1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465
|
# /usr/bin/python
# Last Change: Thu Nov 16 08:00 PM 2006 J
# Module to implement GaussianMixture class.
import numpy as N
from numpy.random import randn, rand
import numpy.linalg as lin
import densities
from misc import _MAX_DBL_DEV
# Right now, two main usages of a Gaussian Model are possible
# - init a Gaussian Model with meta-parameters, and trains it
# - set-up a Gaussian Model to sample it, draw ellipsoides
# of confidences. In this case, we would like to init it with
# known values of parameters. This can be done with the class method
# fromval
# TODO:
# - change bounds methods of GM class instanciations so that it cannot
# be used as long as w, mu and va are not set
# - We have to use scipy now for chisquare pdf, so there may be other
# methods to be used, ie for implementing random index.
# - there is no check on internal state of the GM, that is does w, mu and va values
# make sense (eg singular values)
# - plot1d is still very rhough. There should be a sensible way to
# modify the result plot (maybe returns a dic with global pdf, component pdf and
# fill matplotlib handles). Should be coherent with plot
class GmParamError:
"""Exception raised for errors in gmm params
Attributes:
expression -- input expression in which the error occurred
message -- explanation of the error
"""
def __init__(self, message):
self.message = message
def __str__(self):
return self.message
class GM:
"""Gaussian Mixture class. This is a simple container class
to hold Gaussian Mixture parameters (weights, mean, etc...).
It can also draw itself (confidence ellipses) and samples itself.
"""
# I am not sure it is useful to have a spherical mode...
_cov_mod = ['diag', 'full']
#===============================
# Methods to construct a mixture
#===============================
def __init__(self, d, k, mode = 'diag'):
"""Init a Gaussian Mixture of k components, each component being a
d multi-variate Gaussian, with covariance matrix of style mode.
If you want to build a Gaussian Mixture with knowns weights, means
and variances, you can use GM.fromvalues method directly"""
if mode not in self._cov_mod:
raise GmParamError("mode %s not recognized" + str(mode))
self.d = d
self.k = k
self.mode = mode
# Init to 0 all parameters, with the right dimensions.
# Not sure this is useful in python from an efficiency POV ?
self.w = N.zeros(k)
self.mu = N.zeros((k, d))
if mode == 'diag':
self.va = N.zeros((k, d))
elif mode == 'full':
self.va = N.zeros((k * d, d))
self.is_valid = False
def set_param(self, weights, mu, sigma):
"""Set parameters of the model. Args should
be conformant with metparameters d and k given during
initialisation"""
k, d, mode = check_gmm_param(weights, mu, sigma)
if not k == self.k:
raise GmParamError("Number of given components is %d, expected %d"
% (k, self.k))
if not d == self.d:
raise GmParamError("Dimension of the given model is %d, expected %d"
% (d, self.d))
if not mode == self.mode and not d == 1:
raise GmParamError("Given covariance mode is %s, expected %s"
% (mode, self.mode))
self.w = weights
self.mu = mu
self.va = sigma
self.is_valid = True
@classmethod
def fromvalues(cls, weights, mu, sigma):
"""This class method can be used to create a GM model
directly from its parameters weights, mean and variance
w, mu, va = GM.gen_param(d, k)
gm = GM(d, k)
gm.set_param(w, mu, va)
and
w, mu, va = GM.gen_param(d, k)
gm = GM.fromvalue(w, mu, va)
Are equivalent """
k, d, mode = check_gmm_param(weights, mu, sigma)
res = cls(d, k, mode)
res.set_param(weights, mu, sigma)
return res
#=====================================================
# Fundamental facilities (sampling, confidence, etc..)
#=====================================================
def sample(self, nframes):
""" Sample nframes frames from the model """
if not self.is_valid:
raise GmParamError("""Parameters of the model has not been
set yet, please set them using self.set_param()""")
# State index (ie hidden var)
S = gen_rand_index(self.w, nframes)
# standard gaussian
X = randn(nframes, self.d)
if self.mode == 'diag':
X = self.mu[S, :] + X * N.sqrt(self.va[S,:])
elif self.mode == 'full':
# Faster:
cho = N.zeros((self.k, self.va.shape[1], self.va.shape[1]))
for i in range(self.k):
# Using cholesky looks more stable than sqrtm; sqrtm is not
# available in numpy anyway, only in scipy...
cho[i] = lin.cholesky(self.va[i*self.d:i*self.d+self.d,:])
for s in range(self.k):
tmpind = N.where(S == s)[0]
X[tmpind] = N.dot(X[tmpind], cho[s].transpose()) + self.mu[s]
else:
raise GmParamError('cov matrix mode not recognized, this is a bug !')
return X
def conf_ellipses(self, *args, **kargs):
"""Returns a list of confidence ellipsoids describing the Gmm
defined by mu and va. Check densities.gauss_ell for details
Returns:
-Xe: a list of x coordinates for the ellipses (Xe[i] is
the array containing x coordinates of the ith Gaussian)
-Ye: a list of y coordinates for the ellipses
Example:
Suppose we have w, mu and va as parameters for a mixture, then:
gm = GM(d, k)
gm.set_param(w, mu, va)
X = gm.sample(1000)
Xe, Ye = gm.conf_ellipsoids()
pylab.plot(X[:,0], X[:, 1], '.')
for k in len(w):
pylab.plot(Xe[k], Ye[k], 'r')
Will plot samples X draw from the mixture model, and
plot the ellipses of equi-probability from the mean with
fixed level of confidence 0.39. """
if not self.is_valid:
raise GmParamError("""Parameters of the model has not been
set yet, please set them using self.set_param()""")
Xe = []
Ye = []
if self.mode == 'diag':
for i in range(self.k):
xe, ye = densities.gauss_ell(self.mu[i,:], self.va[i,:],
*args, **kargs)
Xe.append(xe)
Ye.append(ye)
elif self.mode == 'full':
for i in range(self.k):
xe, ye = densities.gauss_ell(self.mu[i,:],
self.va[i*self.d:i*self.d+self.d,:],
*args, **kargs)
Xe.append(xe)
Ye.append(ye)
return Xe, Ye
def check_state(self):
"""
"""
if not self.is_valid:
raise GmParamError("""Parameters of the model has not been
set yet, please set them using self.set_param()""")
if self.mode == 'full':
raise NotImplementedError, "not implemented for full mode yet"
# # How to check w: if one component is negligeable, what shall
# # we do ?
# M = N.max(self.w)
# m = N.min(self.w)
# maxc = m / M
# Check condition number for cov matrix
cond = N.zeros(self.k)
ava = N.absolute(self.va)
for c in range(self.k):
cond[c] = N.amax(ava[c,:]) / N.amin(ava[c,:])
print cond
def gen_param(self, d, nc, varmode = 'diag', spread = 1):
"""Generate valid parameters for a gaussian mixture model.
d is the dimension, nc the number of components, and varmode
the mode for cov matrices.
This is a class method.
Returns: w, mu, va
"""
w = abs(randn(nc))
w = w / sum(w, 0)
mu = spread * randn(nc, d)
if varmode == 'diag':
va = abs(randn(nc, d))
elif varmode == 'full':
va = randn(nc * d, d)
for k in range(nc):
va[k*d:k*d+d] = N.dot( va[k*d:k*d+d],
va[k*d:k*d+d].transpose())
else:
raise GmParamError('cov matrix mode not recognized')
return w, mu, va
gen_param = classmethod(gen_param)
#=======================
# Regularization methods
#=======================
def _regularize(self):
raise NotImplemented("No regularization")
#=================
# Plotting methods
#=================
def plot(self, *args, **kargs):
"""Plot the ellipsoides directly for the model
Returns a list of lines, so that their style can be modified. By default,
the style is red color, and nolegend for all of them.
Does not work for 1d"""
if not self.is_valid:
raise GmParamError("""Parameters of the model has not been
set yet, please set them using self.set_param()""")
k = self.k
Xe, Ye = self.conf_ellipses(*args, **kargs)
try:
import pylab as P
return [P.plot(Xe[i], Ye[i], 'r', label='_nolegend_')[0] for i in range(k)]
#for i in range(k):
# P.plot(Xe[i], Ye[i], 'r')
except ImportError:
raise GmParamError("matplotlib not found, cannot plot...")
def plot1d(self, level = 0.5, fill = 0, gpdf = 0):
"""This function plots the pdfs of each component of the model.
If gpdf is 1, also plots the global pdf. If fill is 1, fill confidence
areas using level argument as a level value
Returns a dictionary h of plot handles so that their properties can
be modified (eg color, label, etc...):
- h['pdf'] is a list of lines, one line per component pdf
- h['gpdf'] is the line for the global pdf
- h['conf'] is a list of filling area
"""
# This is not optimized at all, may be slow. Should not be
# difficult to make much faster, but it is late, and I am lazy
if not self.d == 1:
raise GmParamError("the model is not one dimensional model")
from scipy.stats import norm
nrm = norm(0, 1)
pval = N.sqrt(self.va[:,0]) * nrm.ppf((1+level)/2)
# Compute reasonable min/max for the normal pdf
mc = 3
std = N.sqrt(self.va[:,0])
m = N.amin(self.mu[:, 0] - mc * std)
M = N.amax(self.mu[:, 0] + mc * std)
np = 500
x = N.linspace(m, M, np)
Yf = N.zeros(np)
Yt = N.zeros(np)
# Prepare the dic of plot handles to return
ks = ['pdf', 'conf', 'gpdf']
hp = dict((i,[]) for i in ks)
try:
import pylab as P
for c in range(self.k):
y = self.w[c]/(N.sqrt(2*N.pi) * std[c]) * \
N.exp(-(x-self.mu[c][0])**2/(2*std[c]**2))
Yt += y
h = P.plot(x, y, 'r', label ='_nolegend_')
hp['pdf'].extend(h)
if fill:
#P.axvspan(-pval[c] + self.mu[c][0], pval[c] + self.mu[c][0],
# facecolor = 'b', alpha = 0.2)
id1 = -pval[c] + self.mu[c]
id2 = pval[c] + self.mu[c]
xc = x[:, N.where(x>id1)[0]]
xc = xc[:, N.where(xc<id2)[0]]
Yf = self.w[c]/(N.sqrt(2*N.pi) * std[c]) * \
N.exp(-(xc-self.mu[c][0])**2/(2*std[c]**2))
xc = N.concatenate(([xc[0]], xc, [xc[-1]]))
Yf = N.concatenate(([0], Yf, [0]))
h = P.fill(xc, Yf,
facecolor = 'b', alpha = 0.1, label='_nolegend_')
hp['conf'].extend(h)
#P.fill([xc[0], xc[0], xc[-1], xc[-1]],
# [0, Yf[0], Yf[-1], 0], facecolor = 'b', alpha = 0.2)
if gpdf:
h = P.plot(x, Yt, 'r:', label='_nolegend_')
hp['gpdf'] = h
return hp
except ImportError:
raise GmParamError("matplotlib not found, cannot plot...")
# Syntactic sugar
def __repr__(self):
repr = ""
repr += "Gaussian Mixture:\n"
repr += " -> %d dimensions\n" % self.d
repr += " -> %d components\n" % self.k
repr += " -> %s covariance \n" % self.mode
if self.is_valid:
repr += "Has initial values"""
else:
repr += "Has no initial values yet"""
return repr
def __str__(self):
return self.__repr__()
# Function to generate a random index: this is kept outside any class,
# as the function can be useful for other
def gen_rand_index(p, n):
"""Generate a N samples vector containing random index between 1
and length(p), each index i with probability p(i)"""
# TODO Check args here
# TODO: check each value of inverse distribution is
# different
invcdf = N.cumsum(p)
uni = rand(n)
index = N.zeros(n, dtype=int)
# This one should be a bit faster
for k in range(len(p)-1, 0, -1):
blop = N.where(N.logical_and(invcdf[k-1] <= uni,
uni < invcdf[k]))
index[blop] = k
return index
def check_gmm_param(w, mu, va):
"""Check that w, mu and va are valid parameters for
a mixture of gaussian: w should sum to 1, there should
be the same number of component in each param, the variances
should be positive definite, etc...
Params:
w = vector or list of weigths of the mixture (K elements)
mu = matrix: K * d
va = list of variances (vector K * d or square matrices Kd * d)
returns:
K = number of components
d = dimension
mode = 'diag' if diagonal covariance, 'full' of full matrices
"""
# Check that w is valid
if N.fabs(N.sum(w, 0) - 1) > _MAX_DBL_DEV:
raise GmParamError('weight does not sum to 1')
if not len(w.shape) == 1:
raise GmParamError('weight is not a vector')
# Check that mean and va have the same number of components
K = len(w)
if N.ndim(mu) < 2:
msg = "mu should be a K,d matrix, and a row vector if only 1 comp"
raise GmParamError(msg)
if N.ndim(va) < 2:
msg = """va should be a K,d / K *d, d matrix, and a row vector if
only 1 diag comp"""
raise GmParamError(msg)
(Km, d) = mu.shape
(Ka, da) = va.shape
if not K == Km:
msg = "not same number of component in mean and weights"
raise GmParamError(msg)
if not d == da:
msg = "not same number of dimensions in mean and variances"
raise GmParamError(msg)
if Km == Ka:
mode = 'diag'
else:
mode = 'full'
if not Ka == Km*d:
msg = "not same number of dimensions in mean and variances"
raise GmParamError(msg)
return K, d, mode
if __name__ == '__main__':
# Meta parameters:
# - k = number of components
# - d = dimension
# - mode : mode of covariance matrices
d = 5
k = 4
# Now, drawing a model
mode = 'full'
nframes = 1e3
# Build a model with random parameters
w, mu, va = GM.gen_param(d, k, mode, spread = 3)
gm = GM.fromvalues(w, mu, va)
# Sample nframes frames from the model
X = gm.sample(nframes)
# Plot the data
import pylab as P
P.plot(X[:, 0], X[:, 1], '.', label = '_nolegend_')
# Real confidence ellipses with confidence level
level = 0.50
h = gm.plot(level=level)
# set the first ellipse label, which will appear in the legend
h[0].set_label('confidence ell at level ' + str(level))
P.legend(loc = 0)
P.show()
|