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Directional Statistics
Thomas Verdebout
ULB
Bruxelles, 2013-2014
ULB
Directional Statistics
Introduction
• The theory of errors was first developed by Gauss in relation
to the needs of astronomers
• At that time, everybody thought it was natural to make the
assumption that errors take values in an Euclidean space.
• Of course, the actual topological framework of such errors is
the surface of the Earth
• Directional (or Spherical) Statistics is concerned mainly with
observations which are unit vectors in the plane or in the
three-dimensional space
• The sample space is a circle, a sphere and sometimes an
hypersphere
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Directional Statistics
Introduction
Directional Statistics can be a tool for practitioners in many
different fields:
• Astronomy, Earth Sciences: the surface of the Earth is
approximately a sphere so that spherical data arise readily in
the Earth Sciences and Astronomy.
• Meteorology: wind directions constitute natural circular data
• Biology: study of the moving of animals. Do the animals tend
to take a particular direction or are the directions uniformly
distributed?
• Also in Physics, Psychology, Medicine, Social Sciences
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Directional Statistics
Outline
1
The uniform distribution on hyperspheres
2
Other distributions on hyperspheres
3
Inference for the location based on the spherical mean
4
Depth
5
Depth on hyperspheres
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Directional Statistics
Books, References
• Watson G., Statistics on spheres (The University of Arkansas
lecture notes in the mathematical sciences), John Wiley, NY,
1983
• Fisher NI., Statistical Analysis of Circular Data, Cambridge
University Press, 1993
• Mardia KV. and Jupp P., Directional Statistics (2nd edition),
John Wiley and Sons Ltd., 2000
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Directional Statistics
The uniform distribution on hyperspheres
• One of the most important probability distribution in
Multivariate Statistcs is certainly the uniform distribution on
S k−1
• The first use of the uniform distribution on S 2 can be traced
back to Bernoulli (1734) who won a prize from the French
Academy for an essay on orbits of the planets
• Latter, Lord Rayleigh (1880) was interested in the intensity of
a superposition of a large number of vibrations of the same
frequency but with i.i.d unif(0, 2π) phases.
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Directional Statistics
The uniform distribution on hyperspheres
• Considering a sample of n i.i.d unif(0, 2π) phases θ1 , . . . , θn ,
he was interested in the distribution of the norm of the partial
sum Sn := X1 + . . . , Xn , where Xi := (cos(θi ), sin(θi ))0
• Translating to “our world”, he used the facts that
E(cos(θi )) = E(sin(θi )) = 0, E(cos2 (θi )) = E(sin2 (θi )) = 1/2,
E(cos(θi ) sin(θi )) = 0 togheter with the Central Limit
Theorem to obtain that
(2/n)kSn k2 →d χ22
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Directional Statistics
The uniform distribution on hyperspheres
• Definition. A k-random vector X has a uniform distribution
(X ∼ Unif (S k−1 )) iff X = U/kUk with U ∼ Unif(B k ), where
B k stands for the unit ball in Rk
• The uniform distribution on the hypersphere S k−1 has density
funifk (x) := Γ(k/2)/2π k/2 I[x ∈ S k−1 ],
where Γ(.)
the well-known Gamma function defined by
R ∞is z−1
Γ(z) := 0 t
e −t dt
• For S 1 and S 2 , we recover the well-known funif2 (x) = 1/2π
and funif3 (x) = 1/4π
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Directional Statistics
The uniform distribution on hyperspheres
• Definition. A k-random vector X is rotationally invariant iff
X = OX for all O ∈ SOk := {V, det(V) = 1 and V0 V = Ik }
• We have the following characterization of the rotationally
invariant distributions:
• Proposition. A k-random vector X = (X1 , . . . , Xk )0 is
rotationally invariant iff u0 X =d X1 for all u ∈ S k−1
• Proposition. A k-random vector X is rotationally invariant
on S k−1 iff X ∼ Unif (S k−1 )
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Directional Statistics
The uniform distribution on hyperspheres
• Let X ∼ Unif(S k−1 ), θ = E[X] and Σ := E[(X − θ )(X − θ )0 ]
• Since X ∼ Unif(S k−1 ), X is rotationally symmetric, θ = Oθθ
ΣO0 for any rotation O
and Σ = OΣ
• This implies that θ = 0 and that Σ is proportional to the
identity matrix Ik
• Furthermore, we have that
Σ) = tr(E[XX0 ]) = E[tr(XX0 )] = E[tr(X0 X)] = 1
tr(Σ
• This directly entails that Σ = Ik /k
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Directional Statistics
The uniform distribution on hyperspheres
• The CLT implies that for a sequence X1 , . . . , Xn of i.i.d unit
random vectors Unif(S k−1 ),
n
−1/2
n
X
Xi
i=1
converges weakly to a centered multivariate Gaussian
distribution with covariance matrix Ik /k.
• Therefore,
n
T (n) :=
k X
k
Xi k2
n
i=1
converges to a central chi-square distribution with k degrees
of freedom. We find a multivariate version of the Lord
Rayleigh result.
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Directional Statistics
The uniform distribution on hyperspheres
• The statistic T (n) can be used to test the null hypothesis of a
uniform distribution on the sphere. More precisely, one can
reject the null hypothesis of a uniform distribution on the
sphere when T (n) exceeds the α-upper quantile of a
chi-square distribution with k degrees of freedom
• Nevertheless, one can imagine that there exist several
alternatives under which the asymptotic test based on T (n)
has no power.Typically under non-uniform alternatives with
θ=0
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Directional Statistics
The uniform distribution on hyperspheres
• The uniform distribution on S k−1 plays an important role in
Multivariate Statistics
• In particular, when X ∼ Nk (0, Ik ), X/kXk ∼ Unif(S k−1 )
• In the context of elliptical distributons. A k-random vector X
has an elliptical ditrsibution with location θ and scatter matrix
Σ if it can we written as
X =d θ + d Σ 1/2 U,
where U ∼ Unif(S k−1 ), d > 0 is independent of U
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Directional Statistics
Other distributions on hyperspheres
• Non-uniform distributions came to the attention of
mathematicians and statisticians in the 20th century
• The analysis of spherical statistics essentially started with
R.A. Fisher (1953). From the mid-1950s,Watson further
developed methodologies for spherical (and circular) statistics
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Directional Statistics
Other distributions on hyperspheres
• To describe distributions on hyperspheres, two different
Statistical/Probabilistic ways are possible
• First, one can find distributions which are relatively easy to
handle mathematically and which reasonably “fit data at
hand”
• Another way to find distributions is to define an estimator and
ask for which distribution the estimator is always the MLE
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Directional Statistics
Other distributions on hyperspheres
• For example, we may ask: for which distributions on R with
density ofP
the form f (x − θ) is the sample mean
x̄ := n−1 ni=1 xi always the MLE for θ?
• By differentiation, the problem can be reformulated. Letting
ϕf := f 0 /f , an equivalent question is to ask which density f is
such that
n
X
ϕf (xi − x̄) = 0
i=1
• Gauss showed that the only density for which the equation
just above hold is the gaussian density...
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Directional Statistics
Other distributions on hyperspheres
• An analog of the result obtained by Gauss can be discussed on
the sphere. Consider densities on the hypersphere S k−1 of the
form
f (θθ 0 x)
for some θ ∈ S k−1
• A natural question is for which distributions on S k−1 is the
MLE of θ given by the spherical mean
Pn
i=1 xi /k
Pn
i=1 xi k
?
• Wrapping up, the objective is to find the density f such that
θ̂θ =
Pn
i=1 xi /k
Pn
i=1 xi k
maximize (in θ ) f (θθ 0 x)
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Directional Statistics
Other distributions on hyperspheres
• Consider the MLE θ̂θ f for f (θθ 0 x). Using a Lagragian multiplier
λ for this constrained maximization problem (kθθ k = 1), one
directly obtains that θ̂θ f must satisfy
( P
n
0 θ ) = 2λθ̂
θf
f
i=1 xi ϕf (xi θ̂
0
θθ̂ f θ̂θ f = 1
• From the system just above, we deduce
Pn
θ̂θ f =
k
0 θ)
i=1 xi ϕf (xi θ̂
0 θ )k
i=1 xi ϕf (xi θ̂
Pn
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Directional Statistics
Other distributions on hyperspheres
• Now, consider another density g (θθ 0 x) and the corresponding
MLE θ̂θ g . Duerinckx and Ley (2012) recently obtained that the
equality
Pn
0 θ)
i=1 xi ϕf (xi θ̂
Pn
k i=1 xi ϕf (x0i θ̂θ )k
Pn
= θ̂θ f = θ̂θ g =
k
0 θ)
i=1 xi ϕg (xi θ̂
0 θ )k
i=1 xi ϕg (xi θ̂
Pn
holds for n ≥ 3 if and only if ϕf = κϕg for some positive
constant κ
• In particular, this entails that the function f which
corresponds to the spherical mean is such that ϕf = κ. That
is f (u) = exp(κu)
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Directional Statistics
Other distributions on hyperspheres
• One of the most famous distribution on the sphere is the
Fisher-von Mises-Langevin (FVML) distribution
• Its density is of the form
x 7→ cκ exp[κx0θ ]
• In fact, this density had arisen in Langevin’s (1905) statistical
mechanism discussion of magnetism. Von Mises (1918)
suggested it in a problem related with atomic weights. Then,
it was clearly introduced by the seminal paper of Fisher
(1953)...FVML is fine
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Directional Statistics
Other distributions on hyperspheres
• The parameter κ > 0 is called the concentration parameter of
the FVML distribution
• When κ is big, many observations are expected in the vicinity
of the modal direction θ
• On the contrary, when κ is closed to zero, the data is less
concentrated around θ
• When κ tends to zero, we get closer to the uniform case
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Directional Statistics
Other distributions on hyperspheres
• Another useful density is given by
x 7→ cκ exp[κ cos−1 (x0θ )]
• It was first introduced by Purkayastha (1991) who showed
that it is characterized by the property that the MLE of θ is
the so-called sample median direction
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Directional Statistics
Other distributions on hyperspheres
• There exist many other distributions. Sometimes the
observations are not directions but rather axes. That is, the
vectors x and −x are undistinguishable so that it is +x or −x
which is observed. In this context, it is natural to consider
probability density functions which are antipodally symmetric
in the sense that
f (x) = f (−x)
• A typical example of antipodally symmetric distributions is the
family of Watson distributions
x 7→ cκ exp[κ(x0θ )2 ]
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Directional Statistics
Other distributions on hyperspheres
• A very important property of the FVML distributions is that
they are rotationally symmetric about their modal directions θ
• Saw (1978) has abstracted this property by considering
general distributions with densities of the form f (θθ 0 x)
• The rotationally symmetric distributions enjoy many attractive
mathematical properties
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Directional Statistics
Inference for the location based on the spherical mean
• Consider first the tangent-normal decomposition
X = (X0θ )θθ + (Ik − θθ 0 )X
= (X0θ )θθ + k(Ik − θθ 0 )XkSθ (X),
where Sθ (X) := (Ik − θθ 0 )X/k(Ik − θθ 0 )Xk
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Directional Statistics
Inference for the location based on the spherical mean
• Under any rotationally symetric distribution,
Sθ (X) := (Ik − θθ 0 )X/k(Ik − θθ 0 )Xk is uniformly distributed
on Sθk−2
:= {v, kvk = 1, v0θ = 0}
⊥
• If X has a density f (θθ 0 x), the density of X0θ is given by
t 7→ c f (t)(1 − t 2 )(k−3)/2
• Lemma. Let U = (U1 , . . . , Uk )0 ∼ Unif (S k−1 ). Then the
density of U1 is given by
u 7→
Γ(k/2)
2 k−3
(1
−
u
) 2
π k/2 Γ((k − 1)/2)
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Directional Statistics
Inference for the location based on the spherical mean
• Proof. First, we give a definition of the multivariate Dirichlet
distribution (or multivariate beta distribution)
• Definition. We say that X has a Dirichlet distribution
Dk (p; pnk+1 ), p = (p1 , . . . , pk ) (on o
P
T k := x ∈ Rk , xi > 0, ki=1 xi < 1 ) iff
X =d
Z
,
T
where Z1 , . . . , Zk+1 are independent and such that
P
Zi ∼ Gamma(pi ), Z := (Z1 , . . . , Zk )0 and T := k+1
i=1 Zi
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Directional Statistics
Inference for the location based on the spherical mean
• Recall that a random variable Z has a gamma distribution
Z ∼ Gamma(p) with parameter p iff
fZ (z) = (Γ(p))−1 z p−1 e −z I[z > 0]
• To obtain the density of X = (X1 , . . . , Xk )0 ∼ Dk (p; pk+1 ), we
first consider the joint density of Z1 , . . . , Zk+1 which in view
of the independence is given by
fZ1 ,...,Zk+1 (z1 , . . . , zk+1 ) =
k+1
Y
i=1
!−1
Γ(pi )
k+1
Y
i=1
zi > 0 ∀i
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!
zipi −1
exp[−
k+1
X
i=1
zi ],
Inference for the location based on the spherical mean
• Then, we can obtain the joint distribution of (X1 , . . . , Xk ) and
T =
Pk+1
i=1
Zi
• Use simply the transformation (z1 , . . . , zk+1 ) onto
(x1 , . . . , xk , t), where
zi = txi , i = 1, . . . , k
and zk+1 = t(1 −
k
X
i=1
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xi )
Inference for the location based on the spherical mean
• Computing the Jacobian of this transformation, we obtain
(the absolute value of)

t
0
x1
..

..

.
.
det 
 0
t
xn
P
−t . . . −t 1 − ki=1 xi






 = det 



0 x1
.. 
..
.
. 

0
t xn 
0 ... 0 1
t
= tk
• We obtain that the join pdf of (X1 , . . . , Xk ) and
T =
Pk+1
k+1
Y
i=1
i=1
Zi which is given by
!−1
Γ(pi )
k
Y
!
xipi −1
i=1
1−
n
X
!pk+1 −1
xi
i=1
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Directional Statistics
t
Pk+1
i=1
pi −1 −t
e
Inference for the location based on the spherical mean
• Finally, integrating with respect to t, we obtain
P
Γ( k+1 pi )
fX1 ,...,Xk (x1 , . . . , xk ) = Qk+1i=1
i=1 Γ(pi )
k
Y
!
xipi −1
i=1
1−
n
X
!pk+1 −1
xi
i=1
• Note that the Dirichlet distribution Dk (1; 1) is the uniform
distribution on T k
• Furthermore, we have that when
X = (X1 , . . . , Xk )0 ∼ Dk (p; pk+1 ), then, the marginals are also
Xil )0 ∼ Dl (pi ; q) with
Dirichlet. More precisely,
(Xi1 , . . . , P
P
l
pi = (pi1 , . . . , pil )0 and k+1
i=1 pi −
j=1 pil = q
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Directional Statistics
Inference for the location based on the spherical mean
• Now, we get back to the objective which is to show that he
density of U1 is given by
u 7→
Γ(k/2)
2 k−3
(1
−
u
) 2
π k/2 Γ((k − 1)/2)
• The univariate Gamma distribution si such that y =d χ2m iff
y =d 2z where z ∼ G (m/2).
2 /2
• Consider X1 , . . . , Xk+1 i.i.d. N (0, 1). Then, X12 /2, . . . , Xk+1
are i.i.d. G (1/2).
Pk+1 As2 a direct2consequence,
Pk+1 2 the vector
2
Z = ((X1 / i=1 Xi ), . . . , Xk / i=1 Xi )) is Dk ( 21 1, 12 ).
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Inference for the location based on the spherical mean
• But Z is a k-dimensional “sub-vector” of
P
Pk+1 2
2
2
((X12 / k+1
i=1 Xi ), . . . , Xk+1 /
i=1 Xi )) which has the same
2
2
2 )0 with
distribution as U := (U1 , . . . , Uk+1
0
k
U = (U1 , . . . , Uk+1 ) ∼ Unif(S )
• This directly entails, since the sub-vector of a Dirichlet vector
is Dirichlet vector, that any “sub-vector” of “the square of a
uniformly distributed on the unit sphere” is Dirichlet
• In particular, the square of the first marginal of
U = (U1 , . . . , Uk )0 ∼ Unif(S k−1 ) is D1 ( 12 , 12 (k − 1))
• Its density is given by
x
7→
=
Γ(k/2)
x −1/2 (1 − x)(k−3)/2
(Γ(1/2))k Γ((k − 1)/2)
Γ(k/2)
x −1/2 (1 − x)(k−3)/2
k/2
π Γ((k − 1)/2)
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Directional Statistics
Inference for the location based on the spherical mean
• Finally, the result is obtained by considering the
transformations u 7→
√
√
u and u 7→ − u.
• Remember that for such “many-to-one” transformation,
δxR
L
“fY (y ) = f (xL )| δx
δy | + f (xR )| δy |”
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Directional Statistics
Inference for the location based on the spherical mean
• Then, using the fact that for U ∼ Unif(S k−1 ), z0 U =d U1 for
any z ∈ S k−1 , we have that (let ωk := 2π k/2 /Γ(k/2)) be the
“area” of S k−1 )
Z
S k−1
ωk−1 f (x0θ ) dx = E[f (U0θ )] = E[f (U1 )],
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Directional Statistics
Inference for the location based on the spherical mean
• Using the density of U1 , we obtain that
Z
1
E[f (U1 )] =
Z−1
k−3
Γ(k/2)
(1 − u 2 ) 2 f (u) du
− 1)/2)
π k/2 Γ((k
=
S k−1
ωk−1 f (x0θ ) dx
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Directional Statistics
Inference for the location based on the spherical mean
• Wrapping up, we have the following results in the rotationally
symmetric case
• Under any rotationally symetric distribution,
Sθ (X) := (Ik − θθ 0 )X/k(Ik − θθ 0 )Xk is uniformly distributed
on Sθk−2
:= {v, kvk = 1, v0θ = 0}
⊥
• If X has a density f (θθ 0 x), the density of X0θ is given by
t 7→ c f (t)(1 − t 2 )(k−3)/2
• The sign Sθ (X) and X0θ are independent
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Directional Statistics
Inference for the location based on the spherical mean
• Consider the tangent-normal decomposition
X = (X0θ )θθ + (Ik − θθ 0 )X
= (X0θ )θθ + k(Ik − θθ 0 )XkSθ (X),
• Using results obtained just before, we have that (in the
rotationally symmetric case)
E[X] = E[(X0θ )]θθ
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Directional Statistics
Inference for the location based on the spherical mean
• Then, the CLT implies that n1/2 (X̄ − E[(X0θ )]θθ ) converges
weakly to a Gaussian distribution with mean 0 and covariance
matrix Vf given by
Vf
:= E[XX0 ] − E[X](E[X])0
=
E[(X0θ )2 ]θθθ 0 − E2 [(X0θ )]θθθ 0 + E[1 − (X0θ )2 ]
=
θθ 0 Var(X0θ ) + E[1 − (X0θ )2 ]
(Ik − θθ 0 )
(k − 1)
(Ik − θθ 0 )
(k − 1)
• The asymptotic distribution of the spherical mean X̄/kX̄k can
be obtained using directly the delta method
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Inference for the location based on the spherical mean
• We have that n1/2 (X̄/kX̄k) − θ ) converges weakly to a
Gaussian distribution with mean 0 and covariance matrix
cf (Ik − θθ 0 )
• The constant cf = E[1 − (X0θ )2 ]/(k − 1)E2 [X0θ ] can be
estimated consistantly
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Directional Statistics
Inference for the location based on the spherical mean
• One can use this result to construct asymptotic tests for
H0 : θ = θ 0
• More precisely, it directly follows that under the null,
T n (θθ 0 ) :=
n
(X̄/kX̄k) − θ 0 )0 (Ik − θ 0θ 00 )(X̄/kX̄k) − θ 0 )
ĉf
is asymptotically chi-square with k − 1 degrees of freedom
• One can use T n (θθ ) to construct confidence bands for θ
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Inference for the location based on the spherical mean
• Using the previously presented result, we can also perform
ANOVA...
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Directional Statistics
Depth
• Depth functions represent a recently emerging powerful
methodology in nonparametric multivariate inference. They
provide multivariate notions of order statistics and generate
quantile contours, outlyingness functions, and sign and rank
functions.
• Univariate nonparametric analysis relies heavily on signs and
ranks, order statistics, quantiles,etc
• In Rk , there is no natural order and therefore no
straightforward extension of the above concepts.
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Depth
• For example, whereas the median of a univariate data set
represents a notion of center, the k-vector of coordinatewise
medians can lie outside the convex hull of the data.
• Depth functions constructively solve this problem by
introducing a notion of center as the maximal-depth point and
providing a center-outward ordering of points x in Rk
• Many interesting approaches toward construction of suitable
depth functions have been put forth, beginning with the
seminal paper of Tukey (1975)
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Directional Statistics
Depth
• Let F be a cdf and x ∈ Rk
• The halfspace depth (Tukey 1975): for x ∈ Rk ,
DH (x, F ) = inf{F (H) : x ∈ H closed halfspace},
the minimal probability attached to any closed halfspace with
x on the boundary.
• In particular, the sample halfspace depth of x is the minimum
fraction of data points in any closed halfspace containing x
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Directional Statistics
Depth
• Simplicial Depth: the wide potential scope of depth functions
became clear with the introduction of an important second
one, the simplicial depth (Liu 1988): for x ∈ Rk ,
DS (x, F ) = P(x ∈ S[X1 , ..., Xk+1 ]),
where X1 , ..., Xk+1 represent independent observations from F
and S[X1 , ..., Xk+1 ] denotes the simplex in Rk with vertices
X1 , ..., Xk+1 .
• For a data set in R2 , the sample simplicial depth of a point x
is obtained by considering all triangles formed with three data
points as vertices and taking the fraction of them that cover x.
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Depth
• Some basic properties are desired of any depth function. For
example, affine invariance requires that a depth function
D(x, F ) be independent of the coordinate system.
• When F is symmetric about θ in some sense, D(x, F ) should
also be symmetric about θ as well as maximal at this point.
• Also desirable is that D(x, F ) decrease along each ray outward
from the deepest point.
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Depth
• Not only the underlying pointwise functions D(x, F ), but also
the associated contours, or equivalence classes of points of
equal depth, play special roles.
• Linked with the contours are useful central regions
{x, D(x, F ) ≥ α}, α > 0
• Thus, for example, the univariate boxplot may be extended for
F on Rk by using central regions to describe a middle half or
middle 75% of the population.
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Depth
• A quantile function and a rank function can be associated
with any depth function as discussed above, but also a
quantile function and a rank function.
• For D(x, F ) possessing nested contours enclosing the median
θ Med and bounding central regions {x, D(x, F ) ≥ α}, α > 0,
the depth contours induce a quantile representation
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Depth
• For the median (the point with maximal depth) let it be
Q(0, F ).
• For x 6= θ Med denote it by Q(u, F ) with u = pv where p is the
probability weight of the central region with x on its boundary
and v is the unit vector joining x and θ Med
• In this case, u = R(x, F ) indicates direction toward
x = Q(u, F ) from θ Med
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Directional Statistics
Depth on hyperspheres
• Regina Liu and Kesar Singh (Annals of Statistics, 1992)
introduced concepts of data depth on circles and spheres
• Three different concepts: angular simplicial depth, angular
Tukey’s depth and arc distance depth
• Three medians are derived from these depth concepts
• The concept of depth on spheres leads to a proper notion of
“center” and “center-outward ranking” of directional data
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Directional Statistics
Depth on hyperspheres
• The ranking induced by a notion of depth can be used for
example in Classification problems
• Suppose that you have two training samples (X1 , . . . , Xm ) and
(Y1 , . . . , Yn ) from two different populations on the sphere.
The problem is to classify a new vector Z in one of those two
populations.
• The proposed rule is to classify Z in X if rX /m < rY /m where
rX and rY denote respectively the center-outward ranks of Z
among the Xi ’s and the Yi ’s.
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Depth on hyperspheres
• In general, a depth function gives of a point x ∈ Rk is a
measure of “how central” the point x is relative to a
probability measure.
• In the general Multivariate case, different concepts of depth
are well-known. They have different properties; in general, a
“nice” depth function is (i) monotone relative to any deepest
points, (ii) vanishing at ∞, (iii) maximal at center, etc
• In this couse, I recall the notions of Tukey half space depth
and simplicial depth
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Directional Statistics
Depth on hyperspheres
• We start with the angular simplicial depth. Angular simplicial
depth is an analog for directional data of the simplicial depth
for data on Euclidean spaces
• In Rk , a simplex S(x1 , . . . , xk+1 ) with k + 1 vertices is defined
by the closest convex hull with extremities at these points
• Let F be a cdf and x ∈ Rk . The simplicial depth of x with
respect to F is then defined to be the probability that x
belongs to a simplex S(X1 , . . . , Xk+1 ) where X1 , . . . , Xk+1 are
i.i.d. F .
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Directional Statistics
Depth on hyperspheres
• The edges of a simplex in Rk are the line segments connecting
vertices
• Moving to spheres, the idea is to replace “line segments” by
“shortest curve” joining a pair of points on the sphere
• The shortest curve joining a pair of points x1 , x2 on the sphere
is the short arc joining the points x1 and x2 on the circle
which passes through x1 and x2 and which has the same
center as the sphere (a great circle)
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Directional Statistics
Depth on hyperspheres
• Definition. Let x ∈ S 1 . The angular simplicial depth of x
with respect to a cdf F is defined by
ASD(x) := P[x ∈ arc(X1 , X2 )],
where X1 and X2 are i.i.d. F .
• Definition. Let x ∈ S 2 . The angular simplicial depth of x
with respect to a cdf F is defined by
ASD(x) := P[x ∈ ∆(X1 , X2 , X3 )],
where ∆(x1 , x2 , x3 ) stands or the spherical triangle bounded
by the short arcs arc(x1 , x2 ), arc(x1 , x3 ) and arc(x2 , x3 ) and
where X1 , X2 and X3 are i.i.d. F .
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Depth on hyperspheres
• The generalization to the hypersphere case is evident
• A maximum point for ASD is called an angular spherical
median
• Let F be a cdf on the unit circle S 1 and f be the
corresponding density
• Proposition (Monotonicity of ASD). Suppose that f is
symmetric about θ ∈ S 1 and decreases monotonically on both
sides of θ 0 until the opposite point −θθ . Then, ASD is also
monotonically nonincreasing in both directions from θ to −θθ .
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Directional Statistics
Depth on hyperspheres
• Definition. Let F be the cdf of a random vector X taking
values in S k−1 . F is antipodally symmetric if X =d −X. If F
has a continuous density f , then f (x) = f (−x).
• ASD provides a characterization of the antipodally symmetric
distributions
• Proposition. Assume that f is continuous. Then ASD(x) = c
for some constant c if and only if f (x) = f (−x). Moreover,
1
2
On S 1 , the constant c must be 1/4
On S 2 , the constant c must be 1/8
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Directional Statistics
Depth on hyperspheres
• Proof of 1 (circular case). First, note that It can be shown
that ASD a admits a “differential formula”. Let θ ∈ [0, 2π[,
then (using the abuse of notation ASD(θ))
d
ASD(θ) = 2(Aθ − Cθ )f (θ),
dθ
where Aθ and Cθ stand for the probabilities of the semicircles
joining θ and −θ in the counterclockwise and clockwise
directions respectively. This comes from the fact that for
some perturbation δθ, we have
Z
θ+δθ
ASD(θ + δθ) − ASD(θ) = 2(Aθ − Cθ )
f (u)du + o(δθ)
θ
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Depth on hyperspheres
• To obtain the formula just above, one can use the equality
P(E1 ) − P(E2 ) = P(E1 − E2 ) − P(E2 − E1 ), we have that
ASD(θ + δθ) − ASD(θ) = P(A) − P(B),
where
A := {θ ∈
/ arc(X1 , X2 ), θ + δθ ∈ arc(X1 , X2 )}
and
B := {θ ∈ arc(X1 , X2 ), θ + δθ ∈
/ arc(X1 , X2 )}
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Depth on hyperspheres
• Then, to prove →, assume that ASD(θ) = c for a positive
constant c and for all θ ∈ [0, 2π[. Then, the differential
formula directly entails that Aθ = Cθ for all θ and therefore,
f (θ) = f (−θ) for all θ
• Now, for ←, since f (θ) = f (−θ) for all θ, we have that
Aθ = Cθ and therefore ASD(θ) = c for a positive constant c
for all θ. Now, to show that c = 1/4, we show that
ASD(0) = 1/4. Antipodal symmetry (f (θ) = f (−θ) for all θ)
implies that
Z π
1
ASD(0) = 2
− F (a) f (a)da
2
0
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Depth on hyperspheres
• The result follows easily using the fact that
R
π
0
F (a)f (a)da = 1/8.
• The latter result has immediate statistical implications.
• To test whether the underlying distribution has an antipodal
symmetric distribution or not, one can compare an “empirical
version” of ASD with 1/4.
• More precisely, such an empirical ASD is given by
(n)
ASD
(θ) :=
n
2
−1 X
I[θ ∈ arc(Xi1 , Xi2 )],
∗
P
where ∗ stands for the sum over all the possible pairs
(Xi1 , Xi2 ).
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Depth on hyperspheres
• Then a test can be build for example on
(S 1 )
Tn
:= sup |ASD(n) (θ) − 1/4|
θ
• Large values of Tn
• Needless to say, a reasonable test has to be build using the
fixed-n or the large sample distribution of Tn . Unfortunately,
we do not have any. So one can boostrap or doing something
else or cry
• Of course, the equivalent test on the sphere is given by
(S 2 )
Tn
:= sup |ASD(n) (θ) − 1/8|,
θ
where ASD(n) (θ) is here constructed using spherical triangles
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Depth on hyperspheres
• There exists a link between the “traditional” notion of
simplicial depth and ASD.
• Let F be a cdf and θ some fixed “point” on the unit circle.
• There is a natural “length-preserving” mapping gθ from
[−π, π] to the tangent line Lθ .
• For a “point” φ 6= −θ, |gθ (φ)| is the length of arc(θ, φ) and
the sign of gθ (φ) is − or + depends on whether the direction
in going from θ to φ is counterclockwise or clockwise
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Depth on hyperspheres
• Then, let Fθ denote the cdf of the resulting distribution of the
tangent line Lθ
• The ASD and the simplicial depth (SDθ ) on the tangent line
Lθ with respect to Fθ are linked by
• Proposition. Let F be a continuous cdf on S 1 and
θ ∈ [0, 2π[. Then,
ASD(θ) + ASD(−θ) = SDθ (0).
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• Proof. The following events are clearly equivalent (except for
a null set)
E1 := {0 ∈ segment(g (φ1 ), g (φ2 )}
E2 := {φ1 and φ2 are on two different sides of line(φ1 , φ2 )}
(1)
(2)
E3 := E3 ∪ E3
:= {θ ∈ arc(φ1 , φ2 )} ∪ {−θ ∈ arc(φ1 , φ2 )}
(1)
(2)
The result follows from the fact that E3 ∩ E3
probability 0;
has
SDθ (0) := P(E1 ) = P(E3 ) = ASD(θ) + ASD(−θ)
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Depth on hyperspheres
• A similar result exists on S 2
• We now turn to the angular Tukey depth
• Definition. The angular Tukey depth (ATD) for a given
distribution F on the hypersphere is given by
ATDF (x) := inf PF (S),
S:x∈S
where the infimum is taken over the set of all closed
hemispheres S containing x
• A maximum point is an angular Tukey’s median
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Depth on hyperspheres
• On the circle as well as on the sphere, ATD( ) is bounded
above by 1/2. The value 1/2 is achieved at a point θ on a
sphere if and only if each hemisphere containing θ has
probability greater than or equal to 1/2.
• For a discussion on the robustness aspects of the medians
associated with those depth concepts, see Liu and Singh
(Annals of Statistics, 1992)
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