def check_mean_sigma_keepdims(a, axis):
mu1, sigma1 = mean_sigma(a, axis, keepdims=False)
mu2, sigma2 = mean_sigma(a, axis, keepdims=True)
assert_array_equal(mu1.ravel(), mu2.ravel())
assert_array_equal(sigma1.ravel(), sigma2.ravel())
assert_array_equal(np.broadcast(a, mu2).shape, a.shape)
assert_array_equal(np.broadcast(a, sigma2).shape, a.shape)
def check_mean_sigma_keepdims(a, axis):
mu1, sigma1 = mean_sigma(a, axis, keepdims=False)
mu2, sigma2 = mean_sigma(a, axis, keepdims=True)
assert_array_equal(mu1.ravel(), mu2.ravel())
assert_array_equal(sigma1.ravel(), sigma2.ravel())
assert_array_equal(np.broadcast(a, mu2).shape, a.shape)
assert_array_equal(np.broadcast(a, sigma2).shape, a.shape)
def test_mean_sigma_keepdims(axis):
np.random.seed(0)
a = np.random.random((4, 5, 6))
mu1, sigma1 = mean_sigma(a, axis, keepdims=False)
mu2, sigma2 = mean_sigma(a, axis, keepdims=True)
assert_array_equal(mu1.ravel(), mu2.ravel())
assert_array_equal(sigma1.ravel(), sigma2.ravel())
assert_array_equal(np.broadcast(a, mu2).shape, a.shape)
assert_array_equal(np.broadcast(a, sigma2).shape, a.shape)
def check_mean_sigma(a, axis=None, ddof=0):
mu1, sigma1 = mean_sigma(a, axis=axis, ddof=ddof)
mu2 = np.mean(a, axis=axis)
sigma2 = np.std(a, axis=axis, ddof=ddof)
assert_array_almost_equal(mu1, mu2)
assert_array_almost_equal(sigma1, sigma2)
def test_bootstrap_results():
np.random.seed(0)
x = np.random.normal(0, 1, 100)
distribution = bootstrap(x, 100, np.mean, kwargs=dict(axis=1), random_state=0)
mu, sigma = mean_sigma(distribution)
assert_allclose([mu, sigma], [0.08139846, 0.10465327])
def check_median_sigmaG_approx(a, axis, keepdims, atol=0.15):
med, sigmaG = median_sigmaG(a, axis=axis, keepdims=keepdims)
mu, sigma = mean_sigma(a, axis=axis, ddof=1, keepdims=keepdims)
print np.max(abs(med - mu))
print np.max(abs(sigmaG - sigma))
assert_allclose(med, mu, atol=atol)
assert_allclose(sigmaG, sigma, atol=atol)
def check_median_sigmaG_approx(a, axis, keepdims, atol=0.15):
med, sigmaG = median_sigmaG(a, axis=axis, keepdims=keepdims)
mu, sigma = mean_sigma(a, axis=axis, ddof=1, keepdims=keepdims)
print np.max(abs(med - mu))
print np.max(abs(sigmaG - sigma))
assert_allclose(med, mu, atol=atol)
assert_allclose(sigmaG, sigma, atol=atol)
def test_median_sigmaG_approx(axis, keepdims, atol=0.02):
np.random.seed(0)
a = np.random.normal(0, 1, size=(10, 10000))
med, sigmaG = median_sigmaG(a, axis=axis, keepdims=keepdims)
mu, sigma = mean_sigma(a, axis=axis, ddof=1, keepdims=keepdims)
assert_allclose(med, mu, atol=atol)
assert_allclose(sigmaG, sigma, atol=atol)
def test_bootstrap_results():
np.random.seed(0)
x = np.random.normal(0, 1, 100)
distribution = bootstrap(x, 100, np.mean, kwargs=dict(axis=1),
random_state=0)
mu, sigma = mean_sigma(distribution)
assert_allclose([mu, sigma], [0.08139846, 0.10465327])
def check_mean_sigma(a, axis=None, ddof=0):
mu1, sigma1 = mean_sigma(a, axis=axis,
ddof=ddof)
mu2 = np.mean(a, axis=axis)
sigma2 = np.std(a, axis=axis, ddof=ddof)
assert_array_almost_equal(mu1, mu2)
assert_array_almost_equal(sigma1, sigma2)
def test_mean_sigma(a_shape, axis, ddof):
np.random.seed(0)
a = np.random.random(a_shape)
mu1, sigma1 = mean_sigma(a, axis=axis, ddof=ddof)
mu2 = np.mean(a, axis=axis)
sigma2 = np.std(a, axis=axis, ddof=ddof)
assert_array_almost_equal(mu1, mu2)
assert_array_almost_equal(sigma1, sigma2)
def test_mean_sigma(a_shape, axis, ddof):
np.random.seed(0)
a = np.random.random(a_shape)
mu1, sigma1 = mean_sigma(a, axis=axis,
ddof=ddof)
mu2 = np.mean(a, axis=axis)
sigma2 = np.std(a, axis=axis, ddof=ddof)
assert_array_almost_equal(mu1, mu2)
assert_array_almost_equal(sigma1, sigma2)
sigma1 = 1
sigma2 = 3
np.random.seed(1)
x = np.hstack((np.random.normal(0, sigma1, Npts - N_out),
np.random.normal(0, sigma2, N_out)))
#------------------------------------------------------------
# Compute anderson-darling test
A2, sig, crit = anderson(x)
print("anderson-darling A^2 = {0:.1f}".format(A2))
#------------------------------------------------------------
# Compute non-robust and robust point statistics
mu_sample, sig_sample = mean_sigma(x)
med_sample, sigG_sample = median_sigmaG(x)
#------------------------------------------------------------
# Plot the results
fig, ax = plt.subplots(figsize=(5, 3.75))
# histogram of data
ax.hist(x, 100, histtype='stepfilled', alpha=0.2,
color='k', normed=True)
# best-fit normal curves
x_sample = np.linspace(-15, 15, 1000)
ax.plot(x_sample, norm(mu_sample, sig_sample).pdf(x_sample), '-k',
label='$\sigma$ fit')
ax.plot(x_sample, norm(med_sample, sigG_sample).pdf(x_sample), '--k',
def check_median_sigmaG_approx(a, axis, keepdims, atol=0.15):
med, sigmaG = median_sigmaG(a, axis=axis, keepdims=keepdims)
mu, sigma = mean_sigma(a, axis=axis, ddof=1, keepdims=keepdims)
assert_allclose(med, mu, atol=atol)
assert_allclose(sigmaG, sigma, atol=atol)
def gaussian(x, mu, sigma):
return np.exp(-0.5 * (x - mu) ** 2 / sigma ** 2)
#------------------------------------------------------------
# Draw a random sample from the distribution, and compute
# some quantities
n = 10
xbar = 1
V = 4
sigma_x = np.sqrt(V)
np.random.seed(10)
xi = np.random.normal(xbar, sigma_x, size=n)
mu_mean, sig_mean = mean_sigma(xi, ddof=1)
# compute the analytically expected spread in measurements
mu_std = sig_mean / np.sqrt(n)
sig_std = sig_mean / np.sqrt(2 * (n - 1))
#------------------------------------------------------------
# bootstrap estimates
mu_bootstrap, sig_bootstrap = bootstrap(xi, 1E6, mean_sigma,
kwargs=dict(ddof=1, axis=1))
#------------------------------------------------------------
# Compute analytic posteriors
# distributions for the mean
mu = np.linspace(-3, 5, 1000)
bottom=0.06, top=0.95,
hspace=0.1)
ax = fig.add_subplot(1, 1, 1)
avg = np.mean(qsos_m[:, i])
std = np.std(qsos_m[:, i])
data = (qsos_m[:, i] - avg) / std
x = np.linspace(-5, 5, 1000)
pdf = stats.norm(0, 1).pdf(x)
A2, sig, crit = stats.anderson(data)
D, pD = stats.kstest(data, "norm")
W, pW = stats.shapiro(data)
mu, sigma = mean_sigma(data, ddof=1)
median, sigmaG = median_sigmaG(data)
N = len(data)
Z1 = 1.3 * abs(mu - median) / sigma * np.sqrt(N)
Z2 = 1.1 * abs(sigma / sigmaG - 1) * np.sqrt(N)
print 70 * '_'
print " Kolmogorov-Smirnov test: D = %.2g p = %.2g" % (D, pD)
print " Anderson-Darling test: A^2 = %.2g" % A2
print " significance | critical value "
print " --------------|----------------"
for j in range(len(sig)):
print " %.2f | %.1f%%" % (sig[j], crit[j])
print " Shapiro-Wilk test: W = %.2g p = %.2g" % (W, pW)
print " Z_1 = %.1f" % Z1
# draw underlying points
np.random.seed(0)
Npts = 1E6
x = np.random.normal(loc=0, scale=1, size=Npts)
# add error for each point
e = 3 * np.random.random(Npts)
x += np.random.normal(0, e)
# compute anderson-darling test
A2, sig, crit = anderson(x)
print "anderson-darling A^2 = %.1f" % A2
# compute point statistics
mu_sample, sig_sample = mean_sigma(x, ddof=1)
med_sample, sigG_sample = median_sigmaG(x)
#------------------------------------------------------------
# plot the results
fig, ax = plt.subplots(figsize=(5, 3.75))
ax.hist(x, 100, histtype='stepfilled', alpha=0.2,
color='k', normed=True)
# plot the fitting normal curves
x_sample = np.linspace(-15, 15, 1000)
ax.plot(x_sample, norm(mu_sample, sig_sample).pdf(x_sample),
'-k', label='$\sigma$ fit')
ax.plot(x_sample, norm(med_sample, sigG_sample).pdf(x_sample),
'--k', label='$\sigma_G$ fit')
ax.legend()
#------------------------------------------------------------
# Compute the statistics and plot the results
fig = plt.figure(figsize=(5, 7))
fig.subplots_adjust(left=0.13, right=0.95,
bottom=0.06, top=0.95,
hspace=0.1)
for i in range(2):
ax = fig.add_subplot(2, 1, 1 + i) # 2 x 1 subplot
# compute some statistics
A2, sig, crit = stats.anderson(vals[i])
D, pD = stats.kstest(vals[i], "norm")
W, pW = stats.shapiro(vals[i])
mu, sigma = mean_sigma(vals[i], ddof=1)
median, sigmaG = median_sigmaG(vals[i])
N = len(vals[i])
Z1 = 1.3 * abs(mu - median) / sigma * np.sqrt(N)
Z2 = 1.1 * abs(sigma / sigmaG - 1) * np.sqrt(N)
print 70 * '_'
print " Kolmogorov-Smirnov test: D = %.2g p = %.2g" % (D, pD)
print " Anderson-Darling test: A^2 = %.2g" % A2
print " significance | critical value "
print " --------------|----------------"
for j in range(len(sig)):
print " %.2f | %.1f%%" % (sig[j], crit[j])
print " Shapiro-Wilk test: W = %.2g p = %.2g" % (W, pW)
print " Z_1 = %.1f" % Z1
sigma1 = 1
sigma2 = 3
np.random.seed(1)
x = np.hstack(
(np.random.normal(0, sigma1,
Npts - N_out), np.random.normal(0, sigma2, N_out)))
#------------------------------------------------------------
# Compute anderson-darling test
A2, sig, crit = anderson(x)
print "anderson-darling A^2 = %.1f" % A2
#------------------------------------------------------------
# Compute non-robust and robust point statistics
mu_sample, sig_sample = mean_sigma(x)
med_sample, sigG_sample = median_sigmaG(x)
#------------------------------------------------------------
# Plot the results
fig, ax = plt.subplots(figsize=(5, 3.75))
# histogram of data
ax.hist(x, 100, histtype='stepfilled', alpha=0.2, color='k', normed=True)
# best-fit normal curves
x_sample = np.linspace(-15, 15, 1000)
ax.plot(x_sample,
norm(mu_sample, sig_sample).pdf(x_sample),
'-k',
label='$\sigma$ fit')
# draw underlying points
np.random.seed(0)
Npts = int(1E6)
x = np.random.normal(loc=0, scale=1, size=Npts)
# add error for each point
e = 3 * np.random.random(Npts)
x += np.random.normal(0, e)
# compute anderson-darling test
A2, sig, crit = anderson(x)
print("anderson-darling A^2 = %.1f" % A2)
# compute point statistics
mu_sample, sig_sample = mean_sigma(x, ddof=1)
med_sample, sigG_sample = median_sigmaG(x)
#------------------------------------------------------------
# plot the results
fig, ax = plt.subplots(figsize=(5, 3.75))
ax.hist(x, 100, histtype='stepfilled', alpha=0.2, color='k', density=True)
# plot the fitting normal curves
x_sample = np.linspace(-15, 15, 1000)
ax.plot(x_sample,
norm(mu_sample, sig_sample).pdf(x_sample),
'-k',
label='$\sigma$ fit')
ax.plot(x_sample,
norm(med_sample, sigG_sample).pdf(x_sample),
#------------------------------------------------------------
# Compute the statistics and plot the results
fig = plt.figure(figsize=(5, 7))
fig.subplots_adjust(left=0.13, right=0.95,
bottom=0.06, top=0.95,
hspace=0.1)
for i in range(2):
ax = fig.add_subplot(2, 1, 1 + i) # 2 x 1 subplot
# compute some statistics
A2, sig, crit = stats.anderson(vals[i])
D, pD = stats.kstest(vals[i], "norm")
W, pW = stats.shapiro(vals[i])
mu, sigma = mean_sigma(vals[i], ddof=1)
median, sigmaG = median_sigmaG(vals[i])
N = len(vals[i])
Z1 = 1.3 * abs(mu - median) / sigma * np.sqrt(N)
Z2 = 1.1 * abs(sigma / sigmaG - 1) * np.sqrt(N)
print(70 * '_')
print(" Kolmogorov-Smirnov test: D = %.2g p = %.2g" % (D, pD))
print(" Anderson-Darling test: A^2 = %.2g" % A2)
print(" significance | critical value ")
print(" --------------|----------------")
for j in range(len(sig)):
print(" {0:.2f} | {1:.1f}%".format(sig[j], crit[j]))
print(" Shapiro-Wilk test: W = %.2g p = %.2g" % (W, pW))
print(" Z_1 = %.1f" % Z1)
