Clustering Uncertain Data

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Clustering Uncertain Data

• Speaker Ngai Wang Kay

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• Data Clustering is used to discover any cluster
  patterns in a data set, e.g. the data set may be
  partitioned into several groups, or clusters,
  such that the data within the same cluster are
  closer to each other or more similar (based on
  some distance functions) than the data from any
  other clusters.
• There are many methods for clustering data, and
  K-means clustering is a common one.

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• K-means clustering considers each cluster to have
  a representative and it is the mean of the data
  in the cluster.
• For an example, consider a database of location
  data reported from moving vehicles in a tracking
  system.

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• Given K number of location points (e.g. US White
  House, a school, etc.) around the data for an
  initial guess of the representatives of the
  clusters expected to exist in the data.
• K-means clustering assigns each vehicle to one
  of the K clusters such that its location is
  closer in Euclidean distance to that cluster's
  representative than any others' representatives.

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• Then the representative of each cluster is
  updated to the mean of the locations of the
  vehicles in the cluster. And each vehicle is
  re-assigned to the K clusters with the new
  representatives.
• This process repeats until some objectives is
  met, e.g. no changes of any vehicles' clusters
  between two successive processes.

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• After the clustering, a cluster could be empty. A
  non-empty cluster may have some meaning, e.g. if
  its representative point is very close to US
  White House, the vehicles in the cluster may be
  classified as spies.
• Note that if the vehicles are constantly moving,
  their actual locations may have changed when
  their reported locations data is received.

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• In that case, the data in the database is not
  very accurate. A data will have an "uncertainty"
  region around it where its corresponding
  vehicle's actual location lies within.
• The uncertainty region could be arbitrary or
  simply a circle region using the reported
  location as its center and has a radius of the
  vehicle's maximum speed times the time elapsed
  since the location data is reported.

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• The uncertainty region could also be associated
  with an arbitrary probability density function
  (pdf) for the probability of the vehicle's actual
  location being in a particular point of the
  region.
• For an example, part of the region may be sea, so
  the part may be associated with a total of 0.1
  probability uniformly distributed for the points
  in the part (for the vehicle crashes to any one
  of those points). A probability 0.9 is for the
  vehicle to occur in any points of the remaining
  part of the region.

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• Such kind of data with uncertainty is called
  uncertain data.
• There is only few methods for clustering
  uncertain data. UK-means clustering is a common
  one.
• UK-means clustering is the same as K-means
  clustering except its distance function is using
  the "expected distance" from the data's
  uncertainty region to the representative of the
  candidate cluster to whom it is assigned.

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• For the representative c of the cluster, an
  uncertainty region R with a pdf f, and a
  Euclidean distance function D(p,c) for any two
  points, the expected distance, called ed, is

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• An uncertainty region could have arbitrary shape,
  so the minimum bounding box (MBR) of the region
  is used for R.
• The time complexity in UK-means clustering is
  O(nK) for computing the ed values for each of n
  data and each of K candidate clusters.
• This is very much especially for an uncertainty
  region with arbitrary pdf f that needs to be
  sampled using large number of samples in Monte
  Carlo methods to compute f values for the points.

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• A method called Global-minmax Pruning is
  developed in my research to prune out some
  candidate clusters for a data to save some
  computations of ed values.
• Define lmin(R,M) to be a minimum limit on the ed
  values computed for an uncertainty region R and a
  MBR M of some candidate clusters representative
  points.
• Also define lmax(R,M) to be a maximum limit on
  the ed values computed for an uncertainty region
  R and a MBR M of some candidate clusters
  representative points.

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• Define minmax to be the minimum of lmax values
  among all candidate clusters for a data.
• Then Global-minmax Pruning works like this
• A KD-tree for a given height h is built to index
  the representative points of the candidate
  clusters using p entries in each
  non-leaf node.
• Hence (ph p) / (p 1) non-leaf entries.

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• The entries in each level, except the leaf level,
  are sorted in increasing order in a particular
  dimension alternately. O((h-1) K log(K) ).
• KD-tree is usually small and can be stored in CPU
  cache. Its access time is so little that it is
  ignored.

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• The time complexity for computing the ed values
  is then O(nK-P) if P candidate clusters are
  pruned out, in average, for each object.

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• If the KD-tree uses a capacity p for its non-leaf
  nodes, the worst time complexity for the pruning
  process is O( (h-1) K log(K) n2K (ph p) /
  (p 1) ) when no non-leaf entries are pruned
  out.
• Note this is not better than the time complexity
  O(nK) for computing the ed values in UK-means
  clustering without any pruning if computing an ed
  value is not at least twice slower than computing
  a lmax or lmin value (e.g. for uncertainty
  regions with uniform pdf).
• So another pruning method called Local-minmax
  Pruning is developed in my research to address
  that special case.

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• The worst time complexity for the pruning process
  is then O( (h1) K log(K) n2K 2Q 2(ph
  p) / (p 1) ) when Q leaf entries are pruned
  out before their nodes are visited.
• This is better than t O( (h-1) K log(K) n2K
  Q (ph p) / (p 1) ) of Global-minmax
  Pruning if Q gt (ph p) / (p 1).
• This could even be better than the non-pruning
  methods O(nK) for the earlier special case of
  fast computation of an ed value against a lmin or
  lmax value if Q gt K / 2 (ph p) / (p 1)
  (h-1) K log(K) / 2n .

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• But Local-minmax Pruning is not as effective in
  pruning as Global-minmax Pruning and hence could
  be less efficient when the computations of ed
  values are large overhead as its O(nk P)
  increases.
• Simple way to compute lmin(R,M) is to use the
  Euclidean distance between the two nearest points
  in R and M.
• Simple way to compute lmax(R,M) is to use the
  Euclidean distance between the two farthest
  points in R and M.