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)

• Parzen Window (cont.)

• Kn –Nearest Neighbor Estimation

• The Nearest-Neighbor Rule

Pattern Classification, Chapter 4 (Part 2)

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4Parzen 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.x1

x2

xd

.

.

.

p1

p2

pn

.

.

.

Input unit Input patterns

.

. ..

Modifiable weights (trained)

Wdn

Wd2

W11. .


5

p1

p2

pn

.

.

.

Input patterns

pn

pk... .

.

.

.1

2

c...

Category units

.

.

.

Activations (Emission of nonlinear functions)


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• Training the network

• Algorithm

1.Normalize each pattern x of the training set to 1

2.Place the first training pattern on the input units

3.Set the weights linking the input units and the first pattern units such that: w1 = x1

4.Make a single connection from the first pattern unit to the category unit corresponding to the known class of that pattern

5.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


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• Testing the network

• Algorithm

1.Normalize the test pattern x and place it at the input units

2.Each pattern unit computes the inner product in order to yield the net activation

and emit a nonlinear function

3.Each output unit sums the contributions from all pattern units connected to it

4.Classify by selecting the maximum value of Pn(x | j) (j = 1, …, c)

x.wnet tkk

2k

k

1netexp)net(f

)x|(P)|x(P j

n

1iijn


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• 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 kn samples (kn = f(n))

•kn are called the kn nearest-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


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Illustration

For kn = n = 1 ; the estimate becomes:

Pn(x) = kn / n.Vn = 1 / V1 =1 / 2|x-x1|


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•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

•ki samples amongst k turned out to be labeled i then:

pn(x, i) = ki /n.V

An estimate for pn(i| x) is:k

k

),x(p

),x(p)x|(p i

cj

1jjn

inin


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•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


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•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 x then 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)


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Example:x = (0.68, 0.60)t

Decision: 5 is the label assigned to x

Prototypes Labels A-posteriori probabilities estimated

(0.50, 0.30)

(0.70, 0.65)

2

3

5

6

0.250.75 = P(m | x)

0.700.30


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


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•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


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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

Prototypes Labels

(0.15, 0.35)(0.10, 0.28)(0.09, 0.30)(0.12, 0.20)

1

2

5

2


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254.6 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

kd

i

k

iik baL/1

1

)ba,(


26Distance 1.0 from (0,0,0) for different k


27Scaling the coordinates


28Tanimoto 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

1221

122121

2),(

nnn

nnnSSDTanimoto


294.6.2 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


30100-dim patterns

Shifted s pixelsHand write “8”


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334.8 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.


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36 RCE Training

1. 2. 3. 4. 5. 6. 7. 8.

begin init j=0, λm=max radius

do j = j + 1wij = xi

= arg min D(x, x’) λj = min[ D( , x’) - ε, λm]

if x wk then ajk =1

until j = nend

x̂x̂

iwx

Train weight

Find nearest point not in wi

Set radius

Connect pattern and category


37 RCE Classification

1. 2. 3. 4. 5. 6. 7. 8.

begin init j=0, k=0, x=test pattern, Dt={}

do j = j + 1

if D(x, xj’) < λj then Dt= Dt∪xj’

until j=n

if label of all xj’ Dt is the same

then return label of all xk Dt

else return “ambiguous” label

end

Prototype xj’

Radius of xj’

Set of stored prototypes


38Summary

• 2 Nonparametric estimation approaches1.Densities are estimated (then used for

classification)2.Category is chosen directly

• Densities are estimated • Parzen windows, probabilistic neural networks

• Category is chosen directly• K-nearest-neighbor, reduced coulomb energy

networks


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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)

chapter 4 (part 2): non-parametric classification (sections 4.3-4.5)

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