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 statistics_quadratic_time_mmd():
    from shogun.Features import RealFeatures
    from shogun.Features import MeanShiftDataGenerator
    from shogun.Kernel import GaussianKernel
    from shogun.Statistics import QuadraticTimeMMD
    from shogun.Statistics import BOOTSTRAP, MMD2_SPECTRUM, MMD2_GAMMA, BIASED, UNBIASED
    from shogun.Distance import EuclideanDistance
    from shogun.Mathematics import Statistics, IntVector

    # note that the quadratic time mmd has to store kernel matrices
    # which upper bounds the sample size
    n = 100
    dim = 2
    difference = 0.5

    # 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 median on joint sample
    # alternative is to call a different constructor of QuadraticTimeMMD
    features = gen_p.get_streamed_features(n)
    features = features.create_merged_copy(gen_q.get_streamed_features(n))

    # use data generator class to produce example data
    data = features.get_feature_matrix()

    print "dimension means of X", mean(data.T[0:n].T)
    print "dimension means of Y", mean(data.T[n : 2 * n + 1].T)

    # 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
    # Use a permutation set to temporarily merge features in merged examples
    subset = IntVector.randperm_vec(features.get_num_vectors())
    subset = subset[0:200]
    features.add_subset(subset)
    dist = EuclideanDistance(features, features)
    distances = dist.get_distance_matrix()
    features.remove_subset()
    median_distance = Statistics.matrix_median(distances, True)
    sigma = median_distance ** 2
    print "median distance for Gaussian kernel:", sigma
    kernel = GaussianKernel(10, sigma)

    mmd = QuadraticTimeMMD(kernel, features, n)

    # perform test: compute p-value and test if null-hypothesis is rejected for
    # a test level of 0.05 using different methods to approximate
    # null-distribution
    statistic = mmd.compute_statistic()
    alpha = 0.05

    print "computing p-value using bootstrapping"
    mmd.set_null_approximation_method(BOOTSTRAP)
    # normally, at least 250 iterations should be done, but that takes long
    mmd.set_bootstrap_iterations(10)
    # bootstrapping allows usage of unbiased or biased statistic
    mmd.set_statistic_type(UNBIASED)
    p_value = mmd.compute_p_value(statistic)
    print "p_value:", p_value
    print "p_value <", alpha, ", i.e. test sais p!=q:", p_value < alpha

    # only can do this if SHOGUN was compiled with LAPACK so check
    if "sample_null_spectrum" in dir(QuadraticTimeMMD):
        print "computing p-value using spectrum method"
        mmd.set_null_approximation_method(MMD2_SPECTRUM)
        # normally, at least 250 iterations should be done, but that takes long
        mmd.set_num_samples_sepctrum(50)
        mmd.set_num_eigenvalues_spectrum(n - 10)
        # spectrum method computes p-value for biased statistics only
        mmd.set_statistic_type(BIASED)
        p_value = mmd.compute_p_value(statistic)
        print "p_value:", p_value
        print "p_value <", alpha, ", i.e. test sais p!=q:", p_value < alpha

    print "computing p-value using gamma method"
    mmd.set_null_approximation_method(MMD2_GAMMA)
    # gamma method computes p-value for biased statistics only
    mmd.set_statistic_type(BIASED)
    p_value = mmd.compute_p_value(statistic)
    print "p_value:", p_value
    print "p_value <", alpha, ", i.e. test sais p!=q:", p_value < alpha

    # sample from null distribution (these may be plotted or whatsoever)
    # mean should be close to zero, variance stronly depends on data/kernel
    # bootstrapping, biased statistic
    print "sampling null distribution using bootstrapping"
    mmd.set_null_approximation_method(BOOTSTRAP)
    mmd.set_statistic_type(BIASED)
    mmd.set_bootstrap_iterations(10)
    null_samples = mmd.bootstrap_null()
    print "null mean:", mean(null_samples)
    print "null variance:", var(null_samples)

    # sample from null distribution (these may be plotted or whatsoever)
    # mean should be close to zero, variance stronly depends on data/kernel
    # spectrum, biased statistic
    print "sampling null distribution using spectrum method"
    mmd.set_null_approximation_method(MMD2_SPECTRUM)
    mmd.set_statistic_type(BIASED)
    # 200 samples using 100 eigenvalues
    null_samples = mmd.sample_null_spectrum(50, 10)
    print "null mean:", mean(null_samples)
    print "null variance:", var(null_samples)
Beispiel #3
0
def statistics_quadratic_time_mmd (m,dim,difference):
	from shogun.Features import RealFeatures
	from shogun.Features import MeanShiftDataGenerator
	from shogun.Kernel import GaussianKernel, CustomKernel
	from shogun.Statistics import QuadraticTimeMMD
	from shogun.Statistics import BOOTSTRAP, MMD2_SPECTRUM, MMD2_GAMMA, BIASED, UNBIASED
	from shogun.Mathematics import Statistics, IntVector, RealVector, Math
	
	# init seed for reproducability
	Math.init_random(1)

	# number of examples kept low in order to make things fast

	# streaming data generator for mean shift distributions
	gen_p=MeanShiftDataGenerator(0, dim);
	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_linear_time_mmd (n,dim,difference):
	from shogun.Features import RealFeatures
	from shogun.Features import MeanShiftDataGenerator
	from shogun.Kernel import GaussianKernel
	from shogun.Statistics import LinearTimeMMD
	from shogun.Statistics import BOOTSTRAP, MMD1_GAUSSIAN
	from shogun.Distance import EuclideanDistance
	from shogun.Mathematics 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
	# bootstrapping and gaussian approximation (ony for really large samples)
	alpha=0.05

	#print "computing p-value using bootstrapping"
	mmd.set_null_approximation_method(BOOTSTRAP)
	mmd.set_bootstrap_iterations(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(BOOTSTRAP)
	mmd.set_bootstrap_iterations(10) # normally, far more iterations are needed
	null_samples=mmd.bootstrap_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(BOOTSTRAP)
	mmd.set_bootstrap_iterations(num_null_samples)
	null_samples_boot=mmd.bootstrap_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('Bootstrapped 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)
Beispiel #6
0
def statistics_linear_time_mmd(n, dim, difference):
    from shogun.Features import RealFeatures
    from shogun.Features import MeanShiftDataGenerator
    from shogun.Kernel import GaussianKernel
    from shogun.Statistics import LinearTimeMMD
    from shogun.Statistics import BOOTSTRAP, MMD1_GAUSSIAN
    from shogun.Distance import EuclideanDistance
    from shogun.Mathematics 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
    # bootstrapping and gaussian approximation (ony for really large samples)
    alpha = 0.05

    #print "computing p-value using bootstrapping"
    mmd.set_null_approximation_method(BOOTSTRAP)
    mmd.set_bootstrap_iterations(
        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(BOOTSTRAP)
    mmd.set_bootstrap_iterations(
        10)  # normally, far more iterations are needed
    null_samples = mmd.bootstrap_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(BOOTSTRAP)
    mmd.set_bootstrap_iterations(num_null_samples)
    null_samples_boot = mmd.bootstrap_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('Bootstrapped 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)