Robust methodologies for partition clustering

Computing and Mathematical SciencesLiverpool John Moores University

Robust methodologies for partition clustering

Paulo Lisboa Terence Etchells, Ian Jarman and Simon Chambers

Overview

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Original data projected onto the first 3 eigenvalues of the scatter matrix in the original domain - Clusters r+:1 go:2 ms:3 c*:4 bv:5 kp:6 y.:7 kx:8

Projection onto axis 2

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Partition clustering - critique

Decomposition of the covariance matrix

Landscape mapping of cluster solutions

Validation for two synthetic data sets and metabolic sub-typing 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1

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18 Clustering Performance: Median Cramer V of individual Clusters Vs Invariant J value

Median Cramer V Concordance Value

Invariant

J V

alu

e

2 clusters3 clusters

4 clusters

5 clusters


8 clusters

9 clusters


12 clusters

13 clusters

14 clusters15 clusters.

BioinformaticsNottingham Tenovous Primary

Breast Carcinoma Series

Consecutive series of 1,944 cases of primary operable invasive breast cancer(n=1,076 with all markers present)

Patients presenting during 1986-98

Protein expression comprising

25 immunohistochemical markers related to tumour malignancyderived through high-throughput protein expression using TMA

Abd El-Rehim et al, Int J Cancer, 116, 340-350, 2005.

Partition clustering – relevance to bioinformatics

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Original data projected onto the space of cluster means then onto 2D using scatter matrices - Clusters r+:1 go:2 ms:3 c*:4 bv:5 kp:6 y.:7

C-erbB-2

p53

PgRER

CK 5/6

BRCA1

Identify a suitable algorithm:

Model-based or model-free ?

Hierarchical, K-means, PAM ?

Return {Sa,...,Sz} solutions

Validate & interpret each solution

K-means

i. Assume #K

ii. Initialise #N ?

iii. Sort by optimality ?

iv. Select best for #K ?

v. Select #K(s) ?

vi. Single cluster or ensemble ?

Partition clustering –open issues

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Original data projected onto the space of cluster means then onto 2D using scatter matrices - Clusters r+:1 go:2 ms:3 c*:4 bv:5 kp:6 y.:7

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Scatter matrices

Separation index:Decomposition of the scatter matrix

BWT SSS

Nc

jjTjjB

Nc

j

N

iji

TjiW

N

ii

TiT

mmmmNS

mXmXS

mXmXS

j

1.

1 1

1

.

.

.

S

B

SW

1

SW

2

Invariant separation matrix and index

S

B

SW

1

SW

2

BW SSM .1

BW SStrJ .1

Separation index:Decomposition of the scatter matrix

)()..(

..~

.(.)

.(.)

~.

~

MtrAMTAtrJ

AMTAM

ASTAS

XAX

Nc

i

Ti

C aXXX

1

ˆ.

a1

a2

a3

BW SSM

.1

BW SStrJ

.1

N.B. If |ST|=0 → Project onto subspace of cohort means

UU

UUN

S

where

XX

D

X

T

X

D

X

TT

X

X

..

..

.~

21

21

21

a1

a2

a3

~

~

~

Theorem: is invariant to dimensionality reduction under Mahalanobis rotations

J

Nc

i

iTi aaXX

1

ˆˆ.~

BW SSM~

.~~ 1

1.W BJ tr S S J

~

2xx

x

~ ~

:{ }j jp x x C

~

~

:

min{ }

i

jj

x C

i x p

K-means clustering

~

2xx

x

~ ~ ~ ~

1 2 .x p x p

~ ~ ~

. . ii

x p x

Adaptive Resonance Theory (ART) clustering

Adaptive Resonance Theory (ART) clustering

Concordance measure

22

1 1

N Mij ij

iji j

O E

E

)1,1min(.

2

MNnCV

ClusterMembership

1 … M

1… O11 … O1M

N ON1 … ONM

Optimality principle

Reproducibility with

Best Separation - max(J)

Best Concordance – max(CV)

under repeated initialisations

i. N initialisations

ii. Sort by J

iii. Select top p%

iv. Calculate pairwise CV

v. Retain med(CV)

vi. Plot (J, med_CV)

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Artificial Data Showing 10 Cluster Allocations

Y

Z

Cluster 1

Cluster 2Cluster 3

Cluster 4

Cluster 5

Cluster 6

Cluster 7

Cluster 8Cluster 9

Cluster 10

Synthetic data (10 cohorts)

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Invariant J V

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5 clusters6 clusters7 clusters8 clusters9 clusters10 clustersBest 20 for 5 ClustersBest 20 for 6 ClustersBest 20 for 7 ClustersBest 20 for 8 ClustersBest 20 for 9 ClustersBest 20 for 10 Clusters



Mean Covariance Matrix (i,j)

x y z 11 12 13 21 22 23 31 32 33 N

C1 -0.799 -1.011 -3.336 0.336 0.044 0.074 0.044 0.371 0.210 0.074 0.210 0.582 64

C2 -0.441 -0.569 -2.331 0.428 0.060 -0.002 0.060 0.123 0.157 -0.002 0.157 0.648 42

C3 0.649 -0.344 -4.154 0.620 0.023 -0.035 0.023 0.137 0.070 -0.035 0.070 0.446 61

C4 1.077 0.072 -2.815 0.366 -0.002 0.076 -0.002 0.043 0.104 0.076 0.104 0.563 32

C5 -0.390 -0.242 0.256 0.536 0.013 0.031 0.013 0.348 -0.117 0.031 -0.117 0.689 197

C6 -1.358 -0.658 1.639 0.309 -0.060 -0.055 -0.060 0.245 -0.013 -0.055 -0.013 0.532 131

C7 1.261 0.125 0.862 0.323 0.017 0.027 0.017 0.386 -0.060 0.027 -0.060 0.403 163

C8 -0.593 3.024 -0.498 0.776 0.033 0.175 0.033 0.491 0.003 0.175 0.003 0.695 97

C9 0.251 -0.539 -0.530 0.711 -0.025 0.055 -0.025 0.352 -0.081 0.055 -0.081 0.576 106

C10 0.374 -0.267 1.973 0.390 -0.097 0.041 -0.097 0.343 -0.014 0.041 -0.014 0.322 183

C1 C2 C3 C4 C5 C6 C7 C8 C9

C2 0.7805 . C3 1.2105 1.4828 . C4 1.5054 1.1924 1.0687 . C5 2.4975 1.7636 3.0649 2.3119 . C6 3.3913 2.8294 4.476 3.8029 1.1757 . C7 3.2516 2.5575 3.7002 2.7302 1.2151 2.2233 . C8 2.9776 2.4341 3.0901 2.4774 2.025 2.6082 2.2314 . C9 2.0388 1.2969 2.4543 1.6846 0.7109 1.8176 1.2393 2.2086 .C10 3.7087 3.0487 4.4727 3.5977 1.2717 1.4141 1.233 2.5497 1.6952

Solution with 8 Clusters

Total2 4 7 1 3 5 8 6Originalcohorts

1 58 2 . 4 . . . . 64

2 28 1 . 13 . . . . 42

3 11 50 . . . . . . 61

4 1 26 . 5 . . . . 32

5 . . 109 43 13 16 15 1 197

9 2 . 23 64 . 14 3 . 106

6 . . 25 . 103 . 3 . 131

7 . . 4 4 . 134 21 . 163

10 . . 10 . 16 9 148 . 183

8 . . 1 . . . . 96 97

Total 100 79 172 133 132 173 190 97 1076

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Synthetic data – mixing structure (Sammon Map)

Synthetic data – Visualisation in data space

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Cra

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Median Cramer V of Solution

C ra me r V me a sure s for 5 Cl uste rs

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C ra me r V me a sure s for 1 0 Cl uste rs

Max J

SeCoMax Cv

BioinformaticsNottingham Tenovous Primary

Breast Carcinoma Series

Consecutive series of 1,944 cases of primary operable invasive breast cancer(n=1,076 with all markers present)

Patients presenting during 1986-98

Protein expression comprising

25 immunohistochemical markers related to tumour malignancyderived through high-throughput protein expression using TMA

Abd El-Rehim et al, Int J Cancer, 116, 340-350, 2005.

Marginal distributions

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800cerbb2

Expression value

Fre

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800p53

Expression value

Fre

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Expression value

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Expression value

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Expression value

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Expression valueF

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Landscape map (SeCo)

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Invariant

J V

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4 clusters

5 clusters


8 clusters

9 clusters


12 clusters

13 clusters


Stability index (Cv)

2 3 4 5 6 7 8 9 10 11 12 13 14 150

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1

Blue=Median, Green=MeanBlue=Median, Green=MeanBlue=Median, Green=MeanBlue=Median, Green=MeanBlue=Median, Green=MeanBlue=Median, Green=MeanBlue=Median, Green=MeanBlue=Median, Green=MeanBlue=Median, Green=MeanBlue=Median, Green=MeanBlue=Median, Green=MeanBlue=Median, Green=MeanBlue=Median, Green=MeanBlue=Median, Green=Mean

Box Plots of the Exhaustive Cramer V values

Number of Cluster Centres

Cra

mer

V v

alue

s

ATotal

1 2 3 8 7 5 6 4 1

B 1118 4 0 1 6 1 0 12

142

521 125 0 33 0 0 0 0

179

737 0 122 4 0 2 0 2

167

60 0 29 145 0 0 0 0

174

80 2 6 0 98 0 0 0

106

20 0 6 0 0 94 1 5

106

31 0 0 0 1 0 64 42

108

4 0 0 0 0 1 0 61 32 94

Total 177 131 163 183 106 97 126 93 1076

Landscape map (SeCo)

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Invariant

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4 clusters

5 clusters


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Cluster hierarchy (1)

322

51

244

C3, 459

C2, 373

C1, 244

C1, 781

C2, 295

458

407

246

24

208

24

65

C3, 130

C4, 430

C3, 285

C2, 209

C1, 152

C1, 266

C2, 198

C5, 205

C4, 277

240

103

62

114

201

79116

C1, 121

C4, 230

C5, 192

C3, 215

C2, 109

C6, 209

206

44

107

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169

219

91

C4, 116

C3, 205

C6, 119

C2, 234

C5, 120

C7, 186

C1, 96

160

105

112

67

93

112

26

186

127

107

C4, 94

C7, 167

C1, 142

C8, 106

C2, 106

C5, 179

C6, 174

C3, 108

159

105

94

105

172

51

94

102

44

101

Cluster hierarchy (2)

322

51

243

C3, 459

C2, 373

C1, 244

C1, 781

C2, 295

458

300

209

124

229

C1, 241

C1, 338

C2, 365

C3, 238

C4, 135

C5, 237

C2, 229

C4, 252

C3, 117

134

105

240

228

201

116

117

C1, 119

C5, 184

C2, 198

C6, 120

C4, 209

C3, 246

167

46

161

78

151

113

198

99

C2, 249

C1, 90

C5, 104

C7, 138

C4, 189

C3, 185

C6, 121

172

183

228

114

101

17

62

116

C3, 163

C4, 93

C6, 126

C7, 106

C2, 131

C1, 177

C5, 97

C8, 183

164

120

15

65

125

102

93

116

66

155

91

19

136

20

Solution A

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Solution A

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Solution B

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Solution A

Sub-type profiling

Clusters A Clusters B

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Cluster 1 of 8 (m4)

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Cluster 1 of 8 (m9)

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Cluster 5 of 8 (m4)

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c1 c

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c2

Cluster 2 of 8 (m9)Luminal N

Luminal New 2

Sub-type profiling

300

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Cluster 8 of 8 (m9)


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Cluster 5 of 8 (m9)

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p6

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muc1

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muc2

Cluster 2 of 8 (m4) HER2

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o

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c2

Cluster 6 of 8 (m4) Luminal A

Sub-type profiling

300

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muc1

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Cluster 3 of 8 (m4)

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rom

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Cluster 6 of 8 (m9)

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ap

to

eg

fr

p6

3

fhit

nb

rca1

cerb

b4

cerb

b3

mu

c1

muc1 c

o

mu

c2

Cluster 4 of 8 (m4)

300

250

200

150

100

50

0

e-c

ad

cerb

b2

ck5/6

p53

pgr

ck14

actin

p-c

ad1

ck19

ck7/8

ck18

gcdfper

ar

chro

mo

synapto

egfr

p63

fhit

nbrc

a1

cerb

b4

cerb

b3

muc1

muc1 c

o

muc2

Cluster 4 of 8 (m9)Basal p53 -

Basal muc1 -

Basal muc1 +

Basal p53 +


ClusterA* Clusters in Green at al (2007) Crosstabulation

Count

107 0 0 0 6 0 113

45 4 9 0 4 2 64

0 65 0 0 0 0 65

0 0 58 2 0 1 61

0 0 2 80 0 0 82

0 0 0 0 138 0 138

1 8 0 0 54 2 65

0 0 0 0 0 75 75

153 77 69 82 202 80 663

5

1

2

3

4

6

7

8

Cluster A

Total

2 6 5 4 1 3

Clusters in Green at al (2007)

Total

Consistency with consensus clustering

ClusterB * Clusters in Green at al (2007) Crosstabulation

Count

67 0 7 0 0 2 76

124 7 0 0 0 0 131

1 100 0 0 0 1 102

0 0 60 0 0 0 60

0 0 0 57 43 1 101

0 0 6 24 21 0 51

0 0 0 1 0 75 76

10 46 4 0 5 1 66

202 153 77 82 69 80 663

3

8

2

5

6

4

7

1

Cluster B

Total

1 2 6 4 5 3

Clusters in Green at al (2007)

Total

CoRe 5 Clusters Solution

2 3 1 4 5

Clusters inGreen et al2007

C1 129 4 0 3 66

C2 1 138 0 7 7

C3 14 11 37 16 2

C4 0 0 65 17 0

C5 0 0 56 13 0

C6 1 8 1 37 30

NC 58 72 54 119 110

Molecular sub-typing

Summary

-5000500 -350-300-250-200-150-100-50050100150

-500

-400

-300

-200

-100

0

100

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300

24



5

10

8

18

1

22

1125

23

20

9

19

Pro

ject

ion

onto

axi

s 3


Partition clustering - critique

Decomposition of the covariance matrix

Landscape mapping of cluster solutions

Validation for two synthetic data sets and metabolic sub-typing 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1

2

4

6

8

10

12

14

16



Invariant

J V

alu

e


4 clusters

5 clusters


8 clusters

9 clusters


12 clusters

13 clusters


Ferrara data (n=633)

0 0.2 0.4 0.6 0.8 10

5

10

15

20

25



Invariant

J V

alu

e

2 clusters


5 clusters

6 clusters

7 clusters


10 clusters.

er pr PROLIND neu P53

2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Blue=Median, Green=MeanBlue=Median, Green=MeanBlue=Median, Green=MeanBlue=Median, Green=MeanBlue=Median, Green=MeanBlue=Median, Green=MeanBlue=Median, Green=MeanBlue=Median, Green=MeanBlue=Median, Green=Mean

Box Plots of the Exhaustive Cramer V values

Number of Cluster Centres

Cra

mer

V v

alue

s


SeCo method Total

1 2 3 4 5

Ambrogi et al [7] 1 213 13 0 4 26 256

2 0 203 0 1 3 207

3 0 1 68 0 22 91

4 0 2 0 77 0 79

Total 213 219 68 82 51 633



P5

3

ne

u

PR

OL

IND

pr

er

100

80

60

40

20

0

P5

3

ne

u

PR

OL

IND

pr

er

100

80

60

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0

P5

3

ne

u

PR

OL

IND

pr

er

100

80

60

40

20

0

P5

3

ne

u

PR

OL

IND

pr

er

100

80

60

40

20

0

P53

neu

PR

OLI

NDprer

100

80

60

40

20

0

JMU Cluster 3/5

P53

neu

PR

OLI

NDprer

100

80

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20

0

JMU Cluster 4/5

P53

neu

PR

OLI

NDprer

100

80

60

40

20

0

JMU Cluster 5/5

P53

neu

PR

OLI

NDprer

100

80

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20

0

JMU Cluster 1/5

P53

neu

PR

OLI

NDprer

100

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20

0

JMU Cluster 2/5

-200

-150

-100

-50

0

50

-100-50050100150

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3


4


Pro

ject

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onto

axi

s 3

-200-150

-100-50

050 -100

-50

0

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0

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3

5



4

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ject

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onto

axi

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0 10 20 30 40 50 60 70 80 90 1000

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Pr

P53

er

Cluster 1 (213)

Cluster 2 (219)

Cluster 3 (68)Cluster 4 (82)

Cluster 5 (51)


Robust methodologies for partition clustering

Documents

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