-
Notifications
You must be signed in to change notification settings - Fork 1
/
script.py
339 lines (288 loc) · 10.5 KB
/
script.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
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
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
"""
Minimal working example of a Spectral Mixture Gaussian Process.
The Spectral Mixture Gaussian Process was introduced in:
> Wilson, A., & Adams, R. (2013). Gaussian process kernels for pattern
> discovery and extrapolation. In *International conference on machine learning*
> (pp. 1067-1075). PMLR.
Notation is consistent with the paper.
"""
__date__ = "October 2021"
import matplotlib.pyplot as plt
import numpy as np
import torch
from torch.distributions import Normal, MultivariateNormal, Categorical, \
MixtureSameFamily
from torch.utils.data import TensorDataset, DataLoader
MAX_X = 1.0 # maximum x-value
MAX_FREQ = 20.0 # maximum frequency
EPSILON = 2e-5 # for numerical stability
# Spectral content of true data-generating Gaussian Process:
TRUE_μ = torch.tensor(np.array([5, 8, 13])).to(torch.float)
ΤRUE_LOG_V = torch.tensor(np.log(np.array([0.3, 2.0, 0.8]))).to(torch.float)
TRUE_LOG_W = torch.tensor(np.log(np.array([1, 2, 1]))).to(torch.float)
class SMGP(torch.nn.Module):
"""Spectral Mixture Gaussian Process"""
def __init__(self, q=10):
"""
Parameters
----------
q : int, optional
Number of mixture components
"""
super(SMGP, self).__init__()
self.q = q
# Initialize the GMM components on a linearly-spaced grid with equal
# variance and equal weights.
self.μ = torch.nn.Parameter(torch.linspace(0,MAX_FREQ,q)) # [q]
self.log_v = torch.nn.Parameter(2.0*torch.ones(q)) # [q]
self.log_w = torch.nn.Parameter(torch.zeros(q)) # [q]
# Include a trainable scaling parameter.
self.log_scale = torch.nn.Parameter(torch.zeros(1))
def forward(self, xs, ys):
"""
Calculate the mean data log marginal likelihood over the batch.
Parameters
----------
xs : torch.Tensor
Shape: [t]
ys : torch.Tensor
Shape: [b,t]
Returns
-------
log_like : torch.Tensor
Shape: []
"""
# Calculate the kernel matrix.
kernel_mat = self._get_kernel_mat(xs)
# Contrsuct the Gaussian and evaluate.
gaussian = MultivariateNormal(torch.zeros_like(xs), kernel_mat)
return gaussian.log_prob(ys).mean()
def _get_kernel_mat(self, xs):
"""
Get the kernel matrix.
This function implements Eq. 12 in the paper with P=1.
Parameters
----------
xs : torch.Tensor
Shape: [t]
Returns
-------
kernel_mat : torch.Tensor
Shape: [t,t]
"""
t = xs.shape[0]
# Get the matrix of taus.
τ = torch.abs(xs.unsqueeze(-1) - xs.unsqueeze(-2)) # [t,t]
τ = τ.view(1,t,t) # [1,t,t]
# Evaluate the rest of the terms in Eq. 12.
μ = self.μ.view(self.q,1,1) # [q,1,1]
cos_term = torch.cos(2*np.pi*τ*μ) # [q,t,t]
v = torch.exp(self.log_v).view(self.q,1,1) # [q,1,1]
exp_term = torch.exp(-2*np.pi**2*τ.pow(2)*v)
w = self.log_w.exp().view(self.q,1,1)
kernel_mat = torch.sum(w * exp_term * cos_term, dim=0) # [t,t]
# Scale by a learned parameter and add a small multiple of the identity
# matrix for numerical stability.
kernel_mat = self.log_scale.exp() * kernel_mat + EPSILON * torch.eye(t)
return kernel_mat
def sample(self, xs, n_samples):
"""
Sample from the GP.
Parameters
----------
xs : torch.Tensor
n_samples : int
Returns
-------
samples : torch.Tensor
Shape: [n,t]
"""
with torch.no_grad():
# Calculate the kernel matrix.
kernel_mat = self._get_kernel_mat(xs)
# Construct the GMM.
mix = Categorical(logits=self.log_w)
means = torch.zeros((self.q, xs.shape[0])) # [q,t]
comp = MultivariateNormal(loc=means, covariance_matrix=kernel_mat)
gmm = MixtureSameFamily(mix, comp)
# Sample.
samples = gmm.sample(sample_shape=(n_samples,)) # [n,t]
return samples
def get_psd(self, n_freqs=400):
"""
Get the kernel power spectral density.
Parameters
----------
n_freqs : int, optional
Number of frequencies to evaluate.
Returns
-------
psd : numpy.ndarray
Power spectral density
Shape: [n_freqs]
"""
freqs = torch.linspace(0, MAX_FREQ, n_freqs)
with torch.no_grad():
# Make the spectral GMM. We have to make two copies, one with
# positive μ's and one with negative μ's, because spectral densities
# are symmetric.
mix = Categorical(logits=self.log_w)
comp = Normal(loc=self.μ, scale=(0.5*self.log_v).exp())
gmm = MixtureSameFamily(mix, comp)
psd = (gmm.log_prob(freqs).exp() + gmm.log_prob(-freqs).exp()) / 2
return psd.detach().numpy()
def plot_samples(self, xs, ys, n_samples=2, spacing=10, ax=None):
"""
Plot some real and generated samples.
Parameters
----------
xs : torch.Tensor
Shape: [t]
ys : torch.Tensor
Shape: [b,t]
n_samples : int, optional
spacing : int, optional
Vertical spacing between samples
ax : None or matplotlib.axes._subplots.AxesSubplot
"""
samples = self.sample(xs, n_samples)
x = xs.detach().numpy()
y = ys.detach().numpy()
samples = samples.detach().numpy()
if ax is not None:
plt.sca(ax)
for i in range(n_samples):
label = 'data' if i == 0 else None
plt.plot(x, spacing*i + y[i], c='tab:blue', label=label)
label = 'model' if i == 0 else None
plt.plot(x+MAX_X+0.1, spacing*i + samples[i], c='goldenrod', \
label=label)
plt.axis('off')
plt.title("Generated Samples")
plt.legend()
def plot_forecast(self, xs, ys, idx=0, ax=None):
"""
Plot a GP forecast on the second half of `ys[idx]`.
Parameters
----------
xs : torch.Tensor
Shape: [t]
ys : torch.Tensor
Shape: [b,t]
idx : int, optional
Index for `ys`.
ax : None or matplotlib.axes._subplots.AxesSubplot
"""
with torch.no_grad():
t = xs.shape[0]
temp_xs = torch.cat([xs[:t//2], xs])
# Calculate the kernel matrix.
kernel_mat = self._get_kernel_mat(temp_xs)
# Calculate the forecast by Gaussian conditioning. These formulas
# can be found here, for example, under "Conditional distributions":
# https://en.wikipedia.org/wiki/Multivariate_normal_distribution
k11 = kernel_mat[:t//2,:t//2]
k12 = kernel_mat[:t//2,t//2:]
k22 = kernel_mat[t//2:,t//2:]
# print(k11.shape, k12.shape)
temp = torch.linalg.solve(k11, k12).T
# k11_inv = torch.inverse(k11)
mean = (temp @ ys[idx,:t//2].unsqueeze(-1)).squeeze(-1)
covar = k22 - temp @ k12
stds = torch.diag(covar).sqrt()
xs = xs.detach().numpy()
ys = ys.detach().numpy()
mean = mean.detach().numpy()
stds = stds.detach().numpy()
# Plot.
if ax is not None:
plt.sca(ax)
plt.axvline(x=xs[t//2], c='k', ls='--', alpha=0.5)
plt.plot(xs, mean, c='goldenrod', label="model")
plt.fill_between(xs, mean-stds, mean+stds, fc='goldenrod', alpha=0.2)
plt.plot(xs, ys[idx], c='tab:blue', label="data")
plt.xlabel("x")
plt.ylabel("y")
plt.legend(loc='best')
plt.title("Forecast")
for dir in ['top', 'right']:
plt.gca().spines[dir].set_visible(False)
def plot_psd(self, true_model, n_freqs=400, ax=None):
"""
Plot the kernel frequency representation.
Parameters
----------
true_model : SMGP
n_freqs : int, optional
ax : None or matplotlib.axes._subplots.AxesSubplot
"""
model_psd = self.get_psd(n_freqs=n_freqs)
true_psd = true_model.get_psd(n_freqs=n_freqs)
freqs = torch.linspace(0, MAX_FREQ, n_freqs)
# Plot.
if ax is not None:
plt.sca(ax)
plt.plot(freqs, true_psd, c='tab:blue', label='ground truth')
plt.plot(freqs, model_psd, c='goldenrod', label='model')
plt.xlabel("Frequency")
plt.ylabel("Spectral Density")
plt.legend(loc='best')
plt.title("Power Spectral Densities")
for dir in ['top', 'right']:
plt.gca().spines[dir].set_visible(False)
def generate_data(n=2048, t=200):
"""
Generate synthetic data by drawing samples from a ground-truth GP.
Returns
-------
ground_truth_model : SMGP
xs : torch.Tensor
Shape: [t]
ys : torch.Tensor
Shape: [b,t]
"""
model = SMGP(q=len(TRUE_μ))
model.μ.data = TRUE_μ
model.log_v.data = ΤRUE_LOG_V
model.log_w.data = TRUE_LOG_W
xs = torch.linspace(0, MAX_X, t)
return model, xs, model.sample(xs, n)
if __name__ == '__main__':
# Parameters
load_model = False
save_model = True
epochs = 500
torch.manual_seed(42)
# Get the data and make a Dataloader.
true_model, xs, ys = generate_data() # SMGP, [t], [n,t]
forecast_ys = true_model.sample(xs, 1) # Forecast on held-out data.
dataset = TensorDataset(ys)
loader = DataLoader(dataset, batch_size=128, shuffle=True)
# Make a model.
model = SMGP()
if load_model:
checkpoint = torch.load('state.tar')
model.load_state_dict(checkpoint)
# Enter a training loop to optimize the kernel parameters.
optimizer = torch.optim.Adam(model.parameters())
for i in range(epochs):
epoch_loss = 0.0
for batch in loader:
model.zero_grad()
loss = -model(xs, *batch)
loss.backward()
epoch_loss += loss.item() / len(loader)
optimizer.step()
if i % 10 == 0:
print(f"Epoch {i:03d}, loss: {epoch_loss:.3f}")
# Save the model.
if save_model:
torch.save(model.state_dict(), 'state.tar')
# Make some plots.
fig, axarr = plt.subplots(nrows=3, figsize=(9,7))
model.plot_samples(xs, ys, ax=axarr[0])
model.plot_psd(true_model, ax=axarr[1])
model.plot_forecast(xs, ys, ax=axarr[2])
plt.tight_layout()
plt.savefig('out.jpg', dpi=300)
###