def test_issue_12283():
x = symbols('x')
X = RandomSymbol(x)
Y = RandomSymbol('Y')
Z = RandomMatrixSymbol('Z', 2, 1)
W = RandomMatrixSymbol('W', 2, 1)
RI = RandomIndexedSymbol(Indexed('RI', 3))
assert pspace(Z) == PSpace()
assert pspace(RI) == PSpace()
assert pspace(X) == PSpace()
assert E(X) == Expectation(X)
assert P(Y > 3) == Probability(Y > 3)
assert variance(X) == Variance(X)
assert variance(RI) == Variance(RI)
assert covariance(X, Y) == Covariance(X, Y)
assert covariance(W, Z) == Covariance(W, Z)
def test_issue_19841():
G1 = GUE('U', 2)
G2 = G1.xreplace({2: 2})
assert G1.args == G2.args
X = MatrixSymbol('X', 2, 2)
G = GSE('U', 2)
h_pspace = RandomMatrixPSpace('P', model=density(G))
H = RandomMatrixSymbol('H', 2, 2, pspace=h_pspace)
H2 = RandomMatrixSymbol('H', 2, 2, pspace=None)
assert H.doit() == H
assert (2 * H).xreplace({H: X}) == 2 * X
assert (2 * H).xreplace({H2: X}) == 2 * H
assert (2 * H2).xreplace({H: X}) == 2 * H2
assert (2 * H2).xreplace({H2: X}) == 2 * X
def __new__(cls, sym, dim=None):
sym, dim = _symbol_converter(sym), _sympify(dim)
if dim.is_integer == False:
raise ValueError("Dimension of the random matrices must be "
"integers, received %s instead." % (dim))
self = Basic.__new__(cls, sym, dim)
rmp = RandomMatrixPSpace(sym, model=self)
return RandomMatrixSymbol(sym, dim, dim, pspace=rmp)
def test_GaussianUnitaryEnsemble():
H = RandomMatrixSymbol('H', 3, 3)
G = GUE('U', 3)
assert density(G)(H) == sqrt(2)*exp(-3*Trace(H**2)/2)/(4*pi**(S(9)/2))
i, j = (Dummy('i', integer=True, positive=True),
Dummy('j', integer=True, positive=True))
l = IndexedBase('l')
assert joint_eigen_distribution(G).dummy_eq(
Lambda((l[1], l[2], l[3]),
27*sqrt(6)*exp(-3*(l[1]**2)/2 - 3*(l[2]**2)/2 - 3*(l[3]**2)/2)*
Product(Abs(l[i] - l[j])**2, (j, i + 1, 3), (i, 1, 2))/(16*pi**(S(3)/2))))
s = Dummy('s')
assert level_spacing_distribution(G).dummy_eq(Lambda(s, 32*s**2*exp(-4*s**2/pi)/pi**2))
def test_GaussianSymplecticEnsemble():
H = RandomMatrixSymbol('H', 3, 3)
_H = MatrixSymbol('_H', 3, 3)
G = GSE('O', 3)
assert density(G)(H) == exp(-3*Trace(H**2))/Integral(exp(-3*Trace(_H**2)), _H)
i, j = (Dummy('i', integer=True, positive=True),
Dummy('j', integer=True, positive=True))
l = IndexedBase('l')
assert joint_eigen_distribution(G).dummy_eq(
Lambda((l[1], l[2], l[3]),
162*sqrt(3)*exp(-3*l[1]**2/2 - 3*l[2]**2/2 - 3*l[3]**2/2)*
Product(Abs(l[i] - l[j])**4, (j, i + 1, 3), (i, 1, 2))/(5*pi**(S(3)/2))))
s = Dummy('s')
assert level_spacing_distribution(G).dummy_eq(Lambda(s, S(262144)*s**4*exp(-64*s**2/(9*pi))/(729*pi**3)))
def test_GaussianOrthogonalEnsemble():
H = RandomMatrixSymbol('H', 3, 3)
_H = MatrixSymbol('_H', 3, 3)
G = GOE('O', 3)
assert density(G)(H) == exp(-3*Trace(H**2)/4)/Integral(exp(-3*Trace(_H**2)/4), _H)
i, j = (Dummy('i', integer=True, positive=True),
Dummy('j', integer=True, positive=True))
l = IndexedBase('l')
assert joint_eigen_distribution(G).dummy_eq(
Lambda((l[1], l[2], l[3]),
9*sqrt(2)*exp(-3*l[1]**2/2 - 3*l[2]**2/2 - 3*l[3]**2/2)*
Product(Abs(l[i] - l[j]), (j, i + 1, 3), (i, 1, 2))/(32*pi)))
s = Dummy('s')
assert level_spacing_distribution(G).dummy_eq(Lambda(s, s*pi*exp(-s**2*pi/4)/2))
def GaussianSymplecticEnsemble(sym, dim):
"""
Represents Gaussian Symplectic Ensembles.
Examples
========
>>> from sympy.stats import GaussianSymplecticEnsemble as GSE, density
>>> from sympy import MatrixSymbol
>>> G = GSE('U', 2)
>>> X = MatrixSymbol('X', 2, 2)
>>> density(G)(X)
exp(-2*Trace(X**2))/Integral(exp(-2*Trace(_H**2)), _H)
"""
sym, dim = _symbol_converter(sym), _sympify(dim)
model = GaussianSymplecticEnsembleModel(sym, dim)
rmp = RandomMatrixPSpace(sym, model=model)
return RandomMatrixSymbol(sym, dim, dim, pspace=rmp)
def GaussianUnitaryEnsemble(sym, dim):
"""
Represents Gaussian Unitary Ensembles.
Examples
========
>>> from sympy.stats import GaussianUnitaryEnsemble as GUE, density
>>> from sympy import MatrixSymbol
>>> G = GUE('U', 2)
>>> X = MatrixSymbol('X', 2, 2)
>>> density(G)(X)
exp(-Trace(X**2))/(2*pi**2)
"""
sym, dim = _symbol_converter(sym), _sympify(dim)
model = GaussianUnitaryEnsembleModel(sym, dim)
rmp = RandomMatrixPSpace(sym, model=model)
return RandomMatrixSymbol(sym, dim, dim, pspace=rmp)
def CircularSymplecticEnsemble(sym, dim):
"""
Represents Cicular Symplectic Ensembles.
Examples
========
>>> from sympy.stats import CircularSymplecticEnsemble as CSE
>>> from sympy.stats import joint_eigen_distribution
>>> C = CSE('S', 1)
>>> joint_eigen_distribution(C)
Lambda(t[1], Product(Abs(exp(I*t[_j]) - exp(I*t[_k]))**4, (_j, _k + 1, 1), (_k, 1, 0))/(2*pi))
Note
====
As can be seen above in the example, density of CiruclarSymplecticEnsemble
is not evaluated becuase the exact definition is based on haar measure of
unitary group which is not unique.
"""
sym, dim = _symbol_converter(sym), _sympify(dim)
model = CircularSymplecticEnsembleModel(sym, dim)
rmp = RandomMatrixPSpace(sym, model=model)
return RandomMatrixSymbol(sym, dim, dim, pspace=rmp)
def value(self):
return RandomMatrixSymbol(self.symbol, self.args[2], self.args[3],
self)
def density(self, expr):
n, ZGSE = self.dimension, self.normalization_constant
h_pspace = RandomMatrixPSpace('P', model=self)
H = RandomMatrixSymbol('H', n, n, pspace=h_pspace)
return Lambda(H, exp(-S(n) * Trace(H**2)) / ZGSE)
def CircularEnsemble(sym, dim):
sym, dim = _symbol_converter(sym), _sympify(dim)
model = CircularEnsembleModel(sym, dim)
rmp = RandomMatrixPSpace(sym, model=model)
return RandomMatrixSymbol(sym, dim, dim, pspace=rmp)
ExpectationMatrix, VarianceMatrix, CrossCovarianceMatrix)
j, k = symbols("j,k")
A = MatrixSymbol("A", k, k)
B = MatrixSymbol("B", k, k)
C = MatrixSymbol("C", k, k)
D = MatrixSymbol("D", k, k)
a = MatrixSymbol("a", k, 1)
b = MatrixSymbol("b", k, 1)
A2 = MatrixSymbol("A2", 2, 2)
B2 = MatrixSymbol("B2", 2, 2)
X = RandomMatrixSymbol("X", k, 1)
Y = RandomMatrixSymbol("Y", k, 1)
Z = RandomMatrixSymbol("Z", k, 1)
W = RandomMatrixSymbol("W", k, 1)
R = RandomMatrixSymbol("R", k, k)
X2 = RandomMatrixSymbol("X2", 2, 1)
normal = Normal("normal", 0, 1)
m1 = Matrix([[1, j * Normal("normal2", 2, 1)], [normal, 0]])
def test_multivariate_expectation():
expr = Expectation(a)
