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# LIBRARY IMPORT
from numpy import *
from random import *
import pylab as pylab
from math import *
#
# BRIDGING DENSITY starting from the flat density
#
def density_bridge(x,y,center_coord, std, k,N):
if k == 0:
return 1.0
return pow(density(x,y, center_coord, std), k/N)
#
# mixture of Gaussian: center_coord = centres of the Gaussians
#
def density(x,y, center_coord, std):
N = len(center_coord)
likelihood = 0
for c_x,c_y in center_coord:
likelihood += exp(-(pow(x-c_x,2) + pow(y-c_y,2))/(2*std*std))
return likelihood
#
# 2-dimensional random walk proposals
#
def proposal(x,y,delta):
return (gauss(x,delta), gauss(y,delta))
#
# compute Effective Sampling Size (ESS)
#
def compute_ESS(weights):
ess = sum([w*w for w in weights])
return 1.0/ess
def resampling(particles, weights):
# check that the weights are normalized
if (sum(weights)<0.999) or (sum(weights)>1.001):
print "ERROR: !! weights are not normalized !!"
N = len(particles); particles_new = range(N)
w_sum = weights[0]; current_index = 0; current = random()/N
for k in range(N):
while current > w_sum:
current_index += 1; w_sum += weights[current_index]
particles_new[k] = particles[current_index]
current += 1.0/N
return particles_new
#
# SMC sampler
#
def smc(particles, delta, N_temperature, center_coord, std, save_file = False):
weights = [1.0 for p in particles]
N_particles = len(particles)
for t in arange(1,N_temperature+1):
print "iteration ",t, " out of ", N_temperature
# evolve each particle with pi_t as target
for k in range(N_particles):
x,y = particles[k]
x_new, y_new = proposal(x,y,delta)
#compute log acceptance (for more stability)
L_accept = log(density(x_new,y_new, center_coord, std)) - log(density(x,y, center_coord, std))
if log(random()) < L_accept:
particles[k] = (x_new, y_new)
L_W = log(density(x,y, center_coord, std))
L_W = L_W/N_temperature
weights[k] = weights[k] * exp(L_W)
# renormalise weights
w_sum = sum(weights); weights = [w/w_sum for w in weights]
ESS = compute_ESS(weights)
print " ESS=", ESS
# resampling is ess < N/2
if ESS < N_particles/2:
particles = resampling(particles, weights)
weights = [1.0 for w in weights]
if save_file:
index = t
name = "smc_sampler_" + str(1000+index)
pylab.clf()
radius = 30
frame1 = pylab.gca()
frame1.axes.get_xaxis().set_visible(False)
frame1.axes.get_yaxis().set_visible(False)
pylab.axis([-radius,radius,-radius,radius])
particles_x = [x for (x,y) in particles]
particles_y = [y for (x,y) in particles]
pylab.plot(particles_x, particles_y, "ro", alpha=0.3)
center_x = [x for (x,y) in center_coord]
center_y = [y for (x,y) in center_coord]
pylab.plot(center_x, center_y, "bo", )
pylab.savefig(name)
return resampling(particles, weights)
#
# CREATE TARGET + BRIDGE DENSITIES
#
center_coord = []; std = 1.0; radius = 20
for theta in arange(0, 2*3.1415, 2*3.1415/7):
center_coord.append( (radius*cos(theta), radius*sin(theta)) )
#
# CREATE PARTICLES SYSTEM
#
N_particles = 3000; N_temperature = 100
particles = [(0.0, 0.0) for k in range(N_particles)]
#
# EVOLVE PARTICLES
#
delta = 0.6; save_file = True
particles = smc(particles, delta, N_temperature, center_coord, std, save_file)
particles_x = [x for (x,y) in particles]
particles_y = [y for (x,y) in particles]
#
# PLOT THE RESULTS
#
pylab.figure()
pylab.plot(particles_x, particles_y, "ro",alpha=0.7)
pylab.show()
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