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bocd.py
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bocd.py
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"""============================================================================
Author: Gregory Gundersen
Python implementation of Bayesian online changepoint detection for a normal
model with unknown mean parameter. For algorithm details, see
Adams & MacKay 2007
"Bayesian Online Changepoint Detection"
https://arxiv.org/abs/0710.3742
For Bayesian inference details about the Gaussian, see:
Murphy 2007
"Conjugate Bayesian analysis of the Gaussian distribution"
https://www.cs.ubc.ca/~murphyk/Papers/bayesGauss.pdf
This code is associated with the following blog posts:
http://gregorygundersen.com/blog/2019/08/13/bocd/
http://gregorygundersen.com/blog/2020/10/20/implementing-bocd/
============================================================================"""
import matplotlib.pyplot as plt
from matplotlib.colors import LogNorm
import numpy as np
from scipy.stats import norm
from scipy.special import logsumexp
# -----------------------------------------------------------------------------
def bocd(data, model, hazard):
"""Return run length posterior using Algorithm 1 in Adams & MacKay 2007.
"""
# 1. Initialize lower triangular matrix representing the posterior as
# function of time. Model parameters are initialized in the model class.
#
# When we exponentiate R at the end, exp(-inf) --> 0, which is nice for
# visualization.
#
T = len(data)
log_R = -np.inf * np.ones((T+1, T+1))
log_R[0, 0] = 0 # log 0 == 1
pmean = np.empty(T) # Model's predictive mean.
pvar = np.empty(T) # Model's predictive variance.
log_message = np.array([0]) # log 0 == 1
log_H = np.log(hazard)
log_1mH = np.log(1 - hazard)
for t in range(1, T+1):
# 2. Observe new datum.
x = data[t-1]
# Make model predictions.
pmean[t-1] = np.sum(np.exp(log_R[t-1, :t]) * model.mean_params[:t])
pvar[t-1] = np.sum(np.exp(log_R[t-1, :t]) * model.var_params[:t])
# 3. Evaluate predictive probabilities.
log_pis = model.log_pred_prob(t, x)
# 4. Calculate growth probabilities.
log_growth_probs = log_pis + log_message + log_1mH
# 5. Calculate changepoint probabilities.
log_cp_prob = logsumexp(log_pis + log_message + log_H)
# 6. Calculate evidence
new_log_joint = np.append(log_cp_prob, log_growth_probs)
# 7. Determine run length distribution.
log_R[t, :t+1] = new_log_joint
log_R[t, :t+1] -= logsumexp(new_log_joint)
# 8. Update sufficient statistics.
model.update_params(t, x)
# Pass message.
log_message = new_log_joint
R = np.exp(log_R)
return R, pmean, pvar
# -----------------------------------------------------------------------------
class GaussianUnknownMean:
def __init__(self, mean0, var0, varx):
"""Initialize model.
meanx is unknown; varx is known
p(meanx) = N(mean0, var0)
p(x) = N(meanx, varx)
"""
self.mean0 = mean0
self.var0 = var0
self.varx = varx
self.mean_params = np.array([mean0])
self.prec_params = np.array([1/var0])
def log_pred_prob(self, t, x):
"""Compute predictive probabilities \pi, i.e. the posterior predictive
for each run length hypothesis.
"""
# Posterior predictive: see eq. 40 in (Murphy 2007).
post_means = self.mean_params[:t]
post_stds = np.sqrt(self.var_params[:t])
return norm(post_means, post_stds).logpdf(x)
def update_params(self, t, x):
"""Upon observing a new datum x at time t, update all run length
hypotheses.
"""
# See eq. 19 in (Murphy 2007).
new_prec_params = self.prec_params + (1/self.varx)
self.prec_params = np.append([1/self.var0], new_prec_params)
# See eq. 24 in (Murphy 2007).
new_mean_params = (self.mean_params * self.prec_params[:-1] + \
(x / self.varx)) / new_prec_params
self.mean_params = np.append([self.mean0], new_mean_params)
@property
def var_params(self):
"""Helper function for computing the posterior variance.
"""
return 1./self.prec_params + self.varx
# -----------------------------------------------------------------------------
def generate_data(varx, mean0, var0, T, cp_prob):
"""Generate partitioned data of T observations according to constant
changepoint probability `cp_prob` with hyperpriors `mean0` and `prec0`.
"""
data = []
cps = []
meanx = mean0
for t in range(0, T):
if np.random.random() < cp_prob:
meanx = np.random.normal(mean0, var0)
cps.append(t)
data.append(np.random.normal(meanx, varx))
return data, cps
# -----------------------------------------------------------------------------
def plot_posterior(T, data, cps, R, pmean, pvar):
fig, axes = plt.subplots(2, 1, figsize=(20,10))
ax1, ax2 = axes
ax1.scatter(range(0, T), data)
ax1.plot(range(0, T), data)
ax1.set_xlim([0, T])
ax1.margins(0)
# Plot predictions.
ax1.plot(range(0, T), pmean, c='k')
_2std = 2 * np.sqrt(pvar)
ax1.plot(range(0, T), pmean - _2std, c='k', ls='--')
ax1.plot(range(0, T), pmean + _2std, c='k', ls='--')
ax2.imshow(np.rot90(R), aspect='auto', cmap='gray_r',
norm=LogNorm(vmin=0.0001, vmax=1))
ax2.set_xlim([0, T])
ax2.margins(0)
for cp in cps:
ax1.axvline(cp, c='red', ls='dotted')
ax2.axvline(cp, c='red', ls='dotted')
plt.tight_layout()
plt.show()
# -----------------------------------------------------------------------------
if __name__ == '__main__':
T = 1000 # Number of observations.
hazard = 1/100 # Constant prior on changepoint probability.
mean0 = 0 # The prior mean on the mean parameter.
var0 = 2 # The prior variance for mean parameter.
varx = 1 # The known variance of the data.
data, cps = generate_data(varx, mean0, var0, T, hazard)
model = GaussianUnknownMean(mean0, var0, varx)
R, pmean, pvar = bocd(data, model, hazard)
plot_posterior(T, data, cps, R, pmean, pvar)