GM.plot()
#! Sign, Abs
#! ----------
figure()
N = NormalDistr(0.1,1)
S = sign(N)
A = abs(N)
S.plot(color='r', linewidth=2.0, label="sign(N)")
A.plot(color='g', linewidth=2.0, label="abs(N)")
N.plot(color='b', linewidth=2.0, label="N")
legend()
figure()
params.general.warn_on_dependent = False
demo_distr(S * A)#, theoretical=N)
params.general.warn_on_dependent = True
#! min, max, Conditional distributions
#! ------------------------------------
#!
#!
figure()
E = max(ExponentialDistr() - 1, ZeroDistr())
subplot(211)
E.plot(linewidth=2.0)
E.summary()
subplot(212)
params.general.warn_on_dependent = False
S = E + E + E + E + E + E
params.general.warn_on_dependent = True
#! from __future__ import print_function from functools import partial import numpy from pylab import figure, show from pacal import * from pacal.distr import demo_distr #! Exercise 6.3 d = NormalDistr() * NormalDistr() + NormalDistr() * NormalDistr() figure() demo_distr(d, theoretical = LaplaceDistr()) #! Example 7.3.1 w1 = WeibullDistr(2) w2 = WeibullDistr(3) figure() demo_distr(w1 * w2) #! Example 7.3.2 x1 = BetaDistr(9,3) x2 = BetaDistr(8,3) x3 = BetaDistr(4,2) figure() demo_distr(x1 * x2 * x3) #! Example 8.6.1
from functools import partial
from pylab import *
from mpl_toolkits.axes_grid.inset_locator import zoomed_inset_axes
from mpl_toolkits.axes_grid.inset_locator import mark_inset
from pacal import *
from pacal.distr import demo_distr
if __name__ == "__main__":
#!-------------------------------------------
#! Product of two shifted normal variables
#!-------------------------------------------
#! such a product always has a singularity at 0, but the further the factors' means are from zero, the 'lighter' the singularity becomes
figure()
d = NormalDistr(0, 1) * NormalDistr(0, 1)
demo_distr(d, ymax=1.5, xmin=-5, xmax=5)
#show()
figure()
d = NormalDistr(1, 1) * NormalDistr(1, 1)
demo_distr(d)
#show()
figure()
d = NormalDistr(2, 1) * NormalDistr(2, 1)
demo_distr(d)
#show()
figure()
d = NormalDistr(3, 1) * NormalDistr(3, 1)
d.plot()
#!-------------------
#!
from functools import partial
import numpy
from pylab import figure, show
from pacal import *
from pacal.distr import demo_distr
import time
if __name__ == "__main__":
tic =time.time()
#! Figure 3.1.1
figure()
demo_distr(UniformDistr(0,1) + UniformDistr(0,1),
theoretical = lambda x: x * ((x >= 0) & (x < 1)) + (2-x) * ((x >= 1) & (x <= 2)))
#! Figure 3.1.2
figure()
demo_distr(UniformDistr(0,1) - UniformDistr(0,1),
theoretical = lambda x: (x+1) * ((x >= -1) & (x < 0)) + (1-x) * ((x >= 0) & (x <= 1)))
#!-------------------
#! Section 3.2.2
#!-------------------
figure()
demo_distr(ChiSquareDistr(1) + ChiSquareDistr(1),
theoretical = ExponentialDistr(0.5))
figure()
demo_distr(ChiSquareDistr(1) + ChiSquareDistr(1) + ChiSquareDistr(1),
theoretical = ChiSquareDistr(3))
figure()
from pacal import * from pacal.distr import demo_distr #! #! Inverse of r.v. #! --------------- N = NormalDistr(0,1) d = 1/(N**2)**0.5 #d.summary() C = CauchyDistr() (2/C).summary() (C/2).summary() demo_distr(C/2, 0.5/C) E1 = ChiSquareDistr(1) E2 = ChiSquareDistr(3) figure() demo_distr(log(E1)) figure() demo_distr(log(E2)) figure() demo_distr(log(UniformDistr(0.0,1.0))) figure() demo_distr(exp(UniformDistr(0.0,1.0))) figure() demo_distr(exp(UniformDistr(0.0,1.0))+exp(UniformDistr(0.0,1.0))) figure() demo_distr(log(UniformDistr(0.0,1.0))+log(UniformDistr(0.0,1.0)))
from pacal import *
from pacal.distr import demo_distr
import time
def funEx4_20(m, a, b, x):
return (m + 1) * x / (b**(m + 1) - a**(m + 1))
if __name__ == "__main__":
#!-------------------
#! Section 4.1
#!-------------------
tic = time.time()
figure()
demo_distr(UniformDistr(0, 1) * UniformDistr(0, 1),
theoretical=lambda x: -log(x))
figure()
demo_distr(UniformDistr(0, 1) / UniformDistr(0, 1),
theoretical=lambda x: (x <= 1) * 0.5 + (x > 1) * 0.5 / x**2)
#! Section 4.4.1
def prod_uni_pdf(n, x):
pdf = (-log(x))**(n - 1)
for i in range(2, n):
pdf /= i
return pdf
figure()
demo_distr(UniformDistr(0, 1) * UniformDistr(0, 1) * UniformDistr(0, 1),
theoretical=partial(prod_uni_pdf, 3))
from pacal import *
from pacal.distr import demo_distr
import time
def funEx4_20(m, a, b, x):
return (m + 1) * x / (b ** (m + 1) - a ** (m + 1))
if __name__ == "__main__":
#!-------------------
#! Section 4.1
#!-------------------
tic = time.time()
figure()
demo_distr(UniformDistr(0,1) * UniformDistr(0,1), theoretical = lambda x: -log(x))
figure()
demo_distr(UniformDistr(0,1) / UniformDistr(0,1), theoretical = lambda x: (x<=1) * 0.5 + (x>1) * 0.5 / x**2)
#! Section 4.4.1
def prod_uni_pdf(n, x):
pdf = (-log(x)) ** (n-1)
for i in xrange(2, n):
pdf /= i
return pdf
figure()
demo_distr(UniformDistr(0,1) * UniformDistr(0,1) * UniformDistr(0,1), theoretical = partial(prod_uni_pdf, 3))
pu = UniformDistr(0,1)
for i in xrange(4):
pu *= UniformDistr(0,1)
from pacal import *
from pacal.distr import demo_distr
if __name__ == "__main__":
#! Here we describe inaccurate places in the PaCal project and future topics.
#! Exponential function and logarithm
#! ----------------------------------
#$ The real power or random variable $(X^\alpha)$ is always
#$ well defined, but logarithm and exponent may cause troubles. Bellow we have
#$ some of examples:
#!
figure()
Y = abs(NormalDistr()) ** NormalDistr()
demo_distr(Y, histogram = True, xmin = 0, xmax = 3, ymax = 3, hist_bins = 500)
Y.get_piecewise_cdf().plot(color='b', linewidth = 2.0)
#!
#! As one can see we obtain only six digits accuracy. The main problem
#! concerns singularity at point 1 such kind of singularity is difficult to
#! detect and difficult to interpolate.
#!
Y = UniformDistr(0,1) ** NormalDistr(1,1)
figure()
demo_distr(Y, histogram = True, xmin = 0, xmax = 3, ymax = 3, hist_bins = 500)
Y.get_piecewise_cdf().plot(color='b', linewidth = 2.0)
#!
Y = UniformDistr(0.5, 1.5) ** NormalDistr(0,1)
figure()
demo_distr(Y, histogram = True, xmin = 0, xmax = 3, ymax = 3, hist_bins = 500)
Y.get_piecewise_cdf().plot(color='b', linewidth = 2.0)
t0 = time.time()
def theor_sum_exp(a1, a2, a3, x):
t1 = exp(-3 * a1 * x) / (a2 - a1) / (a3 - a1)
t2 = exp(-3 * a2 * x) / (a1 - a2) / (a3 - a2)
t3 = exp(-3 * a3 * x) / (a1 - a3) / (a2 - a3)
return 3 * a1 * a2 * a3 * (t1 + t2 + t3)
for a1, a2, a3 in [
(1.0, 2.0, 3.0),
(1, 0.01, 100.0),
]:
figure()
d = (ExponentialDistr(a1) + ExponentialDistr(a2) +
ExponentialDistr(a3)) / 3
demo_distr(d, theoretical=partial(theor_sum_exp, a1, a2, a3))
#! Section 9.1.2
#! the L1 statistic for variances
#! Question: are the numerator and denominator independent?
for ns in [
[3, 5, 2], # sample sizes
[4, 5, 10, 7, 3]
]:
print "sample sizes:", ns
N = sum(ns)
num = ChiSquareDistr(ns[0] - 1)
for n in ns[1:]:
num *= ChiSquareDistr(n - 1)
#num.summary()
num **= (1.0 / N)
#!------------------ #! CHAPTERS 6, 7, 8 #!------------------ #! from functools import partial import numpy from pylab import figure, show from pacal import * from pacal.distr import demo_distr #! Exercise 6.3 d = NormalDistr() * NormalDistr() + NormalDistr() * NormalDistr() figure() demo_distr(d, theoretical=LaplaceDistr()) #! Example 7.3.1 w1 = WeibullDistr(2) w2 = WeibullDistr(3) figure() demo_distr(w1 * w2) #! Example 7.3.2 x1 = BetaDistr(9, 3) x2 = BetaDistr(8, 3) x3 = BetaDistr(4, 2) figure() demo_distr(x1 * x2 * x3) #! Example 8.6.1
from functools import partial
import numpy
import time
from pylab import figure, show
from pacal import *
from pacal.distr import demo_distr
if __name__ == "__main__":
tic = time.time()
#! Example 5.1.3
d = NormalDistr() + NormalDistr() * NormalDistr()
demo_distr(d)
#! Example 5.5
d = ExponentialDistr() / (ExponentialDistr() + ExponentialDistr())
figure()
demo_distr(d, xmax=20, ymax=1.5)
#! Exercise 5.5
#! part a
figure()
demo_distr(NormalDistr() / sqrt((NormalDistr()**2 + NormalDistr()**2) / 2),
xmin=-3,
xmax=3)
#! part b
figure()
demo_distr(2 * NormalDistr()**2 / (NormalDistr()**2 + NormalDistr()**2),
from pacal import * from pacal.distr import demo_distr #! #! Inverse of r.v. #! --------------- N = NormalDistr(0,1) d = 1/(N**2)**0.5 #d.summary() C = CauchyDistr() (2/C).summary() (C/2).summary() demo_distr(C/2, 0.5/C) E1 = ChiSquareDistr(1) E2 = ChiSquareDistr(3) figure() demo_distr(log(E1)) figure() demo_distr(log(E2)) figure() demo_distr(log(UniformDistr(0.0,1.0))) figure() demo_distr(exp(UniformDistr(0.0,1.0))) figure() demo_distr(exp(UniformDistr(0.0,1.0))+exp(UniformDistr(0.0,1.0))) figure() demo_distr(log(UniformDistr(0.0,1.0))+log(UniformDistr(0.0,1.0)))
#!
from functools import partial
import numpy
import time
from pylab import figure, show
from pacal import *
from pacal.distr import demo_distr
if __name__ == "__main__":
tic = time.time()
#! Example 5.1.3
d = NormalDistr() + NormalDistr() * NormalDistr()
demo_distr(d)
#! Example 5.5
d = ExponentialDistr() / (ExponentialDistr() + ExponentialDistr())
figure()
demo_distr(d, xmax=20, ymax=1.5)
#! Exercise 5.5
#! part a
figure()
demo_distr(NormalDistr() / sqrt((NormalDistr()**2 + NormalDistr()**2) / 2), xmin=-3, xmax=3)
#! part b
figure()
demo_distr(2 * NormalDistr()**2 / (NormalDistr()**2 + NormalDistr()**2), xmax=20, ymax=2)
#! part c
figure()
from pacal import *
from pacal.distr import demo_distr
if __name__ == "__main__":
#! Here we describe inaccurate places in the PaCal project and future topics.
#! Exponential function and logarithm
#! ----------------------------------
#$ The real power or random variable $(X^\alpha)$ is always
#$ well defined, but logarithm and exponent may cause troubles. Bellow we have
#$ some of examples:
#!
figure()
Y = abs(NormalDistr())**NormalDistr()
demo_distr(Y, histogram=True, xmin=0, xmax=3, ymax=3, hist_bins=500)
Y.get_piecewise_cdf().plot(color='b', linewidth=2.0)
#!
#! As one can see we obtain only six digits accuracy. The main problem
#! concerns singularity at point 1 such kind of singularity is difficult to
#! detect and difficult to interpolate.
#!
Y = UniformDistr(0, 1)**NormalDistr(1, 1)
figure()
demo_distr(Y, histogram=True, xmin=0, xmax=3, ymax=3, hist_bins=500)
Y.get_piecewise_cdf().plot(color='b', linewidth=2.0)
#!
Y = UniformDistr(0.5, 1.5)**NormalDistr(0, 1)
figure()
demo_distr(Y, histogram=True, xmin=0, xmax=3, ymax=3, hist_bins=500)
Y.get_piecewise_cdf().plot(color='b', linewidth=2.0)
from functools import partial
from pylab import *
from mpl_toolkits.axes_grid.inset_locator import zoomed_inset_axes
from mpl_toolkits.axes_grid.inset_locator import mark_inset
from pacal import *
from pacal.distr import demo_distr
if __name__ == "__main__":
#!-------------------------------------------
#! Product of two shifted normal variables
#!-------------------------------------------
#! such a product always has a singularity at 0, but the further the factors' means are from zero, the 'lighter' the singularity becomes
figure()
d = NormalDistr(0,1) * NormalDistr(0,1)
demo_distr(d, ymax=1.5, xmin=-5, xmax=5)
#show()
figure()
d = NormalDistr(1,1) * NormalDistr(1,1)
demo_distr(d)
#show()
figure()
d = NormalDistr(2,1) * NormalDistr(2,1)
demo_distr(d)
#show()
figure()
d = NormalDistr(3,1) * NormalDistr(3,1)
d.plot()
from functools import partial
import numpy
from pylab import figure, show
from pacal import *
from pacal.distr import demo_distr
import time
if __name__ == "__main__":
tic = time.time()
#! Figure 3.1.1
figure()
demo_distr(UniformDistr(0, 1) + UniformDistr(0, 1),
theoretical=lambda x: x * ((x >= 0) & (x < 1)) + (2 - x) *
((x >= 1) & (x <= 2)))
#! Figure 3.1.2
figure()
demo_distr(UniformDistr(0, 1) - UniformDistr(0, 1),
theoretical=lambda x: (x + 1) * ((x >= -1) & (x < 0)) +
(1 - x) * ((x >= 0) & (x <= 1)))
#!-------------------
#! Section 3.2.2
#!-------------------
figure()
demo_distr(ChiSquareDistr(1) + ChiSquareDistr(1),
theoretical=ExponentialDistr(0.5))
figure()
demo_distr(ChiSquareDistr(1) + ChiSquareDistr(1) + ChiSquareDistr(1),
#! Example 9.1.1
#! Implemented elsewhere
#! Example 9.1.2
t0 = time.time()
def theor_sum_exp(a1, a2, a3, x):
t1 = exp(-3*a1*x) / (a2-a1) / (a3-a1)
t2 = exp(-3*a2*x) / (a1-a2) / (a3-a2)
t3 = exp(-3*a3*x) / (a1-a3) / (a2-a3)
return 3*a1*a2*a3 * (t1+t2+t3)
for a1, a2, a3 in [(1.0,2.0,3.0),
(1,0.01,100.0),]:
figure()
d = (ExponentialDistr(a1) + ExponentialDistr(a2) + ExponentialDistr(a3))/3
demo_distr(d, theoretical = partial(theor_sum_exp, a1, a2, a3))
#! Section 9.1.2
#! the L1 statistic for variances
#! Question: are the numerator and denominator independent?
for ns in [[3, 5, 2], # sample sizes
[4, 5, 10, 7, 3]
]:
print "sample sizes:", ns
N = sum(ns)
num = ChiSquareDistr(ns[0] - 1)
for n in ns[1:]:
num *= ChiSquareDistr(n - 1)
#num.summary()
num **= (1.0 / N)
den = ChiSquareDistr(N - len(ns)) / N
