def compare_against_mmd_test():
data = loadmat("../data/02-solar.mat")
X = data["X"]
y = data["y"]
X_train, y_train, X_test, y_test, N, N_test = prepare_dataset(X, y)
kernel = RBF(input_dim=1, variance=0.608, lengthscale=0.207)
m = GPRegression(X_train, y_train, kernel, noise_var=0.283)
m.optimize()
pred_mean, pred_std = m.predict(X_test)
s = GaussianQuadraticTest(None)
gradients = compute_gp_regression_gradients(y_test, pred_mean, pred_std)
U_matrix, stat = s.get_statistic_multiple_custom_gradient(y_test[:, 0], gradients[:, 0])
num_test_samples = 10000
null_samples = bootstrap_null(U_matrix, num_bootstrap=num_test_samples)
# null_samples = sample_null_simulated_gp(s, pred_mean, pred_std, num_test_samples)
p_value_ours = 1.0 - np.mean(null_samples <= stat)
y_rep = np.random.randn(len(X_test)) * pred_std.flatten() + pred_mean.flatten()
y_rep = np.atleast_2d(y_rep).T
A = np.hstack((X_test, y_test))
B = np.hstack((X_test, y_rep))
feats_p = RealFeatures(A.T)
feats_q = RealFeatures(B.T)
width = 1
kernel = GaussianKernel(10, width)
mmd = QuadraticTimeMMD()
mmd.set_kernel(kernel)
mmd.set_p(feats_p)
mmd.set_q(feats_q)
mmd_stat = mmd.compute_statistic()
# sample from null
num_null_samples = 10000
mmd_null_samples = np.zeros(num_null_samples)
for i in range(num_null_samples):
# fix y_rep from above, and change the other one (that would replace y_test)
y_rep2 = np.random.randn(len(X_test)) * pred_std.flatten() + pred_mean.flatten()
y_rep2 = np.atleast_2d(y_rep2).T
A = np.hstack((X_test, y_rep2))
feats_p = RealFeatures(A.T)
width = 1
kernel = GaussianKernel(10, width)
mmd = QuadraticTimeMMD()
mmd.set_kernel(kernel)
mmd.set_p(feats_p)
mmd.set_q(feats_q)
mmd_null_samples[i] = mmd.compute_statistic()
p_value_mmd = 1.0 - np.mean(mmd_null_samples <= mmd_stat)
return p_value_ours, p_value_mmd
def statistics_quadratic_time_mmd(m, dim, difference):
from modshogun import RealFeatures
from modshogun import MeanShiftDataGenerator
from modshogun import GaussianKernel, CustomKernel
from modshogun import QuadraticTimeMMD
from modshogun import BOOTSTRAP, MMD2_SPECTRUM, MMD2_GAMMA, BIASED, UNBIASED
from modshogun import Statistics, IntVector, RealVector, Math
# init seed for reproducability
Math.init_random(1)
random.seed(17)
# number of examples kept low in order to make things fast
# streaming data generator for mean shift distributions
gen_p = MeanShiftDataGenerator(0, dim)
#gen_p.parallel.set_num_threads(1)
gen_q = MeanShiftDataGenerator(difference, dim)
# stream some data from generator
feat_p = gen_p.get_streamed_features(m)
feat_q = gen_q.get_streamed_features(m)
# set kernel a-priori. usually one would do some kernel selection. See
# other examples for this.
width = 10
kernel = GaussianKernel(10, width)
# create quadratic time mmd instance. Note that this constructor
# copies p and q and does not reference them
mmd = QuadraticTimeMMD(kernel, feat_p, feat_q)
# perform test: compute p-value and test if null-hypothesis is rejected for
# a test level of 0.05
alpha = 0.05
# using bootstrapping (slow, not the most reliable way. Consider pre-
# computing the kernel when using it, see below).
# Also, in practice, use at least 250 iterations
mmd.set_null_approximation_method(BOOTSTRAP)
mmd.set_bootstrap_iterations(3)
p_value_boot = mmd.perform_test()
# reject if p-value is smaller than test level
#print "bootstrap: p!=q: ", p_value_boot<alpha
# using spectrum method. Use at least 250 samples from null.
# This is consistent but sometimes breaks, always monitor type I error.
# See tutorial for number of eigenvalues to use .
# Only works with BIASED statistic
mmd.set_statistic_type(BIASED)
mmd.set_null_approximation_method(MMD2_SPECTRUM)
mmd.set_num_eigenvalues_spectrum(3)
mmd.set_num_samples_sepctrum(250)
p_value_spectrum = mmd.perform_test()
# reject if p-value is smaller than test level
#print "spectrum: p!=q: ", p_value_spectrum<alpha
# using gamma method. This is a quick hack, which works most of the time
# but is NOT guaranteed to. See tutorial for details.
# Only works with BIASED statistic
mmd.set_statistic_type(BIASED)
mmd.set_null_approximation_method(MMD2_GAMMA)
p_value_gamma = mmd.perform_test()
# reject if p-value is smaller than test level
#print "gamma: p!=q: ", p_value_gamma<alpha
# compute tpye I and II error (use many more trials in practice).
# Type I error is not necessary if one uses bootstrapping. We do it here
# anyway, but note that this is an efficient way of computing it.
# Also note that testing has to happen on
# difference data than kernel selection, but the linear time mmd does this
# implicitly and we used a fixed kernel here.
mmd.set_null_approximation_method(BOOTSTRAP)
mmd.set_bootstrap_iterations(5)
num_trials = 5
type_I_errors = RealVector(num_trials)
type_II_errors = RealVector(num_trials)
inds = int32(array([x for x in range(2 * m)])) # numpy
p_and_q = mmd.get_p_and_q()
# use a precomputed kernel to be faster
kernel.init(p_and_q, p_and_q)
precomputed = CustomKernel(kernel)
mmd.set_kernel(precomputed)
for i in range(num_trials):
# this effectively means that p=q - rejecting is tpye I error
inds = random.permutation(inds) # numpy permutation
precomputed.add_row_subset(inds)
precomputed.add_col_subset(inds)
type_I_errors[i] = mmd.perform_test() > alpha
precomputed.remove_row_subset()
precomputed.remove_col_subset()
# on normal data, this gives type II error
type_II_errors[i] = mmd.perform_test() > alpha
return type_I_errors.get(), type_I_errors.get(
), p_value_boot, p_value_spectrum, p_value_gamma,
def statistics_quadratic_time_mmd (m,dim,difference): from modshogun import RealFeatures from modshogun import MeanShiftDataGenerator from modshogun import GaussianKernel, CustomKernel from modshogun import QuadraticTimeMMD from modshogun import PERMUTATION, MMD2_SPECTRUM, MMD2_GAMMA, BIASED, BIASED_DEPRECATED from modshogun import Statistics, IntVector, RealVector, Math # init seed for reproducability Math.init_random(1) random.seed(17) # number of examples kept low in order to make things fast # streaming data generator for mean shift distributions gen_p=MeanShiftDataGenerator(0, dim); #gen_p.parallel.set_num_threads(1) gen_q=MeanShiftDataGenerator(difference, dim); # stream some data from generator feat_p=gen_p.get_streamed_features(m); feat_q=gen_q.get_streamed_features(m); # set kernel a-priori. usually one would do some kernel selection. See # other examples for this. width=10; kernel=GaussianKernel(10, width); # create quadratic time mmd instance. Note that this constructor # copies p and q and does not reference them mmd=QuadraticTimeMMD(kernel, feat_p, feat_q); # perform test: compute p-value and test if null-hypothesis is rejected for # a test level of 0.05 alpha=0.05; # using permutation (slow, not the most reliable way. Consider pre- # computing the kernel when using it, see below). # Also, in practice, use at least 250 iterations mmd.set_null_approximation_method(PERMUTATION); mmd.set_num_null_samples(3); p_value_null=mmd.perform_test(); # reject if p-value is smaller than test level #print "bootstrap: p!=q: ", p_value_null<alpha # using spectrum method. Use at least 250 samples from null. # This is consistent but sometimes breaks, always monitor type I error. # See tutorial for number of eigenvalues to use . mmd.set_statistic_type(BIASED); mmd.set_null_approximation_method(MMD2_SPECTRUM); mmd.set_num_eigenvalues_spectrum(3); mmd.set_num_samples_spectrum(250); p_value_spectrum=mmd.perform_test(); # reject if p-value is smaller than test level #print "spectrum: p!=q: ", p_value_spectrum<alpha # using gamma method. This is a quick hack, which works most of the time # but is NOT guaranteed to. See tutorial for details. # Only works with BIASED_DEPRECATED statistic mmd.set_statistic_type(BIASED_DEPRECATED); mmd.set_null_approximation_method(MMD2_GAMMA); p_value_gamma=mmd.perform_test(); # reject if p-value is smaller than test level #print "gamma: p!=q: ", p_value_gamma<alpha # compute tpye I and II error (use many more trials in practice). # Type I error is not necessary if one uses permutation. We do it here # anyway, but note that this is an efficient way of computing it. # Also note that testing has to happen on # difference data than kernel selection, but the linear time mmd does this # implicitly and we used a fixed kernel here. mmd.set_statistic_type(BIASED); mmd.set_null_approximation_method(PERMUTATION); mmd.set_num_null_samples(5); num_trials=5; type_I_errors=RealVector(num_trials); type_II_errors=RealVector(num_trials); inds=int32(array([x for x in range(2*m)])) # numpy p_and_q=mmd.get_p_and_q(); # use a precomputed kernel to be faster kernel.init(p_and_q, p_and_q); precomputed=CustomKernel(kernel); mmd.set_kernel(precomputed); for i in range(num_trials): # this effectively means that p=q - rejecting is tpye I error inds=random.permutation(inds) # numpy permutation precomputed.add_row_subset(inds); precomputed.add_col_subset(inds); type_I_errors[i]=mmd.perform_test()>alpha; precomputed.remove_row_subset(); precomputed.remove_col_subset(); # on normal data, this gives type II error type_II_errors[i]=mmd.perform_test()>alpha; return type_I_errors.get(),type_I_errors.get(),p_value_null,p_value_spectrum,p_value_gamma,
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)
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)
# compare to Lloyd & Gharamani
# sample from GP, and perform MMD two sample test between test data and sampled data
y_rep = np.random.randn(len(X_test)) * pred_std.flatten() + pred_mean.flatten()
y_rep = np.atleast_2d(y_rep).T
# stack together (X_test,y_test) and (X_test, y_pred)
A = np.hstack((X_test, y_test))
B = np.hstack((X_test, y_rep))
# compute MMD between (X_test,y_test) and (X_test, y_pred)
feats_p = RealFeatures(A.T)
feats_q = RealFeatures(B.T)
width = 1
kernel = GaussianKernel(10, width)
mmd = QuadraticTimeMMD()
mmd.set_kernel(kernel)
mmd.set_p(feats_p)
mmd.set_q(feats_q)
mmd_stat = mmd.compute_statistic()
# sample from null
num_null_samples = 10000
mmd_null_samples = np.zeros(num_null_samples)
for i in range(num_null_samples):
# fix y_rep from above, and change the other one (that would replace y_test)
y_rep2 = np.random.randn(len(X_test)) * pred_std.flatten() + pred_mean.flatten()
y_rep2 = np.atleast_2d(y_rep2).T
A = np.hstack((X_test, y_rep2))
feats_p = RealFeatures(A.T)