Functional Nonparametric Unsupervised Classification

1
Functional Nonparametric Unsupervised
Classification

• T-61.6030 Functional Data Analysis
• Antti Sorjamaa
• Time Series Prediction and Chemoinformatics Group
• Laboratory of Computer and Information Science
• Helsinki University of Technology

2
Outline

• Mean, Median and Mode
• Measuring Heterogeneity
• Building a Partition
• Algorithm
• Example
• Comments and Conclusions

3
Mean, Median and Mode

• Well-known statistical symbols describing the
  data
• Mean has problems with robustness, less with
  median and mode
• Usage of these three symbols need the definition
  of a metric (or semi)
• Mode is used as a local maximum instead of the
  global one

4
Mean Example
Speech dataThe solid line is the mean and
other lines the means of the subgroups.Global
mean does not represent much of anything
5
Mean Example (2)
Spectrometric dataThe solid line is the mean
and other lines the means of the
subgroups.Global mean close to submeans with a
vertical shift
6
Measuring Heterogeneity

• Heterogeneity Index
• Subsampled Heterogeneity Index(close to
  Bootstrapping)

7
Splitting Score

• Partitioning Heterogeneity Index
• Splitting Score

8
Building a Partition

• Small Ball Probability and Probability Curves
• Sample partition simply dividing the Probability
  Curves to closest Mode
• Optimal Bandwidth selected by minimizing the
  Entropy, which maximizes the Heterogeneity

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Building a Partition (2)
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Algorithm

• Step 1
• If only 1 Mode ? Stop
• Step 2
• Split the sample and compute SC
• If SC gt t, goto Step 1 for each subclass
• Step 3 Stop

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Example
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Example (2)

• Semi-metric based on the second derivative
• Mean and Mode as the measurements in the
  Heterogeneity Index
• Then the algorithm is used and the data split
  accordingly

13
Example (3)
Resulting division is in 3 groups.Note that
Group 22 is not even tried to divide, because it
is found out to be unimodal
14
Example (4)
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Conclusions

• Good results with the Spectrometric dataset
• Better selection of semi-metrics could improve
  the results
• Metric selection a problem for all methods
• Field is just opened, needs more work
• Many open questions remain

16
Open Questions

• Is it possible to get rates of convergence?
• Convergence of the estimator? Is it strictly
  sample dependent?
• Fully k-means procedure?

17

• Questions?
• Antti.Sorjamaa_at_hut.fi