def _cdf(self, statistic, samples):
# Some simple, exact cases (more in Ruben & Gambino).
if statistic <= 1 / (2 * samples):
return 0.
if statistic >= 1:
return 1.
if statistic <= 1 / samples:
t = 2 * statistic - 1 / samples
return exp(gammaln(samples + 1) + samples * log(t))
if statistic >= 1 - 1 / samples:
return 1 - 2 * (1 - statistic)**samples
# For small sample counts we may use an exact method when needed.
if samples < 150:
# With samples = 150 the matrix calculation takes about 100 ms
# on a ~3 GFLOPS/core processor.
if samples * statistic**2 < 7:
# For a small threshold the Durbin matrix will be small.
return ks_unif_durbin_matrix(samples, statistic)
else:
# Double the one-sided probability; accurate when close to one.
return 1 - 2 * smirnov(samples, statistic)
# Further we need to make a compromise between speed and accuracy.
if samples < 100000 and samples * statistic**1.5 < 1.4:
# The cost of the matrix calculation should still be acceptable.
return ks_unif_durbin_matrix(samples, statistic)
else:
# No options left, but to use an asymptotic approximation.
return ks_unif_pelz_good(samples, statistic)
def test_n_large(self):
# test for large values of n
# Probabilities should go down as n goes up
x = 0.4
pvals = np.array([smirnov(n, x) for n in range(400, 1100, 20)])
dfs = np.diff(pvals)
assert_(np.all(dfs <= 0), msg='Not all diffs negative %s' % dfs)
def _cdf(self, statistic, samples):
# Some simple, exact cases (more in Ruben & Gambino).
if statistic <= 1 / (2 * samples):
return 0.
if statistic >= 1:
return 1.
if statistic <= 1 / samples:
t = 2 * statistic - 1 / samples
return exp(gammaln(samples + 1) + samples * log(t))
if statistic >= 1 - 1 / samples:
return 1 - 2 * (1 - statistic) ** samples
# For small sample counts we may use an exact method when needed.
if samples < 150:
# With samples = 150 the matrix calculation takes about 100 ms
# on a ~3 GFLOPS/core processor.
if samples * statistic ** 2 < 7:
# For a small threshold the Durbin matrix will be small.
return ks_unif_durbin_matrix(samples, statistic)
else:
# Double the one-sided probability; accurate when close to one.
return 1 - 2 * smirnov(samples, statistic)
# Further we need to make a compromise between speed and accuracy.
if samples < 100000 and samples * statistic ** 1.5 < 1.4:
# The cost of the matrix calculation should still be acceptable.
return ks_unif_durbin_matrix(samples, statistic)
else:
# No options left, but to use an asymptotic approximation.
return ks_unif_pelz_good(samples, statistic)
def test_n_large(self):
# test for large values of n
# Probabilities should go down as n goes up
x = 0.4
pvals = np.array([smirnov(n, x) for n in range(400, 1100, 20)])
dfs = np.diff(pvals)
assert_(np.all(dfs <= 0), msg='Not all diffs negative %s' % dfs)
def concentration_pfa(membership, ms_size, trim=False):
membership = membership[membership > 0]
if trim:
ones = np.where(membership > 1 - 1e-5)[0]
membership = np.delete(membership, ones[-min(len(ones), ms_size):])
if len(membership) > 1:
d_min, _ = kstest(membership, 'uniform', alternative='less')
pvalue = smirnov(len(membership), d_min)
else:
pvalue = 1.
if pvalue == 0:
return -300
else:
return np.log10(pvalue)
def _sf(self, statistic, samples):
if statistic >= 1:
# Statistic greater than 1 results in a NaN from Cephes smirnov().
return 0.
if statistic >= 1 - 1 / samples:
# The _cdf code can suffer from some cancellation in this case.
return min(1., 2 * (1 - statistic)**samples)
probability = 1 - self._cdf(statistic, samples)
if probability > 1e-5:
# Not too much precision got lost to cancellation.
return probability
else:
# When the cdf float is very close to one it does not have bits
# of small enough magnitude to express its 1-complement properly.
# Hence, an approximate direct sf calculation may be more precise.
return min(1., 2 * smirnov(samples, statistic))
def test_n_large(self):
# test for large values of n
# Probabilities should go down as n goes up
x = 0.4
pvals = np.array([smirnov(n, x) for n in range(400, 1100, 20)])
dfs = np.diff(pvals)
assert_(np.all(dfs <= 0), msg='Not all diffs negative %s' % dfs)
dataset = [(1000, 1 - 1.0/2000, np.power(2000.0, -1000))]
dataset = np.asarray(dataset)
FuncData(smirnov, dataset, (0, 1), 2, rtol=_rtol).check()
# Check asymptotic behaviour
dataset = [(n, 1.0 / np.sqrt(n), np.exp(-2)) for n in range(1000, 5000, 1000)]
dataset = np.asarray(dataset)
FuncData(smirnov, dataset, (0, 1), 2, rtol=.05).check()
def _sf(self, statistic, samples):
if statistic >= 1:
# Statistic greater than 1 results in a NaN from Cephes smirnov().
return 0.
if statistic >= 1 - 1 / samples:
# The _cdf code can suffer from some cancellation in this case.
return min(1., 2 * (1 - statistic) ** samples)
probability = 1 - self._cdf(statistic, samples)
if probability > 1e-5:
# Not too much precision got lost to cancellation.
return probability
else:
# When the cdf float is very close to one it does not have bits
# of small enough magnitude to express its 1-complement properly.
# Hence, an approximate direct sf calculation may be more precise.
return min(1., 2 * smirnov(samples, statistic))
def test_n_large(self):
# test for large values of n
# Probabilities should go down as n goes up
x = 0.4
pvals = np.array([smirnov(n, x) for n in range(400, 1100, 20)])
dfs = np.diff(pvals)
assert_(np.all(dfs <= 0), msg='Not all diffs negative %s' % dfs)
dataset = [(1000, 1 - 1.0 / 2000, np.power(2000.0, -1000))]
dataset = np.asarray(dataset)
FuncData(smirnov, dataset, (0, 1), 2, rtol=_rtol).check()
# Check asymptotic behaviour
dataset = [(n, 1.0 / np.sqrt(n), np.exp(-2))
for n in range(1000, 5000, 1000)]
dataset = np.asarray(dataset)
FuncData(smirnov, dataset, (0, 1), 2, rtol=.05).check()
def ks1(data, model, x, ord=np.inf):
vals = np.sort(data)
# build the continuous cdf
total = np.sum(model)
ccdf = np.zeros(vals.shape)
for i in xrange(len(vals)):
idx = x <= vals[i]
ccdf[i] = np.sum(model[idx]) / total
# build the discrete cdf
N = len(vals)
dcdf = np.cumsum(np.ones(vals.shape)) / N
d = norm(dcdf-ccdf, ord)
p = smirnov(N, d)
return ccdf, dcdf, d, p
def _sm_smi(n, p):
return smirnov(n, smirnovi(n, p))
def test_nan(self):
assert_(np.isnan(smirnov(1, np.nan)))
def _sm_smi(n, p):
return smirnov(n, smirnovi(n, p))
def test_nan(self):
assert_(np.isnan(smirnov(1, np.nan)))
from scipy.special import smirnov
# Show the probability of a gap at least as big as 0, 0.5 and 1.0 for a sample of size 5
smirnov(5, [0, 0.5, 1.0])
# array([ 1. , 0.056, 0. ])
# Compare a sample of size 5 drawn from a source N(0.5, 1) distribution against
# a target N(0, 1) CDF.
from scipy.stats import norm
n = 5
gendist = norm(0.5, 1) # Normal distribution, mean 0.5, stddev 1
np.random.seed(seed=233423) # Set the seed for reproducibility
x = np.sort(gendist.rvs(size=n))
x
# array([-0.20946287, 0.71688765, 0.95164151, 1.44590852, 3.08880533])
target = norm(0, 1)
cdfs = target.cdf(x)
cdfs
# array([ 0.41704346, 0.76327829, 0.82936059, 0.92589857, 0.99899518])
# # Construct the Empirical CDF and the K-S statistics (Dn+, Dn-, Dn)
ecdfs = np.arange(n + 1, dtype=float) / n
cols = np.column_stack(
[x, ecdfs[1:], cdfs, cdfs - ecdfs[:n], ecdfs[1:] - cdfs])
np.set_printoptions(precision=3)
cols
# array([[ -2.095e-01, 2.000e-01, 4.170e-01, 4.170e-01, -2.170e-01],
# [ 7.169e-01, 4.000e-01, 7.633e-01, 5.633e-01, -3.633e-01],
# [ 9.516e-01, 6.000e-01, 8.294e-01, 4.294e-01, -2.294e-01],
# [ 1.446e+00, 8.000e-01, 9.259e-01, 3.259e-01, -1.259e-01],
