%PDF-
%PDF-
Mini Shell
Mini Shell
from sympy import (symbols, pi, oo, S, exp, sqrt, besselk, Indexed, Sum, simplify,
Rational, factorial, gamma, Piecewise, Eq, Product, Interval,
IndexedBase, RisingFactorial, polar_lift, ProductSet, Range, eye,
Determinant)
from sympy.core.numbers import comp
from sympy.integrals.integrals import integrate
from sympy.matrices import Matrix, MatrixSymbol
from sympy.matrices.expressions.matexpr import MatrixElement
from sympy.stats import density, median, marginal_distribution, Normal, Laplace, E, sample
from sympy.stats.joint_rv_types import (JointRV, MultivariateNormalDistribution,
JointDistributionHandmade, MultivariateT, NormalGamma,
GeneralizedMultivariateLogGammaOmega as GMVLGO, MultivariateBeta,
GeneralizedMultivariateLogGamma as GMVLG, MultivariateEwens,
Multinomial, NegativeMultinomial, MultivariateNormal,
MultivariateLaplace)
from sympy.testing.pytest import raises, XFAIL, skip
from sympy.external import import_module
x, y, z, a, b = symbols('x y z a b')
def test_Normal():
m = Normal('A', [1, 2], [[1, 0], [0, 1]])
A = MultivariateNormal('A', [1, 2], [[1, 0], [0, 1]])
assert m == A
assert density(m)(1, 2) == 1/(2*pi)
assert m.pspace.distribution.set == ProductSet(S.Reals, S.Reals)
raises (ValueError, lambda:m[2])
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)
raises(ValueError, lambda: marginal_distribution(m))
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(-((4*x + y)/(2*x*y)))/(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)
raises (NotImplementedError, lambda: median(m))
# Below tests should work after issue #17267 is resolved
# assert E(X) == mu
# assert variance(X) == sigma
# test symbolic multivariate normal densities
n = 3
Sg = MatrixSymbol('Sg', n, n)
mu = MatrixSymbol('mu', n, 1)
obs = MatrixSymbol('obs', n, 1)
X = MultivariateNormal('X', mu, Sg)
density_X = density(X)
eval_a = density_X(obs).subs({Sg: eye(3),
mu: Matrix([0, 0, 0]), obs: Matrix([0, 0, 0])}).doit()
eval_b = density_X(0, 0, 0).subs({Sg: eye(3), mu: Matrix([0, 0, 0])}).doit()
assert eval_a == sqrt(2)/(4*pi**Rational(3/2))
assert eval_b == sqrt(2)/(4*pi**Rational(3/2))
n = symbols('n', natural=True)
Sg = MatrixSymbol('Sg', n, n)
mu = MatrixSymbol('mu', n, 1)
obs = MatrixSymbol('obs', n, 1)
X = MultivariateNormal('X', mu, Sg)
density_X_at_obs = density(X)(obs)
expected_density = MatrixElement(
exp((S(1)/2) * (mu.T - obs.T) * Sg**(-1) * (-mu + obs)) / \
sqrt((2*pi)**n * Determinant(Sg)), 0, 0)
assert density_X_at_obs == expected_density
def test_MultivariateTDist():
t1 = MultivariateT('T', [0, 0], [[1, 0], [0, 1]], 2)
assert(density(t1))(1, 1) == 1/(8*pi)
assert t1.pspace.distribution.set == ProductSet(S.Reals, S.Reals)
assert integrate(density(t1)(x, y), (x, -oo, oo), \
(y, -oo, oo)).evalf() == 1
raises(ValueError, lambda: MultivariateT('T', [1, 2], [[1, 1], [1, -1]], 1))
t2 = MultivariateT('t2', [1, 2], [[x, 0], [0, y]], 1)
assert density(t2)(1, 2) == 1/(2*pi*sqrt(x*y))
def test_multivariate_laplace():
raises(ValueError, lambda: Laplace('T', [1, 2], [[1, 2], [2, 1]]))
L = Laplace('L', [1, 0], [[1, 0], [0, 1]])
L2 = MultivariateLaplace('L2', [1, 0], [[1, 0], [0, 1]])
assert density(L)(2, 3) == exp(2)*besselk(0, sqrt(39))/pi
L1 = Laplace('L1', [1, 2], [[x, 0], [0, y]])
assert density(L1)(0, 1) == \
exp(2/y)*besselk(0, sqrt((2 + 4/y + 1/x)/y))/(pi*sqrt(x*y))
assert L.pspace.distribution.set == ProductSet(S.Reals, S.Reals)
assert L.pspace.distribution == L2.pspace.distribution
def test_NormalGamma():
ng = NormalGamma('G', 1, 2, 3, 4)
assert density(ng)(1, 1) == 32*exp(-4)/sqrt(pi)
assert ng.pspace.distribution.set == ProductSet(S.Reals, Interval(0, oo))
raises(ValueError, lambda:NormalGamma('G', 1, 2, 3, -1))
assert marginal_distribution(ng, 0)(1) == \
3*sqrt(10)*gamma(Rational(7, 4))/(10*sqrt(pi)*gamma(Rational(5, 4)))
assert marginal_distribution(ng, y)(1) == exp(Rational(-1, 4))/128
assert marginal_distribution(ng,[0,1])(x) == x**2*exp(-x/4)/128
def test_GeneralizedMultivariateLogGammaDistribution():
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*(-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])/(24**n*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_MultivariateBeta():
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]))
assert mb.pspace.distribution.set == ProductSet(Interval(0, 1), Interval(0, 1))
def test_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))
assert ed.pspace.distribution.set == ProductSet(Range(0, 4, 1),
Range(0, 2, 1), Range(0, 2, 1))
# 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_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_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, .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))
assert N.pspace.distribution.set == ProductSet(Range(0, oo, 1),
Range(0, oo, 1), Range(0, oo, 1), Range(0, oo, 1))
def test_JointPSpace_marginal_distribution():
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_JointRV():
x1, x2 = (Indexed('x', i) for i in (1, 2))
pdf = exp(-x1**2/2 + x1 - x2**2/2 - S.Half)/(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(Rational(-1, 2))/(2*sqrt(pi))
def test_expectation():
m = Normal('A', [x, y], [[1, 0], [0, 1]])
assert simplify(E(m[1])) == y
@XFAIL
def test_joint_vector_expectation():
m = Normal('A', [x, y], [[1, 0], [0, 1]])
assert E(m) == (x, y)
def test_sample_numpy():
distribs_numpy = [
MultivariateNormal("M", [3, 4], [[2, 1], [1, 2]]),
MultivariateBeta("B", [0.4, 5, 15, 50, 203]),
Multinomial("N", 50, [0.3, 0.2, 0.1, 0.25, 0.15])
]
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 tuple(sam) in X.pspace.distribution.set
N_c = NegativeMultinomial('N', 3, 0.1, 0.1, 0.1)
raises(NotImplementedError, lambda: sample(N_c, library='numpy'))
def test_sample_scipy():
distribs_scipy = [
MultivariateNormal("M", [0, 0], [[0.1, 0.025], [0.025, 0.1]]),
MultivariateBeta("B", [0.4, 5, 15]),
Multinomial("N", 8, [0.3, 0.2, 0.1, 0.4])
]
size = 3
scipy = import_module('scipy')
if not scipy:
skip('Scipy not installed. Abort tests for _sample_scipy.')
else:
for X in distribs_scipy:
samps = sample(X, size=size)
samps2 = sample(X, size=(2, 2))
for sam in samps:
assert tuple(sam) in X.pspace.distribution.set
for i in range(2):
for j in range(2):
assert tuple(samps2[i][j]) in X.pspace.distribution.set
N_c = NegativeMultinomial('N', 3, 0.1, 0.1, 0.1)
raises(NotImplementedError, lambda: sample(N_c))
def test_sample_pymc3():
distribs_pymc3 = [
MultivariateNormal("M", [5, 2], [[1, 0], [0, 1]]),
MultivariateBeta("B", [0.4, 5, 15]),
Multinomial("N", 4, [0.3, 0.2, 0.1, 0.4])
]
size = 3
pymc3 = import_module('pymc3')
if not pymc3:
skip('PyMC3 is not installed. Abort tests for _sample_pymc3.')
else:
for X in distribs_pymc3:
samps = sample(X, size=size, library='pymc3')
for sam in samps:
assert tuple(sam.flatten()) in X.pspace.distribution.set
N_c = NegativeMultinomial('N', 3, 0.1, 0.1, 0.1)
raises(NotImplementedError, lambda: sample(N_c, library='pymc3'))
def test_sample_seed():
x1, x2 = (Indexed('x', i) for i in (1, 2))
pdf = exp(-x1**2/2 + x1 - x2**2/2 - S.Half)/(2*pi)
X = JointRV('x', pdf)
libraries = ['scipy', 'numpy', 'pymc3']
for lib in libraries:
try:
imported_lib = import_module(lib)
if imported_lib:
s0, s1, s2 = [], [], []
s0 = sample(X, size=10, library=lib, seed=0)
s1 = sample(X, size=10, library=lib, seed=0)
s2 = sample(X, size=10, library=lib, seed=1)
assert all(s0 == s1)
assert all(s1 != s2)
except NotImplementedError:
continue
def test_issue_21057():
m = Normal("x", [0, 0], [[0, 0], [0, 0]])
n = MultivariateNormal("x", [0, 0], [[0, 0], [0, 0]])
p = Normal("x", [0, 0], [[0, 0], [0, 1]])
assert m == n
libraries = ['scipy', 'numpy', 'pymc3']
for library in libraries:
try:
imported_lib = import_module(library)
if imported_lib:
s1 = sample(m, size=8)
s2 = sample(n, size=8)
s3 = sample(p, size=8)
assert tuple(s1.flatten()) == tuple(s2.flatten())
for s in s3:
assert tuple(s.flatten()) in p.pspace.distribution.set
except NotImplementedError:
continue
Zerion Mini Shell 1.0