Necessary and sufficient condition for the existence of a Fréchet mean on the circle

DOI:10.1051/ps/2012015 www.esaim-ps.org

NECESSARY AND SUFFICIENT CONDITION FOR THE EXISTENCE OF A FR´ ECHET MEAN ON THE CIRCLE

Benjamin Charlier

Abstract. Let (S¹, dS¹) be the unit circle in R² endowed with the arclength distance. We give a sufficient and necessary condition for a general probability measureμto admit a well defined Fr´echet mean on (S¹, dS¹). We derive a new sufficient condition of existenceP(α, ϕ) with no restriction on the support of the measure. Then, we study the convergence of the empirical Fr´echet mean to the Fr´echet mean and we give an algorithm to compute it.

Mathematics Subject Classification. 62H11.

Received August 1st, 2011. Revised February 9, 2012.

1. Introduction

1.1. Statistics for non-Euclidean data

In many fields of interest, results of an experiment are objects taking values in non-Euclidean spaces. A rather general framework to model such data is Riemannian geometry and more particularly quotient manifolds. As an illustration, in biology or geology, directional data are often used, seee.g.[14] or [6] and references therein.

In this case, observations take their values in the circle or a sphere, that is, an Euclidean space quotiented by the action of scaling.

The usual definitions of basic statistical concepts were developed in an Euclidean framework. Therefore, these definitions must be adapted for random variables with values in non-Euclidean spaces such as manifolds. To describe the localization of a probability distribution, one needs to define a central value such as a mean or a median. There has been multiple attempts to give a definition of a mean in non Euclidean space, see among many others [2,4,5,8,12,13,16] or [7].

In this paper, we consider the so-called Fr´echet mean, see [7,8,12] or [2] and references therein. We are particularly interested in the study of its uniqueness. The Fr´echet mean is defined on general metric spaces by extending the fact that Euclidean mean minimizes the sum of the squared distance to the data, see equation (1.3) below. To study the well definiteness of the Fr´echet mean on a manifold, two facts must be taken into account:

non uniqueness of geodesics from one point to another (existence of a cut-locus) and the effect of curvature, see e.g. [4] for further discussion. Due to the cut-locus, the distance function is no longer convex and finding conditions to ensure the uniqueness of the Fr´echet mean is not obvious. Two main directions have been explored

Keywords and phrases.Circular data, Fr´echet mean, uniqueness.

1 Institut de Math´ematiques de Toulouse Universit´e de Toulouse et CNRS (UMR 5219), F-31062 Toulouse, France.

benjamin.charlier@math.univ-toulouse.fr

Article published by EDP Sciences c EDP Sciences, SMAI 2013

in the literature: bounding the support of the measure in [3] for then-spheres and in [1,8,10,12] for manifolds, or consider special cases of absolutely continuous radial distributions, see [9] for the unit circle and or [10,11]

for projective spaces. In a sense, these two conditions control the concentration of the probability measure. The philosophy behind these works is to ensure a convexity property of the Fr´echet functional given by equation (1.2) below, seee.g.the introduction of [1] for a review of the above cited papers.

1.2. Fr´ echet mean on the circle

A standard way to extend the definition of the Euclidean mean in non-Euclidean metric space is to use the minimization property of the Euclidean mean. This definition is usually credited to Fr´echet in [7] although some authors credit it to E. Cartan, seee.g.[10]. LetS¹be the unit circle of the plane,

S¹={x²₁+x²₂= 1, (x₁, x₂)∈R²}.

endowed with the arclength distance given for allx= (x₁, x₂), p= (p₁, p₂)∈S¹by d_S1(x, p) = 2 arcsin

x−p 2

, (1.1)

wherex−p=

(x₁−p₁)²+ (x₂−p₂)² is the Euclidean norm inR². In the whole paper,μis a probability measure onS¹ and the Fr´echet functional is defined for allp∈S¹ by

F_μ(p) = 1 2

S¹d²(x, p)dμ(x). (1.2)

The Fr´echet functional is Lipschitz since by the triangle inequality we have|Fμ(p₁)−F_μ(p₂)| ≤2πd_S1(p₁, p₂) for anyp₁, p₂∈S¹. Thus,F_μ attains its minimum in at least one point and the only issue at hand is uniqueness.

Definition 1.1. We say that the Fr´echet mean of a probability measure μ in (S¹, d_S1) is well defined if F_μ admits auniqueargmin. That is, there exists a uniquep^∗∈S¹satisfyingF_μ(p^∗) = min_p∈MF_μ(p), and we note

p^∗= argmin_p∈S1F_μ(p). (1.3)

The argmins of F_μ are also called Riemannian center of mass [16] or intrinsic mean [2] as the (S¹, d_S1) is a simple one dimensional compact Riemannian manifold. The advantage of dealing with a simple object such as the circle is that curvature problems disappear and we only face the cut-locus problem. In this sense, it allows us to completely understand its effect on the non-convexity of the distance functiond_S1, and to give a complete answer about the problem of uniqueness. In what follows, we fully characterize probability measures that admit a well defined Fr´echet mean on the circle (S¹, d_S1). In particularly a necessary and sufficient condition is given in Theorem4.1, which links the existence of a Fr´echet mean for a measure μto the comparison between the distribution μ and the uniform measure λ on S¹. The surprising fact is that λ appears as a benchmark to discriminate measures having a well defined Fr´echet mean. The uniform measureλis the ‘worst’ possible case as all points of the circle is a Fr´echet mean, indeed the Fr´echet functional (1.2) is constant and equals to ^π₃³.

In opposition to what have been done before we do not try to ensure convexity property on the Fr´echet functional. Indeed, the definition of the Fr´echet mean relies on theglobal optimization problem (1.3) which is, in general, non convex. The advantage of our approach is that we do not need to restrict the support or suppose restrictive conditions of symmetry on the density. As the geometry of flat manifold is simple, we can derive explicit form on the Fr´echet functional and its derivative which can be hard to compute in non-flat manifolds such asn-dimensional spheres.

1.3. Organization of the paper

In Section2, we introduce notations that will be used throughout the paper. In Section3, we give explicit expressions for the Fr´echet functional and its derivative and we discuss some properties of critical points of the Fr´echet functional. Section4contains the main result of this paper with the necessary and sufficient condition for the existence of the Fr´echet mean for a general measure (Thm.4.1). We also propose a new sufficient criterion P(α, ϕ) that ensures the well definiteness of the Fr´echet mean for measures with density functions. In Section5, we study the convergence of the empirical Fr´echet mean to the Fr´echet mean, and describe an algorithm to compute the empirical Fr´echet mean.

2. Notations

In what follows,^½_A denotes the indicator function of the set A⊂Rand the notation _b

af(t)dμ_p0(t) stands for the Lebesgue integral

[a,b[f(t)dμ_p0(t) ifa≤band

[b,a[f(t)dμ_p0(t) ifb > a.

2.1. Normal coordinates

Given a base pointp∈S¹, there is a canonical chart called the exponential map defined fromT_pS¹R, the tangent space ofS¹at p, to S¹ and denoted by

e_p:R−→S¹

θ−→e_p(θ) =R_θp, whereR_θ=

cos(θ)−sin(θ) sin(θ) cos(θ)

.

This map is onto but not one to one as it is 2π-periodic. To guaranty the injectivity, we choose to restrict the domain of definition Rofe_p to [−π, π[. Thus, for allp₁, p₂ ∈S¹ there is a now uniqueθ^p_p¹₂ ∈[−π, π[ satisfying e_p1(θ^p_p¹₂) =p₂ and

e_p: [−π, π[−→S¹ and e⁻¹_p :S¹−→[−π, π[, for allp∈S¹.

Such parametrizations are called normal coordinates systems centered at p and θ^p_p¹₂ is nothing else but the coordinate ofp₂read in a normal coordinate system centered inp₁. To simplify the notations, we will omit the exponentp₁ if no confusion is possible and we will writeθ^p_p2¹=θ_p2.

The cut-locus of a point p₀ ∈ S¹ is denoted by ˜p₀ and is equal to the opposite point (in R²) of p₀, that is

˜

p₀=−p0. In a normal coordinates system centered atp∈S¹, the coordinate of ˜p₀isθ_p^p_˜0 =θ_p^p₀−πif 0≤θ_p^p₀ < π orθ_p^p_˜0=θ_p^p₀+πifπ≤θ^p_p0 <0.

2.2. Distance function, probability measures and the Fr´ echet functional

The arclength distance between two points p₁, p₂ ∈ S¹ was defined in (1.1). Given normal coordinates θ^p_p1, θ^p_p2∈[−π, π[ of these points in the chart centered at an arbitrary pointp∈S¹, we have

d_S1(p₁, p₂) =d_R/2πZ(θ^p_p1, θ_p^p₂) := min{θ_p^p₁−θ_p^p₂+ 2πk, k∈Z}. This means that the circleS¹ is (everywhere) locally isometric to the real lineR.

Unless specified, μwill denote a general probability measure on (S¹,B(S¹)) where B(S¹) is the Borel set of S¹⊂R². Given a pointp₀∈S¹,μ_p0 is the image measure ofμthroughe⁻¹_p0 :S¹−→[−π, π[. This is a measure onRwith a support in [−π, π[ which is defined by

μ_p0(A) =μ◦e_p0(A∩[−π, π[), for allA∈ B(R), (2.1) whereB(R) is the Borel set inR. The usual Euclidean mean/expectation and variance ofμ_p0 are denoted

m(μ_p0) =

Rtdμ_p0(t).

Finally, let us introduce for anyp₀∈S¹, the mapF_μp0 : [−π, π[−→Rgiven by F_μp0(θ) :=F_μ(e_p0(θ)) = 1

2 _π

−π

d²_R/2πZ(t, θ)dμ_p0(t)

=1 2

⎧⎨

⎩ _θ−π

−π (t+ 2π−θ)²dμ_p0(t) +_π

θ−π(t−θ)²dμ_p0(t), if 0≤θ < π, _θ+π

−π (t−θ)²dμ_p0(t) +_π

θ+π(t−2π−θ)²dμ_p0(t), if −π≤θ <0.

3. The Fr´ echet functional on the Circle

3.1. The derivative of the Fr´ echet functional

A functionf : [−π, π[−→Ris said left continuous on [−π, π[ if it is left continuous everywhere on ]−π, π[ and with lim_ε→0−f(π+ε) =f(−π). Similarly,f is said to be continuous on [−π, π[ if it is left and right continuous on [−π, π[. We provide here an explicit expression of the derivative ofF_μ.

Proposition 3.1. Let μbe a probability measure on(S¹, d_S1)and fix an arbitraryp₀∈S¹. Then,F_μ :S¹−→R is differentiable in following sense :

1. Letp∈S¹ be a point with a cut-locus ofμ-measure 0, i.e.μ({−p}) = 0. ThenF_μp0 is continuously differen- tiable atθ^p_p⁰ and we have

d

dθF_μp0(θ_p^p0) =

θ_p^p0−2πμ_p0([−π,−π+θ_p^p0[)−m(μ_p0), if 0≤θ^p_p⁰ < π,

θ_p^p0+ 2πμ_p0([π+θ_p^p0, π[)−m(μ_p0), if −π≤θ^p_p⁰ <0. (3.1) 2. The function _dθ^dF_μp0 is left continuous on [−π, π[. Then we extend the definition of the derivative ofF_μ by

setting for allθ∈]−π, π[

d

dθF_μp0(θ) := lim

ε→0⁻

d

dθF_μp0(θ+ε), (3.2)

and _dθ^dF_μp0(−π) = lim_ε→0⁻ _dθ^dF_μp0(π+ε).

3. Let p∈S¹ be a point with a cut-locus of positive measure, i.e.μ({−p})>0. Then, p is a cusp point ofF_μ in the sense thatlim_ε→0− d

dθF_μp0(θ^p_p⁰ +ε)−lim_ε→0+ d

dθF_μp0(θ_p^p0+ε) =−μ({−p}).

Note that the left-continuity comes from our convention on the exponential map which is defined on [−π, π[.

If a measureμ is such that μ({p}) = 0 for allp∈S¹ then F_μ is of classC¹ on [−π, π[. Differentiability issues appear when the measureμ has atoms, see Figure1.

Proof. For convenience we omit in this proof the superscript p₀ by writing θ_p = θ_p^p0 for all p ∈ S¹. In the coordinate system centered atp₀ we have for allθ_p∈[−π, π[

F_μp0(θ_p) = 1 2

Rt²dμ_p0(t)−θ_pm(μ_p0) +1

2θ²_p+ 2π

g⁺_μp

0(θ_p)^½_[0,π[(θ_p) +g⁻_μp

0(θ_p)^½_[−π,0[,(θ_p)

(3.3) whereg⁺_μp

0(θ) =_−π+θ

−π (π+t−θ)dμp0(t) andg⁻_μp

0(θ) =_π

θ+π(π−t+θ)dμ_p0(t).Hence, to prove Proposition3.1, we just have to study the derivative ofg_μ⁺_p

0 andg_μ⁻_p

0.

For allθ_p∈]0, π[ andε∈Rsuch thatθ_p+ε∈]0, π[ we have, 1

ε

g⁺_μp

0(θ_p+ε)−g_μ⁺_p

0(θ_p)

=1 ε

_−π+θp+ε

−π+θp

(π+t−θ_p)dμ_p0(t)−

_−π+θp+ε

−π dμ_p0(t). (3.4)

−3 −2 −1 0 1 2 3

−2

−1.5

−1

−0.5 0 0.5 1 1.5 2 2.5 3

Figure 1. Letμ= ¹₆δ_p1+²₃δ_p∗+¹₆δ_p2 withp₁=R2π

3 p^∗ andp₂ =R₋2π

3 p^∗. In blue: F_μp∗. In green: _dθ^dF_μp∗.

The limit from the left in equation (3.4) is lim_ε→0− 1 ε(g⁺_μp

0(θ_p+ε)−g_μ⁺_p

0(θ_p)) = −μ_p0([−π,−π+θ_p[) when 0 < θ_p < π. The (left) derivative atθ_p = 0 is given by lim_ε→0− 1

ε

g⁻_μp

0(θ_p+ε)−g_μ⁺_p

0(θ_p)

= 0. Similarly, if

−π≤θ_p<0, we have lim_ε→0− 1 ε(g_μ⁻_p

0(θ_p+ε)−g⁻_μp

0(θ_p)) =μ_p0([π+θ_p, π[) and statement 2 is proved.

To prove statement 1, suppose that the cut-locus ofpis ofμ-measure 0. In this case, the limit from the left and from the right in equation (3.4) are equal as lim_ε→0¹_{ε−π+θp+ε}

−π+θp (π+t−θ_p)dμ_p0(t) = 0 sinceμ_p0({θp−π}) = 0.

Thence, formula (3.3) yields _dθ^dg_μ⁺_p

0(θ_p) =−μ_p0([−π,−π+θ_p[), ifθ_p∈[0, π[ and _dθ^dg_μ⁻_p

0(θ_p) =μ_p0([π+θ_p, π[), ifθ_p∈[−π,0[.

Finally, suppose thatp∈S¹has a cut-locus of positive measure. If 0≤θ_p< π, it means thatμ_p0({θp−π})>0 and then, _dθ^dF_μp0(θ_p)−lim_ε→0+ d

dθF_μp0(θ_p+ε) =−lim_ε→0+μ_p0([−π+θ_p,−π+θ_p+ε[) =−μp0({−π+θ_p}).

The case−π≤θ_p<0 is similar and the proof of statement 3 is completed.

3.2. Local minimum of the Fr´ echet functional

The critical points of the Fr´echet functional are the points at which the derivative of F_μ, in the sens of Proposition3.1, is 0. As an immediate consequence of equation (3.1) we have

d

dθF_μp0(θ^p_p⁰) =−m(μp). (3.5)

Thus, critical points are precisely exponential barycenters (i.e. points p∈S¹ at whichm(μ_p) = 0). This fact was already shown in [5] or [12] for general manifolds. Note that a critical point of the Fr´echet functional is not in general an extremum (see an illustration Fig.1) but it is worth noticing that the minima of the Fr´echet functional are regular points in the sens of the following result,

Corollary 3.2. Letμ be a probability measure onS¹. The cut-locus of a (local or global) minimum ofF_μ is of μ-measure 0.

Proof. Let p ∈ S¹ be a point satisfying μ({−p}) > 0. For any p₀ ∈ S¹, Statement 3 of Proposition 3.1 en- sure that the derivative _dθ^dF_μp0 of the Fr´chet functional has a negative jump at θ_p^p0. Hence, the signs of

lim_ε→0− d

dθF_μp0(θ^p_pm⁰ +ε),lim_ε→0+ d

dθF_μp0(θ_p^p_m⁰ +ε)

is either (+,+), (+,−) or (−,−). This means that θ^p_p⁰ cannot be a minimum ofF_μp0 since it would correspond to the case (−,+).

Remark 3.3. Note that assumptions of Corollary 1 in [17] and Theorem 1 in [2] contain a condition of the form μ({˜p}) = 0 to ensure (classical) differentiability of the Fr´echet functional at its minimum. In the case of the circle, Corollary3.2shows us that the Fr´echet functional isautomatically differentiable at its minima.

4. Necessary and sufficient condition for the existence of the Fr´ echet mean

4.1. Main result

Theorem 4.1. Let μ be a probability measure and p^∗ ∈ S¹ be a critical point of F_μ. Then, the following propositions are equivalent,

1. p^∗ is a well defined Fr´echet mean of (S¹, μ).

2. For all0< θ < π _θ

0

λ([−π,−π+t[)−μ_p∗([−π,−π+t[)dt >0, and for all−π≤θ <0,

₀

θ

λ([π+t, π[)−μ_p^∗([π+t, π[)dt >0, whereλis the uniform measure on[−π, π[andμ_p^∗ is defined in (2.1).

Theorem 4.1 gives a necessary and sufficient condition for the existence of the Fr´echet mean of a general measure μ on the circle S¹. This condition is given in terms of comparison between the μ-measure and the uniform measure λ of balls centered at the cut-locus of a global minimum. The important point is that the μ-measure of a (small) neighborhood of the cut-locus of p^∗ cannot be larger than the uniform measure of this neighborhood.

Asμis a probability measure, the functionst−→λ([−π,−π+t[)−μp^∗([−π,−π+t[) andt−→λ(]π−t, π[)− μ_p∗(]π−t, π[) do not need to be always positive fort∈[−π,0[ andt∈[0, π[ respectively. An example where this quantity is always positive is whenμadmits a density which is a decreasing function of the distance to a point p^∗, see [9]. In this case, the density is radially distributed around its mode p^∗ which is, by Theorem4.1, the Fr´echet mean ofμ. Many classical probability distributions used in circular data analysis follow this framework:

von Mises distribution, wrapped normal distribution, geodesic normal distribution [17],etc.

Another well-known example of distributions that admit a well defined Fr´echet mean is distributions with support included in an hemisphere, see [3]. More precisely, suppose that there exists a point ˆp∈S¹ withμ({p∈ S¹, d_S1(p,p)ˆ ≤ ^π₂}) = 1 andμ({p∈S¹, d_S1(p,p)ˆ < ^π₂})>0. In this case, Statement 2of Theorem4.1holds since one can show that the minimump^∗ofF_μis in{p∈S¹, d_S1(p,p)ˆ < ^π₂}and thatF_μp∗(θ)−Fμp∗(0)> ¹₂(θ+π−2θpˆ)² forθ∈[−π, θ_pˆ−^π₂[,F_μp∗(θ)−F_μp∗(0) = ^θ₂² forθ∈[θ_pˆ−^π₂, θ_pˆ+^π₂[ andF_μp∗(θ)−F_μp∗(0)> ¹₂(θ−π−2θ_pˆ)² for allθ ∈[θ_pˆ+ ^π₂, π[. The case of equality corresponds to distributions with support in the boundary of the hemisphere, that is μ_p∗ = (1−ε)δ_θp∗

ˆ

p −^π₂ +εδ_θp∗ ˆ

p +^π₂ with ε= _π¹θ_p^p_ˆ^∗ +¹₂ and in this case, there are two global argmins at 0 and 2θ_pˆ±π, see Figure2.

4.2. Proof of Theorem 4.1

As already noticed, there exists at least one global argminp^∗ ∈S¹ of F_μ sinceF_μ is a continuous function defined on the compact setS¹. Moreover, Proposition3.1and Corollary3.2ensure thatp^∗is a regular critical point ofF_μ,i.e. a zero of the derivative with a cut-locus ofμ-measure 0.

−3 −2 −1 0 1 2 3 1

1.5 2 2.5 3 3.5

(a)

−3 −2 −1 0 1 2 3

1.4 1.6 1.8 2 2.2 2.4

(b)

Figure 2.Letμ_θ= (1−ε)δ_θ−^π₂ +εδ_θ+^π₂ withε= ^θ_π+¹₂. (a)F_μθ withθ= 0.(b)F_μθ withθ= ^3π₁₀. In the normal coordinate system centered atp^∗the functionalG_μp∗(θ) =F_μp∗(θ)−Fμp∗(0) has a particularly simple expression thanks to equation (3.5). Indeed, we have

G_μp∗(θ) =1

2θ²+ 2π^½_[0,π[(θ) _θ−π

−π (π+t−θ)dμ_p^∗(t) + 2π^½_[−π,0[(θ) _π

θ+π

(π−t+θ)dμ_p^∗(t).

It is clear that Statement1is equivalent to the fact that the unique zero ofG_μp∗ is 0. Thence, Theorem 5.1 will be an easy consequence of Lemma4.2below since we haveλ([−π,−π+t[) =_2π^t for all 0≤t < π.

Lemma 4.2. Let μbe a probability measure on S¹ andp^∗∈S¹ be an argmin ofF_μ. Then for anyθ∈[−π, π[

G_μp∗(θ) = 2π

⎧⎨

⎩ _θ

0 t

2π−μ_p∗([−π,−π+t[)dt, if0≤θ < π, ₀

θ −t

2π −μ_p^∗([π+t, π[)dt, if −π≤θ <0 Proof. The probability measureμcan be decomposed as follow,

μ=aμ^d+ (1−a)μ^δ, 0≤a≤1, (4.1)

whereμ^dis a probability measure such thatμ_d({p}) = 0 for allp∈S¹andμ_δ =_+∞

j=1ω_jδ_pj where_+∞

j=0ω_j = 1 and thep_j’s are in S¹. Hence, we consider the two cases separately: first, when the measure is non atomic, and then, when it is purely atomic. The general case follows immediately in view of equation (4.1).

First, assume thatμis an atomless measure ofS¹. Proposition3.1ensures thatF_μis continuously differentiable everywhere and the real functionF_μp∗ is of classC¹ on [−π, π[. Formula (3.1) and the fundamental theorem of calculus gives for allθ∈[−π, π[

G_μp∗(θ) = _θ

0

tdt−2π _θ

0 μ_p^∗([−π,−π+t[)dt, if 0≤θ < π, ₀

θ μ_p^∗([π+t, π[)dt, if −π≤θ <0. (4.2) Consider now the case where μ is a purely atomic measure. First, we treat the case where the number of mass of Dirac in the sum is finite, i.e. μ = _n

j=1ω_jδ_pj, n ∈ N. Recall that F_μp∗ is a Lipschitz function on

[−π, π[. Proposition 3.1 ensures that the derivative is piecewise continuous and formula (3.1) holds for all θ∈[−π, π[\{θ_p˜j}ⁿ_j=1,i.e. points that have a cut-locus ofμ-measure 0. Hence for allθ∈[−π, π[, equation (4.2) holds too.

To treat the case where μ=_+∞

j=1ω_jδ_xj we proceed by approximation. Let φ(n) ={j ∈N| ω_j ≥ ₂¹n} and remark that Card(φ(n)) < +∞ for all n ∈ N since _+∞

j=1ω_j = 1. Then if ν_pⁿ∗ = _c(n)¹

j∈φ(n)ω_jδ_xj, where c(n) =

j∈φ(n)ω_j is a normalizing constant, we have for all θ∈[−π, π[, G_νⁿ

p∗(θ) = _θ

0

tdt−2π _θ

0 ν_pⁿ∗([−π,−π+t[)dt, if 0≤θ < π, ₀

θ ν_pⁿ∗([π+t, π[)dt, if −π≤θ <0.

The sequence (ν_pⁿ∗)_n≥1 converges toμ in total variation. By the dominated convergence Theorem for allθ_p ∈ [−π, π[,G_νn

p∗(θ) converge asn→ ∞ to (4.2).

4.3. The criterion P ( α, ϕ )

Although the necessary and sufficient condition of Theorem4.1 is a key step to understand the problem of non uniqueness of the Fr´echet mean on S¹, it is of little practice interest: we have to know a priori a critical pointp^∗of the Fr´echet functional. In this section, we derive sufficient conditions of existence with no restriction on the support of the probability measure that are easily usable. Let us introduce the following definition:

Definition 4.3. Letf : S¹ −→ R⁺ be a probability density, p ∈ S¹, α∈]0,1] andϕ ∈]0, π[. We say that f satisfies the propertyP(p, α, ϕ) (or to be shortf ∈P(p, α, ϕ)) if for all|θ| ≥ϕ

f_p(θ)≤1−α

2π , (4.3)

where f_p=f ◦e_p: [−π, π[−→R⁺. Moreover, we say thatf ∈P(α, ϕ) if there is a p∈S¹ such that f satisfies P(p, α, ϕ).

The parametersαandϕcontrol the concentration ofμ aroundp. The idea is to control the mass lying in the complementary of the ballB(p, ϕ), see Figure 3. We have the following properties:

Lemma 4.4. Let f :S¹−→Rbe a probability density on the circle. Then 1. f ∈P(p, α₁, ϕ₁) =⇒f ∈P(p, α₂, ϕ₂)if α₁≥α₂ andϕ₁≤ϕ₂.

2. Letp₁, p₂∈S¹ andϕ < ^π₂. Ifd_S1(p₁, p₂)< π−ϕ thenf ∈P(p₁, α, ϕ) =⇒f ∈P(p₂, α, ϕ+d_S1(p₁, p₂)).

3. Iff satisfies P(p, α, ϕ) then|m(μp)| ≤ϕ+^1−α_4π (π−ϕ)². Proof. The first proposition is obvious in view of the Definition4.3.

To prove the second claim suppose that 0< θ_p^p₂¹ =−θ_p^p₁²≤π−ϕ(the other case is similar) and write f_p2(θ) =f_p1(θ+θ_p^p1₂)^½_{[−π,π−θ}^pp21[(θ) +f_p1(θ+θ^p_p¹₂−2π)^½_[π−θ^pp21,π[(θ).

In particular, it implies thatπ > π−θ_p^p₂¹ ≥ϕand sinceP(p₁, α, ϕ) holds, we havef_p2(θ)≤ ^1−α_2π , ifθ≥ϕ−θ^p_p¹₂ or θ ≤ −ϕ−θ_p^p₂¹. This is equivalent to the fact that P(p₂, α,min{| −ϕ−θ_p^p1₂|,|ϕ−θ^p_p¹₂|}) =P(p₂, α, ϕ+θ^p_p2¹) holds. The caseϕ−π≤θ_p^p₂¹ ≤0 is similar and we haveP(p₂, α, ϕ−θ^p_p¹₂). Finally recall that|θ^p_p¹₂|=d_S1(p₁, p₂) and the property is proved.

To show the last claim, we only need to consider the case whereμ_p has its support on [0, π[. Indeed,μ_p = ωμ⁻_p + (1−ω)μ⁺_p where μ⁻_p([−π,0[) =μ⁺_p([0, π[) = 1 and 0≤ω≤1. It yields that m(μ_p) =

Rtd(ωμ⁻_p + (1− ω)μ⁺_p) =

R⁺td(−ωμ⁻_p + (1−ω)μ⁺_p)≤

R⁺tdμ⁺_p. Then, if the densityf_p ofμ_p has its support in [0, π[ we have m(μ_p)≤ϕ

1−

_π

ϕ

f_p(t)dt

+ _π

ϕ

tf_p(t)dt≤ϕ+1−α 2π

_π

ϕ

(t−ϕ)dt,

which gives the result.

−3 −2 −1 0 1 2 3 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Figure 3. An example of distribution satisfyingP(α, ϕ). Plot of the densityf_p in blue with the boundsP(p,0.1,2) in yellow andP(p,0.5,1.6) in red.

If the density f is sufficiently concentrated around a critical point p^∗ ofF_μ then, this point is the Fr´echet mean ofμ. More precisely we have the following result,

Proposition 4.5. Let μ be a probability measure with density f : S¹ −→R⁺ and p^∗ ∈S¹ be a critical point of F_μ. Iff satisfies P(p^∗, α, ϕ)with α∈]0,1]and0< ϕ < ϕ_α=π₁₊^√√^α

α then, μadmits a well defined Fr´echet mean at p^∗.

For allα∈]0,1], we haveϕ₀= 0≤ϕ_α< ^π₂ =ϕ₁. Note that ifα= 1 thenμhas its support included in the ball B(p^∗,^π₂) ={p∈S, d_S1(p^∗, p)≤^π₂} and whenα <1, the support ofμcan be the entire circleS¹.

Proof. Let G_μp∗(θ) = F_μp∗(θ)−F_μp∗(0) for all θ ∈ [−π, π[. As the measure μ admits a density f, G_μp∗

is twice differentiable and equation (3.1) implies that _dθ^d2₂G_μp∗(θ) = 1−2πf(−π+θ), if 0 ≤ θ < π, and

d²

dθ²G_μp∗(θ) = 1−2πf(π+θ), if−π≤θ <0. Sincef ∈P(p^∗, α, ϕ), the functionG_μp∗ is convex on [−π+ϕ, π−ϕ]

and has a unique minimum at 0. Let us show that 0 is the only argmin ofF_μp∗ on [−π, π[. Ifθ∈[π−ϕ, π[, we have thanks to Lemma4.2

G_μp∗(θ) =G_μp∗(π−ϕ) + _θ

π−ϕt−2πμ_p^∗([−π,−π+t[)dt.

Since f ∈ P(p^∗, α, ϕ) we have G_μp∗(π−ϕ) ≥ ^α₂(π−ϕ)² and the second term is bounded from below by _π

π−ϕt−2πν([−π,−π+t[)dt whereν =¹₂(δ_−ϕ+δ_ϕ). It yields forθ∈[π−ϕ, π[, G_μp∗(θ)≥ 1

2(α(π−ϕ)²−ϕ²). (4.4)

The right hand side of (4.4) is strictly positive if ϕ < ϕ_α = π₁₊^√√^α

α. Similarly, the same condition implies

G_μp∗(θ)>0 forθ∈[−π,−π+ϕ[.

We are now able to define a functional class of densities that admit a well defined Fr´echet mean without restriction on the support of the measure.

Table 1. Some values ofα_δ andδϕ_αδ depending onδ∈]0,¹₂[.

δ= 0 ₁₀^{1 1}₅ ¹₃ ¹₂

αδ≤ 0.39 0.46 0.54 0.69 1 δϕα_δ≥ 0 0.12 0.26 0.47 ^π₄

Theorem 4.6. Let 0 < δ < ¹₂ be a parameter of concentration and μ a probability measure with density f ∈P(α, ϕ)(see Def.4.3) with

α_δ ≤α≤1 and ϕ≤δϕ_α

where α_δ be the square of the root of (5−6δ+δ²)X³+ (1−δ²)X²−(2δ+ 1)X −1 that lies in ]0,1] and ϕ_α=π₁₊^√√^α

α. Then μadmits a well defined Fr´echet means.

Firstly, remark that there is no need to know a critical pointa priori. Secondly, the parameterδ controls the concentration off via the inequalityα_δ < αandϕ≤δϕ_α. There is a tradeoff betweenαand the possible value of ϕ: the smaller αis (i.e.the lessf is concentrated) the smaller ϕmust be (i.e. we need to control the value of the density on a bigger interval). As a typical example, takeδ = ¹₃: iff ∈ P(α, ϕ) with 0.69< α ≤1 and ϕ≤0.47 then there is a well defined Fr´echet mean. In Tabular 1 we give examples of numerical values. Note that the column corresponding to δ = 0 is given as a reference only as the set P(α_δ, δϕ_α) is empty for this values ofδ.

Proof. Under the hypothesis of the Theorem4.6, there existsp∈S¹ such thatf ∈P(p, α, ϕ). In this proof, we show that there is a critical pointp^∗ofF_μ satisfyingd_S1(p, p^∗)≤(1−δ)ϕ_α. Thence, by Lemma4.4,f belongs toP(p^∗, α, δϕ_α+ (1−δ)ϕ_α) =P(p^∗, α, ϕ_α) and Proposition4.5ensures thatp^∗ is the Fr´echet mean ofμ.

In the rest of the proof, we show that there existsp^∗∈S¹such that_dθ^dF_μp(θ^p_p∗) = 0 withd_S¹(p, p^∗)≤(1−δ)ϕα. To this end, suppose thatm(μ_p)≥0 (the casem(μ_p)<0 is similar) and recall that equation (3.5) implies that

d

dθF_μp(0) =−m(μp)≤0. We check that, under the hypothesis of Theorem4.6, we have _dθ^dF_μp((1−δ)ϕ_α)≥0.

Since _dθ^dF_μp is a continuous function, the intermediate value Theorem will ensure the existence of a critical pointp^∗ such that|θ^p_p∗| ≤(1−δ)ϕ_α. Equation (3.1) gives

d

dθF_μp((1−δ)ϕ_α) = (1−δ)ϕ_α−2πμ_p([−π,−π+ (1−δ)ϕ_α[)−m(μ_p).

We have−2πμp([−π,−π+ (1−δ)ϕ_α[)≥(α−1)(1−δ)ϕ_α sincef ∈P(p, α, δϕ_α) and|−π+ (1−δ)ϕ_α| ≥δϕ_α. Moreover,−m(μ)≥ −δϕα−^1−α_4π (π−δϕ_α)² by Lemma4.4Statement3. It gives,

d

dθF_μp((1−δ)ϕ_α)≥π(5−6δ+δ²)α√

α+ (1−δ²)α−(2δ+ 1)√ α−1 1 +√

α ·

This quantity is positive as soon as 1 ≥α > α_δ, where √

α_δ is the root of the polynomialX −→ (5−6δ+ δ²)X³+ (1−δ²)X²−(2δ+ 1)X−1 that lies in ]0,1]. It is easy to see thatα1

2 = 1 and numerical experiments show that the (increasing) functionδ−→α_δ takes its value in ]0.39,1[ forδ∈]0,¹₂[.

5. Fr´ echet mean of an empirical measure

5.1. Existence

LetX₁, . . . , X_n be independent and identically distributed random variables with value in (S¹, d_S1) and with probability distributionμ. The empirical measure is defined as usual by

μⁿ= 1 n

n i=1

δ_Xi