-
Notifications
You must be signed in to change notification settings - Fork 12
/
lgcp_model.py
585 lines (485 loc) · 24.7 KB
/
lgcp_model.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
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
# Python 3.6 |Anaconda 3.0 (64-bit)|
# -*- coding: utf-8 -*-
"""
Created: 20-1-2019
Author: Jay Zern Ng (J.Ng.3@warwick.ac.uk)
Description: Implements a univariate log-Gaussian cox process model
with Laplace approx. or Variational Inference to evaluate the posterior.
"""
import numpy as np
import gpflow
import GPy
from probability_model import ProbabilityModel
class LGCPModel(ProbabilityModel):
"""
Statistical model:
Intensity \lambda is an inhomogeneous PP driven by a GP.
f(x) ~ GP(\mu(x), k(x,x'))
y_t | f(x) ~ PP(exp(f(x)))
Inputs at creation:
S1, S2: int;
give the dimensions of the lattice. The one-dimensional case
corresponds to S1=1 and S2=1.
auto_prior_update: boolean
Indicates whether the prior_mean and prior_var should be updated
to the posterior expectation and posterior variance at each time
step. Default is False.
Inputs for model:
prior_signal_variance: float;
stores prior variance hyperparameter of kernel. Default is 1.
prior_lengthscale: float;
stores prior lengthscale hyperparameter of kernel. Default is 1.
custom_kernel: GPFlow/GPy kernel class;
overwrites the inbuilt kernel.
inference_method: string;
specify inference type here, i.e. laplace, variational_inference
or sparse_variational_inference.
refresh_rate: int;
Used for batch optimisation in VI/SVI.
M_pseudo_input_size: int;
size of inducing points |Z| = M
Internal Attributes:
sums: float numpy array; dimension S1xS2x(t+1) at time t
stores y values for each of the S1*S2 series for each possible
run length r=0,1,...t-1, >t-1. at each time point t.
joint_log_probabilities: float numpy array; dimension t+1 at time t
stores the joint log probabilities (y_{1:t}, r_t| q) for this
model q and all potential run lengths r_t = 0,1,...t-1
model_log_evidence: float;
stores in a scalar the evidence for this model, i.e. the joint
probability (y_{1:t}| q) for this model q.
"""
def __init__(self,
prior_signal_variance,
prior_lengthscale,
custom_kernel,
inference_method,
refresh_rate,
S1,
S2,
M_pseudo_input_size=10,
auto_prior_update=False
):
"""Initialize prior quantities for kernel"""
self.prior_signal_variance = prior_signal_variance
self.prior_lengthscale = prior_lengthscale
self.auto_prior_update = auto_prior_update
"""Initialize variables for Gaussian Process and inference method"""
self.m = None
self.kernel = None
self.custom_kernel = custom_kernel
self.likelihood = None
self.mean_function = None
self.inference_method = inference_method
self.M = M_pseudo_input_size
"""Initialize all the quantities that we compute through time"""
self.S1, self.S2 = S1, S2
self.sums = None
self.joint_log_probabilities = 1
self.model_log_evidence = -np.inf
self.retained_run_lengths = None
"""Initalize settings for detector object"""
self.generalized_bayes_rld = "kullback_leibler"
self.has_lags = False
self.hyperparameter_optimization = False
"""Initialize count and refresh rate for batch updating"""
self.count = 0
self.refresh_rate = refresh_rate
def initialization(self, y, cp_model, model_prior):
"""function which is only called ONCE, namely when the very first
lattice of observations y is obtained/observed. Called from the
'update_all_joint_log_probabilities' function within the Detector
object containing the LGCPModel object.
"""
"""Handle zero values for GPy error by adding a small value epsilon
DEBUG: possibly find an alternative way to handle zero values"""
if y == 0:
epsilon = 0.01
y = y + epsilon
y = y.reshape(self.S1, self.S2)
x = np.zeros(1).reshape(self.S1, self.S2)
"""Instantiate the GP using one of the three methods:
laplace, variational_inference, sparse_variational_inference"""
if self.inference_method == 'laplace':
"""Method 1: GPy Laplace Approximation with Poisson Likelihood.
Evaluate the posterior by applying Newton's method to find the
mode (not always optimal!), followed by a second order Taylor
expansion around the mode."""
"""Specify kernel options here. """
if self.custom_kernel is not None:
"""For unusual kernels, i.e. sum, products
of periodic, linear etc, specify it within custom_kernel."""
self.kernel = self.custom_kernel
print("Using custom kernel:")
print(self.kernel)
else:
"""Standard RBF kernel and a bias.
Use RBF for smooth latent functions.
Bias for offsetting default mean at 1"""
self.kernel = GPy.kern.RBF(
input_dim=1,
variance=self.prior_signal_variance,
lengthscale=self.prior_lengthscale
)
"""Optional: specifying prior avoids numerical overflow,
due to kernel not being PSD, hence cholesky decomp doesn't
work"""
self.kernel.set_prior(GPy.priors.Exponential(1), warning=False)
self.kernel += GPy.kern.Bias(1)
print("Using standard kernel")
print(self.kernel)
"""Called once only."""
laplace_inf = GPy.inference.latent_function_inference.Laplace()
"""Specify Poisson likelihood for cox processes."""
self.likelihood = GPy.likelihoods.Poisson()
"""Choose mean function \mu(x;\theta)
Note: Mean function does not work for non-Gaussian likelihood.
Alternatively, choose a Bias kernel to offset mean at 1 or 0."""
self.mean_function = None
"""Build the GP model via the above setting"""
self.m = GPy.core.GP(
x, y,
kernel=self.kernel,
inference_method=laplace_inf,
likelihood=self.likelihood,
mean_function=self.mean_function
)
self.m.optimize()
"""Compute p(y), the objective function of the model being optimised"""
evidence = self.m.log_likelihood()
elif self.inference_method == 'variational_inference':
"""Method 2: GPFlow Variational Inference for posterior.
Minimize the KL-divergence between the the true and approximated
posterior. This is equivalent to maximising the Evidence Lower
Bound (ELBO), because not all log terms in the posterior are
tractable. This implementation is equivalent to svgp with X=Z,
but is more efficient."""
"""Compile and build the model later. Order of this matters because
certain GP settings cannot be changed in GPFlow otherwise."""
with gpflow.defer_build():
if self.custom_kernel is not None:
self.kernel = self.custom_kernel
print("Using custom kernel")
print(self.kernel)
else:
self.kernel = gpflow.kernels.RBF(
input_dim=1,
variance=self.prior_signal_variance,
lengthscales=self.prior_lengthscale
)
"""BUGS:
https://github.com/tensorflow/probability/issues/195
https://github.com/GPflow/GPflow/issues/78
Numerical overflow sometimes when you don't specify prior
Possible solution for kernel not positive semi definite
hence not invertible:
1) specify hyperparameter prior
2) add jitter or white kernel with small variance
3) logistic transform"""
self.kernel.variance.prior = gpflow.priors.Exponential(1)
self.kernel.lengthscales.prior = gpflow.priors.Exponential(
1)
"""Add Bias to offset default mean at 1"""
self.kernel += gpflow.kernels.Bias(input_dim=1)
print("Using standard kernel")
print(self.kernel)
"""Specify Poisson likelihood for cox processes.
GPFlow: For use in a Log Gaussian Cox process (doubly stochastic
model) where the rate function of an inhomogeneous Poisson
process is given by a GP. The intractable likelihood can be
approximated by gridding the space (into bins of size 'binsize')
and using this Poisson likelihood."""
self.likelihood = gpflow.likelihoods.Poisson()
"""Choose mean function \mu(x;\theta)
Note: Mean function does not work for non-Gaussian likelihood.
Alternatively, choose a Bias kernel to offset default mean
at 0 or 1."""
self.mean_function = None
"""Build the GP model using the above settings"""
self.m = gpflow.models.VGP(
X=x, Y=y,
kern=self.kernel,
likelihood=self.likelihood,
mean_function=self.mean_function
)
"""Compile and optimise"""
self.m.compile()
gpflow.train.ScipyOptimizer().minimize(self.m)
"""Compute p(y), the objective function of the model being optimised"""
evidence = self.m.compute_log_likelihood()
elif self.inference_method == 'sparse_variational_inference':
"""Method 3: GPFlow: Sparse Variational Inference
Choose inducing points Z, |Z| = M s.t. M << N
so the time complexity reduces to O(NM^2) instead of O(N^3)
and space from O(N^2) to O(NM)"""
"""Compile and build the model later. Order of this matters because
certain GP settings cannot be changed in GPFlow otherwise."""
with gpflow.defer_build():
if self.custom_kernel is not None:
self.kernel = self.custom_kernel
print("Using custom kernel")
print(self.kernel)
else:
self.kernel = gpflow.kernels.RBF(
input_dim=1,
variance=self.prior_signal_variance,
lengthscales=self.prior_lengthscale
)
"""Optional: specifying prior avoids numerical overflow,
due to kernel not being PSD, hence cholesky decomp doesn't work"""
self.kernel.variance.prior = gpflow.priors.Exponential(1)
self.kernel.lengthscales.prior = gpflow.priors.Exponential(
1)
"""Add Bias to offset default mean at 1"""
self.kernel += gpflow.kernels.Bias(input_dim=1)
print("Using standard kernel")
print(self.kernel)
self.likelihood = gpflow.likelihoods.Poisson()
"""Build the GP model using the above settings"""
self.m = gpflow.models.SVGP(
X=x, Y=y,
kern=self.kernel,
likelihood=self.likelihood,
Z=x
)
"""Compile and optimise GP model"""
self.m.compile()
gpflow.train.ScipyOptimizer().minimize(self.m)
"""Compute p(y), the objective function of the model being optimised"""
evidence = self.m.compute_log_likelihood()
"""Evidence is a singleton array and therefore use the flattened
version by selecting its first element."""
self.model_log_evidence = (np.log(model_prior) + evidence)
"""Get the hazard/probability of CP using *cp_model* as passed
to the probability model from the containing Detector object. Obtain
the *joint_probabilities*. Use the 'pmf' function of the CP model only
at time 0. This 'pmf' function is only used this once and NOWHERE ELSE,
since the hazard function (i.e., the conditional probability of
r_t = r| r_{t-1} = r' can be used from here on"""
"""Ensure that we do not get np.log(0)=np.inf by perturbation"""
if cp_model.pmf_0(1) == 0:
epsilon = 0.005
else:
epsilon = 0
"""Get log-probs for r_1=0 or r_1>0. Typically, we assume that the
first observation corresponds to a CP (i.e. P(r_1 = 0) = 1),
but this need not be the case in general."""
r_equal_0 = (self.model_log_evidence +
np.log(cp_model.pmf_0(0) + epsilon))
r_larger_0 = (self.model_log_evidence +
np.log(cp_model.pmf_0(1) + epsilon))
self.joint_log_probabilities = np.array([r_equal_0, r_larger_0])
"""Update the sufficient statistics.
Note: Following the same naming convention as the PG and MDGP model"""
self.retained_run_lengths = np.array([0, 0])
self.sums = np.array([y, y])
def evaluate_predictive_log_distribution(self, y, t):
"""Returns the log densities of *y* using the predictive posteriors
for all possible run-lengths r=0,1,...,t. Posterior is approximated
depending on the choice of inference method.
The corresponding density is computed for all run-lengths and
returned in a np array"""
"""Handle zero values for GPy error by adding a small value epsilon
DEBUG: possibly find an alternative way to handle zero values"""
if y == 0:
epsilon = 0.01
y = y + epsilon
y = y.reshape(self.S1, self.S2)
"""Method 1:
Evaluate the predictive posteriors of *y* for all possible run-lengths"""
factors = (self.retained_run_lengths + 1)
factors = factors[:, np.newaxis]
factors = factors.astype(np.float64)
sums = np.full((factors.size, 1), y).astype(np.float64)
"""Compute log predictive density
p(y_{*}|D) = p(y_{*}|f_{*})p(f_{*}|\mu_{*}\\sigma^{2}_{*})
Following the notation of Rasmussen & Williams (2006)"""
if self.inference_method == 'laplace':
pred_log_dist = self.m.log_predictive_density(factors, sums)
elif self.inference_method == 'variational_inference':
pred_log_dist = self.m.predict_density(factors, sums)
elif self.inference_method == 'sparse_variational_inference':
pred_log_dist = self.m.predict_density(factors, sums)
"""Evaluate predictive distribution as usual using the GP.
Flip the array because run-lengths are stored in the opposite direction"""
pred_log_dist = np.flip(pred_log_dist)
"""Reshape to [t,] size for every time t"""
return pred_log_dist[:, 0]
def evaluate_log_prior_predictive(self, y, t):
"""Returns the prior log density of the predictive distribution
for all possible run-lengths.
This is the case when no other data has been processed.
"""
"""Handle zero values for GPy error by adding a small value epsilon
DEBUG: possibly find an alternative way to handle zero values"""
if y == 0:
epsilon = 0.01
y = y + epsilon
"""First, reshape the data as always."""
y = y.reshape(self.S1, self.S2)
"""Evaluate pred distr at the most recent point of
the Gaussian Process at time t."""
factors = (self.retained_run_lengths + 1)
factors = factors[-1:, np.newaxis]
factors = factors.astype(np.float64)
"""Compute log predictive density
p(y_{*}|D) = p(y_{*}|f_{*})p(f_{*}|\mu_{*}\\sigma^{2}_{*})
Following the notation of Rasmussen & Williams (2006)"""
if self.inference_method == 'laplace':
pred_log_prior_dist = self.m.log_predictive_density(factors, y)
elif self.inference_method == 'variational_inference':
pred_log_prior_dist = self.m.predict_density(factors, y)
elif self.inference_method == 'sparse_variational_inference':
pred_log_prior_dist = self.m.predict_density(factors, y)
"""Reshape to [1,] size for every time t"""
return pred_log_prior_dist[:, 0]
def update_predictive_distributions(self, y, t, r_evaluations=None):
"""Takes the next observation, *y*, at time *t* and updates the
the lgcp by refitting it and optimising it from its previous state.
"""
"""Handle zero values for GPy error by adding a small value epsilon
DEBUG: possibly find an alternative way to handle zero values"""
if y == 0:
epsilon = 0.01
y = y + epsilon
y = y.reshape(self.S1, self.S2)
"""Inserts *y* at the most recent point of the GP."""
self.sums = np.insert(self.sums, self.sums.size, y, axis=0)
"""Increment run lengths by one, then insert zero at the start"""
self.retained_run_lengths = self.retained_run_lengths + 1
self.retained_run_lengths = np.insert(self.retained_run_lengths, 0, 0)
"""Update observations in the Gaussian Process"""
"""Note:
- Bug in VGP (https://github.com/GPflow/GPflow/issues/809)
Cannot update X and Y, must rebuild from scratch.
- Use self.count for batch optimisation."""
if self.inference_method == 'laplace':
x = self.retained_run_lengths[:, np.newaxis].astype(np.float64)
y = self.sums[:, :, 0]
self.m.set_XY(x, y)
self.m.optimize(max_iters=1000)
elif self.inference_method == 'variational_inference':
self.count = self.count + 1
"""Optimise model for every factor of the refresh_rate"""
if (self.count % self.refresh_rate == 0):
with gpflow.defer_build():
self.m = gpflow.models.VGP(
X=self.retained_run_lengths[:, np.newaxis].astype(np.float64),
Y=self.sums[:, :, 0],
kern=self.kernel,
likelihood=self.likelihood,
mean_function=self.mean_function
)
"""Compile and optimise"""
self.m.compile()
gpflow.train.ScipyOptimizer().minimize(self.m)
elif self.inference_method == 'sparse_variational_inference':
self.count = self.count + 1
"""Optimise model for every factor of the refresh_rate"""
if (self.count % self.refresh_rate == 0):
"""Method 1:Choose Z inducing points uniformly with length M."""
M = self.M
if int(len(np.array(self.retained_run_lengths))) < M:
"""If population size is small, then just take Z=X"""
Z = self.retained_run_lengths
else:
Z = np.round(np.linspace(
0, self.retained_run_lengths[-1], M))
"""Method 2: Randomly choose inducing points with length M = percent * N"""
# percent = 0.2
# M = int(len(np.array(self.retained_run_lengths)) * percent)
# if M == 0:
# """If population size is small, then just take Z=X"""
# Z = self.retained_run_lengths
# else:
# """When population size grows, sample the population times by the precent"""
# Z = np.array(self.retained_run_lengths)
# Z_index = np.random.choice(
# len(np.array(self.retained_run_lengths)),
# size=M,
# replace=False
# )
# Z = Z[Z_index]
"""Method 3: Naively set X=Z, hence M=N and O(NM^2) is really O(N^3)"""
# Z = self.retained_run_lengths
with gpflow.defer_build():
self.m = gpflow.models.SVGP(
X=self.retained_run_lengths[:, np.newaxis].astype(np.float64),
Y=self.sums[:, :, 0],
kern=self.kernel,
likelihood=self.likelihood,
Z=Z[:, np.newaxis]
)
"""Compile and optimise"""
self.m.compile()
gpflow.train.ScipyOptimizer().minimize(self.m)
def get_posterior_expectation(self, t, r_list=None):
"""get the predicted value/expectation from the current posteriors
at time point t, for all possible run-lengths"""
if self.inference_method == 'laplace':
post_mean, post_var = self.m._raw_predict(
self.retained_run_lengths[:, np.newaxis])
elif self.inference_method == 'variational_inference':
post_mean, post_var = self.m.predict_y(
self.retained_run_lengths[:, np.newaxis])
elif self.inference_method == 'sparse_variational_inference':
post_mean, post_var = self.m.predict_y(
self.retained_run_lengths[:, np.newaxis])
return post_mean[:, np.newaxis]
def get_posterior_variance(self, t, r_list=None):
"""get the predicted variance from the current posteriors at
time point t, for all possible run-lengths.
"""
if self.inference_method == 'laplace':
post_mean, post_var = self.m._raw_predict(
self.retained_run_lengths[:, np.newaxis])
elif self.inference_method == 'variational_inference':
post_mean, post_var = self.m.predict_y(
self.retained_run_lengths[:, np.newaxis])
elif self.inference_method == 'sparse_variational_inference':
post_mean, post_var = self.m.predict_y(
self.retained_run_lengths[:, np.newaxis])
return post_var[:, np.newaxis]
def prior_update(self, t, r_list=None):
"""Update the prior variance and lengthscale by setting the optimised
kernel hyperparameters to be the prior for the next update."""
if self.inference_method == 'laplace':
"""Skip. No need to update signal variance and lengthscaleself.
This is already implemented sequentially"""
pass
elif self.inference_method == 'variational_inference':
if self.custom_kernel is None:
self.prior_signal_variance = float(
self.m.kern.kernels[0].variance.value)
self.prior_lengthscale = float(
self.m.kern.kernels[0].lengthscales.value)
elif self.inference_method == 'sparse_variational_inference':
if self.custom_kernel is None:
self.prior_signal_variance = float(
self.m.kern.kernels[0].variance.value)
self.prior_lengthscale = float(
self.m.kern.kernels[0].lengthscales.value)
def prior_log_density(self, y):
"""Unused. Computes the log-density of *y* under the prior."""
if self.inference_method == 'laplace':
"""Note: Need to place a prior over the variance for this to work"""
log_prior = self.m.log_prior()
elif self.inference_method == 'variational_inference':
log_prior = self.m.compute_log_prior()
elif self.inference_method == 'sparse_variational_inference':
log_prior = self.m.compute_log_prior()
return log_prior
def trimmer(self, kept_run_lengths):
"""Trim the relevant quantities"""
self.joint_log_probabilities = self.joint_log_probabilities[kept_run_lengths]
"""Flip the kept_run_lengths"""
self.sums = self.sums[np.flip(kept_run_lengths)]
self.retained_run_lengths = self.retained_run_lengths[np.flip(
kept_run_lengths)]
"""Update p(y), the objective function of the model being optimised"""
if self.inference_method == 'laplace':
self.model_log_evidence = self.m.log_likelihood()
elif self.inference_method == 'variational_inference':
self.model_log_evidence = self.m.compute_log_likelihood()
elif self.inference_method == 'sparse_variational_inference':
self.model_log_evidence = self.m.compute_log_likelihood()