Пример #1
0
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)
Пример #2
0
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