def test_MatrixNormal():
M = MatrixNormal('M', [[5, 6]], [4], [[2, 1], [1, 2]])
assert M.pspace.distribution.set == MatrixSet(1, 2, S.Reals)
X = MatrixSymbol('X', 1, 2)
term1 = exp(-Trace(Matrix([[ S(2)/3, -S(1)/3], [-S(1)/3, S(2)/3]])*(
Matrix([[-5], [-6]]) + X.T)*Matrix([[1/4]])*(Matrix([[-5, -6]]) + X))/2)
assert density(M)(X).doit() == term1/(24*pi)
assert density(M)([[7, 8]]).doit() == exp(-S(1)/3)/(24*pi)
d, n = symbols('d n', positive=True, integer=True)
SM2 = MatrixSymbol('SM2', d, d)
SM1 = MatrixSymbol('SM1', n, n)
LM = MatrixSymbol('LM', n, d)
Y = MatrixSymbol('Y', n, d)
M = MatrixNormal('M', LM, SM1, SM2)
exprd = 4*(2*pi)**(-d*n/2)*exp(-Trace(SM2**(-1)*(-LM.T + Y.T)*SM1**(-1)*(-LM + Y)
)/2)*Determinant(SM1)**(-d)*Determinant(SM2)**(-n)
assert density(M)(Y).doit() == exprd
raises(ValueError, lambda: density(M)(1))
raises(ValueError, lambda: MatrixNormal('M', [1, 2], [[1, 0], [0, 1]], [[1, 0], [2, 1]]))
raises(ValueError, lambda: MatrixNormal('M', [1, 2], [[1, 0], [2, 1]], [[1, 0], [0, 1]]))
raises(ValueError, lambda: MatrixNormal('M', [1, 2], [[1, 0], [0, 1]], [[1, 0], [0, 1]]))
raises(ValueError, lambda: MatrixNormal('M', [1, 2], [[1, 0], [2]], [[1, 0], [0, 1]]))
raises(ValueError, lambda: MatrixNormal('M', [1, 2], [[1, 0], [2, 1]], [[1, 0], [0]]))
raises(ValueError, lambda: MatrixNormal('M', [[1, 2]], [[1, 0], [0, 1]], [[1, 0]]))
raises(ValueError, lambda: MatrixNormal('M', [[1, 2]], [1], [[1, 0]]))
def test_MatrixGamma():
M = MatrixGamma('M', 1, 2, [[1, 0], [0, 1]])
assert M.pspace.distribution.set == MatrixSet(2, 2, S.Reals)
assert isinstance(density(M), MatrixGammaDistribution)
X = MatrixSymbol('X', 2, 2)
num = exp(Trace(Matrix([[-S(1) / 2, 0], [0, -S(1) / 2]]) * X))
assert density(M)(X).doit() == num / (4 * pi * sqrt(Determinant(X)))
assert density(M)([[2, 1], [1, 2]]).doit() == sqrt(3) * exp(-2) / (12 * pi)
X = MatrixSymbol('X', 1, 2)
Y = MatrixSymbol('Y', 1, 2)
assert density(M)([X, Y]).doit() == exp(-X[0, 0] / 2 - Y[0, 1] / 2) / (
4 * pi * sqrt(X[0, 0] * Y[0, 1] - X[0, 1] * Y[0, 0]))
# symbolic
a, b = symbols('a b', positive=True)
d = symbols('d', positive=True, integer=True)
Y = MatrixSymbol('Y', d, d)
Z = MatrixSymbol('Z', 2, 2)
SM = MatrixSymbol('SM', d, d)
M2 = MatrixGamma('M2', a, b, SM)
M3 = MatrixGamma('M3', 2, 3, [[2, 1], [1, 2]])
k = Dummy('k')
exprd = pi**(-d * (d - 1) / 4) * b**(-a * d) * exp(
Trace((-1 / b) * SM**(-1) * Y)) * Determinant(SM)**(-a) * Determinant(
Y)**(a - d / 2 - S(1) / 2) / Product(gamma(-k / 2 + a + S(1) / 2),
(k, 1, d))
assert density(M2)(Y).dummy_eq(exprd)
raises(NotImplementedError, lambda: density(M3 + M)(Z))
raises(ValueError, lambda: density(M)(1))
raises(ValueError, lambda: MatrixGamma('M', -1, 2, [[1, 0], [0, 1]]))
raises(ValueError, lambda: MatrixGamma('M', -1, -2, [[1, 0], [0, 1]]))
raises(ValueError, lambda: MatrixGamma('M', -1, 2, [[1, 0], [2, 1]]))
raises(ValueError, lambda: MatrixGamma('M', -1, 2, [[1, 0], [0]]))
def test_MatrixStudentT():
M = MatrixStudentT('M', 2, [[5, 6]], [[2, 1], [1, 2]], [4])
assert M.pspace.distribution.set == MatrixSet(1, 2, S.Reals)
X = MatrixSymbol('X', 1, 2)
D = pi ** (-1.0) * Determinant(Matrix([[4]])) ** (-1.0) * Determinant(Matrix([[2, 1], [1, 2]])) \
** (-0.5) / Determinant(Matrix([[S(1) / 4]]) * (Matrix([[-5, -6]]) + X)
* Matrix([[S(2) / 3, -S(1) / 3], [-S(1) / 3, S(2) / 3]]) * (
Matrix([[-5], [-6]]) + X.T) + Matrix([[1]])) ** 2
assert density(M)(X) == D
v = symbols('v', positive=True)
n, p = 1, 2
Omega = MatrixSymbol('Omega', p, p)
Sigma = MatrixSymbol('Sigma', n, n)
Location = MatrixSymbol('Location', n, p)
Y = MatrixSymbol('Y', n, p)
M = MatrixStudentT('M', v, Location, Omega, Sigma)
exprd = gamma(v/2 + 1)*Determinant(Matrix([[1]]) + Sigma**(-1)*(-Location + Y)*Omega**(-1)*(-Location.T + Y.T))**(-v/2 - 1) / \
(pi*gamma(v/2)*sqrt(Determinant(Omega))*Determinant(Sigma))
assert density(M)(Y) == exprd
raises(ValueError, lambda: density(M)(1))
raises(
ValueError, lambda: MatrixStudentT('M', 1, [1, 2], [[1, 0], [0, 1]],
[[1, 0], [2, 1]]))
raises(
ValueError, lambda: MatrixStudentT('M', 1, [1, 2], [[1, 0], [2, 1]],
[[1, 0], [0, 1]]))
raises(
ValueError, lambda: MatrixStudentT('M', 1, [1, 2], [[1, 0], [0, 1]],
[[1, 0], [0, 1]]))
raises(
ValueError, lambda: MatrixStudentT('M', 1, [1, 2], [[1, 0], [2]],
[[1, 0], [0, 1]]))
raises(
ValueError,
lambda: MatrixStudentT('M', 1, [1, 2], [[1, 0], [2, 1]], [[1], [2]]))
raises(
ValueError,
lambda: MatrixStudentT('M', 1, [[1, 2]], [[1, 0], [0, 1]], [[1, 0]]))
raises(ValueError, lambda: MatrixStudentT('M', 1, [[1, 2]], [1], [[1, 0]]))
raises(ValueError,
lambda: MatrixStudentT('M', -1, [1, 2], [[1, 0], [0, 1]], [4]))
def test_Wishart():
W = Wishart('W', 5, [[1, 0], [0, 1]])
assert W.pspace.distribution.set == MatrixSet(2, 2, S.Reals)
X = MatrixSymbol('X', 2, 2)
term1 = exp(Trace(Matrix([[-S(1)/2, 0], [0, -S(1)/2]])*X))
assert density(W)(X).doit() == term1 * Determinant(X)/(24*pi)
assert density(W)([[2, 1], [1, 2]]).doit() == exp(-2)/(8*pi)
n = symbols('n', positive=True)
d = symbols('d', positive=True, integer=True)
Y = MatrixSymbol('Y', d, d)
SM = MatrixSymbol('SM', d, d)
W = Wishart('W', n, SM)
k = Dummy('k')
exprd = 2**(-d*n/2)*pi**(-d*(d - 1)/4)*exp(Trace(-(S(1)/2)*SM**(-1)*Y)
)*Determinant(SM)**(-n/2)*Determinant(Y)**(
-d/2 + n/2 - S(1)/2)/Product(gamma(-k/2 + n/2 + S(1)/2), (k, 1, d))
assert density(W)(Y).dummy_eq(exprd)
raises(ValueError, lambda: density(W)(1))
raises(ValueError, lambda: Wishart('W', -1, [[1, 0], [0, 1]]))
raises(ValueError, lambda: Wishart('W', -1, [[1, 0], [2, 1]]))
raises(ValueError, lambda: Wishart('W', 2, [[1, 0], [0]]))
def test_MatrixSet():
M = MatrixSet(2, 2, set=S.Reals)
assert M.shape == (2, 2)
assert M.set == S.Reals
X = Matrix([[1, 2], [3, 4]])
assert X in M
X = ZeroMatrix(2, 2)
assert X in M
raises(TypeError, lambda: A in M)
raises(TypeError, lambda: 1 in M)
M = MatrixSet(n, m, set=S.Reals)
assert A in M
raises(TypeError, lambda: C in M)
raises(TypeError, lambda: X in M)
M = MatrixSet(2, 2, set={1, 2, 3})
X = Matrix([[1, 2], [3, 4]])
Y = Matrix([[1, 2]])
assert (X in M) == S.false
assert (Y in M) == S.false
raises(ValueError, lambda: MatrixSet(2, -2, S.Reals))
raises(ValueError, lambda: MatrixSet(2.4, -1, S.Reals))
raises(TypeError, lambda: MatrixSet(2, 2, (1, 2, 3)))
def set(self):
n, p = self.location_matrix.shape
return MatrixSet(n, p, S.Reals)
def set(self):
k = self.scale_matrix.shape[0]
return MatrixSet(k, k, S.Reals)