Classification

ClassificationClassification

Adriano Joaquim de O Cruz ©2002

NCE/UFRJ

[email protected]

*@2001 Adriano Cruz *NCE e IM - UFRJ Classification 2

ClassificationClassification

Technique that associates samples to Technique that associates samples to classes previously known. classes previously known.

May be Crisp or FuzzyMay be Crisp or Fuzzy SupervisedSupervised

MLP MLP trained trained Non supervisedNon supervised

K-NN e fuzzy K-NN K-NN e fuzzy K-NN not trained not trained


K-NN and Fuzzy K-NNK-NN and Fuzzy K-NN

Classification MethodsClassification Methods

Classes identified by patternsClasses identified by patterns

Classifies by the k nearest neighboursClassifies by the k nearest neighbours

Previous knowledge about the problem Previous knowledge about the problem classesclasses

It is not restricted to a specific distribution of It is not restricted to a specific distribution of the samplesthe samples

Classification

Crisp K-NNCrisp K-NN



Supervised clustering method (Classification Supervised clustering method (Classification method).method).

Classes are defined before hand.Classes are defined before hand. Classes are characterized by sets of elements.Classes are characterized by sets of elements. The number of elements may differ among The number of elements may differ among

classes.classes. The main idea is to associate the sample to the The main idea is to associate the sample to the

class containing more neighbours.class containing more neighbours.



ww22ww11

ww33ww1313

ww1010

ww99

ww44

ww55

ww1414

ww1111 ww1212

ww77

ww88

ww66

ss

Class 1Class 1 Class 2Class 2 Class 3Class 3

Class 4Class 4 Class 5Class 5

3 nearest neighbours, and sample 3 nearest neighbours, and sample ss is closest to is closest to pattern pattern ww66 on class 5. on class 5.



Consider Consider WW={={ww11, w, w22, ..., w, ..., w

tt}} a set of a set of tt labelled labelled

data.data. Each object Each object ww

ii is defined by is defined by ll characteristics characteristics

ww ii=(=(wwi1i1, w, w

i2i2, ..., w, ..., w

ilil).).

Input of Input of yy unclassified elements. unclassified elements. k k the number of closest neighbours of the number of closest neighbours of yy.. EE the set of the set of kk nearest neighbours (NN). nearest neighbours (NN).



Let Let tt be the number of elements that identify be the number of elements that identify the classes.the classes.

Let Let cc be the number of classes. be the number of classes. Let Let WW be the set that contain the be the set that contain the tt elements elements Each cluster is represented by a subset of Each cluster is represented by a subset of

elements from elements from WW..


Crisp K-NN algorithmCrisp K-NN algorithm

setset kk{Calculating the NN}{Calculating the NN}forfor i i = 1 = 1 toto tt

Calculate distance from Calculate distance from yy to to xxii

ifif ii<=<=kk thenthen add add xxii to to EE

elseelse ifif xxii is closer to is closer to yy than any than any previous NNprevious NN

thenthen delete the farthest neighbour delete the farthest neighbour and include and include xxii in the set in the set EE


Crisp K-NN algorithm cont.Crisp K-NN algorithm cont.

Determine the majority class represented in Determine the majority class represented in the set the set E E and include and include yy in this class. in this class.

if if there is a draw, there is a draw, thenthen calculate the sum of distances from calculate the sum of distances from yy to to all neighbours in each class in the drawall neighbours in each class in the drawifif the sums are different the sums are differentthenthen add add xxii to class with smallest to class with smallest sumsumelse else add add xxii to class where last minimum to class where last minimum

was foundwas found

Classification

Fuzzy K-NNFuzzy K-NN



The basis of the algorithm is to assign The basis of the algorithm is to assign membership as a function of the object’s membership as a function of the object’s distance from its K-nearest neighbours and distance from its K-nearest neighbours and the memberships in the possible classes.the memberships in the possible classes.

J. Keller, M. Gray, J. Givens. A Fuzzy K-J. Keller, M. Gray, J. Givens. A Fuzzy K-Nearest Neighbor Algorithm. IEEE Nearest Neighbor Algorithm. IEEE Transactions on Systems, Man and Transactions on Systems, Man and Cybernectics, vol smc-15, no 4, July August Cybernectics, vol smc-15, no 4, July August 1985 1985



w1

w2

w3

w4

w13

w10

w9 w14

w5

w8

w12

w11

w6w7

Classe 1Classe 2

Classe 3

Classe 4 Classe 5



Consider Consider WW={={ww11, w, w

22, ..., w, ..., w

tt}} a set of a set of tt labelled data. labelled data.

Each object Each object wwii is defined by is defined by ll characteristics characteristics ww ii=(=(ww

i1i1, ,

wwi2i2, ..., w, ..., w

ilil).).

Input of Input of yy unclassified elements. unclassified elements. k k the number of closest neighbours of the number of closest neighbours of yy.. EE the set of the set of kk nearest neighbours (NN). nearest neighbours (NN). ii(y)(y) is the membership of is the membership of yy in the class in the class ii ijij is the membership in theis the membership in the itith h class of theclass of the jjth th vector vector

of the of the labelled set (labelled labelled set (labelled wwjj in class i) in class i) ..



Let Let tt be the number of elements that identify be the number of elements that identify the classes.the classes.

Let Let cc be the number of classes. be the number of classes. Let Let WW be the set that contain the be the set that contain the tt elements elements Each cluster is represented by a subset of Each cluster is represented by a subset of

elements from elements from WW..


Fuzzy K-NN algorithmFuzzy K-NN algorithm

setset kk{Calculating the NN}{Calculating the NN}forfor i i = 1 = 1 toto tt

Calculate distance from Calculate distance from yy to to xxii

ifif ii<=<=kk thenthen add add xxii to to EE

elseelse ifif xxii is closer to is closer to yy than than any previous NNany previous NN

thenthen delete the farthest neighbour delete the farthest neighbour and include and include xxii in the set in the set EE


Fuzzy K-NN algorithmFuzzy K-NN algorithm

Calculate Calculate ii(y)(y) using using

forfor i i = 1 = 1 toto c // number of classesc // number of classes

μi( y )=

∑j=1

k

μij ( 1

∥y−x j∥2/(m−1 ))

∑j=1

k

( 1

∥y−x j∥2/ (m−1) )


Computing Computing ijij(y) (y) ijij(y) (y) can be assigned class membership in can be assigned class membership in

several ways.several ways. They can be given complete membership in their They can be given complete membership in their

known class and non membership in all other.known class and non membership in all other. Assign membership based on distance from their Assign membership based on distance from their

class mean.class mean. Assign membership based on the distance from Assign membership based on the distance from

labelled samples of their own class and those labelled samples of their own class and those from other classes.from other classes.

Classification

ICC-KNN SystemICC-KNN System



Non-Parametric Statistical Pattern Non-Parametric Statistical Pattern Recognition SystemRecognition System

Associates FCM, fuzzy KNN and ICCAssociates FCM, fuzzy KNN and ICC

Evaluates data disposed on several class Evaluates data disposed on several class formatsformats



Divided in two modulesDivided in two modules First module (training)First module (training)

chooses the best patterns to use with K-NNchooses the best patterns to use with K-NN chooses the best fuzzy constant and best chooses the best fuzzy constant and best

number of neighbours (K)number of neighbours (K) Second module (classification)Second module (classification)

uses fuzzy k-nn to classifyuses fuzzy k-nn to classify


ICC-KNN First ModuleICC-KNN First Module

Classification ModuleClassification Module Finds structure on data sampleFinds structure on data sample Divided into two phasesDivided into two phases

First phase of trainingFirst phase of training Finds the best patterns for fuzzy K-NNFinds the best patterns for fuzzy K-NN

• FCM – Applied to each class using many numbers FCM – Applied to each class using many numbers of categoriesof categories

• ICC – Finds the best number of categories to ICC – Finds the best number of categories to represent each classrepresent each class


ICC-KNN First PhaseICC-KNN First Phase

Results of applying FCM and ICC Results of applying FCM and ICC

Patterns for K-NN which are the centres of the Patterns for K-NN which are the centres of the chosen run of FCMchosen run of FCM

Number of centres which are all the centres of Number of centres which are all the centres of the number of categories resulting after the number of categories resulting after applying ICC to all FCM runsapplying ICC to all FCM runs


ICC-KNN Second PhaseICC-KNN Second Phase

Second phase of trainingSecond phase of training

Evaluates the best fuzzy constant and the Evaluates the best fuzzy constant and the best number of neighbours so to achieve best number of neighbours so to achieve best performance on the K-NNbest performance on the K-NN

tests several tests several mm and and kk values values

finds finds mm and and kk for the maximum rate of crisp hits for the maximum rate of crisp hits


ICC-KNNICC-KNN

Pattern Recognition ModulePattern Recognition Module Distributes each data to its classDistributes each data to its class

Uses the chosen patterns, m and k to Uses the chosen patterns, m and k to classify dataclassify data


ICC-KNN block diagramICC-KNN block diagram

Class 1

Class s

FCM

FCM

ICC

ICC

Fuzzy K-NN

m k

W, Uw

W Uw

w1

ws

U1cmin

U1cmáx

UScmin

UScmáx

FuzzyK-NN

Classification ModulePattern

RecognitionModule

Not classified Data


ICC-KNNICC-KNN

Let Let R=R={{rr11,r,r22,...,r,...,rnn}} be the set of samples. be the set of samples.

Each sample Each sample rr ii belongs to one of belongs to one of ss known classes. known classes.

Let Let UUicic be the inclusion matrix for the class be the inclusion matrix for the class ii with with cc categories.categories.

Let Let VVicic be the centre matrix for the class be the centre matrix for the class ii with with cc categories.categories.

Let Let wwii be equal to the best be equal to the best VV icic of each class of each class

Let Let WW be the set of sets of centres be the set of sets of centres ww ii


ICC-KNN algorithmICC-KNN algorithm

Classification ModuleClassification Module First phase of trainingFirst phase of training Step 1.Step 1. Set Set mm Step 2.Step 2. Set Set cmin andcmin and cmáxcmáx Step 3.Step 3. For each s known class For each s known class

Generate the set Generate the set RsRs with the points from with the points from RR belonging to the class s belonging to the class sFor each category c in the interval [For each category c in the interval [ cmincmin , , cmáxcmáx ]]

Run Run FCMFCM for c and the set for c and the set RsRs generating generating UscUsc and and VscVsc

Calculate Calculate ICCICC for for RsRs e e UscUsc

EndEnd

Define the patterns Define the patterns ws ws ofof class s as the matrix class s as the matrix VscVsc that that maximizes maximizes ICCICC

Step 4. Step 4. Generate the set W = {w1, ..., ws}Generate the set W = {w1, ..., ws}



Second phase of TrainingSecond phase of Training Step 5. Step 5. Set mmin e mmáx Set mmin e mmáx Step 6. Step 6. SetSet kmin e kmáxkmin e kmáx

For each m from [mmin , mmáx]For each m from [mmin , mmáx]

For each k from [kmin , kmáx]For each k from [kmin , kmáx]

Run fuzzy K-NN for the patterns from the set WRun fuzzy K-NN for the patterns from the set Wgenerating Umk generating Umk

Calculate the number of crisp hits for UmkCalculate the number of crisp hits for Umk

Step 7. Step 7. Choose m and k that yields the best crips hit figuresChoose m and k that yields the best crips hit figures Step 8. Step 8. if there is a drawif there is a draw

If the k’s are differentIf the k’s are different

Choose the smaller kChoose the smaller k

elseelse

Choose the smaller mChoose the smaller m



Pattern Recognition ModulePattern Recognition Module

Step 9. Step 9. Apply fuzzy K-NN using patterns form the set W and the chosen Apply fuzzy K-NN using patterns form the set W and the chosen parameters m and k to the data to be classified.parameters m and k to the data to be classified.


ICC-KNN resultsICC-KNN results


ICC-NN resultsICC-NN results

2000 samples, 4 classes, 500 samples 2000 samples, 4 classes, 500 samples in each classin each class

Classes 1 and 4 – concave classesClasses 1 and 4 – concave classes

Classes 2 and 3 – convex classes, Classes 2 and 3 – convex classes, elliptic formatelliptic format



First phase of trainingFirst phase of training FCM applied to each classFCM applied to each class

Training data 80% Training data 80% 400 samples from each 400 samples from each classclass

cc = 3..7 and = 3..7 and mm = 1,25 = 1,25 ICC applied to resultsICC applied to results

Classes 1 and 4 Classes 1 and 4 4 categories 4 categories Classes 2 and 3 Classes 2 and 3 3 categories 3 categories



Second phase of TrainingSecond phase of Training Running fuzzy K-NNRunning fuzzy K-NN

Patterns from first phasePatterns from first phase Random patternsRandom patterns k = 3 a 7 neighboursk = 3 a 7 neighbours m = {1,1; 1,25; 1,5; 2}m = {1,1; 1,25; 1,5; 2}



Conclusão:K-NN é mais estável em relação ao valor de m para os padrões da PFTConclusão:K-NN é mais estável em relação ao valor de m para os padrões da PFT



Training dataTraining data Lines Lines classes classes Columns Columns classification classification m = 1,5 e k = 3 m = 1,5 e k = 3 96,25% 96,25% m = 1,1 e k = 3 m = 1,1 e k = 3 79,13% (random patterns) 79,13% (random patterns)

34914643972104

733240324376003

103801970379142

12106621320103881

43214321

Padrões AleatóriosPadrões da PFTClasses

Dados de Treinamento



Test dataTest data Lines Lines classes classes Columns Columns classification classification Pad. PFT – 94,75% Pad. Aleat – 79%Pad. PFT – 94,75% Pad. Aleat – 79%

850150991004

1882001090003

00964309342

2002753102971

43214321

Padrões AleatóriosPadrões da PFTClasses

Dados de Testes


ICC-KNN x OthersICC-KNN x Others

FCM, FKCN, GG e GKFCM, FKCN, GG e GK Fase de Treinamento (FTr)Fase de Treinamento (FTr)

Dados de treinamentoDados de treinamento c = 4 e m = {1,1; 1,25; 1,5; 2}c = 4 e m = {1,1; 1,25; 1,5; 2} Associar as categorias às classesAssociar as categorias às classes

• Critério do somatório dos graus de inclusãoCritério do somatório dos graus de inclusãoo Cálculo do somatório dos graus de inclusão dos pontos de Cálculo do somatório dos graus de inclusão dos pontos de

cada classe em cada categoriacada classe em cada categoria

o Uma classe pode ser representada por mais de uma Uma classe pode ser representada por mais de uma categoriacategoria



Fase de TesteFase de Teste Dados de TesteDados de Teste Inicialização dos métodos com os centros da FTrInicialização dos métodos com os centros da FTr Calcula o grau de inclusão dos pontos em cada Calcula o grau de inclusão dos pontos em cada

categoriacategoria

Classe representada por mais de 1 categoriaClasse representada por mais de 1 categoria Grau de inclusão = soma dos graus de inclusão dos Grau de inclusão = soma dos graus de inclusão dos

pontos nas categorias que representam a classepontos nas categorias que representam a classe



GK para m = 2 GK para m = 2 84% 84% FCM e FKCN FCM e FKCN 66% para m = 1,1 e m = 1,25 66% para m = 1,1 e m = 1,25 GG-FCM GG-FCM 69% para m = 1,1 e 1,25 69% para m = 1,1 e 1,25 GG Aleatório GG Aleatório 57,75% para m = 1,1 e 25% para m = 57,75% para m = 1,1 e 25% para m =

1,51,5

18,14s22,66s2,59s2,91s23,11s36,5sT

89,5%69%70,75%70,75%83%95,75%N

84%69%66%66%79%94,75%R

GKGGFKCNFCMKNN A.

ICC-KNN


GKGK


GKGK

Classes

GK

1 2 3 4

1 77 6 0 17

2 6 94 0 0

3 0 0 97 3

4 0 0 32 68

Classification

KNN+Fuzzy Cmeans SystemKNN+Fuzzy Cmeans System


KNN+Fuzzy C-Means algorithmKNN+Fuzzy C-Means algorithm

The idea is an two-layer clustering algorithmThe idea is an two-layer clustering algorithm First an unsupervised tracking of cluster centres is First an unsupervised tracking of cluster centres is

made using K-NN rulesmade using K-NN rules The second layer involves one iteration of the The second layer involves one iteration of the

fuzzy c-means to compute the membership fuzzy c-means to compute the membership degrees and the new fuzzy centres.degrees and the new fuzzy centres.

Ref. N. Zahit et all, Fuzzy Sets and Systems 120 Ref. N. Zahit et all, Fuzzy Sets and Systems 120 (2001) 239-247(2001) 239-247


First Layer (K-NN)First Layer (K-NN)

Let Let XX={={xx11,…,x,…,xnn} be a set of } be a set of nn unlabelled objects. unlabelled objects.

cc is the number of clusters. is the number of clusters. The first layer consists of partitioning The first layer consists of partitioning XX into into cc cells cells

using the fist part of K-NN.using the fist part of K-NN. Each cell Each cell ii is (1<= is (1<=ii<=<=cc) represented as ) represented as EE i i ((yy ii, K-NN , K-NN

of of yy ii, , GGii))

GGii is the center of cell is the center of cell EE ii and defined as and defined as

Gi= ∑xk∈E

i

xk /k+1


KNN-1FCMA settingsKNN-1FCMA settings

Let Let XX={={xx11,…,x,…,xnn} be a set of } be a set of nn unlabelled objects. unlabelled objects.

FixFix c c the number of clusters. the number of clusters. Choose Choose mm>1 (nebulisation factor).>1 (nebulisation factor). Set Set kk = Integer( = Integer(nn//cc –1). –1). LetLet I I={1,2,…,={1,2,…,nn} be the set of all indexes of } be the set of all indexes of XX..


KNN-1FCMA algorithm step 1KNN-1FCMA algorithm step 1

CalculateCalculate GG00

forfor i i = 1 = 1 toto ccSearch in Search in II for the index of the farthest for the index of the farthest object object yyii from from GGi1i1

For For j j = 1 to = 1 to nn Calculate distance from Calculate distance from yyii to to xxjj

if if jj <= <= kkthenthen add add xxjj to to EEii

elseelse ifif xxii is closer to is closer to yy than than any previous NNany previous NN

thenthen delete the farthest neighbour delete the farthest neighbour and include and include xxii in the set in the set EEii


KNN-1FCMA algorithm cont.KNN-1FCMA algorithm cont.

Include Include yyii in the set in the set EEi i ..

Calculate Calculate GGii..

Delete Delete yyii index index and the KNN indexes ofand the KNN indexes of y yii fromfrom I I..

ifif I I

thenthen for each remaining object for each remaining object xx determine the minimum distance determine the minimum distance to to any centre any centre GGii of of EEii..

classify x to the nearest centre.classify x to the nearest centre.update all centres.update all centres.


KNN-1FCMA algorithm step2KNN-1FCMA algorithm step2

Compute the matrix U according to Compute the matrix U according to

Calculate all fuzzy centres usingCalculate all fuzzy centres using

μik=(∑l=1

c

(d ik

d lk )2

m−1)−1

v ij=

∑e=1

n

( μie )m⋅xej

∑e=1

n

( μ ie)m


Results KNN-1FCMAResults KNN-1FCMA

1217163100IRIS

1013142150IRIS23

1913136120S4

1002480S3

801360S2

1100220S1

FCMAKNN-1FCMAFCMA

Number of Iterations avg

Misclassification rate

cElemData

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Classification