6. Introduction to nonparametric clustering

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• 6. Introduction to nonparametric clustering
• Regard feature vectors x1, , xn as sample from
  some density p(x)
• Parametric approach (Cheeseman, McLachlan,
  Raftery)
• Based on premise that each group g is
  represented by density pg that is a member of
  some parametric family gt p(x) is a mixture
• Estimate the parameters of the group densities,
  the mixing proportions, and the number of
  groups from the sample.
• Nonparametric approach (Wishart, Hartigan)
• Based on the premise that distinct groups
  manifest themselves as multiple modes of p(x)
• Estimate modes from sample

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6.1 Describing the modal structure of a
density Consider feature vectors x1 , . , xn as
a sample from some density p(x) . Define level
set L(c p) as the subset of feature space for
which the density p(x) is greater than c. Note
Level sets with multiple connected components
indicate multi-modality There might
not be a single level set that reveals all the
modes
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• The cluster tree of a density
• Modal structure of density is described by
  cluster tree.
• Each node N of cluster tree
• represents a subset D(N) of feature space
• is associated with a density level c(N)
• Root node
• represents the entire feature space
• is associated with density level c(N) 0
• Tree defined recursively to determine
  descendents of node N
• Find lowest level c for which intersection of
  D(N) with L(c p) has two connected components
• If there is no such c then N is leaf of tree
  leaves of tree ltgt modes
• Otherwise, create daughter nodes representing
  the connected components, with associated level
  c

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Goal Estimate the cluster tree of the
underlying density p(x) from the sample feature
vectors x1 , . , xn First step Estimate p(x)
by density estimate p(x) (see below) Second
step Compute cluster tree of p (maybe
approximately)
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6.2 Density estimation Consider feature vectors
x1 , . , xn as a sample from some density
p(x). Goal Estimate p(x) Simplest idea Let
S(x, r) denote a sphere in feature space with
radius r, centered at x. Assuming density is
roughly constant over S(x, r), the expected
number of sample points in S(x, r) is
k n Volume ( S(x, r) ) p(x), giving
p(x) k / (n Volume ( S(x, r) )
Kernel estimate Fix radius r k of
sample feature vectors in S(x, r) K-near-neighbor
estimate Fix count k r smallest radius
for which
S(x, r) contains k sample feature
vectors Many refinements have been suggested
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Example - kernel density estimate in 2-d

• Swept under the rug
• Choice of sphere radius r (for kernel estimate)
  or count k (for near-neighbor estimate) ---
  critical !! There are automatic methods.
• Down-weight observations depending on distance
  from query point
• Adaptive estimation --- vary radius r depending
  on density
• Other types of estimates, etc, etc, etc
  (extensive literature)

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• Computational complexity
• Computing kernel or near-neighbor estimate at
  query point x requires finding nearest neighbors
  of x in sample x1 , . , xn.
• Can find k nearest neighbors of x in time log
  n using spatial partitioning schemes such as k-d
  trees, after n log n pre-processing
• However
• Spatial partitioning most effective if n large
  relative to d.
• Theoretical analysis shows that number of
  nearest neighbors should increase with n and
  decrease with dimensionality d k n (4 / (d
  4)). Relevance ?
• In low dimensions (d lt 4) can use histogram or
  average shifted histogram density estimates based
  on regular binning.
• Evaluation for query point in constant time,
  after pre-processing n
• High dimensionality may present problem

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• 6.3 Recursive algorithms for constructing a
  cluster tree
• For most density estimates p(x), computing level
  sets and finding their connected components is a
  daunting problem --- especially in high
  dimensions.
• Idea Compute sample cluster tree instead
• Each node N of sample cluster tree
• represents a subset X(N) of the sample
• is associated with a density level c(N)
• Root node
• represents the entire sample
• is associated with density level c(N) 0

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• To determine descendents of node N
• Find lowest level c for which the intersection
  of X(N) with L(c p) falls into two
  connected components Note Intersection of
  X(N) with L(c p) consists of those feature
  vectors in the node N for which estimated
  density p(xi) gt c. _at_
• If there is no such c then N is leaf of tree
• Otherwise, create daughter nodes representing
  the connected components, with associated
  level c.
• Note
• _at_ is the critical step. Will in general have to
  rely on heuristic.
• Daughters of a node N do not define a partition
  of X(N). Assigning low density observations in
  X(N) to one of the daughters is supervised
  learning problem

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Illustration
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• Critical step
• Find lowest level c for which observations in
  X(N) with estimated density p(xi) gt c fall into
  two connected components of level set L(c p)
• Heuristic 1 (goes with k-near-neighbor
  density estimate)
• Select feature vectors xi in X(N) with p(xi) gt
  c
• Generate graph connecting each feature vector to
  its k nearest neighbors
• Check whether graph has 1 or 2 connected
  components
• Heuristic 2 (goes with kernel density
  estimate)
• Select feature vectors xi in X(N) with p(xi) gt
  c
• Generate graph connecting feature vectors with
  distance lt r
• Check whether graph has 1 or 2 connected
  components

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• 6.4 Related work / references
• Looking for the connected components of a level
  set --- One-level Mode Analysis --- was first
  suggested by David Wishart (1969).
• Wisharts paper appeared in obscure place ---
  Proceedings of the Colloquium in Numerical
  Taxonomy, St. Andrews, 1968. Nobody in CS cites
  Wishart.
• Idea has been re-invented multiple times ---
  sharpening (Tukey Tukey) DBSCAN (Ester et
  al) Methods differ in heuristics for finding
  connected components of level set.
• Wishart also realized that looking at single
  level set might not be enough to detect all the
  modes gt Hierarchical Mode Analysis. Did
  not think of it as estimating cluster tree.
  Algorithm awkward --- based on iterative
  merging instead of recursive partitioning.
  OPTICS method of Ankerst et al also considers
  level sets for different levels.