def test_ProductPSpace():
X = Normal('X', 0, 1)
Y = Normal('Y', 0, 1)
px = X.pspace
py = Y.pspace
assert pspace(X + Y) == ProductPSpace(px, py)
assert pspace(X + Y) == ProductPSpace(py, px)
def test_given():
X = Normal('X', 0, 1)
Y = Normal('Y', 0, 1)
A = given(X, True)
B = given(X, Y > 2)
assert X == A == B
def test_pspace():
X, Y = Normal('X', 0, 1), Normal('Y', 0, 1)
pytest.raises(ValueError, lambda: pspace(5 + 3))
pytest.raises(ValueError, lambda: pspace(x < 1))
assert pspace(X) == X.pspace
assert pspace(2*X + 1) == X.pspace
assert pspace(2*X + Y) == ProductPSpace(Y.pspace, X.pspace)
def test_random_symbols():
X, Y = Normal('X', 0, 1), Normal('Y', 0, 1)
assert set(random_symbols(2*X + 1)) == {X}
assert set(random_symbols(2*X + Y)) == {X, Y}
assert set(random_symbols(2*X + Y.symbol)) == {X}
assert set(random_symbols(2)) == set()
assert X.is_commutative
def test_rs_swap():
X = Normal('x', 0, 1)
Y = Exponential('y', 1)
XX = Normal('x', 0, 2)
YY = Normal('y', 0, 3)
expr = 2*X + Y
assert expr.subs(rs_swap((X, Y), (YY, XX))) == 2*XX + YY
def test_Density():
X = Die('X', 6)
d = Density(X)
assert d.doit() == density(X)
r = Rational(1, 6)
assert density(2*X).dict == {2: r, 4: r, 6: r, 8: r, 10: r, 12: r}
X = Normal('X', 0, 1)
Y = Normal('Y', 0, 1)
assert str(density(X - Y, Y)(z)) == 'sqrt(2)*E**(-(Y + z)**2/2)/(2*sqrt(pi))'
def test_RandomSymbol():
X = Normal('x', 0, 1)
Y = Normal('x', 0, 2)
assert X.symbol == Y.symbol
assert X != Y
assert X.name == X.symbol.name
X = Normal('lambda', 0, 1) # make sure we can use protected terms
X = Normal('Lambda', 0, 1) # make sure we can use Diofant terms
def test_dependence():
X, Y = Die('X'), Die('Y')
assert independent(X, 2*Y)
assert not dependent(X, 2*Y)
X, Y = Normal('X', 0, 1), Normal('Y', 0, 1)
assert independent(X, Y)
assert dependent(X, 2*X)
# Create a dependency
XX, YY = given(Tuple(X, Y), Eq(X + Y, 3))
assert dependent(XX, YY)
def test_input_value_assertions():
a, b = symbols('a b')
p, q = symbols('p q', positive=True)
pytest.raises(ValueError, lambda: Normal('x', 3, 0))
pytest.raises(ValueError, lambda: Normal('x', a, b))
Normal('X', a, p) # No error raised
pytest.raises(ValueError, lambda: Exponential('x', a))
Exponential('Ex', p) # No error raised
for fn in [Pareto, Weibull, Beta, Gamma]:
pytest.raises(ValueError, lambda: fn('x', a, p))
pytest.raises(ValueError, lambda: fn('x', p, a))
fn('x', p, q) # No error raised
def test_RandomSymbol():
pytest.raises(TypeError, lambda: RandomSymbol(0, 1))
pytest.raises(TypeError, lambda: RandomSymbol(0, Symbol('x')))
X = Normal('x', 0, 1)
Y = Normal('x', 0, 2)
assert X.symbol == Y.symbol
assert X != Y
assert X.name == X.symbol.name
X = Normal('lambda', 0, 1) # make sure we can use protected terms
X = Normal('Lambda', 0, 1) # make sure we can use Diofant terms
pytest.raises(TypeError, lambda: Normal(1, 0, 1))
def test_unevaluated():
X = Normal('x', 0, 1)
assert E(X, evaluate=False) == (
Integral(sqrt(2)*x*exp(-x**2/2)/(2*sqrt(pi)), (x, -oo, oo)))
assert E(X + 1, evaluate=False) == (
Integral(sqrt(2)*x*exp(-x**2/2)/(2*sqrt(pi)), (x, -oo, oo)) + 1)
def test_sample_iter():
X = Normal('X', 0, 1)
expr = X*X + 3
assert all(_ > 3 for _ in sample_iter(expr, numsamples=3))
assert all(_ > 5 for _ in sample_iter(expr, numsamples=3, condition=X > 2))
pytest.raises(ValueError, lambda: tuple(sample_iter(expr, numsamples=3,
condition=X)))
def test_cdf():
X = Normal('x', 0, 1)
d = cdf(X)
assert P(X < 1) == d(1)
assert d(0) == Rational(1, 2)
d = cdf(X, X > 0) # given X>0
assert d(0) == 0
Y = Exponential('y', 10)
d = cdf(Y)
assert d(-5) == 0
assert P(Y > 3) == 1 - d(3)
pytest.raises(ValueError, lambda: cdf(X + Y))
Z = Exponential('z', 1)
f = cdf(Z)
z = Symbol('z')
assert f(z) == Piecewise((1 - exp(-z), z >= 0), (0, True))
U = Uniform('x', 3, 5)
u = cdf(U)
assert u(z) == z/2 - Rational(3, 2)
def test_difficult_univariate():
""" Since using solve in place of deltaintegrate we're able to perform
substantially more complex density computations on single continuous random
variables """
x = Normal('x', 0, 1)
assert density(x**3)
assert density(exp(x**2))
assert density(log(x))
def test_ContinuousRV():
x = Symbol('x')
pdf = sqrt(2)*exp(-x**2/2)/(2*sqrt(pi)) # Normal distribution
# X and Y should be equivalent
X = ContinuousRV(x, pdf)
Y = Normal('y', 0, 1)
assert variance(X) == variance(Y)
assert P(X > 0) == P(Y > 0)
def test_symbolic():
mu1, mu2 = symbols('mu1 mu2', real=True)
s1, s2 = symbols('sigma1 sigma2', real=True, positive=True)
rate = Symbol('lambda', real=True, positive=True)
X = Normal('x', mu1, s1)
Y = Normal('y', mu2, s2)
Z = Exponential('z', rate)
a, b, c = symbols('a b c', real=True)
assert E(X) == mu1
assert E(X + Y) == mu1 + mu2
assert E(a*X + b) == a*E(X) + b
assert variance(X) == s1**2
assert simplify(variance(X + a*Y + b)) == variance(X) + a**2*variance(Y)
assert E(Z) == 1/rate
assert E(a*Z + b) == a*E(Z) + b
assert E(X + a*Z + b) == mu1 + a/rate + b
def test_unevaluated_fail():
X = Normal('x', 0, 1)
assert P(X > 0, evaluate=False) == (
Integral(sqrt(2)*exp(-x**2/2)/(2*sqrt(pi)), (x, 0, oo)))
assert P(X > 0, X**2 < 1, evaluate=False) == (
Integral(sqrt(2)*exp(-x**2/2)/(2*sqrt(pi) *
Integral(sqrt(2)*exp(-x**2/2)/(2*sqrt(pi)),
(x, -1, 1))), (x, 0, 1)))
def test_conditional_1d():
X = Normal('x', 0, 1)
Y = given(X, X >= 0)
assert density(Y)(x) == 2 * density(X)(x)
assert Y.pspace.domain.set == Interval(0, oo, right_open=True)
assert E(Y) == sqrt(2/pi)
assert E(X**2) == E(Y**2)
def test_conditional_1d():
X = Normal('x', 0, 1)
Y = given(X, X >= 0)
assert density(Y) == 2 * density(X)
assert Y.pspace.domain.set == Interval(0, oo)
assert E(Y) == sqrt(2) / sqrt(pi)
assert E(X**2) == E(Y**2)
def test_RandomDomain():
X = Normal('x1', 0, 1)
assert str(where(X > 0)) == "Domain: (0 < x1) & (x1 < oo)"
D = Die('d1', 6)
assert str(where(D > 4)) == "Domain: (Eq(d1, 5)) | (Eq(d1, 6))"
A = Exponential('a', 1)
B = Exponential('b', 1)
assert str(pspace(Tuple(A, B)).domain) == "Domain: (0 <= a) & (0 <= b) & (a < oo) & (b < oo)"
def test_ContinuousDomain():
X = Normal('x', 0, 1)
assert where(X**2 <= 1).set == Interval(-1, 1)
assert where(X**2 <= 1).symbol == X.symbol
where(And(X**2 <= 1, X >= 0)).set == Interval(0, 1)
pytest.raises(ValueError, lambda: where(sin(X) > 1))
Y = given(X, X >= 0)
assert Y.pspace.domain.set == Interval(0, oo, False, True)
def test_multiple_normal():
X, Y = Normal('x', 0, 1), Normal('y', 0, 1)
assert E(X + Y) == 0
assert variance(X + Y) == 2
assert variance(X + X) == 4
assert covariance(X, Y) == 0
assert covariance(2*X + Y, -X) == -2*variance(X)
assert skewness(X) == 0
assert skewness(X + Y) == 0
assert correlation(X, Y) == 0
assert correlation(X, X + Y) == correlation(X, X - Y)
assert moment(X, 2) == 1
assert cmoment(X, 3) == 0
assert moment(X + Y, 4) == 12
assert cmoment(X, 2) == variance(X)
assert smoment(X*X, 2) == 1
assert smoment(X + Y, 3) == skewness(X + Y)
assert E(X, Eq(X + Y, 0)) == 0
assert variance(X, Eq(X + Y, 0)) == Rational(1, 2)
def test_RandomDomain():
X = Normal('x1', 0, 1)
assert str(where(X > 0)) == "Domain: And(0 < x1, x1 < oo)"
D = Die('d1', 6)
assert str(where(D > 4)) == "Domain: Or(Eq(d1, 5), Eq(d1, 6))"
A = Exponential('a', 1)
B = Exponential('b', 1)
assert str(pspace(Tuple(
A, B)).domain) == "Domain: And(0 <= a, 0 <= b, a < oo, b < oo)"
def test_where():
X, Y = Die('X'), Die('Y')
Z = Normal('Z', 0, 1)
assert where(Z**2 <= 1).set == Interval(-1, 1)
assert where(
Z**2 <= 1).as_boolean() == Interval(-1, 1).as_relational(Z.symbol)
assert where(And(X > Y, Y > 4)).as_boolean() == And(
Eq(X.symbol, 6), Eq(Y.symbol, 5))
assert len(where(X < 3).set) == 2
assert 1 in where(X < 3).set
X, Y = Normal('X', 0, 1), Normal('Y', 0, 1)
assert where(And(X**2 <= 1, X >= 0)).set == Interval(0, 1)
XX = given(X, And(X**2 <= 1, X >= 0))
assert XX.pspace.domain.set == Interval(0, 1)
assert XX.pspace.domain.as_boolean() == \
And(0 <= X.symbol, X.symbol**2 <= 1, -oo < X.symbol, X.symbol < oo)
with pytest.raises(TypeError):
XX = given(X, X + 3)
def test_ProductPSpace():
X = Normal('X', 0, 1)
Y = Normal('Y', 0, 1)
px = X.pspace
py = Y.pspace
assert pspace(X + Y) == ProductPSpace(px, py)
assert pspace(X + Y) == ProductPSpace(py, px)
X = Die('X', 2)
Y = Die('Y', 2)
assert (pspace(X + Y).density ==
Dict((frozenset({('X', 1), ('Y', 2)}), Rational(1, 4)),
(frozenset({('X', 1), ('Y', 1)}), Rational(1, 4)),
(frozenset({('X', 2), ('Y', 1)}), Rational(1, 4)),
(frozenset({('X', 2), ('Y', 2)}), Rational(1, 4))))
d = pspace(X + Y).domain
assert ((X.symbol, 1), (Y.symbol, 2)) in d
assert ((X.symbol, 0), (Y.symbol, 2)) not in d
Z = Die('Z', 2)
d1 = pspace(X + Y).domain
assert ProductDomain(d1, Z.pspace.domain) == pspace(X + Y + Z).domain
def test_Sample():
X = Die('X', 6)
Y = Normal('Y', 0, 1)
assert sample(X) in [1, 2, 3, 4, 5, 6]
assert sample(X + Y).is_Float
assert P(X + Y > 0, Y < 0, numsamples=10).is_number
assert P(X > 10, numsamples=10).is_number
assert E(X + Y, numsamples=10).is_number
assert variance(X + Y, numsamples=10).is_number
pytest.raises(TypeError, lambda: P(Y > z, numsamples=5))
assert P(sin(Y) <= 1, numsamples=10, modules=['math']) == 1
assert P(sin(Y) <= 1, cos(Y) < 1, numsamples=10, modules=['math']) == 1
assert all(i in range(1, 7) for i in density(X, numsamples=10))
assert all(i in range(4, 7) for i in density(X, X > 3, numsamples=10))
# Make sure this doesn't raise an error
Y = Normal('Y', 0, 1)
E(Sum(1/z**Y, (z, 1, oo)), Y > 2, numsamples=3, modules='mpmath')
def test_prefab_sampling():
N = Normal('X', 0, 1)
L = LogNormal('L', 0, 1)
E = Exponential('Ex', 1)
P = Pareto('P', 1, 3)
W = Weibull('W', 1, 1)
U = Uniform('U', 0, 1)
B = Beta('B', 2, 5)
G = Gamma('G', 1, 3)
variables = [N, L, E, P, W, U, B, G]
niter = 10
for var in variables:
for i in range(niter):
assert sample(var) in var.pspace.domain.set
def test_single_normal():
mu = Symbol('mu', real=True)
sigma = Symbol('sigma', real=True, positive=True)
X = Normal('x', 0, 1)
Y = X*sigma + mu
assert simplify(E(Y)) == mu
assert simplify(variance(Y)) == sigma**2
pdf = density(Y)
x = Symbol('x')
assert (pdf(x) ==
sqrt(2)*exp(-(x - mu)**2/(2*sigma**2))/(2*sqrt(pi)*sigma))
assert P(X**2 < 1) == erf(sqrt(2)/2)
assert E(X, Eq(X, mu)) == mu
def test_overlap():
X = Normal('x', 0, 1)
Y = Normal('x', 0, 2)
pytest.raises(ValueError, lambda: P(X > Y))
def test_RandomSymbol_diff():
X = Normal('x', 0, 1)
assert (2*X).diff(X)
