def test_JointPSpace_margial_distribution():
from sympy.stats.joint_rv_types import MultivariateT
from sympy import polar_lift
T = MultivariateT('T', [0, 0], [[1, 0], [0, 1]], 2)
assert marginal_distribution(T, T[1])(x) == sqrt(2)*(x**2 + 2)/(
8*polar_lift(x**2/2 + 1)**(S(5)/2))
assert integrate(marginal_distribution(T, 1)(x), (x, -oo, oo)) == 1
def test_JointPSpace_margial_distribution():
from sympy.stats.joint_rv_types import MultivariateT
from sympy import polar_lift
T = MultivariateT('T', [0, 0], [[1, 0], [0, 1]], 2)
assert marginal_distribution(
T, T[1])(x) == sqrt(2) * (x**2 +
2) / (8 * polar_lift(x**2 / 2 + 1)**(5 / 2))
assert integrate(marginal_distribution(T, 1)(x), (x, -oo, oo)) == 1
def test_NormalGamma():
from sympy.stats.joint_rv_types import NormalGamma
from sympy import gamma
ng = NormalGamma('G', 1, 2, 3, 4)
assert density(ng)(1, 1) == 32*exp(-4)/sqrt(pi)
raises(ValueError, lambda:NormalGamma('G', 1, 2, 3, -1))
assert marginal_distribution(ng, 0)(1) == \
3*sqrt(10)*gamma(S(7)/4)/(10*sqrt(pi)*gamma(S(5)/4))
assert marginal_distribution(ng, y)(1) == exp(-S(1)/4)/128
def test_JointPSpace_marginal_distribution():
from sympy.stats.joint_rv_types import MultivariateT
from sympy import polar_lift
T = MultivariateT('T', [0, 0], [[1, 0], [0, 1]], 2)
assert marginal_distribution(T, T[1])(x) == sqrt(2) * (x**2 + 2) / (
8 * polar_lift(x**2 / 2 + 1)**Rational(5, 2))
assert integrate(marginal_distribution(T, 1)(x), (x, -oo, oo)) == 1
t = MultivariateT('T', [0, 0, 0], [[1, 0, 0], [0, 1, 0], [0, 0, 1]], 3)
assert comp(marginal_distribution(t, 0)(1).evalf(), 0.2, .01)
def test_NormalGamma():
from sympy.stats.joint_rv_types import NormalGamma
from sympy import gamma
ng = NormalGamma('G', 1, 2, 3, 4)
assert density(ng)(1, 1) == 32*exp(-4)/sqrt(pi)
raises(ValueError, lambda:NormalGamma('G', 1, 2, 3, -1))
assert marginal_distribution(ng, 0)(1) == \
3*sqrt(10)*gamma(S(7)/4)/(10*sqrt(pi)*gamma(S(5)/4))
assert marginal_distribution(ng, y)(1) == exp(-S(1)/4)/128
def test_MultivariateEwens():
from sympy.stats.joint_rv_types import MultivariateEwens
n, theta, i = symbols('n theta i', positive=True)
# tests for integer dimensions
theta_f = symbols('t_f', negative=True)
a = symbols('a_1:4', positive = True, integer = True)
ed = MultivariateEwens('E', 3, theta)
assert density(ed)(a[0], a[1], a[2]) == Piecewise((6*2**(-a[1])*3**(-a[2])*
theta**a[0]*theta**a[1]*theta**a[2]/
(theta*(theta + 1)*(theta + 2)*
factorial(a[0])*factorial(a[1])*
factorial(a[2])), Eq(a[0] + 2*a[1] +
3*a[2], 3)), (0, True))
assert marginal_distribution(ed, ed[1])(a[1]) == Piecewise((6*2**(-a[1])*
theta**a[1]/((theta + 1)*
(theta + 2)*factorial(a[1])),
Eq(2*a[1] + 1, 3)), (0, True))
raises(ValueError, lambda: MultivariateEwens('e1', 5, theta_f))
# tests for symbolic dimensions
eds = MultivariateEwens('E', n, theta)
a = IndexedBase('a')
j, k = symbols('j, k')
den = Piecewise((factorial(n)*Product(theta**a[j]*(j + 1)**(-a[j])/
factorial(a[j]), (j, 0, n - 1))/RisingFactorial(theta, n),
Eq(n, Sum((k + 1)*a[k], (k, 0, n - 1)))), (0, True))
assert density(eds)(a).dummy_eq(den)
def test_GeneralizedMultivariateLogGammaDistribution():
from sympy.stats.joint_rv_types import GeneralizedMultivariateLogGammaOmega as GMVLGO
from sympy.stats.joint_rv_types import GeneralizedMultivariateLogGamma as GMVLG
h = S.Half
omega = Matrix([[1, h, h, h],
[h, 1, h, h],
[h, h, 1, h],
[h, h, h, 1]])
v, l, mu = (4, [1, 2, 3, 4], [1, 2, 3, 4])
y_1, y_2, y_3, y_4 = symbols('y_1:5', real=True)
delta = symbols('d', positive=True)
G = GMVLGO('G', omega, v, l, mu)
Gd = GMVLG('Gd', delta, v, l, mu)
dend = ("d**4*Sum(4*24**(-n - 4)*(1 - d)**n*exp((n + 4)*(y_1 + 2*y_2 + 3*y_3 "
"+ 4*y_4) - exp(y_1) - exp(2*y_2)/2 - exp(3*y_3)/3 - exp(4*y_4)/4)/"
"(gamma(n + 1)*gamma(n + 4)**3), (n, 0, oo))")
assert str(density(Gd)(y_1, y_2, y_3, y_4)) == dend
den = ("5*2**(2/3)*5**(1/3)*Sum(4*24**(-n - 4)*(-2**(2/3)*5**(1/3)/4 + 1)**n*"
"exp((n + 4)*(y_1 + 2*y_2 + 3*y_3 + 4*y_4) - exp(y_1) - exp(2*y_2)/2 - "
"exp(3*y_3)/3 - exp(4*y_4)/4)/(gamma(n + 1)*gamma(n + 4)**3), (n, 0, oo))/64")
assert str(density(G)(y_1, y_2, y_3, y_4)) == den
marg = ("5*2**(2/3)*5**(1/3)*exp(4*y_1)*exp(-exp(y_1))*Integral(exp(-exp(4*G[3])"
"/4)*exp(16*G[3])*Integral(exp(-exp(3*G[2])/3)*exp(12*G[2])*Integral(exp("
"-exp(2*G[1])/2)*exp(8*G[1])*Sum((-1/4)**n*24**(-n)*(-4 + 2**(2/3)*5**(1/3"
"))**n*exp(n*y_1)*exp(2*n*G[1])*exp(3*n*G[2])*exp(4*n*G[3])/(gamma(n + 1)"
"*gamma(n + 4)**3), (n, 0, oo)), (G[1], -oo, oo)), (G[2], -oo, oo)), (G[3]"
", -oo, oo))/5308416")
assert str(marginal_distribution(G, G[0])(y_1)) == marg
omega_f1 = Matrix([[1, h, h]])
omega_f2 = Matrix([[1, h, h, h],
[h, 1, 2, h],
[h, h, 1, h],
[h, h, h, 1]])
omega_f3 = Matrix([[6, h, h, h],
[h, 1, 2, h],
[h, h, 1, h],
[h, h, h, 1]])
v_f = symbols("v_f", positive=False, real=True)
l_f = [1, 2, v_f, 4]
m_f = [v_f, 2, 3, 4]
omega_f4 = Matrix([[1, h, h, h, h],
[h, 1, h, h, h],
[h, h, 1, h, h],
[h, h, h, 1, h],
[h, h, h, h, 1]])
l_f1 = [1, 2, 3, 4, 5]
omega_f5 = Matrix([[1]])
mu_f5 = l_f5 = [1]
raises(ValueError, lambda: GMVLGO('G', omega_f1, v, l, mu))
raises(ValueError, lambda: GMVLGO('G', omega_f2, v, l, mu))
raises(ValueError, lambda: GMVLGO('G', omega_f3, v, l, mu))
raises(ValueError, lambda: GMVLGO('G', omega, v_f, l, mu))
raises(ValueError, lambda: GMVLGO('G', omega, v, l_f, mu))
raises(ValueError, lambda: GMVLGO('G', omega, v, l, m_f))
raises(ValueError, lambda: GMVLGO('G', omega_f4, v, l, mu))
raises(ValueError, lambda: GMVLGO('G', omega, v, l_f1, mu))
raises(ValueError, lambda: GMVLGO('G', omega_f5, v, l_f5, mu_f5))
raises(ValueError, lambda: GMVLG('G', Rational(3, 2), v, l, mu))
def test_JointRV():
from sympy.stats.joint_rv import JointDistributionHandmade
x1, x2 = (Indexed('x', i) for i in (1, 2))
pdf = exp(-x1**2/2 + x1 - x2**2/2 - S(1)/2)/(2*pi)
X = JointRV('x', pdf)
assert density(X)(1, 2) == exp(-2)/(2*pi)
assert isinstance(X.pspace.distribution, JointDistributionHandmade)
assert marginal_distribution(X, 0)(2) == sqrt(2)*exp(-S(1)/2)/(2*sqrt(pi))
def test_JointRV():
from sympy.stats.joint_rv import JointDistributionHandmade
x1, x2 = (Indexed('x', i) for i in (1, 2))
pdf = exp(-x1**2/2 + x1 - x2**2/2 - S(1)/2)/(2*pi)
X = JointRV('x', pdf)
assert density(X)(1, 2) == exp(-2)/(2*pi)
assert isinstance(X.pspace.distribution, JointDistributionHandmade)
assert marginal_distribution(X, 0)(2) == sqrt(2)*exp(-S(1)/2)/(2*sqrt(pi))
def test_Normal():
m = Normal('A', [1, 2], [[1, 0], [0, 1]])
assert density(m)(1, 2) == 1 / (2 * pi)
raises (ValueError,\
lambda: Normal('M',[1, 2], [[0, 0], [0, 1]]))
n = Normal('B', [1, 2, 3], [[1, 0, 0], [0, 1, 0], [0, 0, 1]])
p = Normal('C', Matrix([1, 2]), Matrix([[1, 0], [0, 1]]))
assert density(m)(x, y) == density(p)(x, y)
assert marginal_distribution(n, 0, 1)(1, 2) == 1 / (2 * pi)
assert integrate(density(m)(x, y), (x, -oo, oo), (y, -oo, oo)).evalf() == 1
raises(ValueError, lambda: Normal('M', [1, 2], [[1, 1], [1, -1]]))
def test_Multinomial():
from sympy.stats.joint_rv_types import Multinomial
n, x1, x2, x3, x4 = symbols('n, x1, x2, x3, x4', nonnegative=True, integer=True)
p1, p2, p3, p4 = symbols('p1, p2, p3, p4', positive=True)
p1_f = symbols('p1_f', negative=True)
M = Multinomial('M', 3, [p1, p2, p3, p4])
M_c = Multinomial('C', 3, 0.5, 0.4, 0.3, 0.2)
f = factorial
assert simplify(density(M)(x1, x2, x3, x4) -
S(6)*p1**x1*p2**x2*p3**x3*p4**x4/(f(x1)*f(x2)*f(x3)*f(x4))) == S(0)
assert marginal_distribution(M_c, M_c[0])(1).round(2) == 7.29
raises(ValueError, lambda: Multinomial('b1', 5, [p1, p2, p3, p1_f]))
raises(ValueError, lambda: Multinomial('b2', n, [p1, p2, p3, p4]))
def test_Normal():
m = Normal('A', [1, 2], [[1, 0], [0, 1]])
assert density(m)(1, 2) == 1 / (2 * pi)
raises (ValueError,\
lambda: Normal('M',[1, 2], [[0, 0], [0, 1]]))
n = Normal('B', [1, 2, 3], [[1, 0, 0], [0, 1, 0], [0, 0, 1]])
p = Normal('C', Matrix([1, 2]), Matrix([[1, 0], [0, 1]]))
assert density(m)(x, y) == density(p)(x, y)
assert marginal_distribution(n, 0, 1)(1, 2) == 1 / (2 * pi)
assert integrate(density(m)(x, y), (x, -oo, oo), (y, -oo, oo)).evalf() == 1
N1 = Normal('N1', [1, 2], [[x, 0], [0, y]])
assert str(density(N1)(0, 0)) == "exp(-(4*x + y)/(2*x*y))/(2*pi*sqrt(x*y))"
raises(ValueError, lambda: Normal('M', [1, 2], [[1, 1], [1, -1]]))
def test_Normal():
m = Normal('A', [1, 2], [[1, 0], [0, 1]])
assert density(m)(1, 2) == 1/(2*pi)
raises (ValueError,\
lambda: Normal('M',[1, 2], [[0, 0], [0, 1]]))
n = Normal('B', [1, 2, 3], [[1, 0, 0], [0, 1, 0], [0, 0, 1]])
p = Normal('C', Matrix([1, 2]), Matrix([[1, 0], [0, 1]]))
assert density(m)(x, y) == density(p)(x, y)
assert marginal_distribution(n, 0, 1)(1, 2) == 1/(2*pi)
assert integrate(density(m)(x, y), (x, -oo, oo), (y, -oo, oo)).evalf() == 1
N1 = Normal('N1', [1, 2], [[x, 0], [0, y]])
assert str(density(N1)(0, 0)) == "exp(-(4*x + y)/(2*x*y))/(2*pi*sqrt(x*y))"
raises (ValueError, lambda: Normal('M', [1, 2], [[1, 1], [1, -1]]))
def test_MultivariateBeta():
from sympy.stats.joint_rv_types import MultivariateBeta
from sympy import gamma
a1, a2 = symbols('a1, a2', positive=True)
a1_f, a2_f = symbols('a1, a2', positive=False)
mb = MultivariateBeta('B', [a1, a2])
mb_c = MultivariateBeta('C', a1, a2)
assert density(mb)(1, 2) == S(2)**(a2 - 1)*gamma(a1 + a2)/\
(gamma(a1)*gamma(a2))
assert marginal_distribution(mb_c, 0)(3) == S(3)**(a1 - 1)*gamma(a1 + a2)/\
(a2*gamma(a1)*gamma(a2))
raises(ValueError, lambda: MultivariateBeta('b1', [a1_f, a2]))
raises(ValueError, lambda: MultivariateBeta('b2', [a1, a2_f]))
raises(ValueError, lambda: MultivariateBeta('b3', [0, 0]))
raises(ValueError, lambda: MultivariateBeta('b4', [a1_f, a2_f]))
def test_NegativeMultinomial():
from sympy.stats.joint_rv_types import NegativeMultinomial
k0, x1, x2, x3, x4 = symbols('k0, x1, x2, x3, x4', nonnegative=True, integer=True)
p1, p2, p3, p4 = symbols('p1, p2, p3, p4', positive=True)
p1_f = symbols('p1_f', negative=True)
N = NegativeMultinomial('N', 4, [p1, p2, p3, p4])
C = NegativeMultinomial('C', 4, 0.1, 0.2, 0.3)
g = gamma
f = factorial
assert simplify(density(N)(x1, x2, x3, x4) -
p1**x1*p2**x2*p3**x3*p4**x4*(-p1 - p2 - p3 - p4 + 1)**4*g(x1 + x2 +
x3 + x4 + 4)/(6*f(x1)*f(x2)*f(x3)*f(x4))) == S(0)
assert marginal_distribution(C, C[0])(1).evalf().round(2) == 0.33
raises(ValueError, lambda: NegativeMultinomial('b1', 5, [p1, p2, p3, p1_f]))
raises(ValueError, lambda: NegativeMultinomial('b2', k0, 0.5, 0.4, 0.3, 0.4))
def test_MultivariateBeta():
from sympy.stats.joint_rv_types import MultivariateBeta
from sympy import gamma
a1, a2 = symbols("a1, a2", positive=True)
a1_f, a2_f = symbols("a1, a2", positive=False, real=True)
mb = MultivariateBeta("B", [a1, a2])
mb_c = MultivariateBeta("C", a1, a2)
assert density(mb)(
1, 2) == S(2)**(a2 - 1) * gamma(a1 + a2) / (gamma(a1) * gamma(a2))
assert marginal_distribution(
mb_c,
0)(3) == S(3)**(a1 - 1) * gamma(a1 + a2) / (a2 * gamma(a1) * gamma(a2))
raises(ValueError, lambda: MultivariateBeta("b1", [a1_f, a2]))
raises(ValueError, lambda: MultivariateBeta("b2", [a1, a2_f]))
raises(ValueError, lambda: MultivariateBeta("b3", [0, 0]))
raises(ValueError, lambda: MultivariateBeta("b4", [a1_f, a2_f]))
def test_Multinomial():
from sympy.stats.joint_rv_types import Multinomial
n, x1, x2, x3, x4 = symbols('n, x1, x2, x3, x4', nonnegative=True, integer=True)
p1, p2, p3, p4 = symbols('p1, p2, p3, p4', positive=True)
p1_f, n_f = symbols('p1_f, n_f', negative=True)
M = Multinomial('M', n, [p1, p2, p3, p4])
C = Multinomial('C', 3, p1, p2, p3)
f = factorial
assert density(M)(x1, x2, x3, x4) == Piecewise((p1**x1*p2**x2*p3**x3*p4**x4*
f(n)/(f(x1)*f(x2)*f(x3)*f(x4)),
Eq(n, x1 + x2 + x3 + x4)), (0, True))
assert marginal_distribution(C, C[0])(x1).subs(x1, 1) ==\
3*p1*p2**2 +\
6*p1*p2*p3 +\
3*p1*p3**2
raises(ValueError, lambda: Multinomial('b1', 5, [p1, p2, p3, p1_f]))
raises(ValueError, lambda: Multinomial('b2', n_f, [p1, p2, p3, p4]))
raises(ValueError, lambda: Multinomial('b3', n, 0.5, 0.4, 0.3, 0.1))
def test_MultivariateEwens():
from sympy.stats.joint_rv_types import MultivariateEwens
n, theta = symbols('n theta', positive=True)
theta_f = symbols('t_f', negative=True)
a = symbols('a_1:4', positive = True, integer = True)
ed = MultivariateEwens('E', 3, theta)
assert density(ed)(a[0], a[1], a[2]) == Piecewise((6*2**(-a[1])*3**(-a[2])*
theta**a[0]*theta**a[1]*theta**a[2]/
(theta*(theta + 1)*(theta + 2)*
factorial(a[0])*factorial(a[1])*
factorial(a[2])), Eq(a[0] + 2*a[1] +
3*a[2], 3)), (0, True))
assert marginal_distribution(ed, ed[1])(a[1]) == Piecewise((6*2**(-a[1])*
theta**a[1]/((theta + 1)*
(theta + 2)*factorial(a[1])),
Eq(2*a[1] + 1, 3)), (0, True))
raises(ValueError, lambda: MultivariateEwens('e1', 5, theta_f))
raises(ValueError, lambda: MultivariateEwens('e1', n, theta))
def test_Normal():
m = Normal('A', [1, 2], [[1, 0], [0, 1]])
assert density(m)(1, 2) == 1/(2*pi)
raises (ValueError, lambda:m[2])
raises (ValueError,\
lambda: Normal('M',[1, 2], [[0, 0], [0, 1]]))
n = Normal('B', [1, 2, 3], [[1, 0, 0], [0, 1, 0], [0, 0, 1]])
p = Normal('C', Matrix([1, 2]), Matrix([[1, 0], [0, 1]]))
assert density(m)(x, y) == density(p)(x, y)
assert marginal_distribution(n, 0, 1)(1, 2) == 1/(2*pi)
assert integrate(density(m)(x, y), (x, -oo, oo), (y, -oo, oo)).evalf() == 1
N = Normal('N', [1, 2], [[x, 0], [0, y]])
assert density(N)(0, 0) == exp(-2/y - 1/(2*x))/(2*pi*sqrt(x*y))
raises (ValueError, lambda: Normal('M', [1, 2], [[1, 1], [1, -1]]))
# symbolic
n = symbols('n', natural=True)
mu = MatrixSymbol('mu', n, 1)
sigma = MatrixSymbol('sigma', n, n)
X = Normal('X', mu, sigma)
assert density(X) == MultivariateNormalDistribution(mu, sigma)
def test_NegativeMultinomial():
from sympy.stats.joint_rv_types import NegativeMultinomial
k0, x1, x2, x3, x4 = symbols("k0, x1, x2, x3, x4",
nonnegative=True,
integer=True)
p1, p2, p3, p4 = symbols("p1, p2, p3, p4", positive=True)
p1_f = symbols("p1_f", negative=True)
N = NegativeMultinomial("N", 4, [p1, p2, p3, p4])
C = NegativeMultinomial("C", 4, 0.1, 0.2, 0.3)
g = gamma
f = factorial
assert (simplify(
density(N)(x1, x2, x3, x4) - p1**x1 * p2**x2 * p3**x3 * p4**x4 *
(-p1 - p2 - p3 - p4 + 1)**4 * g(x1 + x2 + x3 + x4 + 4) /
(6 * f(x1) * f(x2) * f(x3) * f(x4))) is S.Zero)
assert comp(marginal_distribution(C, C[0])(1).evalf(), 0.33, 0.01)
raises(ValueError,
lambda: NegativeMultinomial("b1", 5, [p1, p2, p3, p1_f]))
raises(ValueError,
lambda: NegativeMultinomial("b2", k0, 0.5, 0.4, 0.3, 0.4))
def test_Multinomial():
from sympy.stats.joint_rv_types import Multinomial
n, x1, x2, x3, x4 = symbols("n, x1, x2, x3, x4",
nonnegative=True,
integer=True)
p1, p2, p3, p4 = symbols("p1, p2, p3, p4", positive=True)
p1_f, n_f = symbols("p1_f, n_f", negative=True)
M = Multinomial("M", n, [p1, p2, p3, p4])
C = Multinomial("C", 3, p1, p2, p3)
f = factorial
assert density(M)(x1, x2, x3, x4) == Piecewise(
(
p1**x1 * p2**x2 * p3**x3 * p4**x4 * f(n) /
(f(x1) * f(x2) * f(x3) * f(x4)),
Eq(n, x1 + x2 + x3 + x4),
),
(0, True),
)
assert (marginal_distribution(C, C[0])(x1).subs(x1, 1) == 3 * p1 * p2**2 +
6 * p1 * p2 * p3 + 3 * p1 * p3**2)
raises(ValueError, lambda: Multinomial("b1", 5, [p1, p2, p3, p1_f]))
raises(ValueError, lambda: Multinomial("b2", n_f, [p1, p2, p3, p4]))
raises(ValueError, lambda: Multinomial("b3", n, 0.5, 0.4, 0.3, 0.1))
