def test_sample_discrete():
X, Y, Z = Geometric("X", S.Half), Poisson("Y", 4), Poisson("Z", 1000)
W = Poisson("W", Rational(1, 100))
assert sample(X) in X.pspace.domain.set
assert sample(Y) in Y.pspace.domain.set
assert sample(Z) in Z.pspace.domain.set
assert sample(W) in W.pspace.domain.set
def test_sample_discrete():
X, Y, Z = Geometric('X', S.Half), Poisson('Y', 4), Poisson('Z', 1000)
W = Poisson('W', Rational(1, 100))
assert sample(X) in X.pspace.domain.set
assert sample(Y) in Y.pspace.domain.set
assert sample(Z) in Z.pspace.domain.set
assert sample(W) in W.pspace.domain.set
def test_sample():
X, Y, Z = Geometric('X', S(1) / 2), Poisson('Y', 4), Poisson('Z', 1000)
W = Poisson('W', S(1) / 100)
assert sample(X) in X.pspace.domain.set
assert sample(Y) in Y.pspace.domain.set
assert sample(Z) in Z.pspace.domain.set
assert sample(W) in W.pspace.domain.set
def test_precomputed_characteristic_functions():
import mpmath
def test_cf(dist, support_lower_limit, support_upper_limit):
pdf = density(dist)
t = S('t')
x = S('x')
# first function is the hardcoded CF of the distribution
cf1 = lambdify([t], characteristic_function(dist)(t), 'mpmath')
# second function is the Fourier transform of the density function
f = lambdify([x, t], pdf(x) * exp(I * x * t), 'mpmath')
cf2 = lambda t: mpmath.nsum(lambda x: f(x, t),
[support_lower_limit, support_upper_limit],
maxdegree=10)
# compare the two functions at various points
for test_point in [2, 5, 8, 11]:
n1 = cf1(test_point)
n2 = cf2(test_point)
assert abs(re(n1) - re(n2)) < 1e-12
assert abs(im(n1) - im(n2)) < 1e-12
test_cf(Geometric('g', Rational(1, 3)), 1, mpmath.inf)
test_cf(Logarithmic('l', Rational(1, 5)), 1, mpmath.inf)
test_cf(NegativeBinomial('n', 5, Rational(1, 7)), 0, mpmath.inf)
test_cf(Poisson('p', 5), 0, mpmath.inf)
test_cf(YuleSimon('y', 5), 1, mpmath.inf)
test_cf(Zeta('z', 5), 1, mpmath.inf)
def test_moment_generating_functions():
t = S('t')
geometric_mgf = moment_generating_function(Geometric('g', S.Half))(t)
assert geometric_mgf.diff(t).subs(t, 0) == 2
logarithmic_mgf = moment_generating_function(Logarithmic('l', S.Half))(t)
assert logarithmic_mgf.diff(t).subs(t, 0) == 1 / log(2)
negative_binomial_mgf = moment_generating_function(
NegativeBinomial('n', 5, Rational(1, 3)))(t)
assert negative_binomial_mgf.diff(t).subs(t, 0) == Rational(5, 2)
poisson_mgf = moment_generating_function(Poisson('p', 5))(t)
assert poisson_mgf.diff(t).subs(t, 0) == 5
skellam_mgf = moment_generating_function(Skellam('s', 1, 1))(t)
assert skellam_mgf.diff(t).subs(
t, 2) == (-exp(-2) + exp(2)) * exp(-2 + exp(-2) + exp(2))
yule_simon_mgf = moment_generating_function(YuleSimon('y', 3))(t)
assert simplify(yule_simon_mgf.diff(t).subs(t, 0)) == Rational(3, 2)
zeta_mgf = moment_generating_function(Zeta('z', 5))(t)
assert zeta_mgf.diff(t).subs(t, 0) == pi**4 / (90 * zeta(5))
def test_where():
X = Geometric("X", Rational(1, 5))
Y = Poisson("Y", 4)
assert where(X ** 2 > 4).set == Range(3, S.Infinity, 1)
assert where(X ** 2 >= 4).set == Range(2, S.Infinity, 1)
assert where(Y ** 2 < 9).set == Range(0, 3, 1)
assert where(Y ** 2 <= 9).set == Range(0, 4, 1)
def test_moment_generating_functions():
t = S("t")
geometric_mgf = moment_generating_function(Geometric("g", S.Half))(t)
assert geometric_mgf.diff(t).subs(t, 0) == 2
logarithmic_mgf = moment_generating_function(Logarithmic("l", S.Half))(t)
assert logarithmic_mgf.diff(t).subs(t, 0) == 1 / log(2)
negative_binomial_mgf = moment_generating_function(
NegativeBinomial("n", 5, Rational(1, 3))
)(t)
assert negative_binomial_mgf.diff(t).subs(t, 0) == Rational(5, 2)
poisson_mgf = moment_generating_function(Poisson("p", 5))(t)
assert poisson_mgf.diff(t).subs(t, 0) == 5
skellam_mgf = moment_generating_function(Skellam("s", 1, 1))(t)
assert skellam_mgf.diff(t).subs(t, 2) == (-exp(-2) + exp(2)) * exp(
-2 + exp(-2) + exp(2)
)
yule_simon_mgf = moment_generating_function(YuleSimon("y", 3))(t)
assert simplify(yule_simon_mgf.diff(t).subs(t, 0)) == Rational(3, 2)
zeta_mgf = moment_generating_function(Zeta("z", 5))(t)
assert zeta_mgf.diff(t).subs(t, 0) == pi ** 4 / (90 * zeta(5))
def test_where():
X = Geometric('X', S(1) / 5)
Y = Poisson('Y', 4)
assert where(X**2 > 4).set == Range(3, S.Infinity, 1)
assert where(X**2 >= 4).set == Range(2, S.Infinity, 1)
assert where(Y**2 < 9).set == Range(0, 3, 1)
assert where(Y**2 <= 9).set == Range(0, 4, 1)
def test_conditional():
X = Geometric('X', S(2)/3)
Y = Poisson('Y', 3)
assert P(X > 2, X > 3) == 1
assert P(X > 3, X > 2) == S(1)/3
assert P(Y > 2, Y < 2) == 0
assert P(Eq(Y, 3), Y >= 0) == 9*exp(-3)/2
def test_sample_scipy():
p = S(2) / 3
x = Symbol('x', integer=True, positive=True)
pdf = p * (1 - p)**(x - 1) # pdf of Geometric Distribution
distribs_scipy = [
DiscreteRV(x, pdf, set=S.Naturals),
Geometric('G', 0.5),
Logarithmic('L', 0.5),
NegativeBinomial('N', 5, 0.4),
Poisson('P', 1),
Skellam('S', 1, 1),
YuleSimon('Y', 1),
Zeta('Z', 2)
]
size = 3
numsamples = 5
scipy = import_module('scipy')
if not scipy:
skip('Scipy is not installed. Abort tests for _sample_scipy.')
else:
with ignore_warnings(
UserWarning
): ### TODO: Restore tests once warnings are removed
z_sample = list(
sample(Zeta("G", 7), size=size, numsamples=numsamples))
assert len(z_sample) == numsamples
for X in distribs_scipy:
samps = next(sample(X, size=size, library='scipy'))
samps2 = next(sample(X, size=(2, 2), library='scipy'))
for sam in samps:
assert sam in X.pspace.domain.set
for i in range(2):
for j in range(2):
assert samps2[i][j] in X.pspace.domain.set
def test_sampling_methods():
distribs_numpy = [
Geometric('G', 0.5),
Poisson('P', 1),
Zeta('Z', 2)
]
distribs_scipy = [
Geometric('G', 0.5),
Logarithmic('L', 0.5),
Poisson('P', 1),
Skellam('S', 1, 1),
YuleSimon('Y', 1),
Zeta('Z', 2)
]
distribs_pymc3 = [
Geometric('G', 0.5),
Poisson('P', 1),
]
size = 3
numpy = import_module('numpy')
if not numpy:
skip('Numpy is not installed. Abort tests for _sample_numpy.')
else:
with ignore_warnings(UserWarning):
for X in distribs_numpy:
samps = X.pspace.distribution._sample_numpy(size)
for samp in samps:
assert samp in X.pspace.domain.set
scipy = import_module('scipy')
if not scipy:
skip('Scipy is not installed. Abort tests for _sample_scipy.')
else:
with ignore_warnings(UserWarning):
for X in distribs_scipy:
samps = next(sample(X, size=size))
for samp in samps:
assert samp in X.pspace.domain.set
pymc3 = import_module('pymc3')
if not pymc3:
skip('PyMC3 is not installed. Abort tests for _sample_pymc3.')
else:
with ignore_warnings(UserWarning):
for X in distribs_pymc3:
samps = X.pspace.distribution._sample_pymc3(size)
for samp in samps:
assert samp in X.pspace.domain.set
def test_Poisson():
l = 3
x = Poisson('x', l)
assert E(x) == l
assert variance(x) == l
assert density(x) == PoissonDistribution(l)
assert isinstance(E(x, evaluate=False), Sum)
assert isinstance(E(2 * x, evaluate=False), Sum)
def test_Poisson():
l = 3
x = Poisson('x', l)
assert E(x) == l
assert variance(x) == l
assert density(x) == PoissonDistribution(l)
assert isinstance(E(x, evaluate=False), Sum)
assert isinstance(E(2 * x, evaluate=False), Sum)
assert characteristic_function(x)(0).doit() == 1
def test_conditional():
X = Geometric('X', Rational(2, 3))
Y = Poisson('Y', 3)
assert P(X > 2, X > 3) == 1
assert P(X > 3, X > 2) == Rational(1, 3)
assert P(Y > 2, Y < 2) == 0
assert P(Eq(Y, 3), Y >= 0) == 9 * exp(-3) / 2
assert P(Eq(Y, 3), Eq(Y, 2)) == 0
assert P(X < 2, Eq(X, 2)) == 0
assert P(X > 2, Eq(X, 3)) == 1
def test_Poisson():
l = 3
x = Poisson("x", l)
assert E(x) == l
assert variance(x) == l
assert density(x) == PoissonDistribution(l)
assert isinstance(E(x, evaluate=False), Sum)
assert isinstance(E(2 * x, evaluate=False), Sum)
# issue 8248
assert x.pspace.compute_expectation(1) == 1
def test_Poisson():
l = 3
x = Poisson('x', l)
assert E(x) == l
assert variance(x) == l
assert density(x) == PoissonDistribution(l)
with ignore_warnings(UserWarning): ### TODO: Restore tests once warnings are removed
assert isinstance(E(x, evaluate=False), Expectation)
assert isinstance(E(2*x, evaluate=False), Expectation)
# issue 8248
assert x.pspace.compute_expectation(1) == 1
def test_sample_numpy():
distribs_numpy = [Geometric('G', 0.5), Poisson('P', 1), Zeta('Z', 2)]
size = 3
numpy = import_module('numpy')
if not numpy:
skip('Numpy is not installed. Abort tests for _sample_numpy.')
else:
with ignore_warnings(UserWarning):
for X in distribs_numpy:
samps = next(sample(X, size=size, library='numpy'))
for sam in samps:
assert sam in X.pspace.domain.set
raises(NotImplementedError,
lambda: next(sample(Skellam('S', 1, 1), library='numpy')))
raises(
NotImplementedError, lambda: Skellam('S', 1, 1).pspace.distribution.
sample(library='tensorflow'))
def test_discrete_probability():
X = Geometric('X', S(1)/5)
Y = Poisson('Y', 4)
assert P(Eq(X, 3)) == S(16)/125
assert P(X < 3) == S(9)/25
assert P(X > 3) == S(64)/125
assert P(X >= 3) == S(16)/25
assert P(X <= 3) == S(61)/125
assert P(Ne(X, 3)) == S(109)/125
assert P(Eq(Y, 3)) == 32*exp(-4)/3
assert P(Y < 3) == 13*exp(-4)
assert P(Y > 3).equals(32*(-S(71)/32 + 3*exp(4)/32)*exp(-4)/3)
assert P(Y >= 3).equals(32*(-39/32 + 3*exp(4)/32)*exp(-4)/3)
assert P(Y <= 3) == 71*exp(-4)/3
assert P(Ne(Y, 3)).equals(
13*exp(-4) + 32*(-71/32 + 3*exp(4)/32)*exp(-4)/3)
assert P(X < S.Infinity) is S.One
assert P(X > S.Infinity) is S.Zero
def test_sample_pymc3():
distribs_pymc3 = [
Geometric('G', 0.5),
Poisson('P', 1),
NegativeBinomial('N', 5, 0.4)
]
size = 3
pymc3 = import_module('pymc3')
if not pymc3:
skip('PyMC3 is not installed. Abort tests for _sample_pymc3.')
else:
with ignore_warnings(UserWarning): ### TODO: Restore tests once warnings are removed
for X in distribs_pymc3:
samps = next(sample(X, size=size, library='pymc3'))
for sam in samps:
assert sam in X.pspace.domain.set
raises(NotImplementedError,
lambda: next(sample(Skellam('S', 1, 1), library='pymc3')))
def test_moment_generating_functions():
t = S('t')
geometric_mgf = moment_generating_function(Geometric('g', S(1)/2))(t)
assert geometric_mgf.diff(t).subs(t, 0) == 2
logarithmic_mgf = moment_generating_function(Logarithmic('l', S(1)/2))(t)
assert logarithmic_mgf.diff(t).subs(t, 0) == 1/log(2)
negative_binomial_mgf = moment_generating_function(NegativeBinomial('n', 5, S(1)/3))(t)
assert negative_binomial_mgf.diff(t).subs(t, 0) == S(5)/2
poisson_mgf = moment_generating_function(Poisson('p', 5))(t)
assert poisson_mgf.diff(t).subs(t, 0) == 5
yule_simon_mgf = moment_generating_function(YuleSimon('y', 3))(t)
assert simplify(yule_simon_mgf.diff(t).subs(t, 0)) == S(3)/2
zeta_mgf = moment_generating_function(Zeta('z', 5))(t)
assert zeta_mgf.diff(t).subs(t, 0) == pi**4/(90*zeta(5))
def test_discrete_probability():
X = Geometric('X', Rational(1, 5))
Y = Poisson('Y', 4)
G = Geometric('e', x)
assert P(Eq(X, 3)) == Rational(16, 125)
assert P(X < 3) == Rational(9, 25)
assert P(X > 3) == Rational(64, 125)
assert P(X >= 3) == Rational(16, 25)
assert P(X <= 3) == Rational(61, 125)
assert P(Ne(X, 3)) == Rational(109, 125)
assert P(Eq(Y, 3)) == 32*exp(-4)/3
assert P(Y < 3) == 13*exp(-4)
assert P(Y > 3).equals(32*(Rational(-71, 32) + 3*exp(4)/32)*exp(-4)/3)
assert P(Y >= 3).equals(32*(Rational(-39, 32) + 3*exp(4)/32)*exp(-4)/3)
assert P(Y <= 3) == 71*exp(-4)/3
assert P(Ne(Y, 3)).equals(
13*exp(-4) + 32*(Rational(-71, 32) + 3*exp(4)/32)*exp(-4)/3)
assert P(X < S.Infinity) is S.One
assert P(X > S.Infinity) is S.Zero
assert P(G < 3) == x*(2-x)
assert P(Eq(G, 3)) == x*(-x + 1)**2
def test_sample():
X, Y = Geometric('X', S(1) / 2), Poisson('Y', 4)
assert sample(X) in X.pspace.domain.set
assert sample(Y) in Y.pspace.domain.set
