2012　mdsp pr08 nonparametric approach

Course Calendar Class DATE Contents

1 Sep. 26 Course information & Course overview

2 Oct. 4 Bayes Estimation

3 〃 11 Classical Bayes Estimation - Kalman Filter -

4 〃 18 Simulation-based Bayesian Methods

5 〃 25 Modern Bayesian Estimation ：Particle Filter

6 Nov. 1 HMM(Hidden Markov Model)

Nov. 8 No Class

7 〃 15 Bayesian Decision

8 〃 29 Non parametric Approaches

9 Dec. 6 PCA(Principal Component Analysis)

10 〃 13 ICA(Independent Component Analysis)

11 〃 20 Applications of PCA and ICA

12 〃 27 Clustering, k-means et al.

13 Jan. 17 Other Topics 1 Kernel machine.

14 〃 22(Tue) Other Topics 2

Lecture Plan

Nonparametric Approaches

1. Introduction

1.1 An Example

1.2 Nonparametric Density Estimation Problems

1.3 Histogram density estimation

2. Kernel Density Estimation

3. K-Nearest Neighbor Density Estimation

4. Cross-validation

1. Introduction

3

Automatic Fish-Sorting Process

action １

belt conveyer

Classification/ Decision Theory

p ," sea bass" ,

p ," salmon"

x

x

x1

x2

Decision Boundary

R2 R1

(Duda, Hart, & Stork 2004)

Training phase

Test phase

1.1 An Example -Fish sorting problem-

4

The first step -Training process- The first task is a supervised learning process, thus each observed

sample has its own label either states of nature 𝜔1 𝑜𝑟 𝜔2.

Training(Learning) data

For a given set of N data samples, suppose the following labeled data

(the lightness of a fish) are observed.

1 21 2 1 2

1 2

( ) ( )

1

2

( ) ( )

: , , , , : , , ,

where , are the lightness of i-th samples of (sea bass)

and j-th sample of (salmon) respectively.

We assume , are discrete data as illustrated in Fig.1

N N

i j

i j

x x x y y y

x y

x y

(a)

This joint probability distribution gives the histograms of other

probabilities and densities as shown in Fig.1.

5 These histograms can be used for Bayes decision classification.

(a) Samples drawn from a joint probability over x and ω

Fig.1

iN

N

p x

1P

2P

1p x 2p x

x

x

x x

(b)

(c)

(d) (e)

1

2

6

Density Estimation The approach attempts to estimate the density directly from the

observed data.

The Bayes decision rules discussed in the last lecture have been

developed on the assumption that the relevant probability density

functions and prior probabilities are known. But this is not the case in

practice, we need to estimate the PDFs from a given se of observed data.

given data density distribution

Modeling of density

http://www.unt.edu/benchmarks/archives/2003/february03/rss.htm

7

1.2 Nonparametric approaches

Two approaches for density estimation

Parametric approach :

Assume that the forms of the density functions are known, and the

parameters of its are to be estimated.

Non-parametric approach:

This can be used with arbitrary distributions, and does not assume the

form of density function.

Why nonparametric?*

/ Classical parametric densities are unimodal ones whereas practical

problems involve multimodal densities. Non parametric approaches are

applicable to arbitrary density with few assumptions

*Another idea: a mixed (hybrid) with parametric and non-parametric densities.

8

1.3 Histogram density estimation A single variable (x) case

into distinct intervals (called ) of width (often chosen as

uniform bins ), and the number of data falling in -th bin.

For turning to a normalized probability density w

i

i

i

x

n i

n

Partition bins

count

e put

= at over i-th bin

The density ( ) is approximated by a stepwise function like bar graph.

In multi-dimensional case,

=

where V is

ii

i

i

np x

N

p x

np x

NV

the volume of the bin.

Fig. 2 (RIGHT) Histogram density estimation (from Bishop [3] web site) 50 data points are generated from the distribution shown by the green curve

(1)

9

The feature of the histogram estimation depends on the width (Δi

) of bins as shown in Fig.2.

Small Δ → density tends to have many spikes

Large Δ → density tends to be over-smoothed

Merit: Convenient visualization tool

Problems:

Discontinuities at the bin’s edges

Computational burden in high dimensional space (MD)

2. Kernel Density Estimation

10

-Basic idea of density estimation-

Unknown Density

D

p x

x R

1 2

A set of observations

, , N

N

x x x

p xEstimate

Consider a small regiron surrounding ( ) and define a probability

which means the probability of falling into

The number falling within of overall observe

x x

P p d

x

K N

R

R R

R

R.

d data would be

Suppose can be approximated by a constant over we have

K PN

p x R

generates

(2)

(3)

11

where means the volume of .

Eqs. (2) (3) give the following density estimation form.

Two wayes of exploitation of this:

1) Fix then estimate

P p x V

V

Kp x

NV

V K

R

Kernel Density Estimation

2) Fix then estimate k-Nearest Neighbor Estimation K V

(4)

(5)

12

Kernel Density Estimation

A point : wish to determine the density at this point

Region : small hypercube centered on

the volume of

Find the number of samples K that fall within the region .

For

x

x

V

R

R

R

counting we introduce the kernek function.

[one dimensional kernel function]

11

20

K

uK u

elsewhere

For a given observation , consider

n

n

x

x xK

h

(6)

(7)

13

1

1 2

0

where is called the bandwidth or smoothing parameter.

For a set of observations : 1, ,

gives the number of data which are located

nn

n

Nn

n

hx xx x

Kh

elsewhere

h

x n N

x xK K

h

1

within ,2 2

Substituting (9) into (5)

1

An example graph of is illustrated in Fig.3

Nn

n

h hx x

x xp x K

Nh h

p x

(8)

(9)

(10)

14

Example : Data set {xn} n=1~4

x x1 x2 x3 x4

x x1 x2 x3 x4

11 x xK

h h

41 x xK

h h

4

1

1 i

i

x xK

h h

1

4p x

x1

x2

x3

x4

Fig. 3

15

Discontinuity and Smooth Kernel Function

Kernel Density Estimator will suffer from discontinuities in

estimated density. Smooth kernel function such as Gaussian is used.

This general method is referred to as the kernel density estimator or

Parzen estimator.

Example: Gaussian and 1-D case,

where h is the standard deviation.

Determination of bandwidth h

Small h → spiky p(x)

Large h → over-smoothed p(x)

Defects:

High computational cost

Because of fixed V there may be too few samples in some regions.

2

21

( )1 1exp

22

Nn

n

x xp x

N hh

(11)

16

17

Example: Bayesian decision by the Parzen kernel estimation

Example: Kernel density estimation

Fig 4. Kernel density estimation (from Bishop [3] web site) Apply KDE method to the same 50 data used in Fig. 2

Fig. 5 The decision

boundaries

LEFT : small h

RIGHT: large h

Duda et al [1]

3. K-Nearest Neighbors density estimation

18

KDE approaches use fixed h throughout the data space. But we want

to apply small h for highly dense data region, on the other hand, to set

a larger h for sparse data region.

We come up with the idea of K-Nearest Neighbor (K-NN) approaches.

Expand the region (radius) surrounding the estimation point x until it

encloses K data points.

K

p xNV

Fixed

Determine the minimum

volume V containing K

points in R

Volume of hypersphere with radius ( )

D

D

r x

K Kp x

N V N c r x

(12)

19

x

K-th closest neighbor point

1 2 3

4( 2, , , )

3Dc c c c

Fig. 7 K-nearest neighbor density estimation (from Bishop [3] web site) Apply to the same 50 data points used in Fig. 2 K is the free parameter in K-NN method

r(x)

Problems: 1) integration of p(x) is not bounded, 2) discontinuities, 3)huge computation time and storage

Fig. 6 K-NN algorithm

20

K-NN estimation as a Bayesian classifier

A method to generate decision boundary directly based on a set of

data.

− N training data with class labels (ω1 ~ ωc)

Nl points for l-th class, such that

− Classify a test sample x

− Get a sphere with minimum radius r(x) which encircles K samples.

− The volume of the sphere：

− Kl points for l-th class (ωl)

− Class-conditioned density at x:

− Evidence:

− Prior probabilities:

− Posterior Probabilities (Bayes’ theorem)

1

c

l

l

N N

D

DV c r x

ll

l

Kp x

N V

ll

Np

N

K

p xNV

l l l

l

p x P KP x

p x K

(14)

21

K-NN classifier

− Find the class maximizing the posterior probability (Bayes decision)

− The point x is classified into 𝜔𝑙0

Summary: (Fig.8)

1) Select K data surrounding the estimation point x

2) The point x is assigned the major class of K points in the neighbor.

0 : arg max arg max ll

l l

Kl P x

K

Nearest Neighbor classifier

Let consider K=1 for K-NN classification.

The point x is classified into the class of the nearest point to x.

→ Nearest Neighbor Classifier

Classification boundary of the K-NN with respect to K

small K → tends to make many small class regions (Fig. 9 (b))

large K → few class regions (Fig.9(a))

(14)

22

K-Nearest Neighbor classifier algorithm (K=3)

Nearest Neighbor Classifier algorithm (K=1)

Fig. 9 K-Nearest Neighbor Classifiers (K=3, 1, 31) Bishop [3]

Fig. 8 (Bishop [3])

x

(a) (b) (c)

23

4. Cross-Validation Parameter determination problem:

- Since the classifier have free parameters such as K in K-NN

classifier and h in Kernel density based classifiers, we need to

select the optimal parameters by evaluating the classification

performances.

- Over-fitting problem

- The classifier’s parameter (decision boundary) obtained from

using overall training data will overfit to them

→ Need an appropriate new test data

Cross-Validation

- Given data are split into S parts.

S=4

- Fig. 10

- Use (S-1) parts for training and a rest part is used for testing

- Apply all different parts for the test as shown in Fig. 11 (S=4)

total data

24

training data test data

Experiment 1

Experiment 2

Experiment 3

Experiment 4

Score * 1

Score 2

Score 3

Score 4

×(1/4)＝Averaged score

If we want to determine the best K for K-NN classifier, we

choose the K providing the highest averaged score by the

cross-validation procedure.

*Score= error rate or conditioned risk et al.

Fig. 11

＋）

25

References: [1] R.O. Duda, P.E. Hart, and D. G. Stork, “Pattern Classification”, John Wiley & Sons, 2nd edition, 2004 [2] C. M. Bishop, “Pattern Recognition and Machine Learning”, Springer, 2006 [3] All data files of Bishop’s book are available at the “http://research.microsoft.com/~cmbishop/PRML”