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likelihood_utils.py
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likelihood_utils.py
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import sys
import numpy as np
from scipy.stats import expon, bernoulli, binom
from hawkes_uncertain_simulator import HawkesUncertainModel
def hawkes_log_likelihood_numpy(event_times, intensity, alpha, beta, end_time=None):
"""
Returns Hawkes log likelihood.
:param event_times: (float) list of all event times
:param intensity: (float) intensity of the Hawkes process
:param alpha: (float) alpha of the Hawkes process
:param beta: (float) beta of the Hawkes process
:param end_time: optional (float) end time of the observed interval. Default is the last event time.
"""
if end_time is None and len(event_times) == 0:
sys.exit("Error evaluating Hawkes log-likelihood: end_time must be defined if there is no observed event.")
if end_time is not None and len(event_times) > 0 and end_time < event_times[-1]:
sys.exit("Error evaluating Hawkes log-likelihood: end_time must be >= to the last event time.")
end_time = end_time if end_time is not None else event_times[-1]
num_events = len(event_times)
term1 = 0
if num_events > 0:
a_calc = np.zeros(num_events)
for i in range(1, num_events):
a_calc[i] = np.exp(-1 * beta * (event_times[i] - event_times[i - 1])) * (1 + a_calc[i - 1])
term1 = np.sum(np.log(intensity + alpha * a_calc))
term2 = intensity * end_time
term3 = (alpha / beta) * np.sum(np.exp(-1 * beta * (end_time - event_times))) - num_events
res = term1 - term2 + term3
return res
def poisson_log_likelihood_numpy(num_events, intensity, end_time):
"""
Returns Poisson process log likelihood.
Based on https://stats.stackexchange.com/questions/360814/mle-for-a-homogeneous-poisson-process and
https://math.stackexchange.com/questions/344487/log-likelihood-of-a-realization-of-a-poisson-process
:param num_events: (int) number of event times.
:param intensity: (float) rate/intensity of the Poisson process.
:param end_time: (float) end time of the observed interval. If the last poisson event time < end of the observed
interval, end_time must be set to end of the observed interval.
:return:
"""
return num_events * np.log(intensity) - intensity * end_time
def z_i_posterior_prob(event_time, event_mark, event_times_hist, z_hist, z_prior,
hawkes_params, poisson_lambda,
hawkes_mark_exp_rate, noise_mark_exp_rate, return_prob_z_i_noise=True):
"""
Returns p(z_i=1 | T_1:i, y_i, Z_1:i-1) if `return_prob_z_i_noise`=True, else p(z_i=0 | T_1:i, y_i, Z_1:i-1)
:param event_time: (float) time of the new event i
:param event_mark: (float) mark of the new event i
:param event_times_hist: list of event times from t_0 to t_i-1
:param z_hist: list of booleans to identify event_times_hist events as Hawkes (0/false) or noise/poisson (True)
:param z_prior: prior probability of latent variable Z=1 (prior probability of noise)
:param hawkes_params: a tuple of hawkes parameters (lambda, alpha, beta)
:param poisson_lambda: lambda parameter of the poisson process (noise)
:param hawkes_mark_exp_rate: lambda parameter of the exponential dist for the marks of the Hawkes process events.
:param noise_mark_exp_rate: lambda parameter of the exponential dist for the marks of the noise events.
:param return_prob_z_i_noise: optional (bool), if true returns the probability of z_i = 1, and z_i = 0 otherwise
"""
# noise/hawkes at the end of each variable indicates whether z_i was assumed to be 1 or 0.
z_not_hist = np.logical_not(z_hist)
mark_prob_noise = expon.pdf(event_mark, scale=1./noise_mark_exp_rate)
mark_prob_hawkes = expon.pdf(event_mark, scale=1./hawkes_mark_exp_rate)
hawkes_intensity, hawkes_alpha, hawkes_beta = hawkes_params
hawkes_prob_noise = np.exp(hawkes_log_likelihood_numpy(event_times_hist[z_not_hist],
hawkes_intensity, hawkes_alpha, hawkes_beta, event_time))
hawkes_prob_hawkes = np.exp(hawkes_log_likelihood_numpy(np.append(event_times_hist[z_not_hist], event_time),
hawkes_intensity, hawkes_alpha, hawkes_beta))
poisson_prob_noise = np.exp(poisson_log_likelihood_numpy(np.sum(z_hist) + 1, poisson_lambda, event_time))
poisson_prob_hawkes = np.exp(poisson_log_likelihood_numpy(np.sum(z_hist), poisson_lambda, event_time))
# poisson_prob_noise = 1
# poisson_prob_hawkes = 1
numerator = z_prior * mark_prob_noise * hawkes_prob_noise * poisson_prob_noise
normalizer = ((1 - z_prior) * mark_prob_hawkes * hawkes_prob_hawkes * poisson_prob_hawkes) + numerator
z_i_noise_prob = numerator / normalizer
if return_prob_z_i_noise:
return z_i_noise_prob
return 1 - z_i_noise_prob
def z_i_posterior_log_prob(event_time, event_mark, event_times_hist, z_hist, z_prior,
hawkes_params, poisson_lambda,
hawkes_mark_exp_rate, noise_mark_exp_rate, return_prob_z_i_noise=True):
"""
Returns ln p(z_i=1 | T_1:i, y_i, Z_1:i-1) if `return_prob_z_i_noise`=True, else ln p(z_i=0 | T_1:i, y_i, Z_1:i-1)
Check out z_i_posterior_prob doc.
"""
return np.log(z_i_posterior_prob(event_time, event_mark, event_times_hist, z_hist, z_prior,
hawkes_params, poisson_lambda,
hawkes_mark_exp_rate, noise_mark_exp_rate, return_prob_z_i_noise))
def z_posterior_prob(z, event_times, event_marks, z_prior,
hawkes_params, poisson_lambda,
hawkes_mark_exp_rate, noise_mark_exp_rate):
"""
Returns product of p(z_i=1 | T_1:i, y_i, Z_1:i-1) for i from 0 to len(event_times)
:param z: list of boolean. The list of all z_i's. True is noise, False is Hawkes.
:param event_times: list of all event times
:param event_marks: list of all event marks
:param z_prior: prior probability of latent variable Z=1 (prior probability of noise)
:param hawkes_params: a tuple of hawkes parameters (lambda, alpha, beta)
:param poisson_lambda: lambda parameter of the poisson process (noise)
:param hawkes_mark_exp_rate: lambda parameter of the exponential dist for the marks of the Hawkes process events.
:param noise_mark_exp_rate: lambda parameter of the exponential dist for the marks of the noise events.
"""
z_i_probs = np.zeros(len(event_times))
for i in range(0, len(event_times)):
z_i_probs[i] = z_i_posterior_prob(event_times[i], event_marks[i], event_times[:i], z[:i], z_prior,
hawkes_params, poisson_lambda, hawkes_mark_exp_rate, noise_mark_exp_rate,
return_prob_z_i_noise=False)
print(i, event_times[i], z[i], z_i_probs[i])
return np.prod(z_i_probs)
def z_posterior_log_prob(z, event_times, event_marks, z_prior,
hawkes_params, poisson_lambda,
hawkes_mark_exp_rate, noise_mark_exp_rate):
"""
Returns sum of ln p(z_i=1 | T_1:i, y_i, Z_1:i-1) for i from 0 to len(event_times)
:param z: list of boolean. The list of all z_i's. True is noise, False is Hawkes.
:param event_times: list of all event times
:param event_marks: list of all event marks
:param z_prior: prior probability of latent variable Z=1 (prior probability of noise)
:param hawkes_params: a tuple of hawkes parameters (lambda, alpha, beta)
:param poisson_lambda: lambda parameter of the poisson process (noise)
:param hawkes_mark_exp_rate: lambda parameter of the exponential dist for the marks of the Hawkes process events.
:param noise_mark_exp_rate: lambda parameter of the exponential dist for the marks of the noise events.
"""
z_i_probs = np.zeros(len(event_times))
for i in range(0, len(event_times)):
z_i_probs[i] = z_i_posterior_log_prob(event_times[i], event_marks[i], event_times[:i], z[:i], z_prior,
hawkes_params, poisson_lambda, hawkes_mark_exp_rate, noise_mark_exp_rate,
return_prob_z_i_noise=z[i] == 1)
return np.sum(z_i_probs)
def complete_data_log_likelihood(z, event_times, event_marks, z_prior,
hawkes_params, poisson_lambda,
hawkes_mark_exp_rate, noise_mark_exp_rate):
"""
Returns the complete data log-likelihood for univariate uncertain Hawkes model as derived in the likelihood doc.
ln p(Z/binom) + ln p(mark/exp) + ln prob hawkes + ln prob poisson.
:param z: list of boolean. The list of all z_i's. True is noise, False is Hawkes.
:param event_times: list of all event times
:param event_marks: list of all event marks
:param z_prior: prior probability of latent variable Z=1 (prior probability of noise)
:param hawkes_params: a tuple of hawkes parameters (lambda, alpha, beta)
:param poisson_lambda: lambda parameter of the poisson process (noise)
:param hawkes_mark_exp_rate: lambda parameter of the exponential dist for the marks of the Hawkes process events.
:param noise_mark_exp_rate: lambda parameter of the exponential dist for the marks of the noise events.
"""
z_poisson = z
z_hawkes = np.logical_not(z)
z_log_prob = binom.logpmf(np.sum(z_poisson), len(z_poisson), z_prior)
mark_hawkes_log_prob = np.sum(expon.logpdf(event_marks[z_hawkes], scale=1./hawkes_mark_exp_rate))
mark_noise_log_prob = np.sum(expon.logpdf(event_marks[z_poisson], scale=1./noise_mark_exp_rate))
hawkes_log_prob = hawkes_log_likelihood_numpy(event_times[z_hawkes],
hawkes_params[0], hawkes_params[1], hawkes_params[2],
event_times[-1])
poisson_log_prob = poisson_log_likelihood_numpy(np.sum(z_poisson), poisson_lambda, event_times[-1])
return (z_log_prob +
mark_hawkes_log_prob +
mark_noise_log_prob +
hawkes_log_prob +
poisson_log_prob)
def print_complete_data_log_likelihood(z, event_times, event_marks, z_prior,
hawkes_params, poisson_lambda,
hawkes_mark_exp_rate, noise_mark_exp_rate):
# prints the probability of the last event in a sequence of events to be Hawkes rather than Poisson.
for i in range(len(event_times)):
z_noise_temp = np.append(z[:i], True)
z_hawkes_temp = np.append(z[:i], False)
log_prob_noise = complete_data_log_likelihood(z_noise_temp, event_times[:i + 1], event_marks[:i + 1], z_prior,
hawkes_params, poisson_lambda,
hawkes_mark_exp_rate, noise_mark_exp_rate)
log_prob_hawkes = complete_data_log_likelihood(z_hawkes_temp, event_times[:i + 1], event_marks[:i + 1], z_prior,
hawkes_params, poisson_lambda,
hawkes_mark_exp_rate, noise_mark_exp_rate)
print(i, event_times[i], z[i], log_prob_hawkes / (log_prob_hawkes + log_prob_noise))
def complete_data_log_likelihood_hawkes_only(z, event_times, event_marks, z_prior,
hawkes_params,
hawkes_mark_exp_rate, noise_mark_exp_rate):
"""
Returns log prob as the sum of each individual log prob. Not normalized.
Ignore the prob of poisson.
ln p(Z/bernoulli) + ln p(mark/exp) + ln prob hawkes. This is not normalized, since the denominator is intractable.
:param z: list of boolean. The list of all z_i's. True is noise, False is Hawkes.
:param event_times: list of all event times
:param event_marks: list of all event marks
:param z_prior: prior probability of latent variable Z=1 (prior probability of noise)
:param hawkes_params: a tuple of hawkes parameters (lambda, alpha, beta)
:param hawkes_mark_exp_rate: lambda parameter of the exponential dist for the marks of the Hawkes process events.
:param noise_mark_exp_rate: lambda parameter of the exponential dist for the marks of the noise events.
"""
z_poisson = z
z_hawkes = np.logical_not(z)
z_log_prob = binom.logpmf(np.sum(z_poisson), len(z_poisson), z_prior)
mark_hawkes_log_prob = np.sum(expon.logpdf(event_marks[z_hawkes], scale=1./hawkes_mark_exp_rate))
mark_noise_log_prob = np.sum(expon.logpdf(event_marks[z_poisson], scale=1./noise_mark_exp_rate))
hawkes_log_prob = hawkes_log_likelihood_numpy(event_times[z_hawkes],
hawkes_params[0], hawkes_params[1], hawkes_params[2],
event_times[-1])
return (z_log_prob +
mark_hawkes_log_prob +
mark_noise_log_prob +
hawkes_log_prob)
def print_complete_data_log_likelihood_hawkes_only(z, event_times, event_marks, z_prior,
hawkes_params,
hawkes_mark_exp_rate, noise_mark_exp_rate):
for i in range(len(event_times)):
z_noise_temp = np.append(z[:i], True)
z_hawkes_temp = np.append(z[:i], False)
log_prob_noise = complete_data_log_likelihood_hawkes_only(z_noise_temp, event_times[:i + 1],
event_marks[:i + 1], z_prior,
hawkes_params,
hawkes_mark_exp_rate, noise_mark_exp_rate)
log_prob_hawkes = complete_data_log_likelihood_hawkes_only(z_hawkes_temp, event_times[:i + 1],
event_marks[:i + 1], z_prior,
hawkes_params,
hawkes_mark_exp_rate, noise_mark_exp_rate)
print(i, event_times[i], z[i], log_prob_hawkes / (log_prob_hawkes + log_prob_noise))
# print(i, event_times[i], z[i], log_prob_hawkes, log_prob_noise)
if __name__ == "__main__":
_h_intensity = 0.9
_h_beta = 2
_h_alpha = 1.2
_runtime = 30
_p_intensity = 0.2
_h_exp_rate = 2.5
_p_exp_rate = 1.5
hum = HawkesUncertainModel(h_lambda=_h_intensity, h_alpha=_h_alpha, h_beta=_h_beta, h_exp_rate=_h_exp_rate,
p_lambda=_p_intensity, p_exp_rate=_p_exp_rate,
noise_percentage_ub=0.5, run_time=_runtime, delta=0.01, seed=None)
# Testing out the prob of the full posterior
sim_event_times = hum.mixed_timestamps
sim_event_marks = hum.mixed_expo
sim_true_labels = hum.mixed_labels.astype(np.bool)
sim_true_z_prior = hum.noise_percentage
print(hum.noise_percentage)
hum.plot_hawkes_uncertain()
print("Type 1")
z_posterior_prob(sim_true_labels,
sim_event_times, sim_event_marks, sim_true_z_prior,
(_h_intensity, _h_alpha, _h_beta),
_p_intensity, _h_exp_rate, _p_exp_rate)
print("true labels:", z_posterior_log_prob(sim_true_labels,
sim_event_times, sim_event_marks, sim_true_z_prior,
(_h_intensity, _h_alpha, _h_beta),
_p_intensity, _h_exp_rate, _p_exp_rate))
print("all hawkes", z_posterior_log_prob(np.zeros(len(sim_true_labels)).astype(np.bool),
sim_event_times, sim_event_marks, sim_true_z_prior,
(_h_intensity, _h_alpha, _h_beta),
_p_intensity, _h_exp_rate, _p_exp_rate))
print("all noise", z_posterior_log_prob(np.ones(len(sim_true_labels)).astype(np.bool),
sim_event_times, sim_event_marks, sim_true_z_prior,
(_h_intensity, _h_alpha, _h_beta),
_p_intensity, _h_exp_rate, _p_exp_rate))
print("exact opposite", z_posterior_log_prob(np.logical_not(sim_true_labels),
sim_event_times, sim_event_marks, sim_true_z_prior,
(_h_intensity, _h_alpha, _h_beta),
_p_intensity, _h_exp_rate, _p_exp_rate))
print("Compete Data Log-likelihood")
print_complete_data_log_likelihood(sim_true_labels,
sim_event_times, sim_event_marks, sim_true_z_prior,
(_h_intensity, _h_alpha, _h_beta),
_p_intensity, _h_exp_rate, _p_exp_rate)
print("true labels:", complete_data_log_likelihood(sim_true_labels,
sim_event_times, sim_event_marks, sim_true_z_prior,
(_h_intensity, _h_alpha, _h_beta),
_p_intensity, _h_exp_rate, _p_exp_rate))
print("all hawkes", complete_data_log_likelihood(np.zeros(len(sim_true_labels)).astype(np.bool),
sim_event_times, sim_event_marks, sim_true_z_prior,
(_h_intensity, _h_alpha, _h_beta),
_p_intensity, _h_exp_rate, _p_exp_rate))
print("all noise", complete_data_log_likelihood(np.ones(len(sim_true_labels)).astype(np.bool),
sim_event_times, sim_event_marks, sim_true_z_prior,
(_h_intensity, _h_alpha, _h_beta),
_p_intensity, _h_exp_rate, _p_exp_rate))
print("exact opposite", complete_data_log_likelihood(np.logical_not(sim_true_labels),
sim_event_times, sim_event_marks, sim_true_z_prior,
(_h_intensity, _h_alpha, _h_beta),
_p_intensity, _h_exp_rate, _p_exp_rate))
print("Complete Data Log-likelihood Poisson")
print_complete_data_log_likelihood_hawkes_only(sim_true_labels,
sim_event_times, sim_event_marks, sim_true_z_prior,
(_h_intensity, _h_alpha, _h_beta),
_h_exp_rate, _p_exp_rate)
print("true labels:", complete_data_log_likelihood_hawkes_only(sim_true_labels,
sim_event_times, sim_event_marks, sim_true_z_prior,
(_h_intensity, _h_alpha, _h_beta),
_h_exp_rate, _p_exp_rate))
print("all hawkes", complete_data_log_likelihood_hawkes_only(np.zeros(len(sim_true_labels)).astype(np.bool),
sim_event_times, sim_event_marks, sim_true_z_prior,
(_h_intensity, _h_alpha, _h_beta),
_h_exp_rate, _p_exp_rate))
print("all noise", complete_data_log_likelihood_hawkes_only(np.ones(len(sim_true_labels)).astype(np.bool),
sim_event_times, sim_event_marks, sim_true_z_prior,
(_h_intensity, _h_alpha, _h_beta),
_h_exp_rate, _p_exp_rate))
print("exact opposite", complete_data_log_likelihood_hawkes_only(np.logical_not(sim_true_labels),
sim_event_times, sim_event_marks, sim_true_z_prior,
(_h_intensity, _h_alpha, _h_beta),
_h_exp_rate, _p_exp_rate))