kernel.py 9.63 KB
Newer Older
Malthe Kjær Bisbo's avatar
Malthe Kjær Bisbo committed
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
import numpy as np
from abc import ABC, abstractmethod

from scipy.spatial.distance import pdist, cdist, squareform

class kernel(ABC):
    def __init__(self):
        self._theta = None

    @abstractmethod
    def kernel(self):
        pass

#    @abstractmethod
#    def kernel_vector(self):
#        pass

#    @abstractmethod
#    def kernel_matrix(self):
#        pass

    @abstractmethod
    def kernel_jacobian(self):
        pass

    @abstractmethod
    def kernel_hyperparameter_gradient(self):
        pass

    @property
    def theta(self):
        return self._theta

    @theta.setter
    def theta(self, theta):
        self._theta = theta
    
    def numerical_jacobian(self,x,y, dx=1.e-5):
        if np.ndim(y) == 1:
            y = y.reshape((1,-1))
        nx = len(x)
        ny = y.shape[0]
        f0 = self.kernel(x,y)
        f_jac = np.zeros((ny,nx))
        for i in range(nx):
            x_up = np.copy(x)
            x_down = np.copy(x)
            x_up[i] += 0.5*dx
            x_down[i] -= 0.5*dx
            
            f_up = self.kernel(x_up,y)
            f_down = self.kernel(x_down,y)
            f_jac[:,i] = (f_up - f_down)/dx
        return f_jac

    def numerical_hyperparameter_gradient(self,X, dx=1.e-5):
        N_data = X.shape[0]
        theta = np.copy(self.theta)
        N_hyper = len(theta)
        dK_dTheta = np.zeros((N_hyper, N_data, N_data))
        for i in range(N_hyper):
            theta_up = np.copy(theta)
            theta_down = np.copy(theta)
            theta_up[i] += 0.5*dx
            theta_down[i] -= 0.5*dx
            
            self.theta = theta_up
Malthe Kjær Bisbo's avatar
update    
Malthe Kjær Bisbo committed
68
            K_up = self(X, eval_gradient=False)
Malthe Kjær Bisbo's avatar
Malthe Kjær Bisbo committed
69
            self.theta = theta_down
Malthe Kjær Bisbo's avatar
update    
Malthe Kjær Bisbo committed
70
            K_down = self(X, eval_gradient=False)
Malthe Kjær Bisbo's avatar
Malthe Kjær Bisbo committed
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
            dK_dTheta[i,:,:] = (K_up - K_down)/dx
        return dK_dTheta


class gauss_kernel(kernel):
    def __init__(self, amplitude=10.0, length_scale=10.0, amplitude_bounds=(1e0, 1e3), length_scale_bounds=(1e-1, 1e1)):
        self.amplitude = amplitude
        self.length_scale = length_scale
        self.amplitude_bounds = amplitude_bounds
        self.length_scale_bounds = length_scale_bounds
        self._theta_bounds = [amplitude_bounds, length_scale_bounds]

    def __call__(self, X,Y, eval_gradient=False):
        pass
        
    def kernel(self, X,Y):
        if np.ndim(X) == 1:
            X = X.reshape((1,-1))
        if np.ndim(Y) == 1:
            Y = Y.reshape((1,-1))
        d = cdist(X / self.length_scale,
                  Y / self.length_scale, metric='sqeuclidean')
        K = self.amplitude * np.exp(-0.5 * d)
        return K
        
    def kernel_value(self, x,y):
        d = cdist(x.reshape(1,-1) / self.length_scale,
                  y.reshape(1,-1) / self.length_scale, metric='sqeuclidean')
        K = self.amplitude * np.exp(-0.5 * d)
        return K

    def kernel_vector(self, x,Y):
        d = cdist(x.reshape(1,-1) / self.length_scale,
                  Y / self.length_scale, metric='sqeuclidean')
        K = self.amplitude * np.exp(-0.5 * d)
        return K

    def kernel_matrix(self, X,Y=None):
        if Y is None:
            d = cdist(X / self.length_scale, X / self.length_scale, metric='sqeuclidean')
        else:
            d = cdist(X / self.length_scale, Y / self.length_scale, metric='sqeuclidean')
        K = self.amplitude * np.exp(-0.5 * d)
        return K

    def kernel_jacobian(self, X,Y):
        if np.ndim(X) == 1:
            X = X.reshape((1,-1))
        if np.ndim(Y) == 1:
            Y = Y.reshape((1,-1))
        K = self.kernel(X,Y).T
        dK_dd = -1./(2*self.length_scale**2)*K
        dd_df = 2*(X - Y)

        dk_df = np.multiply(dK_dd, dd_df)
        return dk_df

    @property
    def theta(self):
        self._theta = [self.amplitude, self.length_scale]
        return self._theta

    @theta.setter
    def theta(self, theta):
        self._theta = theta
        self.amplitude = self._theta[0]
        self.length_scale = self._theta[1]

    def dK_da(self, X):
        if np.ndim(X) == 1:
            X = X.reshape((1,-1))
        d = cdist(X / self.length_scale,
                  X / self.length_scale, metric='sqeuclidean')
        dK_da = np.exp(-0.5 * d)
        return dK_da
        
    def dK_dl(self, X):
        if np.ndim(X) == 1:
            X = X.reshape((1,-1))
        d = cdist(X / self.length_scale,
                  X / self.length_scale, metric='sqeuclidean')
        dK_dl = self.amplitude * d/self.length_scale * np.exp(-0.5 * d)
        return dK_dl

    def kernel_hyperparameter_gradient(self, X):
        return np.array([self.dK_da(X), self.dK_dl(X)])
    

class double_gauss_kernel(kernel):
    def __init__(self, amplitude=10., amplitude_bounds=(1e0,1e3),
                 length_scale1=10.0, length_scale1_bounds=(1e0, 1e3),
                 length_scale2=10.0, length_scale2_bounds=(1e0, 1e3),
                 weight=0.01, weight_bounds=(0.01,0.01),
Malthe Kjær Bisbo's avatar
update    
Malthe Kjær Bisbo committed
164
                 noise=1e-5, noise_bounds=(1e-5,1e-5)):
Malthe Kjær Bisbo's avatar
Malthe Kjær Bisbo committed
165
166
167
168
        self.amplitude = amplitude
        self.length_scale1 = length_scale1
        self.length_scale2 = length_scale2
        self.weight = weight
Malthe Kjær Bisbo's avatar
update    
Malthe Kjær Bisbo committed
169
        self.noise = noise
Malthe Kjær Bisbo's avatar
Malthe Kjær Bisbo committed
170
171
172
173
174

        self.amplitude_bounds = amplitude_bounds
        self.length_scale1_bounds = length_scale1_bounds
        self.length_scale2_bounds = length_scale2_bounds
        self.weight_bounds = weight_bounds
Malthe Kjær Bisbo's avatar
update    
Malthe Kjær Bisbo committed
175
        self.noise_bounds = noise_bounds
Malthe Kjær Bisbo's avatar
Malthe Kjær Bisbo committed
176

Malthe Kjær Bisbo's avatar
update    
Malthe Kjær Bisbo committed
177
        self.theta_bounds = np.array([amplitude_bounds, length_scale1_bounds, length_scale2_bounds, weight_bounds, noise_bounds])
Malthe Kjær Bisbo's avatar
Malthe Kjær Bisbo committed
178

Malthe Kjær Bisbo's avatar
update    
Malthe Kjær Bisbo committed
179
180
181
182
183
184
185
186
187
188
189
190
191
192
    def __call__(self, X, eval_gradient=False):
        if np.ndim(X) == 1:
            X = X.reshape((1,-1))
        d1 = cdist(X / self.length_scale1,
                  X / self.length_scale1, metric='sqeuclidean')
        d2 = cdist(X / self.length_scale2,
                  X / self.length_scale2, metric='sqeuclidean')
        K = self.amplitude * (np.exp(-0.5 * d1) + self.weight*np.exp(-0.5 * d2) + self.noise*np.eye(X.shape[0]))

        if eval_gradient:
            K_gradient = self.kernel_hyperparameter_gradient(X)
            return K, K_gradient
        else:
            return K
Malthe Kjær Bisbo's avatar
Malthe Kjær Bisbo committed
193
        
Malthe Kjær Bisbo's avatar
update    
Malthe Kjær Bisbo committed
194
    def kernel(self, X,Y=None):
Malthe Kjær Bisbo's avatar
Malthe Kjær Bisbo committed
195
196
        if np.ndim(X) == 1:
            X = X.reshape((1,-1))
Malthe Kjær Bisbo's avatar
update    
Malthe Kjær Bisbo committed
197
198
199
        if Y is None:
            Y = X
        elif np.ndim(Y) == 1:
Malthe Kjær Bisbo's avatar
Malthe Kjær Bisbo committed
200
201
202
203
204
205
206
207
            Y = Y.reshape((1,-1))
        d1 = cdist(X / self.length_scale1,
                  Y / self.length_scale1, metric='sqeuclidean')
        d2 = cdist(X / self.length_scale2,
                  Y / self.length_scale2, metric='sqeuclidean')
        K = self.amplitude * (np.exp(-0.5 * d1) + self.weight*np.exp(-0.5 * d2))
        return K

Malthe Kjær Bisbo's avatar
update    
Malthe Kjær Bisbo committed
208
209
210
211
    def kernel_value(self, x,y):
        K = self.kernel(x,y)
        return np.asscalar(K)
    
Malthe Kjær Bisbo's avatar
Malthe Kjær Bisbo committed
212
    def kernel_vector(self, x,Y):
Malthe Kjær Bisbo's avatar
update    
Malthe Kjær Bisbo committed
213
        K = self.kernel(x,Y).reshape(-1)
Malthe Kjær Bisbo's avatar
Malthe Kjær Bisbo committed
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
        return K

    def kernel_matrix(self, X,Y=None):
        if Y is None:
            d = cdist(X / self.length_scale, X / self.length_scale, metric='sqeuclidean')
        else:
            d = cdist(X / self.length_scale, Y / self.length_scale, metric='sqeuclidean')
        K = np.exp(-0.5 * d)
        return K

    def kernel_jacobian(self, X,Y):
        """ Jacobian of the kernel with respect to X
        """
        if np.ndim(X) == 1:
            X = X.reshape((1,-1))
        if np.ndim(Y) == 1:
            Y = Y.reshape((1,-1))
        d1 = cdist(X / self.length_scale1,
                   Y / self.length_scale1, metric='sqeuclidean')
        d2 = cdist(X / self.length_scale2,
                   Y / self.length_scale2, metric='sqeuclidean')
        dK1_dd1 = -1/(2*self.length_scale1**2) * np.exp(-0.5 * d1)
        dK2_dd2 = -1/(2*self.length_scale2**2) * np.exp(-0.5 * d2)
        dK_dd = self.amplitude * (dK1_dd1 + self.weight*dK2_dd2)
        dd_df = 2*(X - Y)

        dk_df = np.multiply(dK_dd.T, dd_df)
        return dk_df

    @property
    def theta(self):
Malthe Kjær Bisbo's avatar
update    
Malthe Kjær Bisbo committed
245
246
247
248
        """Returns the log-transformed hyperparameters of the kernel.
        """
        self._theta = np.array([self.amplitude, self.length_scale1, self.length_scale2, self.weight, self.noise])
        return np.log(self._theta)
Malthe Kjær Bisbo's avatar
Malthe Kjær Bisbo committed
249
250
251

    @theta.setter
    def theta(self, theta):
Malthe Kjær Bisbo's avatar
update    
Malthe Kjær Bisbo committed
252
253
254
255
256
        """Sets the hyperparameters of the kernel.

        theta: log-transformed hyperparameters
        """
        self._theta = np.exp(theta)
Malthe Kjær Bisbo's avatar
Malthe Kjær Bisbo committed
257
258
259
260
        self.amplitude = self._theta[0]
        self.length_scale1 = self._theta[1]
        self.length_scale2 = self._theta[2]
        self.weight = self._theta[3]
Malthe Kjær Bisbo's avatar
update    
Malthe Kjær Bisbo committed
261
        self.noise = self._theta[4]
Malthe Kjær Bisbo's avatar
Malthe Kjær Bisbo committed
262
263
264
265
266
267

    def dK_da(self, X):
        d1 = cdist(X / self.length_scale1,
                   X / self.length_scale1, metric='sqeuclidean')
        d2 = cdist(X / self.length_scale2,
                   X / self.length_scale2, metric='sqeuclidean')
Malthe Kjær Bisbo's avatar
update    
Malthe Kjær Bisbo committed
268
        dK_da = np.exp(-0.5 * d1) + self.weight*np.exp(-0.5 * d2) + self.noise*np.eye(X.shape[0])
Malthe Kjær Bisbo's avatar
Malthe Kjær Bisbo committed
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
        return dK_da
        
    def dK_dl1(self, X):
        d1 = cdist(X / self.length_scale1,
                   X / self.length_scale1, metric='sqeuclidean')
        dK_dl1 = self.amplitude*d1/self.length_scale1*np.exp(-0.5 * d1)
        return dK_dl1

    def dK_dl2(self, X):
        d2 = cdist(X / self.length_scale2,
                   X / self.length_scale2, metric='sqeuclidean')
        dK_dl2 = self.amplitude*self.weight*d2/self.length_scale2*np.exp(-0.5 * d2)
        return dK_dl2

    def dK_dw(self, X):
        d2 = cdist(X / self.length_scale2,
                   X / self.length_scale2, metric='sqeuclidean')
        dK_dl2 = self.amplitude*np.exp(-0.5 * d2)
        return dK_dl2

Malthe Kjær Bisbo's avatar
update    
Malthe Kjær Bisbo committed
289
290
291
292
    def dK_dn(self, X):
        dK_dn = self.amplitude * np.eye(X.shape[0])
        return dK_dn

Malthe Kjær Bisbo's avatar
Malthe Kjær Bisbo committed
293
    def kernel_hyperparameter_gradient(self, X):
Malthe Kjær Bisbo's avatar
update    
Malthe Kjær Bisbo committed
294
        return np.array([self.dK_da(X), self.dK_dl1(X), self.dK_dl2(X), self.dK_dw(X), self.dK_dn(X)])