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Chapter 4 (part 2): Non-Parametric Classification (Sections 4.3-4.5)

Pattern Classification All materials in these slides were taken from Pattern Classification (2nd ed) by R. O. Duda, P. E. Hart and D. G. Stork, John Wiley & Sons, 2000 with the permission of the authors and the publisher. Chapter 4 (part 2): Non-Parametric Classification (Sections 4.3-4.5).

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Chapter 4 (part 2): Non-Parametric Classification (Sections 4.3-4.5)

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  1. Pattern ClassificationAll materials in these slides were taken from Pattern Classification (2nd ed) by R. O. Duda, P. E. Hart and D. G. Stork, John Wiley & Sons, 2000with the permission of the authors and the publisher

  2. Chapter 4 (part 2):Non-Parametric Classification (Sections 4.3-4.5) Parzen Window (cont.) Kn –Nearest Neighbor Estimation The Nearest-Neighbor Rule

  3. x1 x2 . . . xd Parzen Windows (cont.) • Parzen Windows – Probabilistic Neural Networks • Compute a Parzen estimate based on n patterns • Patterns with d features sampled from c classes • The input unit is connected to n patterns . . . . . W11 p1 p2 . . . Input patterns Input unit . . Wd2 Wdn pn Modifiable weights (trained)

  4. . . pn . p1 . p2 Input patterns . . . 1 . 2 . . . Category units . pk . . . . c pn Activations (Emission of nonlinear functions)

  5. Training the network • Algorithm • Normalize each pattern x of the training set to 1 • Place the first training pattern on the input units • Set the weights linking the input units and the first pattern units such that: w1 = x1 • Make a single connection from the first pattern unit to the category unit corresponding to the known class of that pattern • Repeat the process for all remaining training patterns by setting the weights such that wk = xk (k = 1, 2, …, n) We finally obtain the following network

  6. Testing the network • Algorithm • Normalize the test pattern x and place it at the input units • Each pattern unit computes the inner product in order to yield the net activation and emit a nonlinear function • Each output unit sums the contributions from all pattern units connected to it • Classify by selecting the maximum value of Pn(x | j) (j = 1, …, c)

  7. Kn - Nearest neighbor estimation • Goal: a solution for the problem of the unknown “best” window function • Let the cell volume be a function of the training data • Center a cell about x and let it grows until it captures knsamples (kn = f(n)) • knare called the knnearest-neighbors of x 2 possibilities can occur: • Density is high near x; therefore the cell will be small which provides a good resolution • Density is low; therefore the cell will grow large and stop until higher density regions are reached We can obtain a family of estimates by setting kn=k1/n and choosing different values for k1

  8. Illustration For kn = n = 1 ; the estimate becomes: Pn(x) = kn / n.Vn = 1 / V1 =1 / 2|x-x1|

  9. Estimation of a-posteriori probabilities • Goal: estimate P(i | x) from a set of n labeled samples • Let’s place a cell of volume V around x and capture k samples • kisamples amongst k turned out to be labeled ithen: pn(x, i) = ki /n.V An estimate for pn(i| x) is:

  10. ki/k is the fraction of the samples within the cell that are labeled i • For minimum error rate, the most frequently represented category within the cell is selected • If k is large and the cell sufficiently small, the performance will approach the best possible

  11. The nearest –neighbor rule • Let Dn = {x1, x2, …, xn} be a set of n labeled prototypes • Let x’  Dn be the closest prototype to a test point xthen the nearest-neighbor rule for classifying x is to assign it the label associated with x’ • The nearest-neighbor rule leads to an error rate greater than the minimum possible: the Bayes rate • If the number of prototype is large (unlimited), the error rate of the nearest-neighbor classifier is never worse than twice the Bayes rate (it can be demonstrated!) • If n  , it is always possible to find x’ sufficiently close so that: P(i | x’)  P(i | x)

  12. Example: x = (0.68, 0.60)t Decision:5is the label assigned to x

  13. If P(m | x)  1, then the nearest neighbor selection is almost always the same as the Bayes selection

  14. The k – nearest-neighbor rule • Goal: Classify x by assigning it the label most frequently represented among the k nearest samples and use a voting scheme

  15. Example: k = 3 (odd value) and x = (0.10, 0.25)t Closest vectors to x with their labels are: {(0.10, 0.28, 2); (0.12, 0.20, 2); (0.15, 0.35,1)} One voting scheme assigns the label 2 to x since 2 is the most frequently represented

  16. 4.6 Metrics and NN Classification • Metrics = “distance” between patterns • Four properties: • Non-negativity D(a, b) ≧0 • Reflexivity D(a, b) = 0 iff a = b • Symmetry D(a, b) = D(b, a) • Triangle inequality D(a, b)+ D(b, c) ≧ D(a, c) • Euclidean distance k=2 • Minkowski metric • Manhattan distance k=1

  17. Distance 1.0 from (0,0,0) for different k

  18. Scaling the coordinates

  19. Tanimoto metric • Use in taxonomy • Identical • D(S1, S2) = 0 • Overlap 50% • D(S1, S2) = (1+1-2*0.5)/(1+1-0.5)=1/1.5=0.666 • No intersection • D(S1, S2) = 1

  20. 4.6.2 Tangent Distance • Transformed patterns to be as similar as possible • Linear approximation to the arbitrary transforms • Perform each of the transformation Fi(x’; ai) on each stored prototype x’. • Tangent vector TVi • TVi=Fi(x’; ai) - x’ • Tangent distance • Dtan(x’, x) = min [ ||( x’ + Ta) – x|| ] a Tangent space

  21. 100-dim patterns Hand write “8” Shifted s pixels

  22. 4.8 Reduced Coulomb Energy Networks • Parzen-window • -> Fixed window • K-NN • -> adjusting the region based on the density • RCE network • -> adjust the size of the window during training according to the distance to the nearest point of a different category.

  23. begininit j=0, λm=max radius do j = j + 1 wij = xi = arg min D(x, x’) λj = min[ D( , x’) - ε, λm] if x  wk then ajk =1 until j = n end RCE Training Train weight Find nearest point not in wi Set radius Connect pattern and category

  24. begininitj=0, k=0, x=test pattern, Dt={} doj = j + 1 if D(x, xj’) < λjthen Dt= Dt∪xj’ until j=n iflabel of all xj’  Dt is the same then return label of all xk Dt else return “ambiguous” label end RCE Classification Set of stored prototypes Prototype xj’ Radius ofxj’

  25. Summary • 2 Nonparametric estimation approaches • Densities are estimated (then used for classification) • Category is chosen directly • Densities are estimated • Parzen windows, probabilistic neural networks • Category is chosen directly • K-nearest-neighbor, reduced coulomb energy networks

  26. Class Exercises • Ex. 13 p.159 • Ex. 3 p.201 • Write a C/C++/Java program that uses a k-nearest neighbor method to classify input patterns. Use the table on p.209 as your training sample. Experiment the program with the following data: • k = 3 x1 = (0.33, 0.58, - 4.8) x2 = (0.27, 1.0, - 2.68) x3 = (- 0.44, 2.8, 6.20) • Do the same thing with k = 11 • Compare the classification results between k = 3 and k = 11 (use the most dominant class voting scheme amongst the k classes)

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