≥ s0) ≤ α 4 s0 P(Im ¯Zn ≤ −s0) ≤ α 4 P(|¯Zn| ≥ √ 2s0) ≤ α Universal, non-asymptotic confidence sets for circular means First construction Thomas Hotz∗ , Florian Kelma & Johannes Wieditz Institut f¨ur Mathematik, Technische Universit¨at Ilmenau Page 10 / 20 First construction (cont.) testing hypothesis E Z = λζ, ζ ∈ S1, λ > 0: 0 λζ ζ Universal, non-asymptotic confidence sets for circular means First construction Thomas Hotz∗ , Florian Kelma & Johannes Wieditz Institut f¨ur Mathematik, Technische Universit¨at Ilmenau Page 11 / 20 First construction (cont.) testing hypothesis E Z = λζ, ζ ∈ S1, λ > 0: 0 λζ ζ P(Re ζ−1 ¯Zn ≤ −s0) ≤ α 4 Universal, non-asymptotic confidence sets for circular means First construction Thomas Hotz∗ , Florian Kelma & Johannes Wieditz Institut f¨ur Mathematik, Technische Universit¨at Ilmenau Page 11 / 20 First construction (cont.) testing hypothesis E Z = λζ, ζ ∈ S1, λ > 0: 0 λζ ζ P(Re ζ−1 ¯Zn ≤ −s0) ≤ α 4 P(Im ζ−1 ¯Zn ≥ sp) ≤ 3 8 α sp with sp = t[−1,1](3 8α, 0) < s0 Universal, non-asymptotic confidence sets for circular means First construction Thomas Hotz∗ , Florian Kelma & Johannes Wieditz Institut f¨ur Mathematik, Technische Universit¨at Ilmenau Page 11 / 20 First construction (cont.) testing hypothesis E Z = λζ, ζ ∈ S1, λ > 0: 0 λζ ζ P(Re ζ−1 ¯Zn ≤ −s0) ≤ α 4 P(Im ζ−1 ¯Zn ≥ sp) ≤ 3 8 α sp P(Im ζ−1 ¯Zn ≤ −sp) ≤ 3 8 α with sp = t[−1,1](3 8α, 0) < s0 Universal, non-asymptotic confidence sets for circular means First construction Thomas Hotz∗ , Florian Kelma & Johannes Wieditz Institut f¨ur Mathematik, Technische Universit¨at Ilmenau Page 11 / 20 First construction (cont.) testing hypothesis E Z = λζ, ζ ∈ S1, λ > 0: 0 λζ ζ P(Re ζ−1 ¯Zn ≤ −s0) ≤ α 4 P(Im ζ−1 ¯Zn ≥ sp) ≤ 3 8 α sp P(Im ζ−1 ¯Zn ≤ −sp) ≤ 3 8 α ¯Zn δH with sp = t[−1,1](3 8α, 0) < s0; critical angle δH for | Im ζ−1 ¯Zn| = sp Universal, non-asymptotic confidence sets for circular means First construction Thomas Hotz∗ , Florian Kelma & Johannes Wieditz Institut f¨ur Mathematik, Technische Universit¨at Ilmenau Page 11 / 20 Confidence set using Hoeffding’s inequality Proposition (TH, FK & JW) CH = S1 if α ≤ 2−n+2 or |¯Zn| ≤ √ 2s0, {ζ ∈ S1: |Arg(ζ−1 ˆµn)| < δH} otherwise, for δH = arcsin(sp/|¯Zn|) is a universal, non-asymptotic, level (1 − α) confidence set for µ. note: if E Z = 0 then P(CH = S1) ≤ exp(−nτ2/8) for some τ > 0 δH = OP(n−1 2 ) δH small for |¯Zn| large but variance perpendicular to µ not taken into account Universal, non-asymptotic confidence sets for circular means First construction Thomas Hotz∗ , Florian Kelma & Johannes Wieditz Institut f¨ur Mathematik, Technische Universit¨at Ilmenau Page 12 / 20 Estimating the variance varying the first construction: for Yk = Im ζ−1Zk consider Vn = 1 n n k=1 Y 2 k ∈ [0, 1] with E Vn = Var Yk = σ2 apply Hoeffding’s inequality again to (numerically) get an upper bound on the variance fulfilling σ2 = Vn + t[0,1](α 4 , 1 − σ2) such that P(σ2 /∈ [0, σ2]) ≤ α 4 “recall” Hoeffding (1963)’s Theorem 3: if W1, . . . , Wn i.i.d. ∼ W with W ∈ (−∞, 1], E W = 0, Var W = ρ2 then P( ¯Wn ≥ w) ≤ 1 + w ρ2 −ρ2−w 1 − w w−1 n 1+ρ2 . for any w ∈ (0, 1) apply to ¯Yn using variance bound σ2 sharper critical value sv = w(α 4 , σ2) perpendicular to µ Universal, non-asymptotic confidence sets for circular means Second construction Thomas Hotz∗ , Florian Kelma & Johannes Wieditz Institut f¨ur Mathematik, Technische Universit¨at Ilmenau Page 13 / 20 Confidence set taking variance into account Proposition (TH, FK & JW) CV = S1 if α ≤ 2−n+2 or |¯Zn| ≤ √ 2s0, {ζ ∈ S1: |Arg(ζ−1 ˆµn)| < δV } otherwise, for δV = arcsin(sv /|¯Zn|) is a universal, non-asymptotic, level (1 − α) confidence set for µ. Universal, non-asymptotic confidence sets for circular means Second construction Thomas Hotz∗ , Florian Kelma & Johannes Wieditz Institut f¨ur Mathematik, Technische Universit¨at Ilmenau Page 14 / 20 Simulation 1: two points of equal mass at ±10◦ 0 50% 50% µ E Z E Z ≈ 0.985 highly concentrated distribution variance perpendicular to µ is maximal nominal coverage 1 − α = 95% n = 400 independent draws in each experiment 10.000 repetitions Universal, non-asymptotic confidence sets for circular means Simulations and applications Thomas Hotz∗ , Florian Kelma & Johannes Wieditz Institut f¨ur Mathematik, Technische Universit¨at Ilmenau Page 15 / 20 Simulation 1: two points of equal mass at ±10◦ Table: Results based on 10,000 repetitions with n = 400 observations each: average observed δH , δV , and δA (with corresponding standard deviation), as well as frequency (with corresponding standard error) with which µ = 1 was covered by CH (first construction), CV (construction taking variance into account), and CA (asymptotic), respectively; the nominal coverage probability was 1 − α = 95%. confidence set mean δ (± s.d.) coverage frequency (± s.e.) CH 8.2◦ (±0.0005◦) 100.0% (±0.0%) CV 2.3◦ (±0.0252◦) 100.0% (±0.0%) CA 1.0◦ (±0.0019◦) 94.8% (±0.2%) CH, CV are conservative variance should be taken into account small standard deviation of δ Universal, non-asymptotic confidence sets for circular means Simulations and applications Thomas Hotz∗ , Florian Kelma & Johannes Wieditz Institut f¨ur Mathematik, Technische Universit¨at Ilmenau Page 16 / 20 Simulation 2: three points placed asymmetrically 0 µ E Z 1% 94%5% E Z = 0.9 highly concentrated distribution nominal coverage 1 − α = 90% n = 100 independent draws in each experiment 10.000 repetitions Universal, non-asymptotic confidence sets for circular means Simulations and applications Thomas Hotz∗ , Florian Kelma & Johannes Wieditz Institut f¨ur Mathematik, Technische Universit¨at Ilmenau Page 17 / 20 Simulation 2: three points placed asymmetrically Table: Results based on 10,000 repetitions with n = 100 observations each: average observed δH , δV , and δA (with corresponding standard deviation), as well as frequency (with corresponding standard error) with which µ = 1 was covered by CH (first construction), CV (construction taking variance into account), and CA (asymptotic), respectively; the nominal coverage probability was 1 − α = 90%. confidence set mean δ (± s.d.) coverage frequency (± s.e.) CH 16.5◦ (±0.8502◦) 100.0% (±0.0%) CV 5.0◦ (±0.3740◦) 100.0% (±0.0%) CA 0.4◦ (±0.2813◦) 62.8% (±0.5%) CH, CV still conservative coverage frequency of CA far from nominal level (90%) Universal, non-asymptotic confidence sets for circular means Simulations and applications Thomas Hotz∗ , Florian Kelma & Johannes Wieditz Institut f¨ur Mathematik, Technische Universit¨at Ilmenau Page 18 / 20 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?) CV has practically useful size (≈ twice that of CA) Outlook: use sharper mass concentration inequalities spheres (etc.) intrinsic means (etc.) Universal, non-asymptotic confidence sets for circular means Summary Thomas Hotz∗ , Florian Kelma & Johannes Wieditz Institut f¨ur Mathematik, Technische Universit¨at Ilmenau Page 19 / 20 Thank you for your attention! Questions? Universal, non-asymptotic confidence sets for circular means Summary Thomas Hotz∗ , Florian Kelma & Johannes Wieditz Institut f¨ur Mathematik, Technische Universit¨at Ilmenau Page 20 / 20