!date
import numpy as np, pandas as pd, matplotlib.pyplot as plt, seaborn as sns
%matplotlib inline
sns.set_context('paper')
sns.set_style('darkgrid')
Sun Jun 7 15:42:45 PDT 2015
import pymc3 as pm, theano.tensor as tt
# simulate data from a known mixture distribution
np.random.seed(12345) # set random seed for reproducibility
k = 3
ndata = 500
spread = 3
centers = np.array([-spread, 0, spread])
# simulate data from mixture distribution
v = np.random.randint(0, k, ndata)
data = centers[v] + np.random.randn(ndata)
plt.hist(data)
(array([ 33., 61., 53., 54., 90., 54., 53., 72., 28., 2.]), array([-5.04042311, -3.9210433 , -2.80166348, -1.68228367, -0.56290385, 0.55647596, 1.67585577, 2.79523559, 3.9146154 , 5.03399522, 6.15337503]), <a list of 10 Patch objects>)
# setup model
model = pm.Model()
with model:
# cluster sizes
a = pm.constant(np.array([1., 1., 1.]))
p = pm.Dirichlet('p', a=a, shape=k)
# ensure all clusters have some points
p_min_potential = pm.Potential('p_min_potential', tt.switch(tt.min(p) < .1, -np.inf, 0))
# cluster centers
means = pm.Normal('means', mu=[0, 0, 0], sd=15, shape=k)
# break symmetry
order_means_potential = pm.Potential('order_means_potential',
tt.switch(means[1]-means[0] < 0, -np.inf, 0)
+ tt.switch(means[2]-means[1] < 0, -np.inf, 0))
# measurement error
sd = pm.Uniform('sd', lower=0, upper=20)
# latent cluster of each observation
category = pm.Categorical('category',
p=p,
shape=ndata)
# likelihood for each observed value
points = pm.Normal('obs',
mu=means[category],
sd=sd,
observed=data)
Cannot compute test value: input 0 (<TensorType(float32, matrix)>) of Op Dot22(<TensorType(float32, matrix)>, <TensorType(float32, matrix)>) missing default value
# make some step-methods particularly suited to the category stochastic
class RandomScanDiscreteMetropolis(pm.step_methods.arraystep.ArrayStep):
def __init__(self, var, model=None, values=[0,1]):
model = pm.modelcontext(model)
self.values = values
super(RandomScanDiscreteMetropolis, self).__init__([var], [model.fastlogp])
def astep(self, q0, logp):
i = np.random.choice(len(q0))
q = np.copy(q0)
q[i] = np.random.choice(self.values)
q_new = pm.step_methods.arraystep.metrop_select(logp(q) - logp(q0), q, q0)
return q_new
class SequentialScanDiscreteMetropolis(pm.step_methods.arraystep.ArrayStep):
def __init__(self, var, model=None, values=[0,1]):
model = pm.modelcontext(model)
self.values = values
self.i = 0
super(SequentialScanDiscreteMetropolis, self).__init__([var], [model.fastlogp])
def astep(self, q0, logp):
q = np.copy(q0)
q[self.i] = np.random.choice(self.values)
self.i = (self.i + 1) % len(q)
q_new = pm.step_methods.arraystep.metrop_select(logp(q) - logp(q0), q, q0)
return q_new
# fit model
with model:
step1 = pm.Metropolis(vars=[p])
step2 = pm.Metropolis(vars=[sd, means])
step3 = SequentialScanDiscreteMetropolis(var=category, values=[0,1,2])
tr = pm.sample(10000, step=[step1, step2] + [step3]*ndata)
[-----------------100%-----------------] 10000 of 10000 complete in 1327.3 sec
# take a look at traceplot for some model parameters
# (with some burn-in and thinning)
pm.plots.traceplot(tr[5000::5], ['p', 'sd', 'means']);
# I prefer autocorrelation plots for serious confirmation of MCMC convergence
pm.autocorrplot(tr[5000::5], ['sd'])
(<matplotlib.figure.Figure at 0x7f897a686150>, array([[<matplotlib.axes._subplots.AxesSubplot object at 0x7f8961e70250>]], dtype=object))
# this currently does not work for vector-valued stochastics :(
# pm.autocorrplot(tr[5000::5], ['sd'])
# the trace of the categorical stochastics is hard to see on the traceplot
# but this is what a single coordinate looks like
i=0
plt.plot(tr['category'][5000::5,i], drawstyle='steps-mid')
plt.axis(ymin=-.1, ymax=2.1)
(0.0, 1000.0, -0.1, 2.1)
def cluster_posterior(i=0):
print 'true cluster:', v[i]
print ' data value:', np.round(data[i],2)
plt.hist(tr['category'][5000::5,i], bins=[-.5,.5,1.5,2.5,], rwidth=.9)
plt.axis(xmin=-.5, xmax=2.5)
plt.xticks([0,1,2])
cluster_posterior(i)
true cluster: 2 data value: 1.29
from IPython.html import widgets
widgets.interact(cluster_posterior, i=[0,ndata])
true cluster: 2 data value: 1.48