def statistics_linear_time_mmd(): from shogun.Features import RealFeatures from shogun.Features import DataGenerator from shogun.Kernel import GaussianKernel from shogun.Statistics import LinearTimeMMD from shogun.Statistics import BOOTSTRAP, MMD1_GAUSSIAN # note that the linear time statistic is designed for much larger datasets n=10000 dim=2 difference=0.5 # use data generator class to produce example data # in pratice, this generate data function could be replaced by a method # that obtains data from a stream data=DataGenerator.generate_mean_data(n,dim,difference) print "dimension means of X", mean(data.T[0:n].T) print "dimension means of Y", mean(data.T[n:2*n+1].T) # create shogun feature representation features=RealFeatures(data) # use a kernel width of sigma=2, which is 8 in SHOGUN's parametrization # which is k(x,y)=exp(-||x-y||^2 / tau), in constrast to the standard # k(x,y)=exp(-||x-y||^2 / (2*sigma^2)), so tau=2*sigma^2 kernel=GaussianKernel(10,8) mmd=LinearTimeMMD(kernel,features, n) # 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=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 gaussian approximation" mmd.set_null_approximation_method(MMD1_GAUSSIAN) 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 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)
def statistics_linear_time_mmd_kernel_choice(): from shogun.Features import RealFeatures, CombinedFeatures from shogun.Features import DataGenerator from shogun.Kernel import GaussianKernel, CombinedKernel from shogun.Statistics import LinearTimeMMD from shogun.Statistics import BOOTSTRAP, MMD1_GAUSSIAN # note that the linear time statistic is designed for much larger datasets n=50000 dim=5 difference=2 # use data generator class to produce example data # in pratice, this generate data function could be replaced by a method # that obtains data from a stream data=DataGenerator.generate_mean_data(n,dim,difference) print "dimension means of X", mean(data.T[0:n].T) print "dimension means of Y", mean(data.T[n:2*n+1].T) # create kernels/features to choose from # here: just a bunch of Gaussian Kernels with different widths # real sigmas are 2^-5, ..., 2^10 sigmas=array([pow(2,x) for x in range(-5,10)]) # shogun has a different parametrization of the Gaussian kernel shogun_sigmas=array([x*x*2 for x in sigmas]) # We will use multiple kernels kernel=CombinedKernel() # two separate feature objects here, could also be one with appended data features=CombinedFeatures() # all kernels work on same features for i in range(len(sigmas)): kernel.append_kernel(GaussianKernel(10, shogun_sigmas[i])) features.append_feature_obj(RealFeatures(data)) mmd=LinearTimeMMD(kernel,features, n) print "start learning kernel weights" mmd.set_opt_regularization_eps(10E-5) mmd.set_opt_low_cut(10E-5) mmd.set_opt_max_iterations(1000) mmd.set_opt_epsilon(10E-7) mmd.optimize_kernel_weights() weights=kernel.get_subkernel_weights() print "learned weights:", weights #pyplot.plot(array(range(len(sigmas))), weights) #pyplot.show() print "index of max weight", weights.argmax()
def statistics_linear_time_mmd_kernel_choice():
from shogun.Features import RealFeatures, CombinedFeatures
from shogun.Features import DataGenerator
from shogun.Kernel import GaussianKernel, CombinedKernel
from shogun.Statistics import LinearTimeMMD
from shogun.Statistics import BOOTSTRAP, MMD1_GAUSSIAN
# note that the linear time statistic is designed for much larger datasets
n = 50000
dim = 5
difference = 2
# use data generator class to produce example data
# in pratice, this generate data function could be replaced by a method
# that obtains data from a stream
data = DataGenerator.generate_mean_data(n, dim, difference)
print "dimension means of X", mean(data.T[0:n].T)
print "dimension means of Y", mean(data.T[n:2 * n + 1].T)
# create kernels/features to choose from
# here: just a bunch of Gaussian Kernels with different widths
# real sigmas are 2^-5, ..., 2^10
sigmas = array([pow(2, x) for x in range(-5, 10)])
# shogun has a different parametrization of the Gaussian kernel
shogun_sigmas = array([x * x * 2 for x in sigmas])
# We will use multiple kernels
kernel = CombinedKernel()
# two separate feature objects here, could also be one with appended data
features = CombinedFeatures()
# all kernels work on same features
for i in range(len(sigmas)):
kernel.append_kernel(GaussianKernel(10, shogun_sigmas[i]))
features.append_feature_obj(RealFeatures(data))
mmd = LinearTimeMMD(kernel, features, n)
print "start learning kernel weights"
mmd.set_opt_regularization_eps(10E-5)
mmd.set_opt_low_cut(10E-5)
mmd.set_opt_max_iterations(1000)
mmd.set_opt_epsilon(10E-7)
mmd.optimize_kernel_weights()
weights = kernel.get_subkernel_weights()
print "learned weights:", weights
#pyplot.plot(array(range(len(sigmas))), weights)
#pyplot.show()
print "index of max weight", weights.argmax()
# for nice plotting that fits into our shogun tutorial import latex_plot_inits # 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=500 # use data generator class to produce example data data=DataGenerator.generate_mean_data(m,dim,difference) # create shogun feature representation features=RealFeatures(data) # 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 # Using all distances here would blow up memory subset=Math.randperm_vec(features.get_num_vectors()) subset=subset[0:200] features.add_subset(subset) dist=EuclideanDistance(features, features) distances=dist.get_distance_matrix()
def statistics_linear_time_mmd (): from shogun.Features import RealFeatures from shogun.Features import DataGenerator 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 # note that the linear time statistic is designed for much larger datasets n=10000 dim=2 difference=0.5 # use data generator class to produce example data # in pratice, this generate data function could be replaced by a method # that obtains data from a stream data=DataGenerator.generate_mean_data(n,dim,difference) print "dimension means of X", mean(data.T[0:n].T) print "dimension means of Y", mean(data.T[n:2*n+1].T) # create shogun feature representation features=RealFeatures(data) # 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 # Using all distances here would blow up memory subset=Math.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=LinearTimeMMD(kernel,features, n) # 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=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 gaussian approximation" mmd.set_null_approximation_method(MMD1_GAUSSIAN) 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 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)
def statistics_quadratic_time_mmd(): from shogun.Features import RealFeatures from shogun.Features import DataGenerator from shogun.Kernel import GaussianKernel from shogun.Statistics import QuadraticTimeMMD from shogun.Statistics import BOOTSTRAP, MMD2_SPECTRUM, MMD2_GAMMA, BIASED, UNBIASED # note that the quadratic time mmd has to store kernel matrices # which upper bounds the sample size n=500 dim=2 difference=0.5 # use data generator class to produce example data data=DataGenerator.generate_mean_data(n,dim,difference) print "dimension means of X", mean(data.T[0:n].T) print "dimension means of Y", mean(data.T[n:2*n+1].T) # create shogun feature representation features=RealFeatures(data) # use a kernel width of sigma=2, which is 8 in SHOGUN's parametrization # which is k(x,y)=exp(-||x-y||^2 / tau), in constrast to the standard # k(x,y)=exp(-||x-y||^2 / (2*sigma^2)), so tau=2*sigma^2 kernel=GaussianKernel(10,8) 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)
def statistics_quadratic_time_mmd():
from shogun.Features import RealFeatures
from shogun.Features import DataGenerator
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, Math
# note that the quadratic time mmd has to store kernel matrices
# which upper bounds the sample size
n = 500
dim = 2
difference = 0.5
# use data generator class to produce example data
data = DataGenerator.generate_mean_data(n, dim, difference)
print "dimension means of X", mean(data.T[0:n].T)
print "dimension means of Y", mean(data.T[n:2 * n + 1].T)
# create shogun feature representation
features = RealFeatures(data)
# 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 = Math.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)
# for nice plotting that fits into our shogun tutorial import latex_plot_inits # 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 = 500 # use data generator class to produce example data data = DataGenerator.generate_mean_data(m, dim, difference) # create shogun feature representation features = RealFeatures(data) # 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 # Using all distances here would blow up memory subset = Math.randperm_vec(features.get_num_vectors()) subset = subset[0:200] features.add_subset(subset) dist = EuclideanDistance(features, features) distances = dist.get_distance_matrix()
def statistics_quadratic_time_mmd (): from shogun.Features import RealFeatures from shogun.Features import DataGenerator 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, Math # note that the quadratic time mmd has to store kernel matrices # which upper bounds the sample size n=500 dim=2 difference=0.5 # use data generator class to produce example data data=DataGenerator.generate_mean_data(n,dim,difference) print "dimension means of X", mean(data.T[0:n].T) print "dimension means of Y", mean(data.T[n:2*n+1].T) # create shogun feature representation features=RealFeatures(data) # 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=Math.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)
