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
def statistics_mmd_kernel_selection_single(m,distance,stretch,num_blobs,angle,selection_method):
from modshogun import RealFeatures
from modshogun import GaussianBlobsDataGenerator
from modshogun import GaussianKernel, CombinedKernel
from modshogun import LinearTimeMMD
from modshogun import MMDKernelSelectionMedian
from modshogun import MMDKernelSelectionMax
from modshogun import MMDKernelSelectionOpt
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
# results for this low number will be bad (unstable, type I error wrong)
m=1000
distance=10
stretch=5
num_blobs=3
angle=pi/4
# streaming data generator
gen_p=GaussianBlobsDataGenerator(num_blobs, distance, 1, 0)
gen_q=GaussianBlobsDataGenerator(num_blobs, distance, stretch, angle)
# stream some data and plot
num_plot=1000
features=gen_p.get_streamed_features(num_plot)
features=features.create_merged_copy(gen_q.get_streamed_features(num_plot))
data=features.get_feature_matrix()
#figure()
#subplot(2,2,1)
#grid(True)
#plot(data[0][0:num_plot], data[1][0:num_plot], 'r.', label='$x$')
#title('$X\sim p$')
#subplot(2,2,2)
#grid(True)
#plot(data[0][num_plot+1:2*num_plot], data[1][num_plot+1:2*num_plot], 'b.', label='$x$', alpha=0.5)
#title('$Y\sim q$')
# create combined kernel with Gaussian kernels inside (shoguns Gaussian kernel is
# different to the standard form, see documentation)
sigmas=[2**x for x in range(-3,10)]
widths=[x*x*2 for x in sigmas]
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
if selection_method=="opt":
selection=MMDKernelSelectionOpt(mmd)
elif selection_method=="max":
selection=MMDKernelSelectionMax(mmd)
elif selection_method=="median":
selection=MMDKernelSelectionMedian(mmd)
# print measures (just for information)
# in case Opt: ratios of MMD and standard deviation
# in case Max: MMDs for each kernel
# Does not work for median method
if selection_method!="median":
ratios=selection.compute_measures()
#print "Measures:", ratios
#subplot(2,2,3)
#plot(ratios)
#title('Measures')
# perform kernel selection
kernel=selection.select_kernel()
kernel=GaussianKernel.obtain_from_generic(kernel)
#print "selected kernel width:", kernel.get_width()
# compute tpye I and II error (use many more trials). Type I error is only
# estimated to check MMD1_GAUSSIAN method for estimating the null
# distribution. Note that testing has to happen on difference data than
# kernel selecting, but the linear time mmd does this implicitly
mmd.set_kernel(kernel)
mmd.set_null_approximation_method(MMD1_GAUSSIAN)
# number of trials should be larger to compute tight confidence bounds
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 kernel,typeIerrors,typeIIerrors
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