1. Non Parametric Classifiers Alexandros Potamianos Dept of ECE, Tech. Univ. of Crete Fall 2004-2005

2. Histograms-Parzen Windows  Main idea: Instead of selecting a parametric distribution (e.g., Gaussian) to describe the properties of the features of a class, compute directly the empirical distribution  class feature histogram

3. Feature Histogram Example X # of samples in each bin  Normalize histogram curve to get feature PDF

4. Parzen Windows: Issues  When compared to parametric methods empirical distributions are: Better because no specific form of the PDF is assumed Worse because over-fitting can easily occur (too small histogram bin)  Parzen proposed rules for adapting bin size based on number of samples in each bin to avoid over- fitting

5. Nearest Neighbor Rule  Main idea (1-NNR): No explicit model (i.e., no training) For each test sample x the “nearest” sample x’ in the training set is found, i.e., argmin x’ d(x, x’) and x is classified to the class where x’ belongs

6. Generalizations k-NNR: Instead of finding the nearest neighbors we find k nearest neighbors from the training set; the sample x is classified to the class where most of the k neighbors belong k-l-NNR: Like k-NNR but at least l of the k nearest neighbor must belong to the same class for a classification decision to be taken (else no decision)

7. Example Training set D 1 = {0,-1,-2} and D 2 = {1,1,1} -2 -1 0 12 3 1-NNR decision boundary 3-NNR decision boundary 3-3-NNR no decision region

8. Computational Efficiency  To speed up NNR classification the training set size can be reduced using the condensing algorithm: The training set is classified using NNR rule misclassified samples are added to the new (condensed) training set one by one until all training samples are correctly classified

9. Conclusions  Non parametric classification algorithms are easy to implement are computationally efficient (in training) don’t make any assumptions are prone to over-fitting are hard to adapt (no detailed model)