def derivative_quant_err(m, p, dist='norm', pw_opt=2):
'''
Compute the derivative of expected variance of quantization error
'''
from scipy.stats import norm, laplace
if dist == 'norm':
cdf_func = norm.cdf(p)
pdf_func = norm.pdf(p)
elif dist == 'laplace':
# https://en.wikipedia.org/wiki/Laplace_distribution
cdf_func = laplace.cdf(p, 0, np.sqrt(0.5))
pdf_func = laplace.pdf(p, 0, np.sqrt(0.5)) # pdf(p, a, b) has variance 2*b^2
else:
raise RuntimeError("Not implemented for distribution: %s !!!" % dist)
## option 1: overlapping
if pw_opt == 1:
# quant_err = [F(p) - F(-p)] * p^2 + 2*[F(m) - F(p)] * m^2
df_dp = 2 * pdf_func * (p * p - m * m) + 2 * p * (2 * cdf_func - 1.0)
## option 2: non-overlapping
else:
# quant_err = [F(p) - F(-p)] * p^2 + 2*[F(m) - F(p)] * (m - p)^2
df_dp = p - 2 * m + 2 * m * cdf_func + m * pdf_func * (2 * p - m)
return df_dp
def laplace(shape, scale):
"""
Standard Laplace noise multiplied by `scale`
Parameters
----------
shape : tuple
Shape of noise.
scale : float
Scale of noise.
"""
rv = laplace(scale=scale, loc=0.)
density = lambda x: np.product(rv.pdf(x))
cdf = lambda x: laplace.cdf(x, loc=0., scale=scale)
pdf = lambda x: laplace.pdf(x, loc=0., scale=scale)
derivative_log_density = lambda x: -np.sign(x) / scale
grad_negative_log_density = lambda x: np.sign(x) / scale
sampler = lambda size: rv.rvs(size=shape + size)
constant = -np.product(shape) * np.log(2 * scale)
return randomization(shape,
density,
cdf,
pdf,
derivative_log_density,
grad_negative_log_density,
sampler,
lipschitz=1. / scale**2,
log_density=lambda x: -np.fabs(np.atleast_2d(x)).
sum(1) / scale - np.log(scale) + constant)
def testLaplace1(seed):
# Test that laplace distribution does what it should
ripl = get_ripl(seed=seed)
# samples
ripl.assume("a", "(laplace -3 2)", label="pid")
observed = collectSamples(ripl, "pid")
# true CDF
laplace_cdf = lambda x: laplace.cdf(x, -3, 2)
return reportKnownContinuous(laplace_cdf, observed)
def laplace_distribution(select_size,
scale=1 / m.sqrt(2),
loc=0,
asked=rvs,
x=0):
if asked == rvs:
return laplace.rvs(size=select_size, scale=scale, loc=loc)
elif asked == pdf:
return laplace.pdf(x, loc=loc, scale=scale)
elif asked == cdf:
return laplace.cdf(x, loc=loc, scale=scale)
return
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 compute_asst_loglik(prec_df, param_df, i, precdist):
theta = param_df.loc[i, 'theta']
loc = param_df.loc[i, 'loc']
scale = param_df.loc[i, 'scale']
if precdist == 'uniform':
cdf_val = uniform.cdf(prec_df['midpoint'], loc, scale)
elif precdist == 'norm':
cdf_val = norm.cdf(prec_df['midpoint'], loc, scale)
elif precdist == 'laplace':
cdf_val = laplace.cdf(prec_df['midpoint'], loc, scale)
elif precdist == 'mixnorm':
cdf_val = 0.5 * norm.cdf(prec_df['midpoint'], loc, scale) + \
0.5 * norm.cdf(prec_df['midpoint'], -loc, scale)
else:
print 'Cannot find cdf given precinct distribution, see ' + \
'compute_asst_loglik().'
assert (False)
return binom.logpmf(prec_df['num_votes_0'], prec_df['tot_votes'],
cdf_val) + np.log(theta)
def test_initialize(self):
def sqrtmi(mat):
"""
Compute matrix inverse square root.
@type mat: array_like
@param mat: matrix for which to compute inverse square root
"""
# find eigenvectors
eigvals, eigvecs = eig(mat)
# eliminate eigenvectors whose eigenvalues are zero
eigvecs = eigvecs[:, eigvals > 0.]
eigvals = eigvals[eigvals > 0.]
# inverse square root
return dot(eigvecs, dot(diag(1. / sqrt(eigvals)), eigvecs.T))
# white data
data = randn(5, 1000)
data = dot(sqrtmi(cov(data)), data)
isa = ISA(5, 10)
isa.initialize(data)
# rows of A should be roughly orthogonal
self.assertLess(sum(square(dot(isa.A, isa.A.T) - eye(5)).flatten()),
1e-3)
p = kstest(
isa.sample_prior(100).flatten(),
lambda x: laplace.cdf(x, scale=1. / sqrt(2.)))[1]
# prior marginals should be roughly Laplace
self.assertGreater(p, 0.0001)
# test initialization with larger subspaces
isa = ISA(5, 10, ssize=2)
isa.initialize(data)
def test_initialize(self): def sqrtmi(mat): """ Compute matrix inverse square root. @type mat: array_like @param mat: matrix for which to compute inverse square root """ # find eigenvectors eigvals, eigvecs = eig(mat) # eliminate eigenvectors whose eigenvalues are zero eigvecs = eigvecs[:, eigvals > 0.] eigvals = eigvals[eigvals > 0.] # inverse square root return dot(eigvecs, dot(diag(1. / sqrt(eigvals)), eigvecs.T)) # white data data = randn(5, 1000) data = dot(sqrtmi(cov(data)), data) isa = ISA(5, 10) isa.initialize(data) # rows of A should be roughly orthogonal self.assertLess(sum(square(dot(isa.A, isa.A.T) - eye(5)).flatten()), 1e-3) p = kstest( isa.sample_prior(100).flatten(), lambda x: laplace.cdf(x, scale=1. / sqrt(2.)))[1] # prior marginals should be roughly Laplace self.assertGreater(p, 0.0001) # test initialization with larger subspaces isa = ISA(5, 10, ssize=2) isa.initialize(data)
import numpy as np from scipy.stats import laplace import matplotlib.pyplot as plt fig, ax = plt.subplots(1, 1) mean, var, skew, kurt = laplace.stats(moments='mvsk') x = np.linspace(laplace.ppf(0.01), laplace.ppf(0.99), 100) ax.plot(x, laplace.pdf(x), 'r-', lw=5, alpha=0.6, label='laplace pdf') rv = laplace() ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf') vals = laplace.ppf([0.001, 0.5, 0.999]) np.allclose([0.001, 0.5, 0.999], laplace.cdf(vals)) r = laplace.rvs(size=1000) #ax.hist(r, normed=True, histtype='stepfilled', alpha=0.2) #ax.legend(loc='best', frameon=False) plt.show()
def func(self, x):
return laplace.cdf(x, loc=self.mu, scale=self.sigma)
def _make_l1_table(self, inc=0.001, grid_type='radius', upper=3):
return diffmethod_table(self.Phi_l1,
f=lambda r: laplace.cdf(np.sqrt(2) * r),
inc=inc,
grid_type=grid_type,
upper=upper)
def cumulative_laplace(x):
return laplace.cdf(x, 0, 1 / np.sqrt(2))
import numpy as np
from scipy.stats import laplace
import matplotlib.pyplot as plt
import sys
t_green = '#009933'
# params
mu = float(sys.argv[1])
b = float(sys.argv[2])
# generate samples
# lots of samples are a lazy-man's smoothing
lap_X = laplace.rvs(loc=mu, scale=b, size=1000000)
x = np.linspace(start=-8*b,stop=8*b,num=1000000)
lap_Q = laplace.cdf(x=x, loc=mu, scale=b)
# plotting pdf!
fig = plt.figure()
h = plt.hist(lap_X, bins=500, alpha=1.0, color=t_green, normed=True, histtype='stepfilled', antialiased=True, linewidth=0)
# to make the plot symmetric, find maximum distance to centre
lower_dx = mu - h[1][0]
upper_dx= h[1][-1] - mu
max_dx = np.max([abs(lower_dx), abs(upper_dx)])
plt.xlim(-max_dx + mu, max_dx +mu)
# labels etc.
plt.title('Laplace PDF with mu = '+str(mu)+' and b = '+str(b), family='monospace', size=16)
plt.xlabel('x', family='monospace', size=16)
plt.ylabel('p(x)', family='monospace', size=16)
plt.tight_layout() # requires Agg
plt.savefig('laplacePDF_mu'+str(mu)+'_b'+str(b)+'.png', dpi=200)
def cumulative_laplace(x):
return laplace.cdf(x, 0, 1 / LAPLACE_COEF)
import matplotlib as mpl
mpl.use('svg')
import matplotlib.pyplot as plt
import matplotlib.mlab as mlab
from scipy.stats import laplace as l
import numpy as np
import math
x = np.linspace(-3, 3, 100)
plt.plot(x, l.pdf(x), linewidth=2.0, label='PDF')
plt.plot(x, l.cdf(x), linewidth=2.0, label='CDF')
plt.legend(bbox_to_anchor=(.35, 1))
plt.savefig('cdf.svg', bbox_inches='tight')
import matplotlib.pyplot as plt from scipy.stats import laplace import numpy as np x = np.linspace(-3, 3, 100) plt.plot(x, laplace.pdf(x),linewidth=2.0, label="laplace PDF") plt.plot(x, laplace.cdf(x),linewidth=2.0, label="laplace CDF") plt.legend(bbox_to_anchor=(.35,1)) plt.show()
uniform_expected = np.zeros(uniform_num_bins-1)
gauss_expected = np.zeros(gauss_num_bins-1)
rayleigh_expected = np.zeros(rayleigh_num_bins-1)
laplace_expected = np.zeros(laplace_num_bins-1)
for k in range(len(uniform_hist[0])):
uniform_expected[k] = num_tests/float(uniform_num_bins-1)
for k in range(len(gauss_hist[0])):
gauss_expected[k] = float(norm.cdf(gauss_hist[1][k+1])-norm.cdf(gauss_hist[1][k]))*num_tests
for k in range(len(rayleigh_hist[0])):
rayleigh_expected[k] = float(rayleigh.cdf(rayleigh_hist[1][k+1])-rayleigh.cdf(rayleigh_hist[1][k]))*num_tests
for k in range(len(laplace_hist[0])):
laplace_expected[k] = float(laplace.cdf(laplace_hist[1][k+1])-laplace.cdf(laplace_hist[1][k]))*num_tests
#*** PLOT HISTOGRAMS AND EXPECTATIONS TAKEN FROM SCIPY ***#
uniform_bins_center = uniform_bins[0:-1]+(uniform_bins[1]-uniform_bins[0])/2.0
gauss_bins_center = gauss_bins[0:-1]+(gauss_bins[1]-gauss_bins[0])/2.0
rayleigh_bins_center = rayleigh_bins[0:-1]+(rayleigh_bins[1]-rayleigh_bins[0])/2.0
laplace_bins_center = laplace_bins[0:-1]+(laplace_bins[1]-laplace_bins[0])/2.0
plt.figure(1)
plt.subplot(2,1,1)
plt.plot(uniform_bins_center,uniform_hist[0],'s--',uniform_bins_center,uniform_expected,'o:')
plt.xlabel('Bins'), plt.ylabel('Count'), plt.title('Uniform: Distribution')
plt.legend(['histogram gr::random','calculation scipy'],loc=1)
from scipy.stats import laplace
from scipy.stats import cauchy
from scipy.stats import uniform
from scipy.stats import poisson
from scipy.stats import norm
import matplotlib.pyplot as plt
from math import sqrt, floor, exp, pi
import math
import numpy as np
fig, ax = plt.subplots(1, 3)
r = laplace.rvs(size=20)
x = np.linspace(min(laplace.ppf(0.01), min(r)), max(laplace.ppf(0.99), max(r)),
100)
ax[0].plot(x, laplace.cdf(x, 0, sqrt(2)))
ax[0].hist(r,
density=True,
bins=floor(len(r)),
histtype='step',
cumulative=True)
ax[0].set_xlabel('x')
ax[0].set_title('Распределение Лапласа, n=20')
r2 = laplace.rvs(size=60)
x2 = np.linspace(min(laplace.ppf(0.01), min(r2)),
max(laplace.ppf(0.99), max(r2)), 100)
ax[1].plot(x2, laplace.cdf(x2, 0, sqrt(2)))
ax[1].hist(r2,
density=True,
bins=floor(len(r2)),
histtype='step',
for k in range(len(uniform_hist[0])):
uniform_expected[k] = num_tests / float(uniform_num_bins - 1)
for k in range(len(gauss_hist[0])):
gauss_expected[k] = float(
norm.cdf(gauss_hist[1][k + 1]) -
norm.cdf(gauss_hist[1][k])) * num_tests
for k in range(len(rayleigh_hist[0])):
rayleigh_expected[k] = float(
rayleigh.cdf(rayleigh_hist[1][k + 1]) -
rayleigh.cdf(rayleigh_hist[1][k])) * num_tests
for k in range(len(laplace_hist[0])):
laplace_expected[k] = float(
laplace.cdf(laplace_hist[1][k + 1]) -
laplace.cdf(laplace_hist[1][k])) * num_tests
#*** PLOT HISTOGRAMS AND EXPECTATIONS TAKEN FROM SCIPY ***#
uniform_bins_center = uniform_bins[0:-1] + (uniform_bins[1] -
uniform_bins[0]) / 2.0
gauss_bins_center = gauss_bins[0:-1] + (gauss_bins[1] - gauss_bins[0]) / 2.0
rayleigh_bins_center = rayleigh_bins[0:-1] + (rayleigh_bins[1] -
rayleigh_bins[0]) / 2.0
laplace_bins_center = laplace_bins[0:-1] + (laplace_bins[1] -
laplace_bins[0]) / 2.0
plt.figure(1)
plt.subplot(2, 1, 1)
from math import sqrt, floor
import math
import numpy as np
fig, ax = plt.subplots(2, 3)
r = laplace.rvs(size=10)
x = np.linspace(min(laplace.ppf(0.01),min(r)), max(laplace.ppf(0.99),max(r)), 100)
ax[0][0].plot(x, laplace.pdf(x,0,sqrt(2)), label='теор.')
ax[0][0].set_title('Распределение Лапласа, n=10')
ax[0][0].hist(r, density=True, bins=floor(sqrt(len(r))),histtype='step',label='практ.')
ax[0][0].legend(loc='best', frameon=False)
ax[0][0].set_xlabel('x')
ax[0][0].set_ylabel('Функция плотности')
ax[0][0].legend()
ax[1][0].plot(x, laplace.cdf(x,0,sqrt(2)), label='теор.')
ax[1][0].hist(r, density=True, bins=floor(sqrt(len(r))),histtype='step',cumulative=True,label='практ.')
ax[1][0].legend(loc='best', frameon=False)
ax[1][0].set_xlabel('x')
ax[1][0].set_ylabel('Функция распределения')
ax[1][0].legend()
r2 = laplace.rvs(size=100)
x2 = np.linspace(min(laplace.ppf(0.01),min(r2)), max(laplace.ppf(0.99),max(r2)), 100)
ax[0][1].plot(x2, laplace.pdf(x2,0,sqrt(2)), label='теор.')
ax[0][1].set_title('Распределение Лапласа, n=100')
ax[0][1].hist(r2, density=True, bins=floor(sqrt(len(r2))),histtype='step',label='практ.')
ax[0][1].legend(loc='best', frameon=False)
ax[0][1].set_xlabel('x')
ax[0][1].set_ylabel('Функция плотности')
ax[0][1].legend()
