def statistics_linear_time_mmd(n, dim, difference):
from modshogun import RealFeatures
from modshogun import MeanShiftDataGenerator
from modshogun import GaussianKernel
from modshogun import LinearTimeMMD
from modshogun import PERMUTATION, MMD1_GAUSSIAN
from modshogun import EuclideanDistance
from modshogun import Statistics, Math
# init seed for reproducability
Math.init_random(1)
# note that the linear time statistic is designed for much larger datasets
# so increase to get reasonable results
# streaming data generator for mean shift distributions
gen_p = MeanShiftDataGenerator(0, dim)
gen_q = MeanShiftDataGenerator(difference, dim)
# 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
# Stream examples and merge them in order to compute median on joint sample
features = gen_p.get_streamed_features(100)
features = features.create_merged_copy(gen_q.get_streamed_features(100))
# compute all pairwise distances
dist = EuclideanDistance(features, features)
distances = dist.get_distance_matrix()
# compute median and determine kernel width (using shogun)
median_distance = Statistics.matrix_median(distances, True)
sigma = median_distance**2
#print "median distance for Gaussian kernel:", sigma
kernel = GaussianKernel(10, sigma)
# mmd instance using streaming features, blocksize of 10000
mmd = LinearTimeMMD(kernel, gen_p, gen_q, n, 10000)
# perform test: compute p-value and test if null-hypothesis is rejected for
# a test level of 0.05
statistic = mmd.compute_statistic()
#print "test statistic:", statistic
# do the same thing using two different way to approximate null-dstribution
# sampling null and gaussian approximation (ony for really large samples)
alpha = 0.05
#print "computing p-value using sampling null"
mmd.set_null_approximation_method(PERMUTATION)
mmd.set_num_null_samples(50) # normally, far more iterations are needed
p_value_boot = mmd.compute_p_value(statistic)
#print "p_value_boot:", p_value_boot
#print "p_value_boot <", alpha, ", i.e. test sais p!=q:", p_value_boot<alpha
#print "computing p-value using gaussian approximation"
mmd.set_null_approximation_method(MMD1_GAUSSIAN)
p_value_gaussian = mmd.compute_p_value(statistic)
#print "p_value_gaussian:", p_value_gaussian
#print "p_value_gaussian <", alpha, ", i.e. test sais p!=q:", p_value_gaussian<alpha
# sample from null distribution (these may be plotted or whatsoever)
# mean should be close to zero, variance stronly depends on data/kernel
mmd.set_null_approximation_method(PERMUTATION)
mmd.set_num_null_samples(10) # normally, far more iterations are needed
null_samples = mmd.sample_null()
#print "null mean:", mean(null_samples)
#print "null variance:", var(null_samples)
# compute type I and type II errors for Gaussian approximation
# number of trials should be larger to compute tight confidence bounds
mmd.set_null_approximation_method(MMD1_GAUSSIAN)
num_trials = 5
alpha = 0.05 # test power
typeIerrors = [0 for x in range(num_trials)]
typeIIerrors = [0 for x in range(num_trials)]
for i in range(num_trials):
# this effectively means that p=q - rejecting is tpye I error
mmd.set_simulate_h0(True)
typeIerrors[i] = mmd.perform_test() > alpha
mmd.set_simulate_h0(False)
typeIIerrors[i] = mmd.perform_test() > alpha
#print "type I error:", mean(typeIerrors), ", type II error:", mean(typeIIerrors)
return statistic, p_value_boot, p_value_gaussian, null_samples, typeIerrors, typeIIerrors
def linear_time_mmd_graphical():
# parameters, change to get different results
m=1000 # set to 10000 for a good test result
dim=2
# setting the difference of the first dimension smaller makes a harder test
difference=1
# number of samples taken from null and alternative distribution
num_null_samples=150
# streaming data generator for mean shift distributions
gen_p=MeanShiftDataGenerator(0, dim)
gen_q=MeanShiftDataGenerator(difference, dim)
# 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]))
# mmd instance using streaming features, blocksize of 10000
block_size=1000
mmd=LinearTimeMMD(combined, gen_p, gen_q, m, block_size)
# kernel selection instance (this can easily replaced by the other methods for selecting
# single kernels
selection=MMDKernelSelectionOpt(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, stream ensures different samples each run
alt_samples=zeros(num_null_samples)
for i in range(len(alt_samples)):
alt_samples[i]=mmd.compute_statistic()
# sample from null distribution
# bootstrapping, biased statistic
mmd.set_null_approximation_method(PERMUTATION)
mmd.set_num_null_samples(num_null_samples)
null_samples_boot=mmd.sample_null()
# fit normal distribution to null and sample a normal distribution
mmd.set_null_approximation_method(MMD1_GAUSSIAN)
variance=mmd.compute_variance_estimate()
null_samples_gaussian=normal(0,sqrt(variance),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()
# 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_gaussian.sort()
thresh_boot=null_samples_boot[floor(len(null_samples_boot)*(1-alpha))];
thresh_gaussian=null_samples_gaussian[floor(len(null_samples_gaussian)*(1-alpha))];
type_one_error_boot=sum(null_samples_boot<thresh_boot)/float(num_null_samples)
type_one_error_gaussian=sum(null_samples_gaussian<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_gaussian)]), max([max(null_samples_boot), max(null_samples_gaussian)])]
# plot null distribution with threshold
subplot(2,3,3)
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_boot, 20, range=hist_range, normed=True);
axvline(thresh_boot, 0, 1, linewidth=2, color='red')
title('Sampled Null Dist.\n' + 'Type I error is ' + str(type_one_error_boot))
# plot null distribution gaussian
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_gaussian, 20, range=hist_range, normed=True);
axvline(thresh_gaussian, 0, 1, linewidth=2, color='red')
title('Null Dist. Gaussian\nType I error is ' + str(type_one_error_gaussian))
# pull plots a bit apart
subplots_adjust(hspace=0.5)
subplots_adjust(wspace=0.5)
def statistics_linear_time_mmd (n,dim,difference): from modshogun import RealFeatures from modshogun import MeanShiftDataGenerator from modshogun import GaussianKernel from modshogun import LinearTimeMMD from modshogun import PERMUTATION, MMD1_GAUSSIAN from modshogun import EuclideanDistance from modshogun import Statistics, Math # init seed for reproducability Math.init_random(1) # note that the linear time statistic is designed for much larger datasets # so increase to get reasonable results # streaming data generator for mean shift distributions gen_p=MeanShiftDataGenerator(0, dim) gen_q=MeanShiftDataGenerator(difference, dim) # 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 # Stream examples and merge them in order to compute median on joint sample features=gen_p.get_streamed_features(100) features=features.create_merged_copy(gen_q.get_streamed_features(100)) # compute all pairwise distances dist=EuclideanDistance(features, features) distances=dist.get_distance_matrix() # compute median and determine kernel width (using shogun) median_distance=Statistics.matrix_median(distances, True) sigma=median_distance**2 #print "median distance for Gaussian kernel:", sigma kernel=GaussianKernel(10,sigma) # mmd instance using streaming features, blocksize of 10000 mmd=LinearTimeMMD(kernel, gen_p, gen_q, n, 10000) # perform test: compute p-value and test if null-hypothesis is rejected for # a test level of 0.05 statistic=mmd.compute_statistic() #print "test statistic:", statistic # do the same thing using two different way to approximate null-dstribution # sampling null and gaussian approximation (ony for really large samples) alpha=0.05 #print "computing p-value using sampling null" mmd.set_null_approximation_method(PERMUTATION) mmd.set_num_null_samples(50) # normally, far more iterations are needed p_value_boot=mmd.compute_p_value(statistic) #print "p_value_boot:", p_value_boot #print "p_value_boot <", alpha, ", i.e. test sais p!=q:", p_value_boot<alpha #print "computing p-value using gaussian approximation" mmd.set_null_approximation_method(MMD1_GAUSSIAN) p_value_gaussian=mmd.compute_p_value(statistic) #print "p_value_gaussian:", p_value_gaussian #print "p_value_gaussian <", alpha, ", i.e. test sais p!=q:", p_value_gaussian<alpha # sample from null distribution (these may be plotted or whatsoever) # mean should be close to zero, variance stronly depends on data/kernel mmd.set_null_approximation_method(PERMUTATION) mmd.set_num_null_samples(10) # normally, far more iterations are needed null_samples=mmd.sample_null() #print "null mean:", mean(null_samples) #print "null variance:", var(null_samples) # compute type I and type II errors for Gaussian approximation # number of trials should be larger to compute tight confidence bounds mmd.set_null_approximation_method(MMD1_GAUSSIAN) num_trials=5; alpha=0.05 # test power typeIerrors=[0 for x in range(num_trials)] typeIIerrors=[0 for x in range(num_trials)] for i in range(num_trials): # this effectively means that p=q - rejecting is tpye I error mmd.set_simulate_h0(True) typeIerrors[i]=mmd.perform_test()>alpha mmd.set_simulate_h0(False) typeIIerrors[i]=mmd.perform_test()>alpha #print "type I error:", mean(typeIerrors), ", type II error:", mean(typeIIerrors) return statistic, p_value_boot, p_value_gaussian, null_samples, typeIerrors, typeIIerrors
