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#!/usr/bin/env python
"""
Multimodal demonstration using gaussian mixture model.
The model is a mixture model representing the probability density from a
product of gaussians.
This example show performance of the algorithm on multimodal densities,
with adjustable number of densities and degree of separation.
The peaks are distributed about the x-y plane so that the marginal densities
in x and y are spaced every 2 units using latin hypercube sampling. For small
peak widths, this means that the densities will not overlap, and the marginal
maximum likelihood for a given x or y value should match the estimated density.
With overlap, the marginal density will over estimate the marginal maximum
likelihood.
Adjust the width of the peaks, *S*, to see the effect of relative diameter of
the modes on sampling. Adjust the height of the peaks, *I*, to see the
effects of the relative height of the modes. Adjust the count *n* to see
the effects of the number of modes.
Note that dream.diffev.de_step adds jitter to the parameters at the 1e-6 level,
so *S* < 1e-4 cannot be modeled reliably.
*draws* is set to 1000 samples per mode. *burn* is set to 100 samples per mode.
Population size *h* is set to 20 per mode. A good choice for number of
sequences *k* is not yet determined.
"""
from __future__ import print_function
import numpy as np
from bumps.dream.model import MVNormal, Mixture
from bumps.names import *
from bumps.util import push_seed
def make_model():
if 1: # Fixed layout of 5 minima
num_modes = 5
S = [0.1]*5
x = [-4, -2, 0, 2, 4]
y = [2, -2, -4, 0, 4]
I = [5, 2.5, 1, 4, 1]
else: # Semirandom layout of n minima
num_modes = 40
S = [0.1]*num_modes
x = np.linspace(-10, 10, num_modes)
y = np.random.permutation(x)
I = 2*np.linspace(-1, 1, num_modes)**2 + 1
## Take only the first two modes
k = 2
S, x, y, I = S[:k], x[:k], y[:k], I[:k]
#S[1] = 1; I[1] = 1; I[0] = 1
dims = 10
centers = [x, y] + [np.random.permutation(x) for _ in range(2, dims)]
centers = np.asarray(centers).T
args = [] # Sequence of density, weight, density, weight, ...
for mu_i, Si, Ii in zip(centers, S, I):
args.extend((MVNormal(mu_i, Si*np.eye(dims)), Ii))
model = Mixture(*args)
if 1:
from bumps.dream.entropy import GaussianMixture
pairs = zip(args[0::2], args[1::2])
triples = ((M.mu, M.sigma, I) for M, I in pairs)
mu, sigma, weight = zip(*triples)
D = GaussianMixture(weight, mu=mu, sigma=sigma)
print("*** Expected entropy: %s bits"%(D.entropy(N=100000)/np.log(2),))
return model
# Need reproducible models if we want to be able to resume a fit
with push_seed(1):
model = make_model()
def plot2d(fn, args=None, range=(-10, 10)):
"""
Return a mesh plotter for the given function.
*args* are the function arguments that are to be meshed (usually the
first two arguments to the function). *range* is the bounding box
for the 2D mesh.
All arguments except the meshed arguments are held fixed.
"""
if args is None:
args = [0, 1]
def plotter(p, view=None):
import pylab
if len(p) == 1:
x = p[0]
r = np.linspace(range[0], range[1], 400)
pylab.plot(x+r, [fn(v) for v in x+r])
pylab.xlabel(args[0])
pylab.ylabel("-log P(%s)"%args[0])
else:
r = np.linspace(range[0], range[1], 20)
x, y = p[args[0]], p[args[1]]
data = np.empty((len(r), len(r)), 'd')
for j, xj in enumerate(x+r):
for k, yk in enumerate(y+r):
p[args[0]], p[args[1]] = xj, yk
data[j, k] = fn(p)
pylab.pcolormesh(x+r, y+r, data)
pylab.plot(x, y, 'o', markersize=6,
markerfacecolor='red', markeredgecolor='black',
markeredgewidth=1, alpha=0.7)
pylab.xlabel(args[0])
pylab.ylabel(args[1])
return plotter
M = VectorPDF(model.nllf, p=[0.]*dims, plot=plot2d(model.nllf))
for _, p in M.parameters().items():
p.range(-10, 10)
problem = FitProblem(M)
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