Existe-t-il une définition acceptée de la médiane d'un échantillon dans l'avion ou d'espaces ordonnés plus élevés?

33

Si oui quoi? Si non pourquoi pas

Pour un échantillon sur la ligne, la médiane minimise l’écart absolu total. Il semblerait naturel d'étendre la définition à R2, etc., mais je ne l'ai jamais vue. Mais ensuite, je suis dans le champ gauche depuis longtemps.

phv3773
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Réponses:

19

Je ne suis pas sûr qu'il existe une définition acceptée pour une médiane multivariée. Celui que je connais bien est le point médian d'Oja , qui minimise la somme des volumes de simplices formés sur des sous-ensembles de points. (Voir le lien pour une définition technique.)

Mise à jour: le site référencé pour la définition d'Oja ci-dessus contient également un bel article couvrant plusieurs définitions d'une médiane à plusieurs variables:

ars
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1
Nice reference: thanks. It comprehensively covers everything mentioned here.
whuber
The same website also contains a nive overview in html: cgm.cs.mcgill.ca/~athens/Geometric-Estimators/intro.html
Aditya
15

As @Ars said there are no accepted definition (and this is a good point). There are general alternatives families of ways to generalize quantiles on Rd, I think the most significant are:

  • Pn(A)AARdλ

    Un(t)=inf(λ(A):Pn(A)tAA)

    Suppose you can find one At that gives you the minimum. Then the set (or an element of the set) A1/2ϵA1/2+ϵ gives you the median when ϵ is made small enough. The definition of the median is recovered when using A=(],x]xR) and λ(],x])=x. Ars answer falls into that framework I guess... tukey's half space location may be obtained using A(a)=(Hx=(tRd:a,tx) and λ(Hx)=x (with xR, aRd).

  • variational definition and M-estimation The idea here is that the α-quantile Qα of a random variable Y in R can be defined through a variational equality.

    • The most common definition is using the quantile regression function ρα (also known as pinball loss, guess why ? ) Qα=arginfxRE[ρα(Yx)]. The case α=1/2 gives ρ1/2(y)=|y| and you can generalize that to higher dimension using l1 distances as done in @Srikant Answer. This is theoretical median but gives you empirical median if you replace expectation by empirical expectation (mean).

    • But Kolshinskii proposes to use Legendre-Fenchel transform: since Qα=Argsups(sαf(s)) where f(s)=12E[|sY||Y|+s] for sR. He gives a lot of deep reasons for that (see the paper ;)). Generalizing this to higher dimensions require working with a vectorial α and replacing sα by s,α but you can take α=(1/2,,1/2).

  • Partial ordering You can generalize the definition of quantiles in Rd as soon as you can create a partial order (with equivalence classes).

Obviously there are bridges between the different formulations. They are not all obvious...

robin girard
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Nice answer, Robin!
ars
12

There are distinct ways to generalize the concept of median to higher dimensions. One not yet mentioned, but which was proposed long ago, is to construct a convex hull, peel it away, and iterate for as long as you can: what's left in the last hull is a set of points that are all candidates to be "medians."

"Head-banging" is another more recent attempt (c. 1980) to construct a robust center to a 2D point cloud. (The link is to documentation and software available at the US National Cancer Institute.)

The principal reason why there are multiple distinct generalizations and no one obvious solution is that R1 can be ordered but R2, R3, ... cannot be.

whuber
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Any measure that coincides with the usual median when restricted to R1 is a candidate generalization. There must be a lot of them.
phv3773
phv:> one can ask for 'the' generalization to preserve (in higher dimensions) some of the interesting properties of the median. This severly limits the number of candidates (see the commenting after Srikant's answer below)
user603
@Whuber:> then notion of ordering can be generalized to R^n for unimodal distributions (see my answer below).
user603
@kwak: could you elaborate a little? The usual mathematical definition of an ordering of a space is independent of any kind of probability distribution, so you must implicitly have some additional assumptions in mind.
whuber
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@Whuber:> You state: "R1 can be ordered but R2, R3, ... cannot be". R2,..,R3 can be ordered in many ways by mapping from Rn to R . One such way is the tukey depth. It has many important properties (robustness to some extend, non parametric, invariance,...) but these only hold for the case of unimodal distributions. Let me know if you want more details.
user603
6

The Tukey halfspace median can be extended to >2 dimensions using DEEPLOC, an algorithm due to Struyf and Rousseeuw; see here for details.

The algorithm is used to approximate the point of greatest depth efficiently; naive methods which attempt to determine this exactly usually run afoul of (the computational version of) "the curse of dimensionality", where the runtime required to calculate a statistic grows exponentially with the number of dimensions of the space.

Gary Campbell
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0

I do not know if any such definition exists but I will try and extend the standard definition of the median to R2. I will use the following notation:

X, Y: the random variables associated with the two dimensions.

mx, my: the corresponding medians.

f(x,y): the joint pdf for our random variables

To extend the definition of the median to R2, we choose mx and my to minimize the following:

E(|(x,y)(mx,my)|

The problem now is that we need a definition for what we mean by:

|(x,y)(mx,my)|

The above is in a sense a distance metric and several possible candidate definitions are possible.

Eucliedan Metric

|(x,y)(mx,my)|=(xmx)2+(ymy)2

Computing the median under the euclidean metric will require computing the expectation of the above with respect to the joint density f(x,y).

Taxicab Metric

|(x,y)(mx,my)|=|xmx|+|ymy|

Computing the median in the case of the taxicab metric involves computing the median of X and Y separately as the metric is separable in x and y.


la source
Srikant:> No. The definition has to have two important feature of the univariate median. a) Invariant to monotone transformation of the data, b) robust to contamination by outliers. None of the extentions you propose have these. The Tukey depth has these qualities.
user603
@kwak What you say makes sense.
@Srikant:> Check the R&S paper cited by Gary Campbell above ;). Best,
user603
@kwak On thinking some more, the taxicab metric does have the features you mentioned as it basically reduces to univariate medians. no?
2
@Srikant:> there are no incorrect answer to phv's questions because there are no 'good answers' either; this area of research is still under development. I simply wanted to point out why it is still an open problem.
user603