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

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.

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:

• Mesures géométriques de la profondeur des données

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 $\mathbb{R}^d$, I think the most significant are:

• $P_n(A)$$A$$\mathbb{A}$$\mathbb{R}^d$$\lambda$

  $U_n(t)=\inf (\lambda(A) : P_n(A)\geq t A\in\mathbb{A})$

  Suppose you can find one $A_{t}$ that gives you the minimum. Then the set (or an element of the set) $A_{1/2-\epsilon}\cap A_{1/2+\epsilon}$ gives you the median when $\epsilon$ is made small enough. The definition of the median is recovered when using $\mathbb{A}=(]-\infty,x] x\in\mathbb{R})$ and $\lambda(]-\infty,x])=x$. Ars answer falls into that framework I guess... tukey's half space location may be obtained using $\mathbb{A}(a)=( H_{x}=(t\in \mathbb{R}^d :\; \langle a, t \rangle \leq x )$ and $\lambda(H_{x})=x$ (with $x\in \mathbb{R}$, $a\in\mathbb{R}^d$).

• variational definition and M-estimation The idea here is that the $\alpha$-quantile $Q_{\alpha}$ of a random variable $Y$ in $\mathbb{R}$ can be defined through a variational equality.

  • The most common definition is using the quantile regression function $\rho_{\alpha}$ (also known as pinball loss, guess why ? ) $Q_{\alpha}=arg\inf_{x\in \mathbb{R}}\mathbb{E}[\rho_{\alpha}(Y-x)]$. The case $\alpha=1/2$ gives $\rho_{1/2}(y)=|y|$ and you can generalize that to higher dimension using $l^1$ 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 $Q_{\alpha}=Arg\sup_s (s\alpha-f(s))$ where $f(s)=\frac{1}{2}\mathbb{E} [|s-Y|-|Y|+s]$ for $s\in \mathbb{R}$. He gives a lot of deep reasons for that (see the paper ;)). Generalizing this to higher dimensions require working with a vectorial $\alpha$ and replacing $s\alpha$ by $\langle s,\alpha\rangle$ but you can take $\alpha=(1/2,\dots,1/2)$.

• Partial ordering You can generalize the definition of quantiles in $\mathbb{R}^d$ 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...

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.

Geometric median is the point with the smallest average euclidian distance from the samples

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.

A definition that comes close to it, for unimodal distributions, is the tukey halfspace median

• http://cgm.cs.mcgill.ca/~athens/Geometric-Estimators/halfspace.html
• http://www.isical.ac.in/~statmath/html/publication/Tukey_tech_rep.pdf
• https://www.isical.ac.in/~statmath/report/11310-15.pdf

I do not know if any such definition exists but I will try and extend the standard definition of the median to $R^2$. I will use the following notation:

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

$m_x$, $m_y$: the corresponding medians.

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

To extend the definition of the median to $R^2$, we choose $m_x$ and $m_y$ to minimize the following:

$E(|(x,y) - (m_x,m_y)|$

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

$|(x,y) - (m_x,m_y)|$

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

Eucliedan Metric

$|(x,y) - (m_x,m_y)| = \sqrt{(x-m_x)^2 + (y-m_y)^2}$

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

$|(x,y) - (m_x,m_y)| = |x-m_x| + |y-m_y|$

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 $y$.