-
Notifications
You must be signed in to change notification settings - Fork 0
/
MarginalDist.py
213 lines (157 loc) · 6.93 KB
/
MarginalDist.py
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
# coding: utf-8
# In[1]:
import autograd.numpy as np
from autograd import grad
import scipy as sc
import pandas as pd
import scipy.io as sio
import matplotlib.pyplot as plt
get_ipython().run_line_magic('matplotlib', 'inline')
import seaborn as sns
import math
from scipy import optimize
# In[4]:
oc1 = sio.loadmat('occam1.mat')
oc2 = sio.loadmat('occam2.mat')
D1=np.array([oc1['x'].T[0],oc1['y'].T[0]])
D2=np.array([oc2['x'].T[0],oc2['y'].T[0]])
# In[5]:
#construct design matrices
def phi_1(x, z=None):
return(np.array([x**0,x,x**2,x**3,x**4,x**5]).T)
def phi_2(x, z):
return(np.array([np.exp(-(x-1)**2/z[0]**2), np.exp(-(x-5)**2/z[1]**2)]).T)
def phi_3(x, z=None):
return(np.array([x,np.cos(2*x)]).T)
phi11 = phi_1(D1[0])
phi12 = phi_2(D1[0],z=[1,1])
phi13 = phi_3(D1[0])
phi21 = phi_1(D2[0])
phi22 = phi_2(D2[0],z=[1,1])
phi23 = phi_3(D2[0])
x1 = D1[0]
x2 = D2[0]
y1 = D1[1]
y2 = D2[1]
z = [1.0,1.0]
# In[6]:
def gradphi(phi,x,z):
dphidz1 = np.array([np.multiply((2*(x-1)**2)/(z[0]**3), phi[:,0]), np.zeros(len(x))]).T
dphidz2 = np.array([np.zeros(len(x)), np.multiply((2*(x-5)**2)/(z[1]**3), phi[:,1])]).T
return(np.array([dphidz1,dphidz2]))
class model():
def __init__(self, x, y, phifunc, z=None):
self.x = x
self.y = y
self.phifunc = phifunc
self.z = z
def fastgrad(self, params):
loga2 = params[0]
logsig2 = params[1]
a2 = np.exp(loga2)
sig2 = np.exp(logsig2)
if(self.z!=None):
self.z = [params[2], params[3]]
phi = self.phifunc(self.x, self.z)
y = self.y
x = self.x
n = len(self.y)
p = phi.shape[1]
sigmap = a2/sig2 * np.dot(phi.T, phi)+np.identity(p)
invsp = np.linalg.inv(sigmap)
#gradient for a2
tracea = 1/sig2*np.trace(np.dot(phi.T, phi))- a2/sig2**2*np.trace(np.dot(phi.T, np.dot(phi, np.dot(invsp, np.dot(phi.T, phi)))))
ysppsy = 1/sig2**2*np.dot(y.T, np.dot(phi, np.dot(phi.T, y)))- 2*a2/(sig2**3)*np.dot(y.T, np.dot(phi, np.dot(phi.T, np.dot(phi, np.dot(invsp, np.dot(phi.T, y))))))+ a2**2/(sig2**4)*np.dot(y.T, np.dot(phi, np.dot(invsp, np.dot(phi.T, np.dot(phi, np.dot(phi.T, np.dot(phi, np.dot(invsp, np.dot(phi.T, y)))))))))
dlda2 = -1/2*(ysppsy-tracea)*a2
#gradient for sigma2
traces = 1/sig2*np.trace(np.identity(n))-a2/sig2**2*np.trace(np.dot(phi.T,np.dot(phi, invsp)))
yssy = 1/sig2**2*np.dot(y.T, y)-2*a2/(sig2**3)*np.dot(y.T, np.dot(phi, np.dot(invsp, np.dot(phi.T, y))))+ a2**2/(sig2**4)*np.dot(y.T, np.dot(phi, np.dot(invsp, np.dot(phi.T, np.dot(phi, np.dot(invsp, np.dot(phi.T, y)))))))
dlds2 = -1/2*(yssy-traces)*sig2
gradients = [dlda2, dlds2]
#gradient for z
dldz = np.zeros(len(z))
if(self.z!=None):
dldz = np.zeros(len(self.z))
dphidz = gradphi(phi, x, z)
for i in range(len(self.z)):
trace = 1/sig2*np.trace(np.dot(phi.T, dphidz[i]))- a2/sig2**2*np.trace(np.dot(phi.T, np.dot(phi, np.dot(invsp, np.dot(phi.T, dphidz[0])))))
trace *=2
yspppsy = 1/sig2**2*np.dot(y.T, np.dot(dphidz[i], np.dot(phi.T, y)))- a2/sig2**3*np.dot(y.T, np.dot(dphidz[i], np.dot(phi.T, np.dot(phi, np.dot(invsp, np.dot(phi.T, y))))))- a2/sig2**3*np.dot(y.T, np.dot(phi, np.dot(invsp, np.dot(phi.T, np.dot(dphidz[i], np.dot(phi.T, y))))))+ a2**2/sig2**4*np.dot(y.T, np.dot(phi, np.dot(invsp, np.dot(phi.T, np.dot(dphidz[i], np.dot(phi.T, np.dot(phi, np.dot(invsp, np.dot(phi.T, y)))))))))
yspppsy *= 2
dldz[i] = a2/2*(yspppsy-trace)
gradients.append(dldz[i])
return np.array(gradients)
def fastlogmarg(self, params):
loga2 = params[0]
logsig2 = params[1]
a2 = np.exp(loga2)
sig2 = np.exp(logsig2)
y = self.y
if(self.z!=None):
self.z = [params[2], params[3]]
phi = self.phifunc(self.x, self.z)
n = len(self.y)
p = phi.shape[1]
sigmap = a2/sig2*np.dot(phi.T, phi)+np.identity(p)
detsp = np.linalg.det(sigmap)
invsp = np.linalg.inv(sigmap)
exponent = 1/(2*sig2)*np.dot(y.T, y)-a2/(2*sig2**2)*np.dot(y.T, np.dot(phi, np.dot(invsp, np.dot(phi.T, y))))
logmarg = -n/2*np.log(2*math.pi)-n/2*np.log(sig2)-1/2*np.log(detsp)-exponent
return(-logmarg)
# In[35]:
tes = model(x1, y1, phi_2, [1.0, 1.0])
print(tes.fastlogmarg([np.log(4), np.log(4), 1.0, 1.0]))
print(tes.fastgrad([np.log(4), np.log(4), 1.0, 1.0]))
# In[7]:
model1 = model(x1, y1, phi_1)
model2 = model(x1, y1, phi_2, [1.0, 1.0])
model3 = model(x1, y1, phi_3)
loghyperparameters = [0.5,1.0]
hyperparameters2 = [1.0, 1.0, 1.0, 1.0]
model1opt = optimize.minimize(model1.fastlogmarg, x0=loghyperparameters, method="BFGS", jac=model1.fastgrad, options={'maxiter':400})
model2opt = optimize.minimize(model2.fastlogmarg, x0=hyperparameters2, method="BFGS", jac=model2.fastgrad, options={'maxiter':400})
model3opt = optimize.minimize(model3.fastlogmarg, x0=loghyperparameters, method="BFGS", jac=model3.fastgrad, options={'maxiter':400})
logmargres = [model1opt.fun, model2opt.fun, model3opt.fun]
print(model1opt.x, model2opt.x[0:2], model3opt.x)
x = ["model1", "model2", "model3"]
plt.bar(x, logmargres/np.sum(logmargres))
# In[8]:
model1b = model(x2, y2, phi_1)
model2b = model(x2, y2, phi_2, [1.0, 1.0])
model3b = model(x2, y2, phi_3)
model1bopt = optimize.minimize(model1b.fastlogmarg, x0=loghyperparameters, method="Nelder-Mead")
model2bopt = optimize.minimize(model2b.fastlogmarg, x0=hyperparameters2, method="Nelder-Mead")
model3bopt = optimize.minimize(model3b.fastlogmarg, x0=loghyperparameters, method="Nelder-Mead")
logmargres2 = [model1bopt.fun, model2bopt.fun, model3bopt.fun]
plt.bar(x, logmargres2/np.sum(logmargres2))
# In[32]:
plt.scatter(x2, y2)
# Calculating posterior mean of the weights for D2
# In[24]:
a2h, sig2h = np.exp(model1bopt.x)
p1 = 6
invpostcov = 1/a2h*np.identity(p1)+1/sig2h*np.dot(phi21.T, phi21)
postcov = np.linalg.inv(invpostcov)
postvar = np.diagonal(postcov)
postmean = (1/sig2h)*np.dot(postcov, np.dot(phi21.T, y2))
print("posterior mean:", postmean)
print("posterior variance:", postvar)
# In[30]:
a2h, sig2h = np.exp(model2bopt.x[0:2])
p2 = 2
phi22 = phi_2(x2, model2bopt.x[2:4])
invpostcov = 1/a2h*np.identity(p2)+1/sig2h*np.dot(phi22.T, phi22)
postcov = np.linalg.inv(invpostcov)
postvar = np.diagonal(postcov)
postmean = (1/sig2h)*np.dot(postcov, np.dot(phi22.T, y2))
print("posterior mean:", postmean)
print("posterior variance:", postvar)
# In[31]:
a2h, sig2h = np.exp(model1bopt.x)
p3 = 2
invpostcov = 1/a2h*np.identity(p3)+1/sig2h*np.dot(phi23.T, phi23)
postcov = np.linalg.inv(invpostcov)
postvar = np.diagonal(postcov)
postmean = (1/sig2h)*np.dot(postcov, np.dot(phi23.T, y2))
print("posterior mean:", postmean)
print("posterior variance:", postvar)