def test_Compound_Distribution():
X = Normal('X', 2, 4)
N = NormalDistribution(X, 4)
C = CompoundDistribution(N)
assert C.is_Continuous
assert C.set == Interval(-oo, oo)
assert C.pdf(x, evaluate=True).simplify() == exp(-x**2/64 + x/16 - S(1)/16)/(8*sqrt(pi))
assert not isinstance(CompoundDistribution(NormalDistribution(2, 3)),
CompoundDistribution)
M = MultivariateNormalDistribution([1, 2], [[2, 1], [1, 2]])
raises(NotImplementedError, lambda: CompoundDistribution(M))
X = Beta('X', 2, 4)
B = BernoulliDistribution(X, 1, 0)
C = CompoundDistribution(B)
assert C.is_Finite
assert C.set == {0, 1}
y = symbols('y', negative=False, integer=True)
assert C.pdf(y, evaluate=True) == Piecewise((S(1)/(30*beta(2, 4)), Eq(y, 0)),
(S(1)/(60*beta(2, 4)), Eq(y, 1)), (0, True))
k, t, z = symbols('k t z', positive=True, real=True)
G = Gamma('G', k, t)
X = PoissonDistribution(G)
C = CompoundDistribution(X)
assert C.is_Discrete
assert C.set == S.Naturals0
assert C.pdf(z, evaluate=True).simplify() == t**z*(t + 1)**(-k - z)*gamma(k \
+ z)/(gamma(k)*gamma(z + 1))
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])
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)
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(-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)
raises(NotImplementedError, lambda: median(m))
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', integer=True, positive=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', integer=True, positive=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_compound_pspace():
X = Normal('X', 2, 4)
Y = Normal('Y', 3, 6)
assert not isinstance(Y.pspace, CompoundPSpace)
N = NormalDistribution(1, 2)
D = PoissonDistribution(3)
B = BernoulliDistribution(0.2, 1, 0)
pspace1 = CompoundPSpace('N', N)
pspace2 = CompoundPSpace('D', D)
pspace3 = CompoundPSpace('B', B)
assert not isinstance(pspace1, CompoundPSpace)
assert not isinstance(pspace2, CompoundPSpace)
assert not isinstance(pspace3, CompoundPSpace)
M = MultivariateNormalDistribution([1, 2], [[2, 1], [1, 2]])
raises(ValueError, lambda: CompoundPSpace('M', M))
Y = Normal('Y', X, 6)
assert isinstance(Y.pspace, CompoundPSpace)
assert Y.pspace.distribution == CompoundDistribution(NormalDistribution(X, 6))
assert Y.pspace.domain.set == Interval(-oo, oo)
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)