This document discusses halfspace depth, a method for defining multivariate outliers and depth contours. It introduces the concept of halfspace depth which defines the depth of a point as the smallest probability of a closed halfspace containing that point. It then discusses how to empirically estimate depth and define depth sets containing points with depth greater than a given value. The document also notes that the empirical depth sets are sensitive to the algorithm used and presents an example comparing two different algorithms applied to the same data. Finally, it discusses potential extensions of this concept such as applying it to functional data through a functional bagplot.

1. Arthur CHARPENTIER, Solvency II’ newspeak
Stress Testing & Reverse Stress Testing
Alexander J. McNeil
arthur.charpentier@univ-rennes1.fr
http ://blogperso.univ-rennes1.fr/arthur.charpentier/index.php/
Financial Risks International Forum ‘Risk Dependencies’, March 2010
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2. Arthur CHARPENTIER, Solvency II’ newspeak
Defining halfspace depth
Given y ∈ Rd , and a direction u ∈ Rd , define the closed half space
Hy,u = {x ∈ Rd such that u x ≤ u y}
and define depth at point y by
depth(y) = inf {P(Hy,u )}
u,u=0
i.e. the smallest probability of a closed half space containing y.
The empirical version is (see Tukey, 1975)
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depth(y) = min 1(X i ∈ Hy,u )
u,u=0 n i=1
For α > 0.5, define the depth set as
Dα = {y ∈ R ∈ Rd such that ≥ 1 − α}.
The empirical version is can be related to the bagplot (Rousseeuw & Ruts, 1999).
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3. Arthur CHARPENTIER, Solvency II’ newspeak
Empirical sets extremely sentive to the algorithm
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where the blue set is the empirical estimation for Dα , α = 0.5.
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4. Arthur CHARPENTIER, Solvency II’ newspeak
The bagplot tool
The depth function introduced here is the multivariate extension of standard
univariate depth measures, e.g.
depth(x) = min{F (x), 1 − F (x− )}
which satisfies depth(Qα ) = min{α, 1 − α}. But one can also consider
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depth(x) = 2 · F (x) · [1 − F (x− )] or depth(x) = 1 − − F (x) .
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Possible extensions to functional bagplot.
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5. Arthur CHARPENTIER, Solvency II’ newspeak
The bagplot tool for mortality models
On the a French dataset, we have the following past outliers,
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(here male log-mortality rates in France from 1899 to 2005).
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6. Arthur CHARPENTIER, Solvency II’ newspeak
The bagplot tool for mortality models
Using functional bagplot techniques it is also possible to identify outliers in
stochastic scenarios,
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7. Arthur CHARPENTIER, Solvency II’ newspeak
Further references
Febrero, N., Galeano, P. & Gonzalez-Manteiga, W. (2007). A functional analysis
of NOx levels : location and scale estimation and outlier detection.
Computational Statistics 22(3), 411-427.
Hyndman, R.J. & Shang, H.L. (2010). Rainbow plots, bagplots and boxplots for
functional data. Journal of Computational and Graphical Statistics. 19(1), 29-45.
Rousseeuw, P.J., Ruts, I. & Tukey, J.W. (2009). The bagplot, a bivariate boxplot.
American Statistician, 53(4), 382-387.
Sood, A., James, G. & Tellis, G. (2009). Functional Regression : A New Model for
Predicting Market Penetration of New Products. Marketing Science, 28(1), 36-51.
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