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Guillaume Suatton
Bayesian Project
Commits
2aff221b
Commit
2aff221b
authored
4 years ago
by
Guillaume Suatton
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Update Variational_Bayesian_Model_Selection_for_Mixture_Distributions.ipynb
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@@ -13,7 +13,7 @@
"metadata": {},
"outputs": [],
"source": [
"
#
%pip install -r requirements.txt\n",
"%pip install -r requirements.txt\n",
"\n",
"# Imports\n",
"import numpy as np\n",
...
...
%% Cell type:markdown id: tags:
### Reproducing the results of the paper
%% Cell type:code id: tags:
```
python
#
%pip install -r requirements.txt
%
pip
install
-
r
requirements
.
txt
# Imports
import
numpy
as
np
from
numpy
import
linalg
as
la
import
matplotlib.pyplot
as
plt
from
matplotlib.patches
import
Ellipse
from
matplotlib.colors
import
LogNorm
from
scipy.special
import
digamma
,
gammaln
,
logsumexp
from
scipy.stats
import
multivariate_normal
def
fit_predict
(
X
,
nb_components
=
10
,
max_iter
=
10
,
seed
=
0
):
# Sizes
n
,
d
=
X
.
shape
# Initialization
rd
=
np
.
random
.
RandomState
(
seed
)
resp
=
rd
.
rand
(
n
,
nb_components
)
/
resp
.
sum
(
axis
=
1
)[:,
np
.
newaxis
]
nu0
,
invW0
,
beta0
=
d
,
np
.
atleast_2d
(
np
.
cov
(
X
.
T
)),
1.
weights
=
np
.
ones
(
nb_components
)
/
nb_components
expectation
=
[
np
.
zeros
((
nb_components
,
d
)),
np
.
array
([
nu0
*
np
.
linalg
.
inv
(
invW0
)
for
_
in
range
(
nb_components
)]),
np
.
zeros
(
nb_components
),
np
.
zeros
((
nb_components
,
d
,
d
))]
# First parameter update
eta
=
resp
.
sum
(
axis
=
0
)
+
10
*
np
.
finfo
(
resp
.
dtype
).
eps
nu
=
nu0
+
eta
S
=
beta0
*
np
.
eye
(
d
)
+
expectation
[
1
]
*
eta
[:,
np
.
newaxis
,
np
.
newaxis
]
inverse_S
=
np
.
linalg
.
inv
(
S
)
inverse_W
=
np
.
zeros
((
nb_components
,
d
,
d
))
m
=
np
.
zeros
((
nb_components
,
d
))
for
k
in
range
(
nb_components
):
m
[
k
]
=
inverse_S
[
k
]
@expectation
[
1
][
k
]
@
(
resp
[:,
k
]
@X
)
s
=
np
.
zeros
((
d
,
d
))
for
i
in
range
(
n
):
s
+=
resp
[
i
,
k
]
*
(
np
.
outer
(
X
[
i
],
X
[
i
])
-
np
.
outer
(
X
[
i
],
expectation
[
0
][
k
])
-
np
.
outer
(
expectation
[
0
][
k
],
X
[
i
])
+
expectation
[
3
][
k
])
inverse_W
[
k
]
=
invW0
+
s
# Initializing the lower bound
lower_bound
=
np
.
zeros
(
max_iter
)
# EM procedure
for
p
in
range
(
max_iter
):
# Computing expectation values
expectation
=
[]
expectation
.
append
(
m
.
copy
())
expectation
.
append
(
nu
[:,
np
.
newaxis
,
np
.
newaxis
]
*
np
.
linalg
.
inv
(
inverse_W
))
expectation
.
append
(
d
*
np
.
log
(
2
)
-
np
.
log
(
np
.
linalg
.
det
(
inverse_W
))
+
np
.
sum
(
digamma
(
0.5
*
(
nu
-
np
.
arange
(
d
)[:,
np
.
newaxis
]))))
inverse_S
=
np
.
linalg
.
inv
(
S
)
expectation
.
append
([
inverse_S
[
i
]
+
np
.
outer
(
m
[
i
],
m
[
i
])
for
i
in
range
(
nb_components
)])
expectation
.
append
([
np
.
trace
(
inverse_S
[
i
])
+
m
[
i
]
@m
[
i
]
for
i
in
range
(
nb_components
)])
# Computing probabilities
log_rho_tilde
=
np
.
zeros
((
n
,
nb_components
))
for
i
in
range
(
n
):
for
j
in
range
(
nb_components
):
log_rho_tilde
[
i
,
j
]
=
0.5
*
expectation
[
2
][
j
]
-
0.5
*
np
.
trace
(
expectation
[
1
][
j
]
@
(
np
.
outer
(
X
[
i
],
X
[
i
])
-
np
.
outer
(
X
[
i
],
expectation
[
0
][
j
])
-
np
.
outer
(
expectation
[
0
][
j
],
X
[
i
])
+
expectation
[
3
][
j
]))
log_rho
=
log_rho_tilde
+
np
.
log
(
weights
+
10
*
np
.
finfo
(
weights
.
dtype
).
eps
)
log_resp
=
log_rho
-
logsumexp
(
log_rho
,
axis
=
1
)[:,
np
.
newaxis
]
# Computing resp
resp
=
np
.
exp
(
log_resp
)
# Updating parameters
eta
=
resp
.
sum
(
axis
=
0
)
+
10
*
np
.
finfo
(
resp
.
dtype
).
eps
nu
=
nu0
+
eta
S
=
beta0
*
np
.
eye
(
d
)
+
expectation
[
1
]
*
eta
[:,
np
.
newaxis
,
np
.
newaxis
]
inverse_S
=
np
.
linalg
.
inv
(
S
)
inverse_W
=
np
.
zeros
((
nb_components
,
d
,
d
))
m
=
np
.
zeros
((
nb_components
,
d
))
for
k
in
range
(
nb_components
):
m
[
k
]
=
inverse_S
[
k
]
@expectation
[
1
][
k
]
@
(
resp
[:,
k
]
@X
)
s
=
np
.
zeros
((
d
,
d
))
for
i
in
range
(
n
):
s
+=
resp
[
i
,
k
]
*
(
np
.
outer
(
X
[
i
],
X
[
i
])
-
np
.
outer
(
X
[
i
],
expectation
[
0
][
k
])
-
np
.
outer
(
expectation
[
0
][
k
],
X
[
i
])
+
expectation
[
3
][
k
])
inverse_W
[
k
]
=
invW0
+
s
# M-step
weights
=
resp
.
sum
(
axis
=
0
)
/
resp
.
sum
()
# Computing the lower bound
ln_p_x
=
np
.
sum
(
resp
*
(
log_rho_tilde
))
ln_p_z
=
np
.
sum
(
resp
*
np
.
log
(
weights
))
ln_p_mu
=
nb_components
*
d
*
np
.
log
(
0.5
*
beta0
/
np
.
pi
)
-
0.5
*
beta0
*
np
.
sum
(
expectation
[
4
])
ln_p_T
=
nb_components
*
log_wishart_B
(
invW0
,
nu0
)
+
0.5
*
(
nu0
-
d
-
1
)
*
expectation
[
2
].
sum
()
-
0.5
*
np
.
trace
(
invW0
*
expectation
[
1
].
sum
())
ln_q_z
=
np
.
sum
(
resp
*
np
.
log
(
resp
))
ln_q_mu
=
-
0.5
*
nb_components
*
d
*
(
1
+
np
.
log
(
2
*
np
.
pi
))
+
0.5
*
np
.
sum
(
np
.
log
(
np
.
linalg
.
det
(
S
)))
ln_q_T
=
np
.
sum
([
log_wishart_B
(
inverse_W
[
k
],
nu
[
k
])
for
k
in
range
(
nb_components
)])
+
np
.
sum
(
0.5
*
(
nu
-
d
-
1
)
*
expectation
[
2
])
-
np
.
sum
(
0.5
*
np
.
trace
(
inverse_W
@
expectation
[
1
],
axis1
=
1
,
axis2
=
2
))
lower_bound
[
p
]
=
ln_p_x
+
ln_p_z
+
ln_p_mu
+
ln_p_T
+
ln_q_z
+
ln_q_mu
+
ln_q_T
# Displaying every ten iterations the result
if
p
%
10
==
0
:
covs
=
inverse_W
/
nu
[:,
np
.
newaxis
,
np
.
newaxis
]
display_2D
(
m
,
covs
,
nb_components
,
weights
,
X
)
plt
.
title
(
'
iteration
'
+
str
(
p
))
plt
.
show
()
# Final parameter update
covs
=
inverse_W
/
nu
[:,
np
.
newaxis
,
np
.
newaxis
]
display_2D
(
m
,
covs
,
nb_components
,
weights
,
X
)
plt
.
title
(
'
iteration
'
+
str
(
p
))
plt
.
show
()
def
log_wishart_B
(
inverse_W
,
nu
):
d
=
len
(
inverse_W
)
return
+
0.5
*
nu
*
np
.
log
(
np
.
linalg
.
det
(
inverse_W
))
-
0.5
*
nu
*
d
*
np
.
log
(
2
)
-
0.25
*
d
*
(
d
-
1
)
*
np
.
log
(
np
.
pi
)
-
gammaln
(
0.5
*
(
nu
-
np
.
arange
(
d
))).
sum
()
def
display_2D
(
m
,
covs
,
nb_components
,
weights
,
X
):
# Grid
xmin
,
xmax
,
ymin
,
ymax
=
X
[:,
0
].
min
(),
X
[:,
0
].
max
(),
X
[:,
1
].
min
(),
X
[:,
1
].
max
()
mx
,
my
=
.
1
*
(
xmax
-
xmin
),
.
1
*
(
ymax
-
ymin
)
# margins
xmin
,
xmax
,
ymin
,
ymax
=
xmin
-
mx
,
xmax
+
mx
,
ymin
-
my
,
ymax
+
my
x
=
np
.
linspace
(
xmin
,
xmax
,
200
)
y
=
np
.
linspace
(
ymin
,
ymax
,
200
)
x
,
y
=
np
.
meshgrid
(
x
,
y
)
pos
=
np
.
empty
(
x
.
shape
+
(
2
,))
pos
[:,:,
0
]
=
x
;
pos
[:,:,
1
]
=
y
# Distributions
rvs
=
[
multivariate_normal
(
m
[
k
],
covs
[
k
])
for
k
in
range
(
nb_components
)]
Z
=
sum
([
weights
[
k
]
*
rvs
[
k
].
pdf
(
pos
)
for
k
in
range
(
nb_components
)])
# Figure
plt
.
figure
(
figsize
=
(
6
,
6
))
# Dataset plot
plt
.
plot
(
*
X
.
T
,
'
.
'
,
c
=
'
k
'
)
plt
.
xlim
(
xmin
,
xmax
)
plt
.
ylim
(
ymin
,
ymax
)
# Component distributions
ax
=
plt
.
gca
()
for
k
in
range
(
nb_components
):
if
weights
[
k
]
>=
1e-5
:
plot_confidence_ellipse
(
m
[
k
],
covs
[
k
],
0.9
,
ax
=
ax
,
ec
=
'
black
'
,
alpha
=
max
(
0.3
,
weights
[
k
]
/
max
(
weights
)))
def
plot_confidence_ellipse
(
mu
,
cov
,
alph
,
ax
,
clabel
=
None
,
label_bg
=
'
white
'
,
**
kwargs
):
c
=
-
2
*
np
.
log
(
1
-
alph
)
# quantile at alpha of the chi_squarred distr. with df = 2
Lambda
,
Q
=
la
.
eig
(
cov
)
# eigenvalues and eigenvectors (col. by col.)
# Compute the attributes of the ellipse
width
,
heigth
=
2
*
np
.
sqrt
(
c
*
Lambda
)
# compute the value of the angle theta (in degree)
theta
=
180
*
np
.
arctan
(
Q
[
1
,
0
]
/
Q
[
0
,
0
])
/
np
.
pi
if
cov
[
1
,
0
]
else
0
# Create the ellipse
if
'
fc
'
not
in
kwargs
.
keys
():
kwargs
[
'
fc
'
]
=
'
None
'
level_line
=
Ellipse
(
mu
,
width
,
heigth
,
angle
=
theta
,
**
kwargs
)
return
ax
.
add_patch
(
level_line
)
```
%% Cell type:code id: tags:
```
python
# Dataset creation
dataset_1
=
np
.
random
.
multivariate_normal
(
mean
=
[
1
,
1
],
cov
=
[[
0.01
,
0
],[
0
,
0.01
]],
size
=
(
100
))
dataset_2
=
np
.
random
.
multivariate_normal
(
mean
=
[
-
1
,
-
1
],
cov
=
[[
0.01
,
0
],[
0
,
0.01
]],
size
=
(
100
))
dataset
=
np
.
vstack
((
dataset_1
,
dataset_2
))
# Training and visualization of the iterations
fit_predict
(
dataset
,
nb_components
=
5
,
max_iter
=
100
)
```
%% Output
%% Cell type:code id: tags:
```
python
```
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