gouws python 2010
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A Python implementation of graphical models
by
Almero Gouws
March 2010
Thesis presented in partial fulfilment of the requirements for the degree
of Master of Science in Engineering at Stellenbosch University
Supervisor: Professor Ben Herbst
Department of Applied Mathematics, Faculty of Engineering
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x6
x5
x4
x3 x2
p(x1)
p(x2)
p(x3) p(x2)
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n
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p(x1)
p(x2|x1)
1(x1, x2)
2(x2, x3)
1(x1, x2) 1(x1, x2, x3)
2(x2, x3) 2(x1, x2, x3)
3(x1, x2, x3) 1(x1, x2, x3) 2(x1, x2, x3)
E3(x1, x2)
3(x1, x2, x3)
x3= 0
E3(x1, x3) 3(x1, x2, x3) x2= 1
1 x1 x2
p(x1) p(x2)
1 2
2x2(x2)
[1(x1, x2) + x21(x2)]
1x1(x1)
p(x1)
p(x2|x1)
(x1, x2) ES S(x1, x2)
ES S(x1, x1)
E(x1|x2= 1)
M(x1|x2= 1)
R
(x1, x2)
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N
x = {x1,...,xN} x
x N x = {x1,...,xN}
N
x
x
xi
{x1, x2, x3, x4, x5, x6}
x2
x1 x2 p(x2|x1)
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xi
p(xi|i) i xi
N
p(x) =N
i=1
p(xi|i),
i xi
p(x) =p(x1)p(x2|x1)p(x3|x1)p(x4|x2)p(x5|x3)p(x6|x2, x5).
xi
xi
x1
p(x1)
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x1 p(x1)
x1 = 0 x1= 10.4 0.6
p(x1)
x2 x1 p(x2|x1)
x2
x1 p(x2|x1)
x1= 0 x1 = 1x2 = 0 0.7 0.6x2 = 1 0.3 0.4
p(x2|x1)
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N x = {x1,...,xN} x
N
x = {x1,...,xN} N x
{x1, x2, x3, x4, x5, x6}
x2 x5 x6
x2 x5
x6 x2 x5
x6
xs (xs) C
C
p(x) = 1Z
cC
c(xc),
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xc c 1
Z
Z=x
cC
c(xc).
p(x) = 1
Z(x1, x2)(x1, x3)(x2, x4)(x2, x5, x6).
(xs)
xs
xs
{x1, x2} {x2, x3}
1(x1, x2) 2(x2, x3)
1(x1, x2) 2(x2, x3)
p(x) = 1
Z
(x1, x2)(x2, x3).
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x2= 0 x2 = 1
x1 = 0 0.8 0.1x1 = 1 0.4 0.9
1(x1, x2)
x3= 0 x3 = 1x2 = 0 0.7 0.5x2 = 1 0.5 0.3
2(x2, x3)
1(x1, x2) 2(x2, x3)
1(x1, x2)
2(x2, x3) 1(x1, x2)
2(x2, x3) 3(x1, x2, x3)
3(x1, x2, x3) =(x1, x2) +(x2, x3).
3(x1, x2, x3)
1(x1, x2) 2(x2, x3)
x1 x2 x3 1(x1, x2) x3
1(x1, x2) 1(x1, x2, x3)
1(x1, x2, x3 = 0) = 1(x1, x2, x3= 1) =1(x1, x2).
2(x2, x3)
x1 2(x1, x2, x3)
2(x2, x3, x1 = 0) = 2(x2, x3, x1= 1) =2(x2, x3).
1(x1, x2)
1(x1, x2, x3)
2(x2, x3) 2(x1, x2, x3)
x3 = 0 x2= 0 x2 = 1 x3 = 1 x2= 0 x2 = 1x1 = 0 0.8 0.1 x1 = 0 0.8 0.1x1 = 1 0.4 0.9 x1 = 1 0.4 0.9
1(x1, x2) 1(x1, x2, x3)
3(x1, x2, x3) 1(x1, x2, x3) 2(x1, x2, x3)
1(x1, x2, x3) 2(x1, x2, x3)
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x3 = 0 x2= 0 x2 = 1 x3 = 1 x2= 0 x2 = 1
x1 = 0 0.7 0.5 x1 = 0 0.5 0.3x1 = 1 0.7 0.5 x1 = 1 0.5 0.3
2(x2, x3) 2(x1, x2, x3)
x3 = 0 x2= 0 x2 = 1 x3 = 1 x2= 0 x2 = 1x1 = 0 1.5 0.6 x1 = 0 1.3 0.4x1 = 1 1.1 1.4 x1 = 1 0.9 1.2
3(x1, x2, x3) 1(x1, x2, x3)
2(x1, x2, x3)
3(x1, x2, x3) x3 = 0
3(x1, x2, x3)
x3 = 0 E3(x1, x2)
3(x1, x2, x3) x3= 0 E3(x1, x2)
E3(x1, x3) 3(x1, x2, x3) x2 = 1
x2= 0 x2 = 1x1 = 0 1.5 0.6x1 = 1 1.1 1.4
E3(x1, x2) 3(x1, x2, x3) x3= 0
x3= 0 x3 = 1x1 = 0 0.6 0.4x1 = 1 1.4 1.2
E3(x1, x3) 3(x1, x2, x3) x2= 1
p(x)
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p(x) = 1
Z1(x1, x3)2(x2, x3)3(x3, x4)4(x3, x5).
1(x1, x3) x1 x3
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x2 x3 x4 x5
p(x2|x1) p(x3|x1) p(x4|x2) p(x5|x3)
p(x2|x1) x1 x2 x1 x2
p(x2|x1)
(xc, xp) =p(xc|xp).
x6 x6 x2 x5
p(x6|x2, x5) x2 x5
x6
x2 x5 x2
x5
p(x6|x2, x5)
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(xc,xp) =p(xc|xp).
xc xp xc
x1 p(x1) p(x1)
(xc) =p(xc).
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N
p(xn) xn
p(x) = 1
Z(x1, x2)(x1, x3)(x2, x4)(x3, x5)(x2, x5, x6).
p(x1, x2, x3, x4, x5)
p(x1, x2, x3, x4, x5) =x6
1
Z
(x1, x2)(x1, x3)(x2, x4)(x3, x5)(x2, x5, x6).
k
k6
O(k6)
k6
x6
x6
p(x) = 1
Z(x1, x2)(x1, x3)(x2, x4)(x3, x5)
x6
(x2, x5, x6).
(x2, x5, x6) k3
O(k3)
O(k6)
p(x1)
p(x1) = 1
Z
x2,x3,x4,x5,x6
(x1, x2)(x1, x3)(x2, x4)(x3, x5)(x2, x5, x6).
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p(x1) = 1Z
x2
(x1, x2)x3
(x1, x3)x4
(x2, x4)x5
(x3, x5)x6
(x2, x5, x6).
m6(x2, x5) =
x6(x2, x4, x6)
p(x1) = 1
Z
x2
(x1, x2)x3
(x1, x3)x4
(x2, x4)x5
(x3, x5)m6(x2, x5),
m6(x2, x5) (x2, x5, x6) x6
x2 x5 x6
x6
x6
x6
m5(x2, x3) =
x5
(x3, x5)m6(x2, x5)
p(x1) = 1
Z
x2
(x1, x2)x3
(x1, x3)m5(x2, x3)x4
(x2, x4).
x5
m5(x2, x3) x2 x3
x2
x3
x5 x2 x3
x5
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p(x1) = 1
Z
x2
(x1, x2)m4(x2)x3
(x1, x3)m5(x2, x3),
= 1
Z
x2
(x1, x2)m4(x2)m3(x1, x2),
= 1
Zm2(x1),
x4 x3 x2
x1
x4
x3 x2
p(x1|x6)
x6 x1
x6
E(x2, x5) E
p(x1|x6) = 1
Z
x2
(x1, x2)x3
(x1, x3)x4
(x2, x4)x5
(x3, x5)E(x2, x5).
x6 x6
p(x1)
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p(x1) = 1
Z
x2
(x1, x2)x3
(x1, x3)x4
(x2, x4)x6
x5
(x3, x5)(x2, x5, x6),
= 1
Z
x2
(x1, x2)x3
(x1, x3)x4
(x2, x4)x6
m5(x2, x3, x6),
= 1
Z
x2
(x1, x2)m4(x2)x3
(x1, x3)m6(x2, x3),
= 1Z
x2
(x1, x2)m4(x2)m3(x1, x2),
= 1
Zm2(x1).
x6
(x2, x5, x6) k3
O(k3)
x5 (x2, x5, x6)(x3, x5)
k4
O(k4)
p(x1)
p(x1) p(x2)
p(x1) p(x2)
p(x2)
p(x1)
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p(x1)
p(x2)
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p(x) = 1
Z(x1, x2)(x2, x3)(x3, x4)(x4, x5)(x5, x6),
= 1
Z
5i=1
(xi, xi+1),
1Z
p(x1)
(6, 5, 4, 3, 2, 1)
x1
p(x3)
x3
(6, 5, 4, 2, 1, 3)
p(x3) = 1
Z
x1,x2,x4,x5,x6
(x1, x2)(x2, x3)(x3, x4)(x4, x5)(x5, x6),
= 1
Z
x1,x2
(x1, x2)(x2, x3)x4
(x3, x4)x5
(x4, x5)x6
(x5, x6).
x2 (x1, x2)(x2, x3) O(k3)
x1
x2
(6, 5, 4, 1, 2, 3)
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p(x3) = 1Z
x2
(x2, x3)x1
(x1, x2)x4
(x3, x4)x5
(x4, x5)x6
(x5, x6),
O(k2)
p(x3) = 1
Z
x2,x1
1i=2
(xi, xi+1)
x4,x5,x6
5i=3
(xi, xi+1)
.
x3
p(x3) = 1
Z(x3)(x3),
(x3) = x2,x11
i=2 (xi, xi+1) (x3) = x4,x5,x65
i=3 (xi, xi+1).
(x3) x2 x3 (x3)
x4 x3
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(x3)
(x3) =x2
(x2, x3)x1
(x1, x2),
=x2
(x2, x3)(x2).
(xn)
(xn1) xn xn1
xn1 x1 (x2) =
x1 (x1, x2) (xn)
(x3) =
x4,x5,x6
5i=3
(xi, xi+1),
=x4
(x3, x4)x5,x6
5i=4
(xi, xi+1),
=x4
(x3, x4)(x4).
(xn) (xn+1)
xn xn+1 xn+1
x6 (x5) =
x6
(x5, x6)
p(x1)
x1
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xn xm
xm
x3 x1
x2 x4 x5
x1
x1
x4
x3
p(x1)
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p(x1)
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3(x2) =x3
2(x2, x3),
4(x2) =x4
3(x2, x4),
2(x1) =x2
1(x1, x2)3(x2)4(x2).
x2 x1
x2
1 x21(x2)
x21(x2)
x2 3(x2) 4(x2)
x21(x2) =3(x2)4(x2).
1 x1 1
x21(x2) 1x1(x1)
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x1
1x1(x1) 2(x1)
1x1(x1) = 2(x1),
=x2
1(x1, x2)3(x2)4(x2),
1x1(x1) =x2
1(x1, x2)1x1(x1).
1 x1
1x1(x1)
x1 x2
xm s
xms(xm) =
rne(xm)\s
rxm(xm),
r ne(xm)\s xm s
s xm
sxm(xm) =xs\xm
s(xs)
rxs\xm
xrs(xr),
xs\xm s xm
x(x) = 1,
x(x) =(x).
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p(x3)
x3
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p(x2)
x2
p(x3)
p(x2)
p(x3) p(x2)
x32 2x2
x2 2 p(x3) p(x3)
x3
p(xn) xn
N
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2(N1)
(N1)
2(N1)
N
N(N 1)
p(x)
x
1
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x2= 0 x2 = 1
x1 = 0 0.4 0.6x1 = 1 0.4 0.1
1 x1 x2
p(xn) xn
p(x1) p(x2)
p(x1= 0) p(x1= 1) p(x2 = 0) p(x2 = 1)
p(x1) p(x2)
x1 = 0 x2= 0
x1 = 0 x2 = 1
xmax = argmax
x
p(x),
p(xmax) = maxx
p(x).
p(xmax)
p(x) = 1
Z1(x1, x2)2(x2, x3),
p(xmax) = 1
Zmax
x1(x1, x2)2(x2, x3).
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p(xmax) = 1
Zmax
x1max
x2max
x31(x1, x2)2(x2, x3).
max(ab, ac) =a max(b, c), a >0.
(x1, x2) x3
p(xmax) = 1
Zmax
x1max
x21(x1, x2)max
x32(x2, x3).
log(p(xmax)) = maxx
log(p(x)).
log(p(x)) = log(1
Z) + log(1(x1, x2)) + log(2(x2, x3)).
max(a+ b, a + c) =a+ max(b, c),
log(p(xmax)) = log(1Z
) + maxx1
maxx2
log 1(x1, x2) + maxx3
log 2(x2, x3).
xm s
xms(xm) =
rne(xm)\srxm(xm),
r ne(xm)\s xm s
s xm
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sxm(xm) = maxxs\xm
[log(s(xs)) +
rxs\xm
xrs(xr)],
xs\xm s xm
x(x) = 0,
x(x) = log((x)).
p(x)
p(x)
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x1
x1max = argmax
x1
1x1(x1).
1 x2= 0 x2 = 1 2 x3= 0 x3 = 1x1 = 0 x2 = 0 x1 = 1 x2 = 1
1 2
x1 x3
x32(x3) = 0.
2 x2
2x2(x2) = maxx3
[2(x2, x3) +x32(x3)],
= maxx3
2(x2, x3).
x2 = 0 x2= 1
2x2(x2)
x3
x2 = 0 x2 = 1 2
x2 = 0 x3 = 0
x2
x3
2
x2 = 1 x3 = 1
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x2 x3
x2 x2
1 1
2
x21(x2) =2x2(x2).
1 x1
1x1(x1) = maxx2
[1(x1, x2) +x21(x2)].
[1(x1, x2) + x21(x2)]
x2= 0 x2 = 1x1 = 0 x1 = 1
[1(x1, x2) + x21(x2)]
1x1(x1)
x1 = 0 x1= 1
1x1(x1)
x1 = 0 x2 = 1
x1 = 1 x2 = 0
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x1
x1 = 0 x2 = 1 x2 = 1 x3 = 1 x1 = 1
x2= 0 x2= 0 x3 = 0
s xm
xm
sxm(xm) = argmaxxs\xm
[log(s(xs)) + rxs\xm
xrs(xr)].
x5
x2 x3
x1 x2 x3 x4 x1
x2 x4
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{x1, x2, x3} {x2, x3, x5} {x2, x5, x6}
{x2, x4}
Ci
p(x) = 1
Z1(x1, x2, x3)2(x2, x3, x5)3(x2, x4)4(x2, x5, x6),
Si
Si i(Xs) Xs
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p(x) =
i i(Xi)i j(xj)
.
1Z
Z
Z
Ci Cj CiCj x5
x5
x5
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p(x1)
C1
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Cs Sm
CsSm(xm) =xs\xm
s(xs)
rne(Cs)\Sm
SrCs(xr),
xm m xs
s xs\xm xs xm ne(Cs)\Sm
Cs Sm
s s s
s(xs) =s(xs)
rne(Cs)\Sm
SrCs(xr),
Cs Sm
CsSm(xm) = xs\xms(xs).
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Sm Cs
SmCs(xm) =m(xm)
rne(Sm)\Cs
CrSm(xr),
xm m xr r ne(Sm)\Cs Sm Cs
m
Sm m m
m(xm) =m(xm)
rne(Sm)\Cs
CrSm(xr).
Sm Cs
SmCs(xm) =m(xm).
s(xs) =s(xs)
rne(Cs)\Sm
r(xr),
Cs
s
Sm
Cs
s (xs) =s(xs)
rne(Cs)\Sm
r (xr)rne(Cs)\Sm
r(xr).
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s s
Cs
s(xs) =s(xs)
rne(Cs)\Sm
r(xr)rne(Cs)\Sm
r(xr).
rne(Cs)\Sm
r(xr)
Cs Sm Sm
p(xn) xn
xn
xn p(x2)
p(x2) = 1
Z 2(x2),
Z=x2
2(x2).
p(x) =
i
i (xi)
i i (xi)
.
C4 x5
C3 C4 C3
C2 C3 C2 x5
C4
C3
x5
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xi
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Cs Sm
CsSm(xm) = maxxs\xm
log(s(xs)) +
rne(Cs)\Sm
SrCs(xr),
xm m xs s xs\xm xs xm ne(Cs)\Sm Cs Sm
Sm Cs
SmCs(xm) =m(xm) +
rne(Sm)\Cs
CrSm(xr),
xm m xr
r ne(Sm)\Cs Sm Cs
m
log(1) = 0
C1
C2 S1
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C2S1(x2, x3) = maxx4
log(2(x2, x3, x4)).
x2
x3
x4
x2 x3
C2S1
S1 C2
S1C1(x2, x3) =C2S1(x2, x3),
C1
(x1max
, x2max
, x3max
) = argmaxx1,x2,x3 [log(1(x1, x2, x3)) +S1C1(x2, x3)].
x4
x2
max
x3max
x4
CsSm(xm) = argmaxxs\xm
log(s(xs)) +
rne(Cs)\Sm
SrCs(xr),
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p(x1)
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C1 x1 x2
n(xn) xn
xn
xn
xm
Cs m(xm)
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n
xmCs(xm) =m(xm).
s(xs) =s(xs)
mne(Cs)
m(xm),
s
(xs) s(xs) ne(Cs)
Cs
Cs
xm
xm
Csxm(xm) =xs\xm
s(xs).
Cs
Cs Csxm(xm)
Cs
s
xm
m(xm) =m(xm)
Csxm(xm)
Csxm(xm).
2 1
3
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p(xn) xn xn
xn
-
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-
8/10/2019 Gouws Python 2010
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N
S N
S
-
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x1 x2
p(x1) p(x2|x1)
p(x1) x1
x1 S p(x1)
x1
x1
x1 {0, 0, 1, 0} x1 = 0 x1 = 1 p(x1 = 0) = 3/4
p(x1 = 1) = 1/4 p(x1)
x1 = 0 x1= 10.75 0.25
p(x1)
p(x2|x1)
p(x2|x1) p(x1)
x1 x2
(x1, x2) {(0, 0), (0, 1), (1, 1), (0, 0)}
p(x2 = 0|x1 = 0) = 2/4 p(x2= 0|x1= 1) = 1/4 p(x2= 1|x1= 1) = 1/4 p(x2= 1|x1= 0) = 0/4
p(x2|x1)
x1= 0 x1 = 1x2 = 0 0.66 0x2 = 1 0.33 1
p(x2|x1)
p(xi|i)
-
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p(xi|i) p(xi|i)
xi i
(xs)
xs
S
N
S
x1 x2
-
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(x1, x2)
(x1, x2)
ES S(x1, x2)
(x1, x2) x1= 0 x1 = 1 ES S(x1, x2) x1= 0 x1 = 1x2 = 0 x2 = 0 x2 = 1 x2 = 1
(x1, x2) ES S(x1, x2)
{0, 0}
{1, 1}
{0, 0}
(x1, x2)
ES S(x1, x2) x1= 0 x1= 1x2 = 0 x2 = 1
ES S(x1, x1)
(Hidden, 1)
(x1, x2)
(x1, x2)
E(x1|x2 = 1)
E(x1|x2 = 0) x1 = 0 x1= 1x2 = 1
E(x1|x2= 1)
-
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x1
E(x1|x2 = 1) M(x1|x2 = 1)
M(x1|x2 = 0) x1 = 0 x1= 1x2 = 1
M(x1|x2 = 1)
M(x1|x2 = 0)
(x1, x2)
R
(x1, x2)
R(x1, x2) x1= 0 x1 = 1x2 = 0 x2 = 1
R(x1, x2)
R(x1, x2) ES S(x1, x2)
ES S(x1, x2) x1= 0 x1= 1x2 = 0 x2 = 1
(x1, x2)
(x1, x2) x1= 0 x1 = 1 ES S(x1, x2) x1= 0 x1 = 1x2 = 0 x2 = 0 x2 = 1 x2 = 1
(x1, x2) ES S(x1, x2)
(x1, x2)
-
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(Xs)
(Xs)
(Xs) ES S(Xs)
Sj S
ES S(Xs)
Sj
(Xs)
ES S(Xs)
ES S(Xs)
ES S(Xs) (Xs)
ES S(Xs)
-
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i(Xs)
ES Si(Xs)
Sj S
Sj
i(Xs)
i(Xs)
ES Si(Xs)
i(Xs)
i(Xs)
ES Si(Xs)
ES Si(Xs)
i(Xs)
ES Si(Xs) i(Xs)
-
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ES Si(Xs)
-
8/10/2019 Gouws Python 2010
68/139
-
8/10/2019 Gouws Python 2010
69/139
{xn, yn}
-
8/10/2019 Gouws Python 2010
70/139
-
8/10/2019 Gouws Python 2010
71/139
xi {0, 2, 5}
{x0, x2, x5}
xi
k
xi= k
-
8/10/2019 Gouws Python 2010
72/139
xi
k
xi= k
mlc[i] =k
xi k
-
8/10/2019 Gouws Python 2010
73/139
N
N
N
N
i
j
-
8/10/2019 Gouws Python 2010
74/139
x1 x2 x3 x4 x5 x6
x1 x2 x3 x4 x5 x6
xi xi
{x1 : 2, x2 : 4, x3 : 3, x4 : 5, x5 : 2, x6 : 10} [2, 4, 3, 5, 2, 10]
x4
xi xj
xi k
-
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xi
-
8/10/2019 Gouws Python 2010
76/139
k
xi = k xi
104
-
8/10/2019 Gouws Python 2010
77/139
232
-
8/10/2019 Gouws Python 2010
78/139
k
xi = k xi
-
8/10/2019 Gouws Python 2010
79/139
-
8/10/2019 Gouws Python 2010
80/139
N
xi xj 1
x1 x2 x3 x4 x5 x6x1 x2 x3 x4 x5 x6
xi
-
8/10/2019 Gouws Python 2010
81/139
xi xj
xi k
xi
-
8/10/2019 Gouws Python 2010
82/139
k
-
8/10/2019 Gouws Python 2010
83/139
xi = k xi
104
232
-
8/10/2019 Gouws Python 2010
84/139
k
xi = k xi
-
8/10/2019 Gouws Python 2010
85/139
-
8/10/2019 Gouws Python 2010
86/139
xi xp
xi {xi,xp}
p(x2|x1) {x1, x2}
-
8/10/2019 Gouws Python 2010
87/139
{x1, x2}
x1
x2
= [x2, x5, x6]
= [2, 4, 3]
x2
x6
-
8/10/2019 Gouws Python 2010
88/139
-
8/10/2019 Gouws Python 2010
89/139
xi
k
xi=k
xi
-
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90/139
(x1, x2, x3) x3= 1
E(x1, x2)
x3 x3 = 0
E(x1, x2) (x1, x2, x3)
Xj Xi Xi nbr nbr[j][0]
Xi Xj
-
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{XiXj} nbr[j][1] Xi Xj
(Xi Xj) 2(x2, x3, x5) 2(x2, x4, x5) 3(x2, x3) 4(x2, x5, x6)
2(x2, x3, x5) nbrs
nbrs[1][0] = [2, 3]
nbrs[1][1]
1(x2, x3)
nbrs[3][0] = [2]
nbrs[3][1]
2(x2)
nbrs[4][0] = [2, 5]
nbrs[4][1]
3(x2, x5)
nbrs
xj
-
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k
xdomain[i]=k
-
8/10/2019 Gouws Python 2010
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xi
-
8/10/2019 Gouws Python 2010
94/139
-
8/10/2019 Gouws Python 2010
95/139
xi
k xi=k
xi
-
8/10/2019 Gouws Python 2010
96/139
{x2, x3}
x2
x3
= [x1, x3, x4] = [3, 7, 2] x3 x4
xj
-
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97/139
-
8/10/2019 Gouws Python 2010
98/139
xPOTENTIAL.domain[i] k xPOTENTIAL.domain[i]= k
-
8/10/2019 Gouws Python 2010
99/139
-
8/10/2019 Gouws Python 2010
100/139
xkey
xkey
-
8/10/2019 Gouws Python 2010
101/139
-
8/10/2019 Gouws Python 2010
102/139
-
8/10/2019 Gouws Python 2010
103/139
i j
xi
xi
-
8/10/2019 Gouws Python 2010
104/139
-
8/10/2019 Gouws Python 2010
105/139
-
8/10/2019 Gouws Python 2010
106/139
-
8/10/2019 Gouws Python 2010
107/139
-
8/10/2019 Gouws Python 2010
108/139
-
8/10/2019 Gouws Python 2010
109/139
xi k
xi=k
-
8/10/2019 Gouws Python 2010
110/139
-
8/10/2019 Gouws Python 2010
111/139
xi k
xi=k
-
8/10/2019 Gouws Python 2010
112/139
-
8/10/2019 Gouws Python 2010
113/139
-
8/10/2019 Gouws Python 2010
114/139
-
8/10/2019 Gouws Python 2010
115/139
-
8/10/2019 Gouws Python 2010
116/139
-
8/10/2019 Gouws Python 2010
117/139
-
8/10/2019 Gouws Python 2010
118/139
xi k
xi=k
-
8/10/2019 Gouws Python 2010
119/139
-
8/10/2019 Gouws Python 2010
120/139
-
8/10/2019 Gouws Python 2010
121/139
-
8/10/2019 Gouws Python 2010
122/139
xi k
xi=k
-
8/10/2019 Gouws Python 2010
123/139
-
8/10/2019 Gouws Python 2010
124/139
-
8/10/2019 Gouws Python 2010
125/139
xi k
xi=k
-
8/10/2019 Gouws Python 2010
126/139
-
8/10/2019 Gouws Python 2010
127/139
-
8/10/2019 Gouws Python 2010
128/139
xi xj
xi xj
-
8/10/2019 Gouws Python 2010
129/139
xi xj
xi xj
-
8/10/2019 Gouws Python 2010
130/139
-
8/10/2019 Gouws Python 2010
131/139
-
8/10/2019 Gouws Python 2010
132/139
i
xi xi
yi xi
yi i
{xi, xj}
(xi, xj) =e(xixj)
2
i ,
i
{xi, yi}
(xi, yi) =e(xiyi)
2
n ,
-
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n
-
8/10/2019 Gouws Python 2010
134/139
(B1, Bt,t1) B1
Bt,t1
-
8/10/2019 Gouws Python 2010
135/139
-
8/10/2019 Gouws Python 2010
136/139
-
8/10/2019 Gouws Python 2010
137/139
-
8/10/2019 Gouws Python 2010
138/139
-
8/10/2019 Gouws Python 2010
139/139
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