Пример #1
0
def hsic_graphical():
    # parameters, change to get different results
    m = 250
    difference = 3

    # setting the angle lower makes a harder test
    angle = pi / 30

    # number of samples taken from null and alternative distribution
    num_null_samples = 500

    # use data generator class to produce example data
    data = DataGenerator.generate_sym_mix_gauss(m, difference, angle)

    # create shogun feature representation
    features_x = RealFeatures(array([data[0]]))
    features_y = RealFeatures(array([data[1]]))

    # 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
    subset = int32(array([x for x in range(features_x.get_num_vectors())
                          ]))  # numpy
    subset = random.permutation(subset)  # numpy permutation
    subset = subset[0:200]
    features_x.add_subset(subset)
    dist = EuclideanDistance(features_x, features_x)
    distances = dist.get_distance_matrix()
    features_x.remove_subset()
    median_distance = np.median(distances)
    sigma_x = median_distance**2
    features_y.add_subset(subset)
    dist = EuclideanDistance(features_y, features_y)
    distances = dist.get_distance_matrix()
    features_y.remove_subset()
    median_distance = np.median(distances)
    sigma_y = median_distance**2
    print "median distance for Gaussian kernel on x:", sigma_x
    print "median distance for Gaussian kernel on y:", sigma_y
    kernel_x = GaussianKernel(10, sigma_x)
    kernel_y = GaussianKernel(10, sigma_y)

    # create hsic instance. Note that this is a convienience constructor which copies
    # feature data. features_x and features_y are not these used in hsic.
    # This is only for user-friendlyness. Usually, its ok to do this.
    # Below, the alternative distribution is sampled, which means
    # that new feature objects have to be created in each iteration (slow)
    # However, normally, the alternative distribution is not sampled
    hsic = HSIC(kernel_x, kernel_y, features_x, features_y)

    # sample alternative distribution
    alt_samples = zeros(num_null_samples)
    for i in range(len(alt_samples)):
        data = DataGenerator.generate_sym_mix_gauss(m, difference, angle)
        features_x.set_feature_matrix(array([data[0]]))
        features_y.set_feature_matrix(array([data[1]]))

        # re-create hsic instance everytime since feature objects are copied due to
        # useage of convienience constructor
        hsic = HSIC(kernel_x, kernel_y, features_x, features_y)
        alt_samples[i] = hsic.compute_statistic()

    # sample from null distribution
    # permutation, biased statistic
    hsic.set_null_approximation_method(PERMUTATION)
    hsic.set_num_null_samples(num_null_samples)
    null_samples_boot = hsic.sample_null()

    # fit gamma distribution, biased statistic
    hsic.set_null_approximation_method(HSIC_GAMMA)
    gamma_params = hsic.fit_null_gamma()
    # sample gamma with parameters
    null_samples_gamma = array([
        gamma(gamma_params[0], gamma_params[1])
        for _ in range(num_null_samples)
    ])

    # plot
    figure()

    # plot data x and y
    subplot(2, 2, 1)
    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
    grid(True)
    plot(data[0], data[1], 'o')
    title('Data, rotation=$\pi$/' + str(1 / angle * pi) + '\nm=' + str(m))
    xlabel('$x$')
    ylabel('$y$')

    # compute threshold for test level
    alpha = 0.05
    null_samples_boot.sort()
    null_samples_gamma.sort()
    thresh_boot = null_samples_boot[floor(
        len(null_samples_boot) * (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_gamma = sum(
        null_samples_gamma < thresh_boot) / float(num_null_samples)

    # plot alternative distribution with threshold
    subplot(2, 2, 2)
    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
    grid(True)
    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_gamma)]),
        max([max(null_samples_boot),
             max(null_samples_gamma)])
    ]

    # plot null distribution with threshold
    subplot(2, 2, 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
    grid(True)
    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 gamma
    subplot(2, 2, 4)
    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
    grid(True)
    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))
    grid(True)

    # pull plots a bit apart
    subplots_adjust(hspace=0.5)
    subplots_adjust(wspace=0.5)
Пример #2
0
def hsic_graphical():
	# parameters, change to get different results
	m=250
	difference=3

	# setting the angle lower makes a harder test
	angle=pi/30

	# number of samples taken from null and alternative distribution
	num_null_samples=500

	# use data generator class to produce example data
	data=DataGenerator.generate_sym_mix_gauss(m,difference,angle)

	# create shogun feature representation
	features_x=RealFeatures(array([data[0]]))
	features_y=RealFeatures(array([data[1]]))

	# 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
	subset=int32(array([x for x in range(features_x.get_num_vectors())])) # numpy
	subset=random.permutation(subset) # numpy permutation
	subset=subset[0:200]
	features_x.add_subset(subset)
	dist=EuclideanDistance(features_x, features_x)
	distances=dist.get_distance_matrix()
	features_x.remove_subset()
	median_distance=np.median(distances)
	sigma_x=median_distance**2
	features_y.add_subset(subset)
	dist=EuclideanDistance(features_y, features_y)
	distances=dist.get_distance_matrix()
	features_y.remove_subset()
	median_distance=np.median(distances)
	sigma_y=median_distance**2
	print "median distance for Gaussian kernel on x:", sigma_x
	print "median distance for Gaussian kernel on y:", sigma_y
	kernel_x=GaussianKernel(10,sigma_x)
	kernel_y=GaussianKernel(10,sigma_y)

	# create hsic instance. Note that this is a convienience constructor which copies
	# feature data. features_x and features_y are not these used in hsic.
	# This is only for user-friendlyness. Usually, its ok to do this.
	# Below, the alternative distribution is sampled, which means
	# that new feature objects have to be created in each iteration (slow)
	# However, normally, the alternative distribution is not sampled
	hsic=HSIC(kernel_x,kernel_y,features_x,features_y)

	# sample alternative distribution
	alt_samples=zeros(num_null_samples)
	for i in range(len(alt_samples)):
		data=DataGenerator.generate_sym_mix_gauss(m,difference,angle)
		features_x.set_feature_matrix(array([data[0]]))
		features_y.set_feature_matrix(array([data[1]]))

		# re-create hsic instance everytime since feature objects are copied due to
		# useage of convienience constructor
		hsic=HSIC(kernel_x,kernel_y,features_x,features_y)
		alt_samples[i]=hsic.compute_statistic()

	# sample from null distribution
	# permutation, biased statistic
	hsic.set_null_approximation_method(PERMUTATION)
	hsic.set_num_null_samples(num_null_samples)
	null_samples_boot=hsic.sample_null()

	# fit gamma distribution, biased statistic
	hsic.set_null_approximation_method(HSIC_GAMMA)
	gamma_params=hsic.fit_null_gamma()
	# sample gamma with parameters
	null_samples_gamma=array([gamma(gamma_params[0], gamma_params[1]) for _ in range(num_null_samples)])

	# plot
	figure()

	# plot data x and y
	subplot(2,2,1)
	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
	grid(True)
	plot(data[0], data[1], 'o')
	title('Data, rotation=$\pi$/'+str(1/angle*pi)+'\nm='+str(m))
	xlabel('$x$')
	ylabel('$y$')

	# compute threshold for test level
	alpha=0.05
	null_samples_boot.sort()
	null_samples_gamma.sort()
	thresh_boot=null_samples_boot[floor(len(null_samples_boot)*(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_gamma=sum(null_samples_gamma<thresh_boot)/float(num_null_samples)

	# plot alternative distribution with threshold
	subplot(2,2,2)
	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
	grid(True)
	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_gamma)]), max([max(null_samples_boot), max(null_samples_gamma)])]

	# plot null distribution with threshold
	subplot(2,2,3)
	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
	grid(True)
	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 gamma
	subplot(2,2,4)
	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
	grid(True)
	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))
	grid(True)

	# pull plots a bit apart
	subplots_adjust(hspace=0.5)
	subplots_adjust(wspace=0.5)
Пример #3
0
def statistics_hsic(n, difference, angle):
    from modshogun import RealFeatures
    from modshogun import DataGenerator
    from modshogun import GaussianKernel
    from modshogun import HSIC
    from modshogun import PERMUTATION, HSIC_GAMMA
    from modshogun import EuclideanDistance
    from modshogun import Statistics, Math

    # for reproducable results (the numpy one might not be reproducible across
    # different OS/Python-distributions
    Math.init_random(1)
    np.random.seed(1)

    # note that the HSIC has to store kernel matrices
    # which upper bounds the sample size

    # use data generator class to produce example data
    data = DataGenerator.generate_sym_mix_gauss(n, difference, angle)
    #plot(data[0], data[1], 'x');show()

    # create shogun feature representation
    features_x = RealFeatures(np.array([data[0]]))
    features_y = RealFeatures(np.array([data[1]]))

    # 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
    subset = np.random.permutation(features_x.get_num_vectors()).astype(
        np.int32)
    subset = subset[0:200]
    features_x.add_subset(subset)
    dist = EuclideanDistance(features_x, features_x)
    distances = dist.get_distance_matrix()
    features_x.remove_subset()
    median_distance = np.median(distances)
    sigma_x = median_distance**2
    features_y.add_subset(subset)
    dist = EuclideanDistance(features_y, features_y)
    distances = dist.get_distance_matrix()
    features_y.remove_subset()
    median_distance = np.median(distances)
    sigma_y = median_distance**2
    #print "median distance for Gaussian kernel on x:", sigma_x
    #print "median distance for Gaussian kernel on y:", sigma_y
    kernel_x = GaussianKernel(10, sigma_x)
    kernel_y = GaussianKernel(10, sigma_y)

    hsic = HSIC(kernel_x, kernel_y, features_x, features_y)

    # 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 = hsic.compute_statistic()
    #print "HSIC:", statistic
    alpha = 0.05

    #print "computing p-value using sampling null"
    hsic.set_null_approximation_method(PERMUTATION)
    # normally, at least 250 iterations should be done, but that takes long
    hsic.set_num_null_samples(100)
    # sampling null allows usage of unbiased or biased statistic
    p_value_boot = hsic.compute_p_value(statistic)
    thresh_boot = hsic.compute_threshold(alpha)
    #print "p_value:", p_value_boot
    #print "threshold for 0.05 alpha:", thresh_boot
    #print "p_value <", alpha, ", i.e. test sais p and q are dependend:", p_value_boot<alpha

    #print "computing p-value using gamma method"
    hsic.set_null_approximation_method(HSIC_GAMMA)
    p_value_gamma = hsic.compute_p_value(statistic)
    thresh_gamma = hsic.compute_threshold(alpha)
    #print "p_value:", p_value_gamma
    #print "threshold for 0.05 alpha:", thresh_gamma
    #print "p_value <", alpha, ", i.e. test sais p and q are dependend:", p_value_gamma<alpha

    # sample from null distribution (these may be plotted or whatsoever)
    # mean should be close to zero, variance stronly depends on data/kernel
    # sampling null, biased statistic
    #print "sampling null distribution using sample_null"
    hsic.set_null_approximation_method(PERMUTATION)
    hsic.set_num_null_samples(100)
    null_samples = hsic.sample_null()
    #print "null mean:", np.mean(null_samples)
    #print "null variance:", np.var(null_samples)
    #hist(null_samples, 100); show()

    return p_value_boot, thresh_boot, p_value_gamma, thresh_gamma, statistic, null_samples
Пример #4
0
def statistics_hsic (n, difference, angle):
	from modshogun import RealFeatures
	from modshogun import DataGenerator
	from modshogun import GaussianKernel
	from modshogun import HSIC
	from modshogun import BOOTSTRAP, HSIC_GAMMA
	from modshogun import EuclideanDistance
	from modshogun import Math, Statistics, IntVector

	# init seed for reproducability
	Math.init_random(1)

	# note that the HSIC has to store kernel matrices
	# which upper bounds the sample size

	# use data generator class to produce example data
	data=DataGenerator.generate_sym_mix_gauss(n,difference,angle)
	#plot(data[0], data[1], 'x');show()

	# create shogun feature representation
	features_x=RealFeatures(array([data[0]]))
	features_y=RealFeatures(array([data[1]]))

	# 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
	subset=IntVector.randperm_vec(features_x.get_num_vectors())
	subset=subset[0:200]
	features_x.add_subset(subset)
	dist=EuclideanDistance(features_x, features_x)
	distances=dist.get_distance_matrix()
	features_x.remove_subset()
	median_distance=Statistics.matrix_median(distances, True)
	sigma_x=median_distance**2
	features_y.add_subset(subset)
	dist=EuclideanDistance(features_y, features_y)
	distances=dist.get_distance_matrix()
	features_y.remove_subset()
	median_distance=Statistics.matrix_median(distances, True)
	sigma_y=median_distance**2
	#print "median distance for Gaussian kernel on x:", sigma_x
	#print "median distance for Gaussian kernel on y:", sigma_y
	kernel_x=GaussianKernel(10,sigma_x)
	kernel_y=GaussianKernel(10,sigma_y)

	hsic=HSIC(kernel_x,kernel_y,features_x,features_y)

	# 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=hsic.compute_statistic()
	#print "HSIC:", statistic
	alpha=0.05

	#print "computing p-value using bootstrapping"
	hsic.set_null_approximation_method(BOOTSTRAP)
	# normally, at least 250 iterations should be done, but that takes long
	hsic.set_bootstrap_iterations(100)
	# bootstrapping allows usage of unbiased or biased statistic
	p_value_boot=hsic.compute_p_value(statistic)
	thresh_boot=hsic.compute_threshold(alpha)
	#print "p_value:", p_value_boot
	#print "threshold for 0.05 alpha:", thresh_boot
	#print "p_value <", alpha, ", i.e. test sais p and q are dependend:", p_value_boot<alpha

	#print "computing p-value using gamma method"
	hsic.set_null_approximation_method(HSIC_GAMMA)
	p_value_gamma=hsic.compute_p_value(statistic)
	thresh_gamma=hsic.compute_threshold(alpha)
	#print "p_value:", p_value_gamma
	#print "threshold for 0.05 alpha:", thresh_gamma
	#print "p_value <", alpha, ", i.e. test sais p and q are dependend::", p_value_gamma<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"
	hsic.set_null_approximation_method(BOOTSTRAP)
	hsic.set_bootstrap_iterations(100)
	null_samples=hsic.bootstrap_null()
	#print "null mean:", mean(null_samples)
	#print "null variance:", var(null_samples)
	#hist(null_samples, 100); show()

	return p_value_boot, thresh_boot, p_value_gamma, thresh_gamma, statistic, null_samples