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
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
|
import numpy as np
from scipy import optimize
from scipy.spatial.distance import pdist, squareform
try:
from sklearn.preprocessing import OrdinalEncoder
UseOrdinalEncoder = True
except ImportError:
UseOrdinalEncoder = False
import pytest
from numpy.testing import assert_array_almost_equal, assert_array_equal
from skopt.learning.gaussian_process import GaussianProcessRegressor
from skopt.learning.gaussian_process.kernels import (
RBF,
ConstantKernel,
DotProduct,
ExpSineSquared,
HammingKernel,
Matern,
RationalQuadratic,
WhiteKernel,
)
KERNELS = []
for length_scale in [np.arange(1, 6), [0.2, 0.3, 0.5, 0.6, 0.1]]:
KERNELS.extend(
[
RBF(length_scale=length_scale),
Matern(length_scale=length_scale, nu=0.5),
Matern(length_scale=length_scale, nu=1.5),
Matern(length_scale=length_scale, nu=2.5),
RationalQuadratic(alpha=2.0, length_scale=2.0),
ExpSineSquared(length_scale=2.0, periodicity=3.0),
ConstantKernel(constant_value=1.0),
WhiteKernel(noise_level=2.0),
Matern(length_scale=length_scale, nu=2.5) ** 3.0,
RBF(length_scale=length_scale) + Matern(length_scale=length_scale, nu=1.5),
RBF(length_scale=length_scale) * Matern(length_scale=length_scale, nu=1.5),
DotProduct(sigma_0=2.0),
]
)
# Copied (shamelessly) from sklearn.gaussian_process.kernels
def _approx_fprime(xk, f, epsilon, args=()):
f0 = f(*((xk,) + args))
grad = np.zeros((f0.shape[0], f0.shape[1], len(xk)), float)
ei = np.zeros((len(xk),), float)
for k in range(len(xk)):
ei[k] = 1.0
d = epsilon * ei
grad[:, :, k] = (f(*((xk + d,) + args)) - f0) / d[k]
ei[k] = 0.0
return grad
def kernel_X_Y(x, y, kernel):
X = np.expand_dims(x, axis=0)
Y = np.expand_dims(y, axis=0)
return kernel(X, Y)[0][0]
def numerical_gradient(X, Y, kernel, step_size=1e-4):
grad = []
for y in Y:
num_grad = optimize.approx_fprime(X, kernel_X_Y, step_size, y, kernel)
grad.append(num_grad)
return np.asarray(grad)
def check_gradient_correctness(kernel, X, Y, step_size=1e-4):
X_grad = kernel.gradient_x(X, Y)
num_grad = numerical_gradient(X, Y, kernel, step_size)
assert_array_almost_equal(X_grad, num_grad, decimal=3)
@pytest.mark.fast_test
@pytest.mark.parametrize("kernel", KERNELS)
def test_gradient_correctness(kernel):
rng = np.random.RandomState(0)
X = rng.randn(5)
Y = rng.randn(10, 5)
check_gradient_correctness(kernel, X, Y)
@pytest.mark.fast_test
@pytest.mark.parametrize("random_state", [0, 1])
@pytest.mark.parametrize("kernel", KERNELS)
def test_gradient_finiteness(random_state, kernel):
"""When x is the same as X_train, gradients might become undefined because they are
divided by d(x, X_train).
Check they are equal to numerical gradients at such points.
"""
rng = np.random.RandomState(random_state)
X = rng.randn(5).tolist()
Y = [X]
check_gradient_correctness(kernel, X, Y, 1e-6)
@pytest.mark.fast_test
def test_distance_string():
# Inspired by test_hamming_string_array in scipy.tests.test_distance
a = np.array(
[
'eggs',
'spam',
'spam',
'eggs',
'spam',
'spam',
'spam',
'spam',
'spam',
'spam',
'spam',
'eggs',
'eggs',
'spam',
'eggs',
'eggs',
'eggs',
'eggs',
'eggs',
'spam',
],
dtype='|S4',
)
b = np.array(
[
'eggs',
'spam',
'spam',
'eggs',
'eggs',
'spam',
'spam',
'spam',
'spam',
'eggs',
'spam',
'eggs',
'spam',
'eggs',
'spam',
'spam',
'eggs',
'spam',
'spam',
'eggs',
],
dtype='|S4',
)
true_values = np.array([[0, 0.45], [0.45, 0]])
X = np.vstack((a, b))
hm = HammingKernel()
assert_array_almost_equal(-np.log(hm(X)) / 20.0, true_values)
@pytest.mark.fast_test
def test_isotropic_kernel():
rng = np.random.RandomState(0)
X = rng.randint(0, 4, (5, 3))
hm = HammingKernel()
# Scipy calulates the mean. We need exp(-sum)
hamming_distance = squareform(pdist(X, metric='hamming'))
scipy_dist = np.exp(-hamming_distance * X.shape[1])
assert_array_almost_equal(scipy_dist, hm(X))
@pytest.mark.fast_test
def test_anisotropic_kernel():
rng = np.random.RandomState(0)
X = rng.randint(0, 4, (5, 3))
hm = HammingKernel()
X_kernel = hm(X)
hm_aniso = HammingKernel(length_scale=[1.0, 1.0, 1.0])
X_kernel_aniso = hm_aniso(X)
assert_array_almost_equal(X_kernel, X_kernel_aniso)
hm = HammingKernel(length_scale=2.0)
X_kernel = hm(X)
hm_aniso = HammingKernel(length_scale=[2.0, 2.0, 2.0])
X_kernel_aniso = hm_aniso(X)
assert_array_almost_equal(X_kernel, X_kernel_aniso)
@pytest.mark.fast_test
def test_kernel_gradient():
rng = np.random.RandomState(0)
hm = HammingKernel(length_scale=2.0)
X = rng.randint(0, 4, (5, 3))
K, K_gradient = hm(X, eval_gradient=True)
assert_array_equal(K_gradient.shape, (5, 5, 1))
def eval_kernel_for_theta(theta, kernel):
kernel_clone = kernel.clone_with_theta(theta)
K = kernel_clone(X, eval_gradient=False)
return K
K_gradient_approx = _approx_fprime(hm.theta, eval_kernel_for_theta, 1e-10, (hm,))
assert_array_almost_equal(K_gradient_approx, K_gradient, 4)
hm = HammingKernel(length_scale=[1.0, 1.0, 1.0])
K_gradient_approx = _approx_fprime(hm.theta, eval_kernel_for_theta, 1e-10, (hm,))
K, K_gradient = hm(X, eval_gradient=True)
assert_array_equal(K_gradient.shape, (5, 5, 3))
assert_array_almost_equal(K_gradient_approx, K_gradient, 4)
X = rng.randint(0, 4, (3, 2))
hm = HammingKernel(length_scale=[0.1, 2.0])
K_gradient_approx = _approx_fprime(hm.theta, eval_kernel_for_theta, 1e-10, (hm,))
K, K_gradient = hm(X, eval_gradient=True)
assert_array_equal(K_gradient.shape, (3, 3, 2))
assert_array_almost_equal(K_gradient_approx, K_gradient, 4)
@pytest.mark.fast_test
def test_Y_is_not_None():
rng = np.random.RandomState(0)
hm = HammingKernel()
X = rng.randint(0, 4, (5, 3))
hm = HammingKernel(length_scale=[1.0, 1.0, 1.0])
assert_array_equal(hm(X), hm(X, X))
@pytest.mark.fast_test
def test_gp_regressor():
rng = np.random.RandomState(0)
X = np.asarray(
[["ham", "spam", "ted"], ["ham", "ted", "ted"], ["ham", "spam", "spam"]]
)
y = rng.randn(3)
hm = HammingKernel(length_scale=[1.0, 1.0, 1.0])
if UseOrdinalEncoder:
enc = OrdinalEncoder()
enc.fit(X)
gpr = GaussianProcessRegressor(hm)
if UseOrdinalEncoder:
gpr.fit(enc.transform(X), y)
assert_array_almost_equal(gpr.predict(enc.transform(X)), y)
assert_array_almost_equal(gpr.predict(enc.transform(X[:2])), y[:2])
else:
gpr.fit(X, y)
assert_array_almost_equal(gpr.predict(X), y)
assert_array_almost_equal(gpr.predict(X[:2]), y[:2])
|
