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