1. Pattern recognition Professor Aly A. Farag
Lecture 15 – Non Parametric Methods
Professor Aly A. Farag
Computer Vision and Image Processing Laboratory
University of Louisville
URL: ;
Planned for ECE 620 and ECE Summer 2011
TA/Grader: Melih Aslan; CVIP Lab Rm 6,

2. Introduction
All Parametric densities are unimodal (have a single local maximum), whereas many practical problems involve multi-modal densities
Nonparametric procedures can be used with arbitrary distributions and without the assumption that the forms of the underlying densities are known
There are two types of nonparametric methods:
Estimating P(x | j )
Bypass probability and go directly to a-posteriori probability estimation

3. Pattern Classification, Ch4 (Part 1)
Density Estimation
Basic idea:
Probability that a vector x will fall in region R is:
P is a smoothed (or averaged) version of the density function p(x) if we have a sample of size n; therefore, the probability that k points fall in R is then:
and the expected value for k is:
E(k) = nP (3)
Pattern Classification, Ch4 (Part 1)

4. Pattern Classification, Ch4 (Part 1)
ML estimation of P = 
is reached for
Therefore, the ratio k/n is a good estimate for the probability P and hence for the density function p.
p(x) is continuous and that the region R is so small that p does not vary significantly within it, we can write:
where is a point within R and V the volume enclosed by R.
Pattern Classification, Ch4 (Part 1)

5. Pattern Classification, Ch4 (Part 1)
Combining equation (1) , (3) and (4) yields:
Pattern Classification, Ch4 (Part 1)

6. Density Estimation (cont.)
Justification of equation (4)
We assume that p(x) is continuous and that region R is so small that p does not vary significantly within R. Since p(x) = constant, it is not a part of the sum.
Pattern Classification, Ch4 (Part 1)

7. Pattern Classification, Ch4 (Part 1)
Where: (R) is: a surface in the Euclidean space R2
a volume in the Euclidean space R3
a hypervolume in the Euclidean space Rn
Since p(x)  p(x’) = constant, therefore in the Euclidean space R3:
Pattern Classification, Ch4 (Part 1)

8. Pattern Classification, Ch4 (Part 1)
Condition for convergence
The fraction k/(nV) is a space averaged value of p(x).
p(x) is obtained only if V approaches zero.
This is the case where no samples are included in R: it is an uninteresting case!
In this case, the estimate diverges: it is an uninteresting case!
Pattern Classification, Ch4 (Part 1)

9. Pattern Classification, Ch4 (Part 1)
The volume V needs to approach 0 anyway if we want to use this estimation
Practically, V cannot be allowed to become small since the number of samples is always limited
One will have to accept a certain amount of variance in the ratio k/n
Theoretically, if an unlimited number of samples is available, we can circumvent this difficulty
To estimate the density of x, we form a sequence of regions
R1, R2,…containing x: the first region contains one sample, the second two samples and so on.
Let Vn be the volume of Rn, kn the number of samples falling in Rn and pn(x) be the nth estimate for p(x):
pn(x) = (kn/n)/Vn (7)
Pattern Classification, Ch4 (Part 1)

10. Pattern Classification, Ch4 (Part 1)
Three necessary conditions should apply if we want pn(x) to converge to p(x):
There are two different ways of obtaining sequences of regions that satisfy these conditions:
(a) Shrink an initial region where Vn = 1/n and show that
This is called “the Parzen-window estimation method”
(b) Specify kn as some function of n, such as kn = n; the volume Vn is
grown until it encloses kn neighbors of x. This is called “the kn-nearest
neighbor estimation method”
Pattern Classification, Ch4 (Part 1)

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13. Pattern Classification, Ch4 (Part 1)
Parzen Windows
Parzen-window approach to estimate densities assume that the region Rn is a d-dimensional hypercube
((x-xi)/hn) is equal to unity if xi falls within the hypercube of volume Vn centered at x and equal to zero otherwise.
Pattern Classification, Ch4 (Part 1)

14. Pattern Classification, Ch4 (Part 1)
The number of samples in this hypercube is:
By substituting kn in equation (7), we obtain the following estimate:
Pn(x) estimates p(x) as an average of functions of x and
the samples (xi) (i = 1,… ,n). These functions  can be general!
Pattern Classification, Ch4 (Part 1)

15. Pattern Classification, Ch4 (Part 1)
Illustration
The behavior of the Parzen-window method
Case where p(x) N(0,1)
Let (u) = (1/(2) exp(-u2/2) and hn = h1/n (n>1)
(h1: known parameter)
Thus:
is an average of normal densities centered at the
samples xi.
Pattern Classification, Ch4 (Part 1)

16. Pattern Classification, Ch4 (Part 1)
Numerical results:
For n = 1 and h1=1
For n = 10 and h = 0.1, the contributions of the individual samples are clearly observable !
Pattern Classification, Ch4 (Part 1)

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19. Analogous results are also obtained in two dimensions as illustrated:
Pattern Classification, Ch4 (Part 1)

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21. Pattern Classification, Ch4 (Part 1)
Case where p(x) = 1.U(a,b) + 2.T(c,d) (unknown density) (mixture of a uniform and a triangle density)
Pattern Classification, Ch4 (Part 1)

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Classification example
In classifiers based on Parzen-window estimation:
We estimate the densities for each category and classify a test point by the label corresponding to the maximum posterior
The decision region for a Parzen-window classifier depends upon the choice of window function as illustrated in the following figure.
Pattern Classification, Ch4 (Part 1)

24. Pattern Classification, Ch4 (Part 1)

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

26. 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 x1 x2 xd . p1 p2 .
Input unit
Input
patterns . .
Wd2
Wdn pn
Pattern Classification, Chapter 4 (Part 2)
Modifiable weights (trained)

27. (Emission of nonlinear functions). . pn . p1 . p2
Input
patterns . 1 . 2 .
Category
units pk . . . c pn
Activations
(Emission of nonlinear functions)
Pattern Classification, Chapter 4 (Part 2)

28. 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
Pattern Classification, Chapter 4 (Part 2)

29. Pattern Classification, Chapter 4 (Part 2)

30. 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)
Pattern Classification, Chapter 4 (Part 2)

31. 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
Pattern Classification, Chapter 4 (Part 2)

32. Illustration For kn = n = 1 ; the estimate becomes:
Pn(x) = kn / n.Vn = 1 / V1 =1 / 2|x-x1|
Pattern Classification, Chapter 4 (Part 2)

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35. 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:
Pattern Classification, Chapter 4 (Part 2)

36. 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
Pattern Classification, Chapter 4 (Part 2)

37. 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)
Pattern Classification, Chapter 4 (Part 2)

38. A-posteriori probabilities estimated
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
0.25
0.75 = P(m | x)
0.70
0.30
Pattern Classification, Chapter 4 (Part 2)

39. If P(m | x)  1, then the nearest neighbor selection is almost always the same as the Bayes selection
Pattern Classification, Chapter 4 (Part 2)

40. Pattern Classification, Chapter 4 (Part 2)

41. 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
Pattern Classification, Chapter 4 (Part 2)

42. Pattern Classification, Chapter 4 (Part 2)

43. Example: Prototypes Labels (0.15, 0.35) (0.10, 0.28) (0.09, 0.30)
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
Pattern Classification, Chapter 4 (Part 2)