# Stream examples and merge them in order to compute median on joint sample
# alternative is to call a different constructor of QuadraticTimeMMD
features = gen_p.get_streamed_features(m)
features = features.create_merged_copy(gen_q.get_streamed_features(m))
# use biased statistic
mmd = QuadraticTimeMMD(kernel, features, m)
mmd.set_statistic_type(BIASED)
# sample alternative distribution
alt_samples = zeros(num_null_samples)
for i in range(len(alt_samples)):
# Stream examples and merge them in order to replace in MMD
features = gen_p.get_streamed_features(m)
features = features.create_merged_copy(gen_q.get_streamed_features(m))
mmd.set_p_and_q(features)
alt_samples[i] = mmd.compute_statistic()
# sample from null distribution
# bootstrapping, biased statistic
mmd.set_null_approximation_method(BOOTSTRAP)
mmd.set_statistic_type(BIASED)
mmd.set_bootstrap_iterations(num_null_samples)
null_samples_boot = mmd.bootstrap_null()
# sample from null distribution
# spectrum, biased statistic
if "sample_null_spectrum" in dir(QuadraticTimeMMD):
mmd.set_null_approximation_method(MMD2_SPECTRUM)
mmd.set_statistic_type(BIASED)
null_samples_spectrum = mmd.sample_null_spectrum(num_null_samples, m - 10)
def quadratic_time_mmd_graphical():
# parameters, change to get different results
m=100
dim=2
# setting the difference of the first dimension smaller makes a harder test
difference=0.5
# number of samples taken from null and alternative distribution
num_null_samples=500
# streaming data generator for mean shift distributions
gen_p=MeanShiftDataGenerator(0, dim)
gen_q=MeanShiftDataGenerator(difference, dim)
# Stream examples and merge them in order to compute MMD on joint sample
# alternative is to call a different constructor of QuadraticTimeMMD
features=gen_p.get_streamed_features(m)
features=features.create_merged_copy(gen_q.get_streamed_features(m))
# use the median kernel selection
# create combined kernel with Gaussian kernels inside (shoguns Gaussian kernel is
# compute median data distance in order to use for Gaussian kernel width
# 0.5*median_distance normally (factor two in Gaussian kernel)
# However, shoguns kernel width is different to usual parametrization
# Therefore 0.5*2*median_distance^2
# Use a subset of data for that, only 200 elements. Median is stable
sigmas=[2**x for x in range(-3,10)]
widths=[x*x*2 for x in sigmas]
print "kernel widths:", widths
combined=CombinedKernel()
for i in range(len(sigmas)):
combined.append_kernel(GaussianKernel(10, widths[i]))
# create MMD instance, use biased statistic
mmd=QuadraticTimeMMD(combined,features, m)
mmd.set_statistic_type(BIASED)
# kernel selection instance (this can easily replaced by the other methods for selecting
# single kernels
selection=MMDKernelSelectionMax(mmd)
# perform kernel selection
kernel=selection.select_kernel()
kernel=GaussianKernel.obtain_from_generic(kernel)
mmd.set_kernel(kernel);
print "selected kernel width:", kernel.get_width()
# sample alternative distribution (new data each trial)
alt_samples=zeros(num_null_samples)
for i in range(len(alt_samples)):
# Stream examples and merge them in order to replace in MMD
features=gen_p.get_streamed_features(m)
features=features.create_merged_copy(gen_q.get_streamed_features(m))
mmd.set_p_and_q(features)
alt_samples[i]=mmd.compute_statistic()
# sample from null distribution
# bootstrapping, biased statistic
mmd.set_null_approximation_method(BOOTSTRAP)
mmd.set_statistic_type(BIASED)
mmd.set_bootstrap_iterations(num_null_samples)
null_samples_boot=mmd.bootstrap_null()
# sample from null distribution
# spectrum, biased statistic
if "sample_null_spectrum" in dir(QuadraticTimeMMD):
mmd.set_null_approximation_method(MMD2_SPECTRUM)
mmd.set_statistic_type(BIASED)
null_samples_spectrum=mmd.sample_null_spectrum(num_null_samples, m-10)
# fit gamma distribution, biased statistic
mmd.set_null_approximation_method(MMD2_GAMMA)
mmd.set_statistic_type(BIASED)
gamma_params=mmd.fit_null_gamma()
# sample gamma with parameters
null_samples_gamma=array([gamma(gamma_params[0], gamma_params[1]) for _ in range(num_null_samples)])
# to plot data, sample a few examples from stream first
features=gen_p.get_streamed_features(m)
features=features.create_merged_copy(gen_q.get_streamed_features(m))
data=features.get_feature_matrix()
# plot
figure()
title('Quadratic Time MMD')
# plot data of p and q
subplot(2,3,1)
grid(True)
gca().xaxis.set_major_locator( MaxNLocator(nbins = 4) ) # reduce number of x-ticks
gca().yaxis.set_major_locator( MaxNLocator(nbins = 4) ) # reduce number of x-ticks
plot(data[0][0:m], data[1][0:m], 'ro', label='$x$')
plot(data[0][m+1:2*m], data[1][m+1:2*m], 'bo', label='$x$', alpha=0.5)
title('Data, shift in $x_1$='+str(difference)+'\nm='+str(m))
xlabel('$x_1, y_1$')
ylabel('$x_2, y_2$')
# histogram of first data dimension and pdf
subplot(2,3,2)
grid(True)
gca().xaxis.set_major_locator( MaxNLocator(nbins = 3) ) # reduce number of x-ticks
gca().yaxis.set_major_locator( MaxNLocator(nbins = 3 )) # reduce number of x-ticks
hist(data[0], bins=50, alpha=0.5, facecolor='r', normed=True)
hist(data[1], bins=50, alpha=0.5, facecolor='b', normed=True)
xs=linspace(min(data[0])-1,max(data[0])+1, 50)
plot(xs,normpdf( xs, 0, 1), 'r', linewidth=3)
plot(xs,normpdf( xs, difference, 1), 'b', linewidth=3)
xlabel('$x_1, y_1$')
ylabel('$p(x_1), p(y_1)$')
title('Data PDF in $x_1, y_1$')
# compute threshold for test level
alpha=0.05
null_samples_boot.sort()
null_samples_spectrum.sort()
null_samples_gamma.sort()
thresh_boot=null_samples_boot[floor(len(null_samples_boot)*(1-alpha))];
thresh_spectrum=null_samples_spectrum[floor(len(null_samples_spectrum)*(1-alpha))];
thresh_gamma=null_samples_gamma[floor(len(null_samples_gamma)*(1-alpha))];
type_one_error_boot=sum(null_samples_boot<thresh_boot)/float(num_null_samples)
type_one_error_spectrum=sum(null_samples_spectrum<thresh_boot)/float(num_null_samples)
type_one_error_gamma=sum(null_samples_gamma<thresh_boot)/float(num_null_samples)
# plot alternative distribution with threshold
subplot(2,3,4)
grid(True)
gca().xaxis.set_major_locator( MaxNLocator(nbins = 3) ) # reduce number of x-ticks
gca().yaxis.set_major_locator( MaxNLocator(nbins = 3) ) # reduce number of x-ticks
hist(alt_samples, 20, normed=True);
axvline(thresh_boot, 0, 1, linewidth=2, color='red')
type_two_error=sum(alt_samples<thresh_boot)/float(num_null_samples)
title('Alternative Dist.\n' + 'Type II error is ' + str(type_two_error))
# compute range for all null distribution histograms
hist_range=[min([min(null_samples_boot), min(null_samples_spectrum), min(null_samples_gamma)]), max([max(null_samples_boot), max(null_samples_spectrum), max(null_samples_gamma)])]
# plot null distribution with threshold
subplot(2,3,3)
gca().xaxis.set_major_locator( MaxNLocator(nbins = 3) ) # reduce number of x-ticks
gca().yaxis.set_major_locator( MaxNLocator(nbins = 3 )) # reduce number of x-ticks
hist(null_samples_boot, 20, range=hist_range, normed=True);
axvline(thresh_boot, 0, 1, linewidth=2, color='red')
title('Bootstrapped Null Dist.\n' + 'Type I error is ' + str(type_one_error_boot))
grid(True)
# plot null distribution spectrum
subplot(2,3,5)
grid(True)
gca().xaxis.set_major_locator( MaxNLocator(nbins = 3) ) # reduce number of x-ticks
gca().yaxis.set_major_locator( MaxNLocator(nbins = 3) ) # reduce number of x-ticks
hist(null_samples_spectrum, 20, range=hist_range, normed=True);
axvline(thresh_spectrum, 0, 1, linewidth=2, color='red')
title('Null Dist. Spectrum\nType I error is ' + str(type_one_error_spectrum))
# plot null distribution gamma
subplot(2,3,6)
grid(True)
gca().xaxis.set_major_locator( MaxNLocator(nbins = 3) ) # reduce number of x-ticks
gca().yaxis.set_major_locator( MaxNLocator(nbins = 3) ) # reduce number of x-ticks
hist(null_samples_gamma, 20, range=hist_range, normed=True);
axvline(thresh_gamma, 0, 1, linewidth=2, color='red')
title('Null Dist. Gamma\nType I error is ' + str(type_one_error_gamma))
# pull plots a bit apart
subplots_adjust(hspace=0.5)
subplots_adjust(wspace=0.5)
# Stream examples and merge them in order to compute median on joint sample # alternative is to call a different constructor of QuadraticTimeMMD features=gen_p.get_streamed_features(m) features=features.create_merged_copy(gen_q.get_streamed_features(m)) # use biased statistic mmd=QuadraticTimeMMD(kernel,features, m) mmd.set_statistic_type(BIASED) # sample alternative distribution alt_samples=zeros(num_null_samples) for i in range(len(alt_samples)): # Stream examples and merge them in order to replace in MMD features=gen_p.get_streamed_features(m) features=features.create_merged_copy(gen_q.get_streamed_features(m)) mmd.set_p_and_q(features) alt_samples[i]=mmd.compute_statistic() # sample from null distribution # bootstrapping, biased statistic mmd.set_null_approximation_method(BOOTSTRAP) mmd.set_statistic_type(BIASED) mmd.set_bootstrap_iterations(num_null_samples) null_samples_boot=mmd.bootstrap_null() # sample from null distribution # spectrum, biased statistic if "sample_null_spectrum" in dir(QuadraticTimeMMD): mmd.set_null_approximation_method(MMD2_SPECTRUM) mmd.set_statistic_type(BIASED) null_samples_spectrum=mmd.sample_null_spectrum(num_null_samples, m-10)