def test_chi_squared():
k = Symbol("k", integer=True)
X = ChiSquared('x', k)
assert density(X)(x) == 2**(-k / 2) * x**(k / 2 - 1) * exp(-x / 2) / gamma(
k / 2)
assert cdf(X)(x) == Piecewise(
(lowergamma(k / 2, x / 2) / gamma(k / 2), x >= 0), (0, True))
X = ChiSquared('x', 15)
assert cdf(X)(3) == -14873 * sqrt(6) * exp(
-S(3) / 2) / (5005 * sqrt(pi)) + erf(sqrt(6) / 2)
def test_precomputed_characteristic_functions():
import mpmath
def test_cf(dist, support_lower_limit, support_upper_limit):
pdf = density(dist)
t = Symbol('t')
x = Symbol('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.quad(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(Beta('b', 1, 2), 0, 1)
test_cf(Chi('c', 3), 0, mpmath.inf)
test_cf(ChiSquared('c', 2), 0, mpmath.inf)
test_cf(Exponential('e', 6), 0, mpmath.inf)
test_cf(Logistic('l', 1, 2), -mpmath.inf, mpmath.inf)
test_cf(Normal('n', -1, 5), -mpmath.inf, mpmath.inf)
test_cf(RaisedCosine('r', 3, 1), 2, 4)
test_cf(Rayleigh('r', 0.5), 0, mpmath.inf)
test_cf(Uniform('u', -1, 1), -1, 1)
test_cf(WignerSemicircle('w', 3), -3, 3)
def test_sample_scipy():
distribs_scipy = [
Beta("B", 1, 1),
BetaPrime("BP", 1, 1),
Cauchy("C", 1, 1),
Chi("C", 1),
Normal("N", 0, 1),
Gamma("G", 2, 7),
GammaInverse("GI", 1, 1),
GaussianInverse("GUI", 1, 1),
Exponential("E", 2),
LogNormal("LN", 0, 1),
Pareto("P", 1, 1),
StudentT("S", 2),
ChiSquared("CS", 2),
Uniform("U", 0, 1)
]
size = 3
scipy = import_module('scipy')
if not scipy:
skip('Scipy is not installed. Abort tests for _sample_scipy.')
else:
for X in distribs_scipy:
samps = sample(X, size=size, library='scipy')
samps2 = 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_sample_numpy():
distribs_numpy = [
Beta("B", 1, 1),
Normal("N", 0, 1),
Gamma("G", 2, 7),
Exponential("E", 2),
LogNormal("LN", 0, 1),
Pareto("P", 1, 1),
ChiSquared("CS", 2),
Uniform("U", 0, 1)
]
size = 3
numpy = import_module('numpy')
if not numpy:
skip('Numpy is not installed. Abort tests for _sample_numpy.')
else:
for X in distribs_numpy:
samps = sample(X, size=size, library='numpy')
for sam in samps:
assert sam in X.pspace.domain.set
raises(NotImplementedError,
lambda: sample(Chi("C", 1), library='numpy'))
raises(
NotImplementedError,
lambda: Chi("C", 1).pspace.distribution.sample(library='tensorflow'))
def test_sample_pymc3():
distribs_pymc3 = [
Beta("B", 1, 1),
Cauchy("C", 1, 1),
Normal("N", 0, 1),
Gamma("G", 2, 7),
GaussianInverse("GI", 1, 1),
Exponential("E", 2),
LogNormal("LN", 0, 1),
Pareto("P", 1, 1),
ChiSquared("CS", 2),
Uniform("U", 0, 1)
]
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(Chi("C", 1), library='pymc3')))
def test_prob_neq():
E = Exponential('E', 4)
X = ChiSquared('X', 4)
assert P(Ne(E, 2)) == 1
assert P(Ne(X, 4)) == 1
assert P(Ne(X, 4)) == 1
assert P(Ne(X, 5)) == 1
assert P(Ne(E, x)) == 1
def test_chi_squared():
k = Symbol("k", integer=True)
X = ChiSquared('x', k)
assert density(X)(x) == 2**(-k/2)*x**(k/2 - 1)*exp(-x/2)/gamma(k/2)
assert cdf(X)(x) == Piecewise((lowergamma(k/2, x/2)/gamma(k/2), x >= 0), (0, True))
assert E(X) == k
assert variance(X) == 2*k
X = ChiSquared('x', 15)
assert cdf(X)(3) == -14873*sqrt(6)*exp(-S(3)/2)/(5005*sqrt(pi)) + erf(sqrt(6)/2)
k = Symbol("k", integer=True, positive=False)
raises(ValueError, lambda: ChiSquared('x', k))
k = Symbol("k", integer=False, positive=True)
raises(ValueError, lambda: ChiSquared('x', k))
def test_chi_squared():
k = Symbol("k", integer=True)
X = ChiSquared('x', k)
# Tests the characteristic function
assert characteristic_function(X)(x) == ((-2*I*x + 1)**(-k/2))
assert density(X)(x) == 2**(-k/2)*x**(k/2 - 1)*exp(-x/2)/gamma(k/2)
assert cdf(X)(x) == Piecewise((lowergamma(k/2, x/2)/gamma(k/2), x >= 0), (0, True))
assert E(X) == k
assert variance(X) == 2*k
X = ChiSquared('x', 15)
assert cdf(X)(3) == -14873*sqrt(6)*exp(Rational(-3, 2))/(5005*sqrt(pi)) + erf(sqrt(6)/2)
k = Symbol("k", integer=True, positive=False)
raises(ValueError, lambda: ChiSquared('x', k))
k = Symbol("k", integer=False, positive=True)
raises(ValueError, lambda: ChiSquared('x', k))
def test_precomputed_cdf():
x = symbols("x", real=True, finite=True)
mu = symbols("mu", real=True, finite=True)
sigma, xm, alpha = symbols("sigma xm alpha", positive=True, finite=True)
n = symbols("n", integer=True, positive=True, finite=True)
distribs = [
Normal("X", mu, sigma),
Pareto("P", xm, alpha),
ChiSquared("C", n),
Exponential("E", sigma),
# LogNormal("L", mu, sigma),
]
for X in distribs:
compdiff = cdf(X)(x) - simplify(X.pspace.density.compute_cdf()(x))
compdiff = simplify(compdiff.rewrite(erfc))
assert compdiff == 0
def test_sample_scipy():
distribs_scipy = [
Beta("B", 1, 1),
BetaPrime("BP", 1, 1),
Cauchy("C", 1, 1),
Chi("C", 1),
Normal("N", 0, 1),
Gamma("G", 2, 7),
GammaInverse("GI", 1, 1),
GaussianInverse("GUI", 1, 1),
Exponential("E", 2),
LogNormal("LN", 0, 1),
Pareto("P", 1, 1),
StudentT("S", 2),
ChiSquared("CS", 2),
Uniform("U", 0, 1)
]
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
g_sample = list(
sample(Gamma("G", 2, 7), size=size, numsamples=numsamples))
assert len(g_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_chi_squared():
k = Symbol("k", integer=True)
X = ChiSquared('x', k)
assert density(X)(x) == 2**(-k/2)*x**(k/2 - 1)*exp(-x/2)/gamma(k/2)
def test_moment_generating_function():
t = symbols('t', positive=True)
# Symbolic tests
a, b, c = symbols('a b c')
mgf = moment_generating_function(Beta('x', a, b))(t)
assert mgf == hyper((a, ), (a + b, ), t)
mgf = moment_generating_function(Chi('x', a))(t)
assert mgf == sqrt(2)*t*gamma(a/2 + S.Half)*\
hyper((a/2 + S.Half,), (Rational(3, 2),), t**2/2)/gamma(a/2) +\
hyper((a/2,), (S.Half,), t**2/2)
mgf = moment_generating_function(ChiSquared('x', a))(t)
assert mgf == (1 - 2 * t)**(-a / 2)
mgf = moment_generating_function(Erlang('x', a, b))(t)
assert mgf == (1 - t / b)**(-a)
mgf = moment_generating_function(ExGaussian("x", a, b, c))(t)
assert mgf == exp(a * t + b**2 * t**2 / 2) / (1 - t / c)
mgf = moment_generating_function(Exponential('x', a))(t)
assert mgf == a / (a - t)
mgf = moment_generating_function(Gamma('x', a, b))(t)
assert mgf == (-b * t + 1)**(-a)
mgf = moment_generating_function(Gumbel('x', a, b))(t)
assert mgf == exp(b * t) * gamma(-a * t + 1)
mgf = moment_generating_function(Gompertz('x', a, b))(t)
assert mgf == b * exp(b) * expint(t / a, b)
mgf = moment_generating_function(Laplace('x', a, b))(t)
assert mgf == exp(a * t) / (-b**2 * t**2 + 1)
mgf = moment_generating_function(Logistic('x', a, b))(t)
assert mgf == exp(a * t) * beta(-b * t + 1, b * t + 1)
mgf = moment_generating_function(Normal('x', a, b))(t)
assert mgf == exp(a * t + b**2 * t**2 / 2)
mgf = moment_generating_function(Pareto('x', a, b))(t)
assert mgf == b * (-a * t)**b * uppergamma(-b, -a * t)
mgf = moment_generating_function(QuadraticU('x', a, b))(t)
assert str(mgf) == (
"(3*(t*(-4*b + (a + b)**2) + 4)*exp(b*t) - "
"3*(t*(a**2 + 2*a*(b - 2) + b**2) + 4)*exp(a*t))/(t**2*(a - b)**3)")
mgf = moment_generating_function(RaisedCosine('x', a, b))(t)
assert mgf == pi**2 * exp(a * t) * sinh(b * t) / (b * t *
(b**2 * t**2 + pi**2))
mgf = moment_generating_function(Rayleigh('x', a))(t)
assert mgf == sqrt(2)*sqrt(pi)*a*t*(erf(sqrt(2)*a*t/2) + 1)\
*exp(a**2*t**2/2)/2 + 1
mgf = moment_generating_function(Triangular('x', a, b, c))(t)
assert str(mgf) == ("(-2*(-a + b)*exp(c*t) + 2*(-a + c)*exp(b*t) + "
"2*(b - c)*exp(a*t))/(t**2*(-a + b)*(-a + c)*(b - c))")
mgf = moment_generating_function(Uniform('x', a, b))(t)
assert mgf == (-exp(a * t) + exp(b * t)) / (t * (-a + b))
mgf = moment_generating_function(UniformSum('x', a))(t)
assert mgf == ((exp(t) - 1) / t)**a
mgf = moment_generating_function(WignerSemicircle('x', a))(t)
assert mgf == 2 * besseli(1, a * t) / (a * t)
# Numeric tests
mgf = moment_generating_function(Beta('x', 1, 1))(t)
assert mgf.diff(t).subs(t, 1) == hyper((2, ), (3, ), 1) / 2
mgf = moment_generating_function(Chi('x', 1))(t)
assert mgf.diff(t).subs(t, 1) == sqrt(2) * hyper(
(1, ), (Rational(3, 2), ), S.Half) / sqrt(pi) + hyper(
(Rational(3, 2), ),
(Rational(3, 2), ), S.Half) + 2 * sqrt(2) * hyper(
(2, ), (Rational(5, 2), ), S.Half) / (3 * sqrt(pi))
mgf = moment_generating_function(ChiSquared('x', 1))(t)
assert mgf.diff(t).subs(t, 1) == I
mgf = moment_generating_function(Erlang('x', 1, 1))(t)
assert mgf.diff(t).subs(t, 0) == 1
mgf = moment_generating_function(ExGaussian("x", 0, 1, 1))(t)
assert mgf.diff(t).subs(t, 2) == -exp(2)
mgf = moment_generating_function(Exponential('x', 1))(t)
assert mgf.diff(t).subs(t, 0) == 1
mgf = moment_generating_function(Gamma('x', 1, 1))(t)
assert mgf.diff(t).subs(t, 0) == 1
mgf = moment_generating_function(Gumbel('x', 1, 1))(t)
assert mgf.diff(t).subs(t, 0) == EulerGamma + 1
mgf = moment_generating_function(Gompertz('x', 1, 1))(t)
assert mgf.diff(t).subs(t, 1) == -e * meijerg(((), (1, 1)),
((0, 0, 0), ()), 1)
mgf = moment_generating_function(Laplace('x', 1, 1))(t)
assert mgf.diff(t).subs(t, 0) == 1
mgf = moment_generating_function(Logistic('x', 1, 1))(t)
assert mgf.diff(t).subs(t, 0) == beta(1, 1)
mgf = moment_generating_function(Normal('x', 0, 1))(t)
assert mgf.diff(t).subs(t, 1) == exp(S.Half)
mgf = moment_generating_function(Pareto('x', 1, 1))(t)
assert mgf.diff(t).subs(t, 0) == expint(1, 0)
mgf = moment_generating_function(QuadraticU('x', 1, 2))(t)
assert mgf.diff(t).subs(t, 1) == -12 * e - 3 * exp(2)
mgf = moment_generating_function(RaisedCosine('x', 1, 1))(t)
assert mgf.diff(t).subs(t, 1) == -2*e*pi**2*sinh(1)/\
(1 + pi**2)**2 + e*pi**2*cosh(1)/(1 + pi**2)
mgf = moment_generating_function(Rayleigh('x', 1))(t)
assert mgf.diff(t).subs(t, 0) == sqrt(2) * sqrt(pi) / 2
mgf = moment_generating_function(Triangular('x', 1, 3, 2))(t)
assert mgf.diff(t).subs(t, 1) == -e + exp(3)
mgf = moment_generating_function(Uniform('x', 0, 1))(t)
assert mgf.diff(t).subs(t, 1) == 1
mgf = moment_generating_function(UniformSum('x', 1))(t)
assert mgf.diff(t).subs(t, 1) == 1
mgf = moment_generating_function(WignerSemicircle('x', 1))(t)
assert mgf.diff(t).subs(t, 1) == -2*besseli(1, 1) + besseli(2, 1) +\
besseli(0, 1)
