pattern recognition professor aly a. farag computer vision and image processing laboratory...

PATTERN RECOGNITION

Professor Aly A. FaragComputer Vision and Image Processing Laboratory

University of LouisvilleURL: www.cvip.uofl.edu ; E-mail: [email protected]

Planned for ECE 620 and ECE 655 - Summer 2011TA/Grader: Melih Aslan; CVIP Lab Rm 6, [email protected]

Lecture 15 – Non Parametric Methods

http://www.cvip.uofl.edu/

mailto:[email protected]

mailto:[email protected]

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

Pattern Classification, Ch4 (Part 1) 3

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)

(1) 'dx)'x(pP

(2) )P1(P k

nP knk

k


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.

)|P(Max k

Pn

kˆ

(4) V)x(p'dx)'x(p


Combining equation (1) , (3) and (4) yields:

V

n/k)x(p


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.

(4) V)x(p'dx)'x(p


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:

)()'x(p'dx)x(1)'x(p'dx)'x(p'dx)'x(p

nV

k)x(p and

V).x(p'dx)'x(p


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!

fixed)n (if 0)x(plim0k ,0V

)x(plim0k ,0V


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 difficultyTo 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)


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”

0n/klim )3

klim )2

0Vlim )1

nn

nn

nn

)x(p)x(pn

n


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.

otherwise0

d , 1,...j 2

1u 1

(u)

:functionwindow following the be (u) Let

) ofedge the of length :(h hV

j

nndnn


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!

ni

1i n

in h

xxk

n

ini

1i nn h

xx

V1

n1

)x(p


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.

n

ini

1i nn h

xx

h

1

n

1)x(p


Numerical results:

For n = 1 and h1=1

For n = 10 and h = 0.1, the contributions of the individual samples are clearly observable !

)1,x(N)xx(e2

1 )xx()x(p 1

21

2/111


Analogous results are also obtained in two dimensions as illustrated:


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


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.

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) 26

Parzen Windows (cont.)

Parzen Windows – Probabilistic Neural Networks

Compute a Parzen estimate based on n patterns

Patterns with d features sampled from c classesThe input unit is connected to n patterns

.x1

x2

xd

.

.

.

p1

p2

pn

.

.

.

Input unit Input patterns

.

. ..

Modifiable weights (trained)

Wdn

Wd2

W11. .


p1

p2

pn

.

.

.

Input patterns

pn

pk... .

.

.

.1

2

c...

Category units

.

.

.

Activations (Emission of nonlinear functions)


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


Testing the network

Algorithm

1. Normalize the test pattern x and place it at the input units2. 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

2

kk

1netexp)net(f

)x|(P)|x(P j

n

1iijn


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 dataCenter 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


Illustration

For kn = n = 1 ; the estimate becomes:

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


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 sampleski 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


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


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)


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)

23

56

0.250.75 = P(m | x)

0.700.30


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


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


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)

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