Statistical approach to numerical databases: clustering using normalised Minkowski metrics

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Statistical approach to numerical
databasesclustering using normalised Minkowski
metrics
D.T. Pham Y.I. Prostov Maria Suarez Alvarez
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Clustering

• Clustering is the unsupervised classification
  of data points into groups (clusters) based
  on some similarity or distance measure. Each
  group consists of data points that are similar
  between themselves and dissimilar to data
  points of other groups.
• Typical Applications
• As a stand-alone tool to get insight into data
  distribution, to observe the characteristics of
  each cluster and to focus on a particular set of
  clusters for analysis.
• As a preprocessing step for other
  algorithms such as classification, which will
  operate on the selected clusters.

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Similarity

• It is very common to calculate the dissimilarity
  between two features using a distance measure.
  The most popular metric for continuous features
  is the Euclidean distance.
• It is very intuitive to employ it because it is
  in everyday use.

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Similarity Measures

• Euclidean metric (distance) (or L2 metric)

TheTchebysheff (Chebyshev) or maximum norm
The Minkowski distance (or metric)
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Statistical approach to normalisationof feature
vectors

• Each record (row) of a database may be regarded
  as a random sample of the population under
  consideration, i.e. one has a database of N
  observations (samples) and each sample (record)
  is a realisation of possible values of the
  feature
• vector A.
• It is reasonable to expect that the mean
  contributions of individual features to the
  overall similarity measure should be equal.

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Statistical characteristics of an attribute

• Let the j-th feature component be distributed
  randomly with a density function
• The variance of the distribution is and its
  mean is

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Normalisation of featurevectors for Minkowski
metrics

• The expectation of the contribution of the j-th
  attribute to the Minkowski metric is

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Euclidean distance (p2)

• For the following normalised variables
• the mean contributions of all normalised
  components and to the square Euclidean
  distance are the same

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Normalised Minkowski metrics

• Introducing a constant for each normal
  distribution
• The modified Minkowski metric for a
• database is
• where

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Conclusion

• A numerical database is regarded as a random
  sample of objects for the domain under
  consideration.
• A statistical approach is applied to
  normalisation of all attributes of the feature
  vectors of data sets.
• Using the Minkowski distance as an example of a
  similarity metric, new normalised metrics are
  introduced such that the means of contributions
  of all attributes are the same. Contributions of
  the features to similarity measures are
  approximately equalised.
• Such a normalisation is achieved by scaling of
  the numerical attributes.