转自 http://www.cnblogs.com/yymn/p/4716107.html
如果我手里只有一把锤子，我就会到处找钉子。
pip install emcee
np_test1.py
import matplotlib.pyplot as plt
import numpy as np
np.random.seed(42)
theta_true = (25, 0.5)
xdata = 100 * np.random.random(20)
ydata = theta_true[0] + theta_true[1] * xdata
# add scatter to points
xdata = np.random.normal(xdata, 10)
ydata = np.random.normal(ydata, 10)
plt.plot(xdata, ydata, 'ok')
plt.xlabel('x')
plt.ylabel('y')
plt.show()
import matplotlib.pyplot as plt
import numpy as np
fig, ax = plt.subplots(subplot_kw=dict(aspect='equal'))
x = np.linspace(-1, 1)
for slope in np.arange(0, 10, 0.1):
plt.plot(x, slope * x, '-k')
ax.axis([-1, 1, -1, 1], aspect='equal')
plt.show()
# -*- coding: cp936 -*-
import matplotlib.pyplot as plt
import numpy as np
import emcee
np.random.seed(42)
theta_true = (25, 0.5)
xdata = 100 * np.random.random(20)
ydata = theta_true[0] + theta_true[1] * xdata
# add scatter to points
xdata = np.random.normal(xdata, 10)
ydata = np.random.normal(ydata, 10)
# Create some convenience routines for plotting
# 让我们做一些辅助工作来可视化数据
def compute_sigma_level(trace1, trace2, nbins=20):
"""From a set of traces, bin by number of standard deviations"""
L, xbins, ybins = np.histogram2d(trace1, trace2, nbins)
L[L == 0] = 1E-16
logL = np.log(L)
shape = L.shape
L = L.ravel()
# obtain the indices to sort and unsort the flattened array
i_sort = np.argsort(L)[::-1]
i_unsort = np.argsort(i_sort)
L_cumsum = L[i_sort].cumsum()
L_cumsum /= L_cumsum[-1]
xbins = 0.5 * (xbins[1:] + xbins[:-1])
ybins = 0.5 * (ybins[1:] + ybins[:-1])
return xbins, ybins, L_cumsum[i_unsort].reshape(shape)
def plot_MCMC_trace(ax, xdata, ydata, trace, scatter=False, **kwargs):
"""Plot traces and contours"""
xbins, ybins, sigma = compute_sigma_level(trace[0], trace[1])
ax.contour(xbins, ybins, sigma.T, levels=[0.683, 0.955], **kwargs)
if scatter:
ax.plot(trace[0], trace[1], ',k', alpha=0.1)
ax.set_xlabel(r'$\alpha$')
ax.set_ylabel(r'$\beta$')
def plot_MCMC_model(ax, xdata, ydata, trace):
"""Plot the linear model and 2sigma contours"""
ax.plot(xdata, ydata, 'ok')
alpha, beta = trace[:2]
xfit = np.linspace(-20, 120, 10)
yfit = alpha[:, None] + beta[:, None] * xfit
mu = yfit.mean(0)
sig = 2 * yfit.std(0)
ax.plot(xfit, mu, '-k')
ax.fill_between(xfit, mu - sig, mu + sig, color='lightgray')
ax.set_xlabel('x')
ax.set_ylabel('y')
def plot_MCMC_results(xdata, ydata, trace, colors='k'):
"""Plot both the trace and the model together"""
fig, ax = plt.subplots(1, 2, figsize=(10, 4))
plot_MCMC_trace(ax[0], xdata, ydata, trace, True, colors=colors)
plot_MCMC_model(ax[1], xdata, ydata, trace)
# Define our posterior using Python functions
# for clarity, I've separated-out the prior and likelihood
# but this is not necessary. Note that emcee requires log-posterior
# 用Python函数定义后验，我把先验和似然估计分开写了，其实没必要，主要是显得更简洁。
# 注意emcee需要对数后验证
def log_prior(theta):
alpha, beta, sigma = theta
if sigma < 0:
return -np.inf # log(0)
else:
return -1.5 * np.log(1 + beta ** 2) - np.log(sigma)
def log_likelihood(theta, x, y):
alpha, beta, sigma = theta
y_model = alpha + beta * x
return -0.5 * np.sum(np.log(2 * np.pi * sigma ** 2) + (y - y_model) ** 2 / sigma ** 2)
def log_posterior(theta, x, y):
return log_prior(theta) + log_likelihood(theta, x, y)
# Here we'll set up the computation. emcee combines multiple "walkers",
# each of which is its own MCMC chain. The number of trace results will
# be nwalkers * nsteps
# 设置计算参数。emcee组合了多个"walkers"，每个都有自己的MCMC链。
# 跟踪结果的数量为 nwalkers * nsteps
ndim = 3 # number of parameters in the model
nwalkers = 50 # number of MCMC walkers
nburn = 1000 # "burn-in" period to let chains stabilize
nsteps = 2000 # number of MCMC steps to take
# set theta near the maximum likelihood, with
np.random.seed(0)
starting_guesses = np.random.random((nwalkers, ndim))
# Here's the function call where all the work happens:
# we'll time it using IPython's %time magic
# 这就是所有调用内容。
# 这里用IPythn的%time方法计时。
sampler = emcee.EnsembleSampler(nwalkers, ndim, log_posterior, args=[xdata, ydata])
sampler.run_mcmc(starting_guesses, nsteps)
print("done")
# sampler.chain is of shape (nwalkers, nsteps, ndim)
# we'll throw-out the burn-in points and reshape:
# sampler.chain返回数组维度为(nwalkers, nsteps, ndim)
sampler.chain
emcee_trace = sampler.chain[:, nburn:, :].reshape(-1, ndim).T
plot_MCMC_results(xdata, ydata, emcee_trace)
plt.show()