def Cauchy_Entropy(w0, l0, w1, l1):
if w0 == 0 and l0 == 0:
x0 = 0.5
else:
x0 = w0/(w0 + l0) # Mode of beta
sig0 = np.sqrt( (w0+1)*(l0+1)/( (w0+l0+2)**2 * (w0+l0+3) ) ) # Std of beta
if w1 == 0 and l1 ==0:
x1 = 0.5
else:
x1 = w1/(w1 + l1)
sig1 = np.sqrt( (w1+1)*(l1+1)/( (w1+l1+2)**2 * (w1+l1+3) ) )
global MC_samples
seq.reset()
x = seq.get(MC_samples)
x = np.reshape(x, MC_samples)
pdf0 = cauchy.pdf(x, x0, sig0)
pdf1 = cauchy.pdf(x, x1, sig1)
cdf0 = cauchy.cdf(x, x0, sig0)
cdf1 = cauchy.cdf(x, x1, sig1)
rho = pdf0 * cdf1 + pdf1 * cdf0
integral = np.mean( -rho * np.log( rho ) )
return integral
def cauchy_distribution(select_size, asked=rvs, x=0):
if asked == rvs:
return cauchy.rvs(size=select_size)
elif asked == pdf:
return cauchy.pdf(x)
elif asked == cdf:
return cauchy.cdf(x)
def distribution_function(x, mu, sigma, distribution):
if distribution == Distribution.NORMAL:
return norm.cdf(x, mu, sigma)
elif distribution == Distribution.CAUCHY:
return cauchy.cdf(x, mu, sigma)
elif distribution == Distribution.LAPLACE:
return laplace.cdf(x, mu, sigma)
elif distribution == Distribution.POISSON:
return poisson.cdf(x, mu, sigma)
elif distribution == Distribution.UNIFORM:
return uniform.cdf(x, mu, sigma)
else:
return None
def cdf(self, x, df):
return cauchy.cdf(x, self.loc, self.scl)
# prior constraints: # smooth, median(theta)=0, mode(theta)=1, theta can be (-inf,1),(-1,0),(0,1),(1,inf) with p=0.25 # taking a gaussian prior which satisfies the above constraints prior_var = 2.19 prior_mu = 0 prior_std = math.sqrt(prior_var) p_range = norm.cdf(1, prior_mu, prior_std) - norm.cdf(-1, prior_mu, prior_std) assert np.allclose(p_range, 0.5, 1e-2) # Computing posterior mean using gaussian prior post_var = 1 / (1 / obs_var + 1 / prior_var) post_mean = post_var * (prior_mu / prior_var + obs_x / obs_var) assert np.allclose(post_mean, 3.43, 1e-2) # taking a cauchy prior which satisfies the above constraints loc = 0 scale = 1 p_range = cauchy.cdf(1, loc, scale) - cauchy.cdf(-1, loc, scale) assert np.allclose(p_range, 0.5, 1e-2) # Computing posterior mean using cauchy prior inf = 5.2 lik = lambda theta: norm.pdf(obs_x, theta, obs_std) prior = lambda theta: cauchy.pdf(obs_x, theta, obs_std) post = lambda theta: lik(theta) * prior(theta) Z = integrate.quad(post, -inf, inf)[0] post_mean = integrate.quad(lambda theta: theta * post(theta) / Z, -inf, inf)[0] assert np.allclose(post_mean, 4.56, 1e-2)
def func(self, x):
return cauchy.cdf(x, loc=self.mu, scale=self.sigma)
def __call__(self, x, loc=0, scale=1):
return cauchy.cdf(x, loc=loc, scale=scale)
# Display the probability density function (``pdf``): x = np.linspace(cauchy.ppf(0.01), cauchy.ppf(0.99), 100) ax.plot(x, cauchy.pdf(x), 'r-', lw=5, alpha=0.6, label='cauchy pdf') # Alternatively, the distribution object can be called (as a function) # to fix the shape, location and scale parameters. This returns a "frozen" # RV object holding the given parameters fixed. # Freeze the distribution and display the frozen ``pdf``: rv = cauchy() ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf') # Check accuracy of ``cdf`` and ``ppf``: vals = cauchy.ppf([0.001, 0.5, 0.999]) np.allclose([0.001, 0.5, 0.999], cauchy.cdf(vals)) # True # Generate random numbers: r = cauchy.rvs(size=1000) # And compare the histogram: ax.hist(r, normed=True, histtype='stepfilled', alpha=0.2) ax.legend(loc='best', frameon=False) plt.show()
def main():
usage = 'usage: %prog [options] arg'
parser = OptionParser(usage)
parser.add_option('-o', dest='out_dir', default='sad_norm')
parser.add_option(
'-s',
dest='sample',
default=100000,
type='int',
help='Number of SNPs to sample for fit [Default: %default]')
(options, args) = parser.parse_args()
if len(args) != 1:
parser.error('Must provide SAD HDF5 path')
else:
sad_h5_path = args[0]
# retrieve chromosome SAD HDF5 files
chr_sad_h5_files = sorted(glob.glob('%s/*/sad.h5' % sad_h5_path))
assert (len(chr_sad_h5_files) > 0)
# clean out any existing fits
# count SNPs across chromosomes
num_snps = 0
for chr_sad_h5_file in chr_sad_h5_files:
chr_sad_h5 = h5py.File(chr_sad_h5_file, 'r+')
# delete fit params
if 'target_cauchy_fit_loc' in chr_sad_h5.keys():
del chr_sad_h5['target_cauchy_fit_loc']
del chr_sad_h5['target_cauchy_fit_scale']
# delete norm params
if 'target_cauchy_norm_loc' in chr_sad_h5.keys():
del chr_sad_h5['target_cauchy_norm_loc']
del chr_sad_h5['target_cauchy_norm_scale']
# count SNPs
num_snps += chr_sad_h5['SAD'].shape[0]
num_targets = chr_sad_h5['SAD'].shape[-1]
chr_sad_h5.close()
# sample SNPs across chromosomes
sad = sample_sad(chr_sad_h5_files, options.sample, num_snps, num_targets)
# initialize fit parameters
target_cauchy_fit_loc = np.zeros(num_targets)
target_cauchy_fit_scale = np.zeros(num_targets)
# fit parameters
for ti in range(num_targets):
print('Fitting t%d' % ti, flush=True)
cp = cauchy.fit(sad[:, ti])
target_cauchy_fit_loc[ti] = cp[0]
target_cauchy_fit_scale[ti] = cp[1]
del sad
# write across chromosomes
for chr_sad_h5_file in chr_sad_h5_files:
chr_sad_h5 = h5py.File(chr_sad_h5_file, 'r+')
chr_sad_h5.create_dataset('target_cauchy_fit_loc',
data=target_cauchy_fit_loc)
chr_sad_h5.create_dataset('target_cauchy_fit_scale',
data=target_cauchy_fit_scale)
chr_sad_h5.close()
# compute normalization parameters
for chr_sad_h5_file in chr_sad_h5_files:
chr_sad5 = SAD5(chr_sad_h5_file)
# QC fit table
if not os.path.isdir(options.out_dir):
os.mkdir(options.out_dir)
fit_out = open('%s/fits.txt' % options.out_dir, 'w')
for ti in range(num_targets):
print('%-4d %7.1e %7.1e' %
(ti, target_cauchy_fit_loc[ti], target_cauchy_fit_scale[ti]),
file=fit_out)
fit_out.close()
# QC quantiles
quantile_dir = '%s/quantiles' % options.out_dir
if not os.path.isdir(quantile_dir):
os.mkdir(quantile_dir)
sad_qc = sample_sad(chr_sad_h5_files, 2048, num_snps, num_targets)
for ti in np.linspace(0, num_targets - 1, 64, dtype='int'):
# compute cauchy and argsort quantiles
cauchy_q = cauchy.cdf(sad_qc[:, ti],
loc=target_cauchy_fit_loc[ti],
scale=target_cauchy_fit_scale[ti])
sort_i = np.argsort(sad_qc[:, ti])
quantile_pdf = '%s/t%d.pdf' % (quantile_dir, ti)
jointplot(np.linspace(0, 1, len(sort_i)),
cauchy_q[sort_i],
quantile_pdf,
square=True,
cor=None,
x_label='Empirical',
y_label='Cauchy')
# QC plots
norm_dir = '%s/norm' % options.out_dir
if not os.path.isdir(norm_dir):
os.mkdir(norm_dir)
chr_sad5 = SAD5(chr_sad_h5_files[0])
qc_sample = 2048
if qc_sample < chr_sad5.num_snps:
ri = sorted(
np.random.choice(np.arange(chr_sad5.num_snps),
size=qc_sample,
replace=False))
else:
ri = np.arange(chr_sad5.num_snps)
qc_sad_raw = chr_sad5.sad_matrix[ri]
qc_sad_norm = chr_sad5[ri]
for ti in np.linspace(0, num_targets - 1, 32, dtype='int'):
plt.figure()
sns.jointplot(qc_sad_raw[:, ti],
qc_sad_norm[:, ti],
joint_kws={
'alpha': 0.5,
's': 10
})
plt.savefig('%s/t%d.pdf' % (norm_dir, ti))
plt.close()
ax[2].plot(x3, laplace.cdf(x3, 0, sqrt(2)))
ax[2].hist(r3,
density=True,
bins=floor(len(r3)),
histtype='step',
cumulative=True)
ax[2].set_xlabel('x')
ax[2].set_title('Распределение Лапласа, n=100')
#plt.show()#Лапласа
fig2, ay = plt.subplots(1, 3)
r = cauchy.rvs(size=20)
y = np.linspace(min(cauchy.ppf(0.01), min(r)), max(cauchy.ppf(0.99), max(r)),
100)
ay[0].plot(y, cauchy.cdf(y, 0, 1))
ay[0].hist(r,
density=True,
bins=floor(len(r)),
histtype='step',
cumulative=True)
ay[0].set_xlabel('x')
ay[0].set_title('Распределение Коши, n=20')
r2 = cauchy.rvs(size=60)
y2 = np.linspace(min(cauchy.ppf(0.01), min(r2)), max(cauchy.ppf(0.99),
max(r2)), 100)
ay[1].plot(y2, cauchy.cdf(y2, 0, 1))
ay[1].hist(r2,
density=True,
bins=floor(len(r2)),
def main():
usage = "usage: %prog [options] arg"
parser = OptionParser(usage)
parser.add_option("-o", dest="out_dir", default="sad_norm")
parser.add_option(
"-s",
dest="sample",
default=100000,
type="int",
help="Number of SNPs to sample for fit [Default: %default]",
)
(options, args) = parser.parse_args()
if len(args) != 1:
parser.error("Must provide SAD HDF5 path")
else:
sad_h5_path = args[0]
# retrieve chromosome SAD HDF5 files
chr_sad_h5_files = sorted(glob.glob("%s/*/sad.h5" % sad_h5_path))
assert len(chr_sad_h5_files) > 0
# clean out any existing fits
# count SNPs across chromosomes
num_snps = 0
for chr_sad_h5_file in chr_sad_h5_files:
chr_sad_h5 = h5py.File(chr_sad_h5_file, "r+")
# delete fit params
if "target_cauchy_fit_loc" in chr_sad_h5.keys():
del chr_sad_h5["target_cauchy_fit_loc"]
del chr_sad_h5["target_cauchy_fit_scale"]
# delete norm params
if "target_cauchy_norm_loc" in chr_sad_h5.keys():
del chr_sad_h5["target_cauchy_norm_loc"]
del chr_sad_h5["target_cauchy_norm_scale"]
# count SNPs
num_snps += chr_sad_h5["SAD"].shape[0]
num_targets = chr_sad_h5["SAD"].shape[-1]
chr_sad_h5.close()
# sample SNPs across chromosomes
sad = sample_sad(chr_sad_h5_files, options.sample, num_snps, num_targets)
# initialize fit parameters
target_cauchy_fit_loc = np.zeros(num_targets)
target_cauchy_fit_scale = np.zeros(num_targets)
# fit parameters
for ti in range(num_targets):
print("Fitting t%d" % ti, flush=True)
cp = cauchy.fit(sad[:, ti])
target_cauchy_fit_loc[ti] = cp[0]
target_cauchy_fit_scale[ti] = cp[1]
del sad
# write across chromosomes
for chr_sad_h5_file in chr_sad_h5_files:
chr_sad_h5 = h5py.File(chr_sad_h5_file, "r+")
chr_sad_h5.create_dataset("target_cauchy_fit_loc", data=target_cauchy_fit_loc)
chr_sad_h5.create_dataset(
"target_cauchy_fit_scale", data=target_cauchy_fit_scale
)
chr_sad_h5.close()
# compute normalization parameters
for chr_sad_h5_file in chr_sad_h5_files:
chr_sad5 = SAD5(chr_sad_h5_file)
# QC fit table
if not os.path.isdir(options.out_dir):
os.mkdir(options.out_dir)
fit_out = open("%s/fits.txt" % options.out_dir, "w")
for ti in range(num_targets):
print(
"%-4d %7.1e %7.1e"
% (ti, target_cauchy_fit_loc[ti], target_cauchy_fit_scale[ti]),
file=fit_out,
)
fit_out.close()
# QC quantiles
quantile_dir = "%s/quantiles" % options.out_dir
if not os.path.isdir(quantile_dir):
os.mkdir(quantile_dir)
sad_qc = sample_sad(chr_sad_h5_files, 2048, num_snps, num_targets)
for ti in np.linspace(0, num_targets - 1, 64, dtype="int"):
# compute cauchy and argsort quantiles
cauchy_q = cauchy.cdf(
sad_qc[:, ti],
loc=target_cauchy_fit_loc[ti],
scale=target_cauchy_fit_scale[ti],
)
sort_i = np.argsort(sad_qc[:, ti])
quantile_pdf = "%s/t%d.pdf" % (quantile_dir, ti)
jointplot(
np.linspace(0, 1, len(sort_i)),
cauchy_q[sort_i],
quantile_pdf,
square=True,
cor=None,
x_label="Empirical",
y_label="Cauchy",
)
# QC plots
norm_dir = "%s/norm" % options.out_dir
if not os.path.isdir(norm_dir):
os.mkdir(norm_dir)
chr_sad5 = SAD5(chr_sad_h5_files[0])
qc_sample = 2048
if qc_sample < chr_sad5.num_snps:
ri = sorted(
np.random.choice(
np.arange(chr_sad5.num_snps), size=qc_sample, replace=False
)
)
else:
ri = np.arange(chr_sad5.num_snps)
qc_sad_raw = chr_sad5.sad_matrix[ri]
qc_sad_norm = chr_sad5[ri]
for ti in np.linspace(0, num_targets - 1, 32, dtype="int"):
plt.figure()
sns.jointplot(
qc_sad_raw[:, ti], qc_sad_norm[:, ti], joint_kws={"alpha": 0.5, "s": 10}
)
plt.savefig("%s/t%d.pdf" % (norm_dir, ti))
plt.close()
font = ImageFont.truetype(font_loc, 12)
one_sided = False
for i in range(17):
ix = zigzag2(i, 0, 10)
im = Image.new("RGB", (512, 512), "black")
draw = ImageDraw.Draw(im, 'RGBA')
if one_sided:
fn = lambda x: 300 - norm.cdf(x, 250, std) * 100
else:
fn_tmp = lambda x: min(norm.cdf(x, 250, std), norm.sf(x, 250, std))
fn = lambda x: 300 - (fn_tmp(x)) * 200
draw_curve(fn, draw, rgba=(255, 165, 0))
if one_sided:
fn1 = lambda x: 300 - cauchy.cdf(x, 250, std) * 100
else:
fn_tmp1 = lambda x: min(cauchy.cdf(x, 250, std), cauchy.sf(
x, 250, std))
fn1 = lambda x: 300 - (fn_tmp1(x)) * 200
draw_curve(fn1, draw, rgba=(222, 49, 99))
draw.line((250, 200, 250, 300), "white")
draw.line((0, 300, 512, 300), "yellow")
y1 = 300
x1 = 50 + ix / 10 * (450 - 50)
draw.ellipse((x1 - 3, y1 - 3, x1 + 3, y1 + 3),
outline=(255, 255, 0),
fill=(255, 255, 0, 150))
x2 = 250
y1 = fn(x1)
from scipy.stats import cauchy print(cauchy.cdf(2,3,3))
ax[1][2].legend()
plt.show()#Лапласа
fig2, ay = plt.subplots(2, 3)
r = cauchy.rvs(size=10)
y = np.linspace(min(cauchy.ppf(0.01),min(r)), max(cauchy.ppf(0.99),max(r)), 100)
ay[0][0].plot(y, cauchy.pdf(y,0,1), label='теор.')
ay[0][0].set_title('Распределение Коши, n=10')
ay[0][0].hist(r, density=True, bins=floor(sqrt(len(r))),histtype='step',label='практ.')
ay[0][0].legend(loc='best', frameon=False)
ay[0][0].set_xlabel('x')
ay[0][0].set_ylabel('Функция плотности')
ay[0][0].legend()
ay[1][0].plot(y, cauchy.cdf(y,0,1), label='теор.')
ay[1][0].hist(r, density=True, bins=floor(sqrt(len(r))),histtype='step',cumulative=True,label='практ.')
ay[1][0].legend(loc='best', frameon=False)
ay[1][0].set_xlabel('x')
ay[1][0].set_ylabel('Функция распределения')
ay[1][0].legend()
r2 = cauchy.rvs(size=100)
y2 = np.linspace(min(cauchy.ppf(0.01),min(r2)), max(cauchy.ppf(0.99),max(r2)), 100)
ay[0][1].plot(y2, cauchy.pdf(y2,0,1), label='теор.')
ay[0][1].set_title('Распределение Коши, n=100')
ay[0][1].hist(r2, density=True, bins=floor(sqrt(len(r2))),histtype='step',label='практ.')
ay[0][1].legend(loc='best', frameon=False)
ay[0][1].set_xlabel('x')
ay[0][1].set_ylabel('Функция плотности')
ay[0][1].legend()