An Efficient Distance Calculation Method for Uncertain Objects

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An Efficient Distance Calculation Method for
Uncertain Objects

• Edward Hung
• csehung_at_comp.polyu.edu.hk
• Hong Kong Polytechnic University
• 2007 CIDM, Hawaii, USA, Apr 1-5, 2007

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Outline

• Why we care about uncertain objects and their
  distances?
• Analytic Solutions for Uniform and Gaussian
  Distributions
• Five Approximation Methods (DM, PRS, GAPS, PGM,
  ASG) for Arbitrary Distributions
• Equivalence of PRS, PGM and ASG
• Performance Study
• Conclusion

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Uncertain Objects From Where?

• Sources
• Readings from sensors
• Classification results of image processing using
  statistical classifiers
• Results from predictive programs used for stock
  market
• Weather prediction
• Etc

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Uncertain Objects How to Represent?

• Representation
• An exact value with margins of error
• E.g., 1560.5, 23.8, 24.9
• An uncertainty domain with a probability
  distribution/density function (PDF/pdf)
• Discrete E.g., for object o1, UD(o1)
  5.1,5.2,5.3, P1(5.1) 0.3, P1(5.2)0.4,
  P1(5.3)0.3
• Continuous E.g., for object o2 with uniform
  distribution, UD(o2) 6,11, p2(x) 0.2 where
  6 x 11

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Uncertain Objects handled traditionally

• Transformed into exact values to store in
  traditional databases
• Weighted average or mean
• Value of highest frequency or possibility
• Why bad??
• Intermediate and final results of mining or
  queries will also be approximate and may be wrong
• E.g., deviation of cluster centroids and wrong
  assignment of some data
• Shown in experimental results later

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Distance Why Important?

• Various queries and data mining tasks, e.g.,
• Nearest-neighbor queries
• Clustering (e.g., K-means clustering)

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Distance Why Expensive?

• An uncertain object has more than one possible
  location
• Discrete E.g., o1 (o2) has n1 (n2) possible
  locations
• n1n2 possible pair-wise combinations of their
  locations to calculate distances
• Probability of each location may be different

o1
o2
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Distance Why Expensive?

• Continuous E.g., take n samples on each uncertain
  object
• More samples in region of higher probability
  density
• Each sample has the same probability

o1
o2
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Distance Why Expensive?

• Approximation by a grid of a finite number of
  cells formed on the uncertainty domain (region)1
• A grid of 14X14 cells
• Probability of each cell determined by sampling
• All combinations of cells of two objects ?
  196X196 distance calculations

1e.g., used in Ngai, et al., Efficient
clustering of uncertain data, in the 2006 IEEE
International Conference on Data Mining (ICDM).
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Why Expected Distance?

• All possible pair-wise combinations ? a distance
  function di,j(x) to return the probability (or
  density) that the distance between objects oi and
  oj is x
• VERY expensive (previous slides)
• Expected distance weighted average of all
  combinations distances
• Could be much cheaper IF we do NOT need to try
  all combinations
• Squared Euclidean distance chosen
• Easier integration compared with Euclidean
  distance or Manhattan distance

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Analytic Solutions

• Uniform pdf
• Gaussian pdf

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Uniform pdf

• c2(a2-abb2)/3
• C2r2/2
• C23r2/5
• C2(r12r22)/3

(5) C2(r12r22)/2 (6) C2r12/23r22/5 (7)
C23(r12r22)/5
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Gaussian pdf

• For objects oi with Gaussian pdf N(µi,Si), where
  µi is a dX1 mean vector, Si is a dXd covariance
  matrix,
• Expected distance between objects oi, oj is
• EDAS(oi, oj) µi - µj2 trace(Si)
  trace(Sj)
• where trace(Si) is sum of all diagonal elements
  in Si

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Approximation Methods for Arbitrary pdf

• 5 methods proposed
• Distance between Means (DM)
• Pair-wise between Random Samples (PRS)
• Grid Approximation and Pair-wise between Samples
  (GAPS)
• Pair-wise between Gaussian Mixture (PGM)
• Approximation by Single Gaussian (ASG)

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1. Distance between Means (DM)

• EDDM(oi, oj) µi - µj2

o1
o2
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2. Pair-wise between Random Samples (PRS)

• take n samples on each uncertain object
• More samples in region of higher probability
  density each sample has the same probability

o1
o2
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3. Grid Approximation and Pair-wise between
Samples (GAPS)

• Approximation by a grid of vs X vs cells formed
  on the uncertainty domain
• Probability of each cell determined by sampling

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4. Pair-wise between Gaussian Mixture (PGM)

• Approximate an uncertain object oi by a mixture
  of Gaussian distributions ?uCi Ai,uN(µi,u,Si,u)
• use K-means to cluster samples into a few
  clusters)
• EDPGM(oi, oj) ?uCi ?vCj Ai,uAj,v(µi,u
  µj,v2 trace(Si,u) trace(Sj,v))

o1
o2
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5. Approximation by Single Gaussian (ASG)

• Approximate an uncertain object oi by a single
  Gaussian distributions
• N(µi,Si)
• EDASG(oi, oj) µi - µj2 trace(Si)
  trace(Sj)
• Complexity O((ninj)d)

o1
o2
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Equivalence of PRS, PGM and ASG

• Theorem
• Given any uncertain objects oi, oj and their
  samples xi,1,,xi,ni, xj,1,,xj,nj,
  EDPRS(oi,oj)EDPGM(oi,oj)EDASG(oi,oj)
• Theoretically ASG is the most inexpensive
  compared with all other methods (except DM) with
  the same results as PRS and PGM
• What about compared with DM and GAPS?

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Performance Study

• Experimental results also show that ASG is
• much more accurate than DM with comparable speed
• much faster than GAPS with higher or comparable
  accuracy
• grid cells samples

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Experiment 1

• 100 uncertain objects (4 Gaussian pdfs, variances
  in 1,10)

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Experiment 1
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Experiment 1

• ASG 0.02ms

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Experiment 2

• Data generated in the way as
• Ngai, et al., Efficient clustering of uncertain
  data, in the 2006 IEEE International Conference
  on Data Mining (ICDM)
• A grid of 14X14 cells
• Probability of each cell randomly generated
• normalized

GAPS produces the correct solution, but how close
is ASG?
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Experiment 2
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Experiment 2

• ASG 0.02ms

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Experiment 3

• ASG also approximates well objects with uniform
  pdf
• 10 objects with radius in 1,5, random located
  in 100X100 2D space
• ASG takes 100 samples, and repeats for 6 times
• Accuracy
• Worst case gt 0.98
• Average gt 0.99

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Experiment 4

• Scalability w.r.t. Dimensions
• 2/3/4-D
• 256/216/256 samples/cells
• ASG
• Accuracy 0.97 0.99
• Time 0.02ms or less

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Experiment 4
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Experiment 4
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Conclusion

• Importance of expected distance calculation in
  queries and data mining applications on uncertain
  data
• Analytic solutions of special cases
  (uniform/Gaussian pdf)
• ASG can obtain highly accurate results quickly
• ASG can replace GAPS used in recent research work