Método de WARD

Método de WARD
Individuo X1 X2
A
10
5
B
20 20
C
30 10
D
30 15
E
5 10
Nivel 1
Combinaciones posibles:
Partición
(A, B), C, D, E
(A, C), B, D, E
(A, D), B, C, E
(A, E), B, C, D
(B, C), A, D, E
(B, D), A, C, E
(B, E), A, C, D
(C, D), A, B, E
(C, E), A, B, D
(D, E), A, B, C
5
2
= 10
Ek
EAB = 162.5
CAB = (15, 12.5)
EC = ED = EE = 0
EAC = 212.5
CAC = (20, 7.5)
EB = ED = EE = 0
EAD = 250
CAD = (20, 10)
EB = EC = EE = 0
EAE = 25
CAE = (7.5, 7.5)
EB = EC = ED = 0
EBC = 100
CBC = (25, 15)
EA = ED = EE = 0
EBD = 62.5
CBD = (25, 17.5)
EA = EC = EE = 0
EBE = 162.5
CBE = (12.5, 15)
EA = EC = ED = 0
ECD = 12.5
CCD = (30, 12.5)
EA = EB = EE = 0
ECE = 312.5
CCE = (17.5; 10)
EA = EB = ED = 0
EDE = 325
CDE = (17.5; 12.5)
EA = EB = EC = 0
Centroides
E
ΔE
162.5 162.5
212.5 212.5
250
250
25
25
100
100
62.5
62.5
162.5 162.5
12.5
12.5
312.5 312.5
325
325
Cluster (C, D), A, B, E
EAB = (10 − 15)2 + (5 − 12.5)2 + (20 − 15)2 + (20 − 12.5)2 = 162.5
EAB = 102 + 52 + 202 + 202 − 2(152 + 12.52 ) = 162.5
Nivel 2
Combinaciones posibles:
Partición
((C, D), A), B, E
((C, D), B), A, E
((C, D), E), A, B
(A, B), (C, D), E
(A, E), (C, D), B
(B, E), (C, D), A
4
2
=6
Ek
ECDA = 316.66
CCDA = (23.33, 10)
EB = EE = 0
ECDB = 116.66
CCDB = (26.66, 15)
EA = EE = 0
ECDE = 433.33
CCDE = (21.66, 11.66)
EA = EB = 0
EAB = 162.5
CAB = (15, 12.5)
ECD = 12.5
CCD = (30, 12.5)
EE = 0
EAE = 25
CAE = (7.5, 7.5)
ECD = 12.5
CCD = (30, 12.5)
EB = 0
EBE = 162.5
CBE = (12.5, 15)
ECD = 12.5
CCD = (30, 12.5)
EA = 0
Centroides
E
ΔE
316.66 304.16
116.66 104.16
433.33 420.83
175
162.5
37.5
25
175
162.5
Cluster (A, E), (C, D), B
Cálculos para la partición ((C, D), A), B, E:
m
(C,D),A
E(C,D),A
2(mCD ) + mA
2(30, 12, 5) + (10, 5)
=
=
= (23.33, 10)
3
3
3
2
(C,D),A
(C,D),A 2
=
(xil
− ml
) =
i=1 l=1
= (30 − 23.33)2 + (10 − 10)2 + (30 − 23.33)2 + (15 − 10)2 + (10 − 23.33)2 + (5 − 10)2 = 316.66
3 2
2
(C,D),A 2
(C,D),A 2
E(C,D),A =
(xil
) −3
(ml
) =
i=1 l=1
l=1
= 302 + 102 + 302 + 152 + 102 + 52 − 3(23.332 + 102 ) = 316.66
ΔE(C,D),A = E(C,D),A − ECD − EA = 316.66 − 12.5 − 0 = 304.16
2
nCD nA CD
1 · 2
2
ΔE(C,D),A =
(ml − mA
(30 − 10)2 + (12.5 − 5)2 = 304.16
l ) =
nCD + nA l=1
3
1
ΔE(C,D),A =
(nC + nA )ΔECA + (nD + nA )ΔEDA − nA ΔECD =
nCD + nA
1
=
(2 · 212.5 + 2 · 250 − 12.5) = 304.16
2+1
Nivel 3
Combinaciones posibles:
Partición
3
2
=3
Centroides
((A, E), (C, D)), B CAECD = (18.75, 10)
((A, E), B), (C, D)
((C, D), B), (A, E)
CAEB = (11.66, 11.66)
CCD = (30, 12.5)
CCDB = (26.66, 15)
CAE = (7.5, 7.5)
Ek
EAECD = 568.75
EB = 0
EAEB = 233.33
ECD = 12.5
ECDB = 116.66
EAE = 25
E
568.75 531.25
245.8
Nivel 4
Partición
Centroide
E ΔE
(((C, D), B), (A, E)) CCDBAE = (19, 12) 650 508.34
A
E
C
104.16
508.34
Ward
D
B
208.3
141.66 104.16
Cluster ((C, D), B), (A, E)
12.5
25
ΔE