Geometric and combinatorial issues in data depth

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Geometric and combinatorial issues in data depth

• Greg Aloupis
• Université Libre de Bruxelles

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What is data depth?

• A quantitative measurement of how central a point
  is with respect to a data set.
• Goals to be able to rank data points, and to
  find the center of the data cloud.

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Some geometric bivariate medians

• Convex hull peeling (Tukey 70s)
• 85 Chazelle ?(nlogn)
• Halfspace median (Tukey 74)
• 01 Langerman-Steiger O(nlog 3 n), 03 Chan
  O(nlog n) -randomized
• Oja median (Oja 83)
• 01 G.A.-Langerman-Soss-Toussaint O(nlog 3 n)
• Simplicial median (Liu 88)
• 01 ALST O(n4)

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Convex hull peeling
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Convex hull peeling
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Convex hull peeling
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Convex hull peeling
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Halfspace, simplicial and Oja depthsof a point ?
in bivariate data set S

• Each median is a point with max/min depth

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Halfspace, simplicial and Oja depthsof a point ?
in bivariate data set S

• (Tukey) halfspace depth
• For every line through ?, count points
  above/below.
• Return minimum number counted over all lines.

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Halfspace, simplicial and Oja depthsof a point ?
in bivariate data set S

• (Tukey) halfspace depth
• For every line through ?, count points
  above/below.
• Return minimum number counted over all lines.

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Halfspace, simplicial and Oja depthsof a point ?
in bivariate data set S

• (Tukey) halfspace depth
• For every line through ?, count points
  above/below.
• Return minimum number counted over all lines.

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Halfspace, simplicial and Oja depthsof a point ?
in bivariate data set S

• (Tukey) halfspace depth
• For every line through ?, count points above.
• Return minimum number counted over all lines.

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Halfspace, simplicial and Oja depthsof a point ?
in bivariate data set S

• (Liu) simplicial depth
• Count the closed triangles in S that contain ?.

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Halfspace, simplicial and Oja depthsof a point ?
in bivariate data set S

• (Liu) simplicial depth
• Count the closed triangles in S that contain ?.

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Halfspace, simplicial and Oja depthsof a point ?
in bivariate data set S

• (Liu) simplicial depth
• Count the closed triangles in S that contain ?.

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Halfspace, simplicial and Oja depthsof a point ?
in bivariate data set S

• (Liu) simplicial depth
• Count the closed triangles in S that contain ?.

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Halfspace, simplicial and Oja depthsof a point ?
in bivariate data set S

• (Liu) simplicial depth
• Count the closed triangles in S that contain ?.

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Halfspace, simplicial and Oja depthsof a point ?
in bivariate data set S

• (Liu) simplicial depth
• Count the closed triangles in S that contain ?.
• etc

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Halfspace, simplicial and Oja depthsof a point ?
in bivariate data set S

• Oja depth
• Sum areas of all triangles with vertices (?,si
  ,sj)

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Halfspace, simplicial and Oja depthsof a point ?
in bivariate data set S

• Oja depth
• Sum areas of all triangles with vertices (?,si
  ,sj)

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Halfspace, simplicial and Oja depthsof a point ?
in bivariate data set S

• Oja depth
• Sum areas of all triangles with vertices (?,si
  ,sj)

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Halfspace, simplicial and Oja depthsof a point ?
in bivariate data set S

• Oja depth
• Sum areas of all triangles with vertices (?,si
  ,sj)
• etc

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Halfspace, simplicial and Oja depthsof a point ?
in bivariate data set S

• O(nlog n) Khuller-Mitchell 89,
  Gil-Steiger-Wigderson 92, Roussewu-Ruts 96
• W(nlog n) G.A.-Cortes-Gomez-Soss-Toussaint 01,
  Langerman-Steiger 01, G.A.-McLeish 05

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Issue 1What is the complexity of computing the
depth k of a point if k is known to be
small/large?

• If the peel median has depth kgt1 then can we
  compute it faster? (GSW92)
• !!! this just in simplicial depth in O(nnlog
  (1 k/n))
• Elmasry-Elbassioni ? CCCG last week
• Is there a lower bound, sensitive to parameter k?
• Something similar for halfspace depth?
• Current attempts for O(nlog k)

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Issue 2 (Improve) simplicial median computation

• Remember, that horrible n4 result a few slides
  back

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Easy observation

• I the set of line segments between pairs in S.
• The simplicial median is on an intersection of
  two segments in I.

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Outline of a method

• Preprocessing O(n3) brute-force, actually O(n2)
• Count number of points above/below each segment.
• Compute depth of all points.
• For each segment,
• sort all intersections with other segments.
• O(n2log n).
• Calculate depth of each intersection in O(1)
  time
• O(n2)
• Overall O(n4log n)

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• Constant time to update depth as we walk

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• Instead of sorting intersection points and
  processing each segment alone, we can use
  topological sweep.
• The time complexity becomes O(n4) and the space
  used is O(n2).
• Can we improve this?
• i.e. find some structure in this depth function

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Conjecture

• A point of maximum simplicial depth can always be
  found on the intersection of two halving segments
• (weak) experiments have not contradicted this

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Desirable properties of data depth functions

• Affine invariance (at the very least)
• Robustness
• Outliers should not influence the center.
• Monotonicity
• Center should move in same direction as
  perturbations

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monotonicity
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Robustness to outliers

• breakdown point fraction of data that must be
  moved/added so that median is placed at infinity.
• Oja median was considered to be robust, but
  finally it was shown that the breakdown point can
  be near zero for certain configurations. (planar
  case) (Niinimaa,Oja,Tableman 90)
• simplicial median dont know. But the data
  point of maximum depth can be moved away with few
  corrupting points (GSW 92) (planar case)
• halfspace median great! 1/(d1)
  (Donoho,Gasko 92)

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Robustness to outliers

• breakdown point fraction of data that must be
  moved/added so that median is placed at infinity.
• Max breakdown ½
• In 1D, only the median is affine invariant,
  monotonic and has max breakdown
• Is there such an estimator in higher dimensions?

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Issue 3How does the breakdown point depend on
the depth of the median?

• Convex peeling breakdown is zero, unless depth
  is linear (GSW92)
• Halfspace breakdown is higher (1/3) for
  centrosymmetric data distributions, where depth
  is roughly 1/2
• Instead of 1/(d1)
• So what can we say about other estimators?
• For deepest point in plane
• For deepest data point

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Issue 4Non-strategic breakdown

• All work so far involved carefully placing
  outliers (erroneous or corrupt data), to move an
  estimator far away.
• (is corrupt data really placed carefully in
  practice?)
• What about
• average outliers (random or evenly spaced
  placement)
• strong breakdown (should work regardless of
  direction at infinity)
• special-case outliers (axis-parallel, or radial
  extension, or ?)

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Issue 5Computing/analyzing other estimators

• Projection outlyingness of q
  (Donoho-Gasko 92)
• Take max of the following, over all
  projections
• q-Median / (median deviation from
  median)
• Find an algorithm for the least outlying point.
• Gil-Steiger-Wigderson
• superposition of unit vectors to data points
  v(ai)
• median is a (data) point with v(ai) R lt 1
  ???
• computation in o(n2) ? Properties?
• Zonoid depth, Delaunay depth

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Issue 6Points of high depth

• A point w/ Tukey depth ? n/(d1) is a
  centerpoint.
• Guaranteed to exist, by Hellys thm.
• O(n) time computation (Jadhav-Mukhopadhyay 94)
• Can be considered to be a median generalization.
• ¼ (n 3) ? simplicial depth ? 2/9 (n 3)
  (Boros-Furedi 82)
• (in R2 , ignoring quadratic terms)
• Can we compute a high depth point quickly?
• Tverberg points in R2 have depth ? 1/27 (n 3)
  and can be computed in O(n) time. Anything
  better?
• Is there a point with high Oja depth?
  (normalized)

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Things I may have mentioned in the abstract but
forgot to include here

• Is it faster to locate a deep point without
  computing its depth?
• How many points have depthgtk ?
• When do simplicial depth levels become
  disconnected?

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merci