Rainer Dyckerhoff a Xiaohui Liu b Karl Mosler a Pavlo Mozharovskyi c
a Institute of Econometrics and Statistics, University of Cologne
b School of Statistics,
Jiangxi University of Finance and Economics;
Research Center of Applied Statistics, Jiangxi University of Finance and Economics
c LTCI, Telecom Paris, Institut Polytechnique de Paris
LTCI Data Science seminar
Paris, June 6, 2019
Data depth
Tukey depth: definition Tukey trimmed regions Applications
Computation of the Tukey depth Theoretical background Algorithm and simulations
Computation of Tukey trimmed regions Existing approaches
The proposed algorithm Tukey median
Outlook for approximations
Conclusions
Data depth
Tukey depth: definition Tukey trimmed regions Applications
Computation of the Tukey depth Theoretical background Algorithm and simulations
Computation of Tukey trimmed regions Existing approaches
The proposed algorithm Tukey median
Outlook for approximations
Conclusions
Database
Tables containing data:
- rows = objects - columns = properties
Subject 1
Subject 161 Subject 2
Subject 162
Weight Age 1350
1500
32 32
1320 28
1150 27
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Data depth
Tukey depth: definition Tukey trimmed regions Applications
Computation of the Tukey depth Theoretical background Algorithm and simulations
Computation of Tukey trimmed regions Existing approaches
The proposed algorithm Tukey median
Outlook for approximations
Conclusions
the “center” of a distribution. For x ∈ R d and a d -variate random vector X distributed as P ∈ P, a data depth is a function
D : R d × P → [0, 1], (x , P ) 7→ D(x |P ) that is:
I affine invariant: D(Ax + b|AX + b) = D(x |X );
I vanishing at infinity: lim ||x ||→∞ D(x |X ) = 0;
I monotone w.r.t. the deepest point: for any
x ∗ ∈ argmax x∈R d D(x |X ), any x ∈ R d , and any 0 ≤ α ≤ 1 it holds: D(x|X ) ≤ D(x ∗ + α(x − x ∗ )|X );
I upper semicontinuous in x: the upper-level sets D τ (X ) = {x ∈ R d : D(x |X ) ≥ τ } are closed for all τ ;
I (quasiconcave in x): the upper-level sets are convex for all τ .
Tukey depth of x ∈ R d w.r.t. a d -variate random vector X distributed as P is defined as the smallest probability mass of a closed halfspace containing x:
D T (x|X ) = inf{P (H) : H is a closed halfspace, x ∈ H}, and w.r.t. a data set X = {x 1 , ..., x n } ⊂ R d :
D T(n) (x|X ) = 1 n min
u∈S d−1
]{i : u 0 x i ≥ u 0 x }.
Other depth notions: Mahalanobis (’36), projection (Stahel, ’81;
Donoho, ’82), simplicial volume (Oja, ’83), simplicial (Liu, ’90), zonoid (Koshevoy, Mosler, ’97), spatial (Vardi, Zhang, ’00;
Serfling, ’02) depth.
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800 1000 1200 1400
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800 1000 1200 1400
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800 1000 1200 1400
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800 1000 1200 1400
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800 1000 1200 1400
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800 1000 1200 1400
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800 1000 1200 1400
20 25 30 35
Weight, in grams
Age , in w eeks
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157 / 161
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800 1000 1200 1400
20 25 30 35
Weight, in grams
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152 / 161
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800 1000 1200 1400
20 25 30 35
Weight, in grams
Age , in w eeks
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