def test_discreteuniform():
# Symbolic
a, b, c, t = symbols('a b c t')
X = DiscreteUniform('X', [a, b, c])
assert E(X) == (a + b + c)/3
assert simplify(variance(X)
- ((a**2 + b**2 + c**2)/3 - (a/3 + b/3 + c/3)**2)) == 0
assert P(Eq(X, a)) == P(Eq(X, b)) == P(Eq(X, c)) == S('1/3')
Y = DiscreteUniform('Y', range(-5, 5))
# Numeric
assert E(Y) == S('-1/2')
assert variance(Y) == S('33/4')
for x in range(-5, 5):
assert P(Eq(Y, x)) == S('1/10')
assert P(Y <= x) == S(x + 6)/10
assert P(Y >= x) == S(5 - x)/10
assert dict(density(Die('D', 6)).items()) == \
dict(density(DiscreteUniform('U', range(1, 7))).items())
assert characteristic_function(X)(t) == exp(I*a*t)/3 + exp(I*b*t)/3 + exp(I*c*t)/3
assert moment_generating_function(X)(t) == exp(a*t)/3 + exp(b*t)/3 + exp(c*t)/3
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_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 E(X, Eq(X + Y, 0)) == 0
assert variance(X, Eq(X + Y, 0)) == S.Half
def test_literal_probability():
X = Normal('X', 2, 3)
Y = Normal('Y', 3, 4)
Z = Poisson('Z', 4)
W = Poisson('W', 3)
x, y, w, z = symbols('x, y, w, z')
assert Probability(X > 0).doit() == probability(X > 0)
assert Probability(X > x).doit() == probability(X > x)
assert Expectation(X).doit() == expectation(X)
assert Expectation(X**2).doit() == expectation(X**2)
assert Expectation(x*X) == x*Expectation(X)
assert Expectation(2*X + 3*Y + z*X*Y) == 2*Expectation(X) + 3*Expectation(Y) + z*Expectation(X*Y)
assert Expectation(2*X + 3*Y + z*X*Y, evaluate=False).args == (2*X + 3*Y + z*X*Y,)
assert Expectation(sin(X)) == Expectation(sin(X), evaluate=False)
assert Expectation(2*x*sin(X)*Y + y*X**2 + z*X*Y) == 2*x*Expectation(sin(X)*Y) + y*Expectation(X**2) + z*Expectation(X*Y)
assert Variance(w) == 0
assert Variance(X).doit() == variance(X)
assert Variance(X + z) == Variance(X)
assert Variance(X*Y).args == (Mul(X, Y),)
assert type(Variance(X*Y)) == Variance
assert Variance(z*X) == z**2*Variance(X)
assert Variance(X + Y) == Variance(X) + Variance(Y) + 2*Covariance(X, Y)
assert Variance(X + Y + Z + W) == (Variance(X) + Variance(Y) + Variance(Z) + Variance(W) +
2 * Covariance(X, Y) + 2 * Covariance(X, Z) + 2 * Covariance(X, W) +
2 * Covariance(Y, Z) + 2 * Covariance(Y, W) + 2 * Covariance(W, Z))
assert Variance(X**2).doit() == variance(X**2)
assert Variance(X**2, evaluate=False) == Variance(X**2)
assert Variance(x*X**2) == x**2*Variance(X**2)
assert Variance(sin(X)).args == (sin(X),)
assert Variance(sin(X), evaluate=False) == Variance(sin(X))
assert Variance(x*sin(X)) == x**2*Variance(sin(X))
assert Covariance(w, z) == 0
assert Covariance(X, w) == 0
assert Covariance(w, X) == 0
assert Covariance(X, Y).args == (X, Y)
assert type(Covariance(X, Y)) == Covariance
assert Covariance(z*X + 3, Y) == z*Covariance(X, Y)
assert Covariance(X, X) == Variance(X)
assert Covariance(z*X + 3, w*Y + 4) == w*z*Covariance(X,Y)
assert Covariance(X, Y) == Covariance(Y, X)
assert Covariance(X + Y, Z + W) == Covariance(W, X) + Covariance(W, Y) + Covariance(X, Z) + Covariance(Y, Z)
assert Covariance(x*X + y*Y, z*Z + w*W) == (x*w*Covariance(W, X) + w*y*Covariance(W, Y) +
x*z*Covariance(X, Z) + y*z*Covariance(Y, Z))
assert Covariance(x*X**2 + y*sin(Y), z*Y*Z**2 + w*W) == (w*x*Covariance(W, X**2) + w*y*Covariance(sin(Y), W) +
x*z*Covariance(Y*Z**2, X**2) + y*z*Covariance(Y*Z**2, sin(Y)))
assert Covariance(X, X**2) == Covariance(X, X**2, evaluate=False)
assert Covariance(X, sin(X)) == Covariance(sin(X), X, evaluate=False)
assert Covariance(X**2, sin(X)*Y) == Covariance(sin(X)*Y, X**2, evaluate=False)
def test_bernoulli():
p, a, b = symbols("p a b")
X = Bernoulli(p, a, b, symbol="B")
assert E(X) == a * p + b * (-p + 1)
assert density(X)[a] == p
assert density(X)[b] == 1 - p
X = Bernoulli(p, 1, 0, symbol="B")
assert E(X) == p
assert variance(X) == -p ** 2 + p
E(a * X + b) == a * E(X) + b
variance(a * X + b) == a ** 2 * variance(X)
def test_bernoulli():
p, a, b = symbols('p a b')
X = Bernoulli('B', p, a, b)
assert E(X) == a*p + b*(-p + 1)
assert density(X)[a] == p
assert density(X)[b] == 1 - p
X = Bernoulli('B', p, 1, 0)
assert E(X) == p
assert simplify(variance(X)) == p*(1 - p)
E(a*X + b) == a*E(X) + b
variance(a*X + b) == a**2 * variance(X)
def test_rademacher():
X = Rademacher('X')
assert E(X) == 0
assert variance(X) == 1
assert density(X)[-1] == S.Half
assert density(X)[1] == S.Half
def test_rayleigh():
sigma = Symbol("sigma", positive=True)
X = Rayleigh('x', sigma)
assert density(X)(x) == x*exp(-x**2/(2*sigma**2))/sigma**2
assert E(X) == sqrt(2)*sqrt(pi)*sigma/2
assert variance(X) == -pi*sigma**2/2 + 2*sigma**2
def test_pareto_numeric():
xm, beta = 3, 2
alpha = beta + 5
X = Pareto('x', xm, alpha)
assert E(X) == alpha*xm/S(alpha - 1)
assert variance(X) == xm**2*alpha / S(((alpha - 1)**2*(alpha - 2)))
def test_weibull():
a, b = symbols('a b', positive=True)
X = Weibull('x', a, b)
assert simplify(E(X)) == simplify(a * gamma(1 + 1/b))
assert simplify(variance(X)) == simplify(a**2 * gamma(1 + 2/b) - E(X)**2)
assert simplify(skewness(X)) == (2*gamma(1 + 1/b)**3 - 3*gamma(1 + 1/b)*gamma(1 + 2/b) + gamma(1 + 3/b))/(-gamma(1 + 1/b)**2 + gamma(1 + 2/b))**(S(3)/2)
def test_bernoulli():
p, a, b, t = symbols('p a b t')
X = Bernoulli('B', p, a, b)
assert E(X) == a*p + b*(-p + 1)
assert density(X)[a] == p
assert density(X)[b] == 1 - p
assert characteristic_function(X)(t) == p * exp(I * a * t) + (-p + 1) * exp(I * b * t)
assert moment_generating_function(X)(t) == p * exp(a * t) + (-p + 1) * exp(b * t)
X = Bernoulli('B', p, 1, 0)
assert E(X) == p
assert simplify(variance(X)) == p*(1 - p)
assert E(a*X + b) == a*E(X) + b
assert simplify(variance(a*X + b)) == simplify(a**2 * variance(X))
def test_Logarithmic():
p = S.One / 2
x = Logarithmic('x', p)
assert E(x) == -p / ((1 - p) * log(1 - p))
assert variance(x) == -1/log(2)**2 + 2/log(2)
assert E(2*x**2 + 3*x + 4) == 4 + 7 / log(2)
assert isinstance(E(x, evaluate=False), Sum)
def test_gamma():
k = Symbol("k", positive=True)
theta = Symbol("theta", positive=True)
X = Gamma('x', k, theta)
assert density(X) == Lambda(_x,
_x**(k - 1)*theta**(-k)*exp(-_x/theta)/gamma(k))
assert cdf(X, meijerg=True) == Lambda(_z, Piecewise(
(-k*lowergamma(k, 0)/gamma(k + 1) + k*lowergamma(k, _z/theta)/gamma(k + 1), _z >= 0), (0, True)))
assert variance(X) == (-theta**2*gamma(k + 1)**2/gamma(k)**2 +
theta*theta**(-k)*theta**(k + 1)*gamma(k + 2)/gamma(k))
k, theta = symbols('k theta', real=True, bounded=True, positive=True)
X = Gamma('x', k, theta)
assert simplify(E(X)) == k*theta
# can't get things to simplify on this one so we use subs
assert variance(X).subs(k, 5) == (k*theta**2).subs(k, 5)
def test_maxwell():
a = Symbol("a", positive=True)
X = Maxwell("x", a)
assert density(X)(x) == (sqrt(2) * x ** 2 * exp(-x ** 2 / (2 * a ** 2)) / (sqrt(pi) * a ** 3))
assert E(X) == 2 * sqrt(2) * a / sqrt(pi)
assert simplify(variance(X)) == a ** 2 * (-8 + 3 * pi) / pi
def test_weibull_numeric():
# Test for integers and rationals
a = 1
bvals = [S.Half, 1, S(3) / 2, 5]
for b in bvals:
X = Weibull("x", a, b)
assert simplify(E(X)) == simplify(a * gamma(1 + 1 / S(b)))
assert simplify(variance(X)) == simplify(a ** 2 * gamma(1 + 2 / S(b)) - E(X) ** 2)
def test_rayleigh():
sigma = Symbol("sigma", positive=True)
x = Symbol("x")
X = Rayleigh(sigma, symbol=x)
assert density(X) == Lambda(_x, _x*exp(-_x**2/(2*sigma**2))/sigma**2)
assert E(X) == sqrt(2)*sqrt(pi)*sigma/2
assert variance(X) == -pi*sigma**2/2 + 2*sigma**2
def test_Poisson():
l = 3
x = Poisson('x', l)
assert E(x) == l
assert variance(x) == l
assert density(x) == PoissonDistribution(l)
assert isinstance(E(x, evaluate=False), Sum)
assert isinstance(E(2*x, evaluate=False), Sum)
def test_negative_binomial():
r = 5
p = S(1) / 3
x = NegativeBinomial('x', r, p)
assert E(x) == p*r / (1-p)
assert variance(x) == p*r / (1-p)**2
assert E(x**5 + 2*x + 3) == S(9207)/4
assert isinstance(E(x, evaluate=False), Sum)
def test_symbolic():
mu1, mu2 = symbols("mu1 mu2", real=True, bounded=True)
s1, s2 = symbols("sigma1 sigma2", real=True, bounded=True, positive=True)
rate = Symbol("lambda", real=True, positive=True, bounded=True)
X = Normal("x", mu1, s1)
Y = Normal("y", mu2, s2)
Z = Exponential("z", rate)
a, b, c = symbols("a b c", real=True, bounded=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_symbolic():
mu1, mu2 = symbols('mu1 mu2', real=True, finite=True)
s1, s2 = symbols('sigma1 sigma2', real=True, finite=True, positive=True)
rate = Symbol('lambda', real=True, positive=True, finite=True)
X = Normal('x', mu1, s1)
Y = Normal('y', mu2, s2)
Z = Exponential('z', rate)
a, b, c = symbols('a b c', real=True, finite=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_Poisson():
l = 3
x = Poisson('x', l)
assert E(x) == l
assert variance(x) == l
assert density(x) == PoissonDistribution(l)
assert isinstance(E(x, evaluate=False), Sum)
assert isinstance(E(2*x, evaluate=False), Sum)
assert characteristic_function(x)(0).doit() == 1
def test_nakagami():
mu = Symbol("mu", positive=True)
omega = Symbol("omega", positive=True)
X = Nakagami("x", mu, omega)
assert density(X)(x) == (2 * x ** (2 * mu - 1) * mu ** mu * omega ** (-mu) * exp(-x ** 2 * mu / omega) / gamma(mu))
assert simplify(E(X, meijerg=True)) == (sqrt(mu) * sqrt(omega) * gamma(mu + S.Half) / gamma(mu + 1))
assert simplify(variance(X, meijerg=True)) == (
omega - omega * gamma(mu + S(1) / 2) ** 2 / (gamma(mu) * gamma(mu + 1))
)
def test_rademacher():
X = Rademacher('X')
t = Symbol('t')
assert E(X) == 0
assert variance(X) == 1
assert density(X)[-1] == S.Half
assert density(X)[1] == S.Half
assert characteristic_function(X)(t) == exp(I*t)/2 + exp(-I*t)/2
assert moment_generating_function(X)(t) == exp(t) / 2 + exp(-t) / 2
def test_maxwell():
a = Symbol("a", positive=True)
x = Symbol("x")
X = Maxwell(a, symbol=x)
assert density(X) == (Lambda(_x, sqrt(2)*_x**2*exp(-_x**2/(2*a**2))/
(sqrt(pi)*a**3)))
assert E(X) == 2*sqrt(2)*a/sqrt(pi)
assert simplify(variance(X)) == a**2*(-8 + 3*pi)/pi
def test_beta():
a, b = symbols('alpha beta', positive=True)
B = Beta('x', a, b)
assert pspace(B).domain.set == Interval(0, 1)
dens = density(B)
x = Symbol('x')
assert dens(x) == x**(a - 1)*(1 - x)**(b - 1) / beta(a, b)
assert simplify(E(B)) == a / (a + b)
assert simplify(variance(B)) == a*b / (a**3 + 3*a**2*b + a**2 + 3*a*b**2 + 2*a*b + b**3 + b**2)
# Full symbolic solution is too much, test with numeric version
a, b = 1, 2
B = Beta('x', a, b)
assert expand_func(E(B)) == a / S(a + b)
assert expand_func(variance(B)) == (a*b) / S((a + b)**2 * (a + b + 1))
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)) == S.Half
def test_maxwell():
a = Symbol("a", positive=True)
X = Maxwell('x', a)
assert density(X)(x) == (sqrt(2)*x**2*exp(-x**2/(2*a**2))/
(sqrt(pi)*a**3))
assert E(X) == 2*sqrt(2)*a/sqrt(pi)
assert simplify(variance(X)) == a**2*(-8 + 3*pi)/pi
assert cdf(X)(x) == erf(sqrt(2)*x/(2*a)) - sqrt(2)*x*exp(-x**2/(2*a**2))/(sqrt(pi)*a)
assert diff(cdf(X)(x), x) == density(X)(x)
def test_binomial_symbolic():
n = 10 # Because we're using for loops, can't do symbolic n
p = symbols('p', positive=True)
X = Binomial('X', n, p)
assert simplify(E(X)) == n*p == simplify(moment(X, 1))
assert simplify(variance(X)) == n*p*(1 - p) == simplify(cmoment(X, 2))
assert cancel((skewness(X) - (1-2*p)/sqrt(n*p*(1-p)))) == 0
# Test ability to change success/failure winnings
H, T = symbols('H T')
Y = Binomial('Y', n, p, succ=H, fail=T)
assert simplify(E(Y) - (n*(H*p + T*(1 - p)))) == 0
def test_discreteuniform():
# Symbolic
a, b, c = symbols('a b c')
X = DiscreteUniform('X', [a,b,c])
assert E(X) == (a+b+c)/3
assert variance(X) == (a**2+b**2+c**2)/3 - (a/3+b/3+c/3)**2
assert P(Eq(X, a)) == P(Eq(X, b)) == P(Eq(X, c)) == S('1/3')
Y = DiscreteUniform('Y', range(-5, 5))
# Numeric
assert E(Y) == S('-1/2')
assert variance(Y) == S('33/4')
assert skewness(Y) == 0
for x in range(-5, 5):
assert P(Eq(Y, x)) == S('1/10')
assert P(Y <= x) == S(x+6)/10
assert P(Y >= x) == S(5-x)/10
assert density(Die('D', 6)) == density(DiscreteUniform('U', range(1,7)))
def test_uniform():
l = Symbol("l", real=True, finite=True)
w = Symbol("w", positive=True, finite=True)
X = Uniform("x", l, l + w)
assert simplify(E(X)) == l + w / 2
assert simplify(variance(X)) == w ** 2 / 12
# With numbers all is well
X = Uniform("x", 3, 5)
assert P(X < 3) == 0 and P(X > 5) == 0
assert P(X < 4) == P(X > 4) == S.Half
def test_DiscreteMarkovChain():
# pass only the name
X = DiscreteMarkovChain("X")
assert isinstance(X.state_space, Range)
assert X.index_set == S.Naturals0
assert isinstance(X.transition_probabilities, MatrixSymbol)
t = symbols('t', positive=True, integer=True)
assert isinstance(X[t], RandomIndexedSymbol)
assert E(X[0]) == Expectation(X[0])
raises(TypeError, lambda: DiscreteMarkovChain(1))
raises(NotImplementedError, lambda: X(t))
nz = Symbol('n', integer=True)
TZ = MatrixSymbol('M', nz, nz)
SZ = Range(nz)
YZ = DiscreteMarkovChain('Y', SZ, TZ)
assert P(Eq(YZ[2], 1), Eq(YZ[1], 0)) == TZ[0, 1]
raises(ValueError, lambda: sample_stochastic_process(t))
raises(ValueError, lambda: next(sample_stochastic_process(X)))
# pass name and state_space
# any hashable object should be a valid state
# states should be valid as a tuple/set/list/Tuple/Range
sym, rainy, cloudy, sunny = symbols('a Rainy Cloudy Sunny', real=True)
state_spaces = [(1, 2, 3), [Str('Hello'), sym, DiscreteMarkovChain],
Tuple(1, exp(sym), Str('World'), sympify=False), Range(-1, 5, 2),
[rainy, cloudy, sunny]]
chains = [DiscreteMarkovChain("Y", state_space) for state_space in state_spaces]
for i, Y in enumerate(chains):
assert isinstance(Y.transition_probabilities, MatrixSymbol)
assert Y.state_space == state_spaces[i] or Y.state_space == FiniteSet(*state_spaces[i])
assert Y.number_of_states == 3
with ignore_warnings(UserWarning): # TODO: Restore tests once warnings are removed
assert P(Eq(Y[2], 1), Eq(Y[0], 2), evaluate=False) == Probability(Eq(Y[2], 1), Eq(Y[0], 2))
assert E(Y[0]) == Expectation(Y[0])
raises(ValueError, lambda: next(sample_stochastic_process(Y)))
raises(TypeError, lambda: DiscreteMarkovChain("Y", dict((1, 1))))
Y = DiscreteMarkovChain("Y", Range(1, t, 2))
assert Y.number_of_states == ceiling((t-1)/2)
# pass name and transition_probabilities
chains = [DiscreteMarkovChain("Y", trans_probs=Matrix([[]])),
DiscreteMarkovChain("Y", trans_probs=Matrix([[0, 1], [1, 0]])),
DiscreteMarkovChain("Y", trans_probs=Matrix([[pi, 1-pi], [sym, 1-sym]]))]
for Z in chains:
assert Z.number_of_states == Z.transition_probabilities.shape[0]
assert isinstance(Z.transition_probabilities, ImmutableDenseMatrix)
# pass name, state_space and transition_probabilities
T = Matrix([[0.5, 0.2, 0.3],[0.2, 0.5, 0.3],[0.2, 0.3, 0.5]])
TS = MatrixSymbol('T', 3, 3)
Y = DiscreteMarkovChain("Y", [0, 1, 2], T)
YS = DiscreteMarkovChain("Y", ['One', 'Two', 3], TS)
assert YS._transient2transient() == None
assert YS._transient2absorbing() == None
assert Y.joint_distribution(1, Y[2], 3) == JointDistribution(Y[1], Y[2], Y[3])
raises(ValueError, lambda: Y.joint_distribution(Y[1].symbol, Y[2].symbol))
assert P(Eq(Y[3], 2), Eq(Y[1], 1)).round(2) == Float(0.36, 2)
assert (P(Eq(YS[3], 2), Eq(YS[1], 1)) -
(TS[0, 2]*TS[1, 0] + TS[1, 1]*TS[1, 2] + TS[1, 2]*TS[2, 2])).simplify() == 0
assert P(Eq(YS[1], 1), Eq(YS[2], 2)) == Probability(Eq(YS[1], 1))
assert P(Eq(YS[3], 3), Eq(YS[1], 1)) == TS[0, 2]*TS[1, 0] + TS[1, 1]*TS[1, 2] + TS[1, 2]*TS[2, 2]
TO = Matrix([[0.25, 0.75, 0],[0, 0.25, 0.75],[0.75, 0, 0.25]])
assert P(Eq(Y[3], 2), Eq(Y[1], 1) & TransitionMatrixOf(Y, TO)).round(3) == Float(0.375, 3)
with ignore_warnings(UserWarning): ### TODO: Restore tests once warnings are removed
assert E(Y[3], evaluate=False) == Expectation(Y[3])
assert E(Y[3], Eq(Y[2], 1)).round(2) == Float(1.1, 3)
TSO = MatrixSymbol('T', 4, 4)
raises(ValueError, lambda: str(P(Eq(YS[3], 2), Eq(YS[1], 1) & TransitionMatrixOf(YS, TSO))))
raises(TypeError, lambda: DiscreteMarkovChain("Z", [0, 1, 2], symbols('M')))
raises(ValueError, lambda: DiscreteMarkovChain("Z", [0, 1, 2], MatrixSymbol('T', 3, 4)))
raises(ValueError, lambda: E(Y[3], Eq(Y[2], 6)))
raises(ValueError, lambda: E(Y[2], Eq(Y[3], 1)))
# extended tests for probability queries
TO1 = Matrix([[Rational(1, 4), Rational(3, 4), 0],[Rational(1, 3), Rational(1, 3), Rational(1, 3)],[0, Rational(1, 4), Rational(3, 4)]])
assert P(And(Eq(Y[2], 1), Eq(Y[1], 1), Eq(Y[0], 0)),
Eq(Probability(Eq(Y[0], 0)), Rational(1, 4)) & TransitionMatrixOf(Y, TO1)) == Rational(1, 16)
assert P(And(Eq(Y[2], 1), Eq(Y[1], 1), Eq(Y[0], 0)), TransitionMatrixOf(Y, TO1)) == \
Probability(Eq(Y[0], 0))/4
assert P(Lt(X[1], 2) & Gt(X[1], 0), Eq(X[0], 2) &
StochasticStateSpaceOf(X, [0, 1, 2]) & TransitionMatrixOf(X, TO1)) == Rational(1, 4)
assert P(Lt(X[1], 2) & Gt(X[1], 0), Eq(X[0], 2) &
StochasticStateSpaceOf(X, [None, 'None', 1]) & TransitionMatrixOf(X, TO1)) == Rational(1, 4)
assert P(Ne(X[1], 2) & Ne(X[1], 1), Eq(X[0], 2) &
StochasticStateSpaceOf(X, [0, 1, 2]) & TransitionMatrixOf(X, TO1)) is S.Zero
assert P(Ne(X[1], 2) & Ne(X[1], 1), Eq(X[0], 2) &
StochasticStateSpaceOf(X, [None, 'None', 1]) & TransitionMatrixOf(X, TO1)) is S.Zero
assert P(And(Eq(Y[2], 1), Eq(Y[1], 1), Eq(Y[0], 0)), Eq(Y[1], 1)) == 0.1*Probability(Eq(Y[0], 0))
# testing properties of Markov chain
TO2 = Matrix([[S.One, 0, 0],[Rational(1, 3), Rational(1, 3), Rational(1, 3)],[0, Rational(1, 4), Rational(3, 4)]])
TO3 = Matrix([[Rational(1, 4), Rational(3, 4), 0],[Rational(1, 3), Rational(1, 3), Rational(1, 3)],[0, Rational(1, 4), Rational(3, 4)]])
Y2 = DiscreteMarkovChain('Y', trans_probs=TO2)
Y3 = DiscreteMarkovChain('Y', trans_probs=TO3)
assert Y3._transient2absorbing() == None
raises (ValueError, lambda: Y3.fundamental_matrix())
assert Y2.is_absorbing_chain() == True
assert Y3.is_absorbing_chain() == False
TO4 = Matrix([[Rational(1, 5), Rational(2, 5), Rational(2, 5)], [Rational(1, 10), S.Half, Rational(2, 5)], [Rational(3, 5), Rational(3, 10), Rational(1, 10)]])
Y4 = DiscreteMarkovChain('Y', trans_probs=TO4)
w = ImmutableMatrix([[Rational(11, 39), Rational(16, 39), Rational(4, 13)]])
assert Y4.limiting_distribution == w
assert Y4.is_regular() == True
TS1 = MatrixSymbol('T', 3, 3)
Y5 = DiscreteMarkovChain('Y', trans_probs=TS1)
assert Y5.limiting_distribution(w, TO4).doit() == True
assert Y5.stationary_distribution(condition_set=True).subs(TS1, TO4).contains(w).doit() == S.true
TO6 = Matrix([[S.One, 0, 0, 0, 0],[S.Half, 0, S.Half, 0, 0],[0, S.Half, 0, S.Half, 0], [0, 0, S.Half, 0, S.Half], [0, 0, 0, 0, 1]])
Y6 = DiscreteMarkovChain('Y', trans_probs=TO6)
assert Y6._transient2absorbing() == ImmutableMatrix([[S.Half, 0], [0, 0], [0, S.Half]])
assert Y6._transient2transient() == ImmutableMatrix([[0, S.Half, 0], [S.Half, 0, S.Half], [0, S.Half, 0]])
assert Y6.fundamental_matrix() == ImmutableMatrix([[Rational(3, 2), S.One, S.Half], [S.One, S(2), S.One], [S.Half, S.One, Rational(3, 2)]])
assert Y6.absorbing_probabilities() == ImmutableMatrix([[Rational(3, 4), Rational(1, 4)], [S.Half, S.Half], [Rational(1, 4), Rational(3, 4)]])
# test for zero-sized matrix functionality
X = DiscreteMarkovChain('X', trans_probs=Matrix([[]]))
assert X.number_of_states == 0
assert X.stationary_distribution() == Matrix([[]])
# test communication_class
# see https://drive.google.com/drive/folders/1HbxLlwwn2b3U8Lj7eb_ASIUb5vYaNIjg?usp=sharing
# tutorial 2.pdf
TO7 = Matrix([[0, 5, 5, 0, 0],
[0, 0, 0, 10, 0],
[5, 0, 5, 0, 0],
[0, 10, 0, 0, 0],
[0, 3, 0, 3, 4]])/10
Y7 = DiscreteMarkovChain('Y', trans_probs=TO7)
tuples = Y7.communication_classes()
classes, recurrence, periods = list(zip(*tuples))
assert classes == ([1, 3], [0, 2], [4])
assert recurrence == (True, False, False)
assert periods == (2, 1, 1)
TO8 = Matrix([[0, 0, 0, 10, 0, 0],
[5, 0, 5, 0, 0, 0],
[0, 4, 0, 0, 0, 6],
[10, 0, 0, 0, 0, 0],
[0, 10, 0, 0, 0, 0],
[0, 0, 0, 5, 5, 0]])/10
Y8 = DiscreteMarkovChain('Y', trans_probs=TO8)
tuples = Y8.communication_classes()
classes, recurrence, periods = list(zip(*tuples))
assert classes == ([0, 3], [1, 2, 5, 4])
assert recurrence == (True, False)
assert periods == (2, 2)
TO9 = Matrix([[2, 0, 0, 3, 0, 0, 3, 2, 0, 0],
[0, 10, 0, 0, 0, 0, 0, 0, 0, 0],
[0, 2, 2, 0, 0, 0, 0, 0, 3, 3],
[0, 0, 0, 3, 0, 0, 6, 1, 0, 0],
[0, 0, 0, 0, 5, 5, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 10, 0, 0, 0, 0],
[4, 0, 0, 5, 0, 0, 1, 0, 0, 0],
[2, 0, 0, 4, 0, 0, 2, 2, 0, 0],
[3, 0, 1, 0, 0, 0, 0, 0, 4, 2],
[0, 0, 4, 0, 0, 0, 0, 0, 3, 3]])/10
Y9 = DiscreteMarkovChain('Y', trans_probs=TO9)
tuples = Y9.communication_classes()
classes, recurrence, periods = list(zip(*tuples))
assert classes == ([0, 3, 6, 7], [1], [2, 8, 9], [5], [4])
assert recurrence == (True, True, False, True, False)
assert periods == (1, 1, 1, 1, 1)
# test custom state space
Y10 = DiscreteMarkovChain('Y', [1, 2, 3], TO2)
tuples = Y10.communication_classes()
classes, recurrence, periods = list(zip(*tuples))
assert classes == ([1], [2, 3])
assert recurrence == (True, False)
assert periods == (1, 1)
# testing miscellaneous queries
T = Matrix([[S.Half, Rational(1, 4), Rational(1, 4)],
[Rational(1, 3), 0, Rational(2, 3)],
[S.Half, S.Half, 0]])
X = DiscreteMarkovChain('X', [0, 1, 2], T)
assert P(Eq(X[1], 2) & Eq(X[2], 1) & Eq(X[3], 0),
Eq(P(Eq(X[1], 0)), Rational(1, 4)) & Eq(P(Eq(X[1], 1)), Rational(1, 4))) == Rational(1, 12)
assert P(Eq(X[2], 1) | Eq(X[2], 2), Eq(X[1], 1)) == Rational(2, 3)
assert P(Eq(X[2], 1) & Eq(X[2], 2), Eq(X[1], 1)) is S.Zero
assert P(Ne(X[2], 2), Eq(X[1], 1)) == Rational(1, 3)
assert E(X[1]**2, Eq(X[0], 1)) == Rational(8, 3)
assert variance(X[1], Eq(X[0], 1)) == Rational(8, 9)
raises(ValueError, lambda: E(X[1], Eq(X[2], 1)))
raises(ValueError, lambda: DiscreteMarkovChain('X', [0, 1], T))
# testing miscellaneous queries with different state space
X = DiscreteMarkovChain('X', ['A', 'B', 'C'], T)
assert P(Eq(X[1], 2) & Eq(X[2], 1) & Eq(X[3], 0),
Eq(P(Eq(X[1], 0)), Rational(1, 4)) & Eq(P(Eq(X[1], 1)), Rational(1, 4))) == Rational(1, 12)
assert P(Eq(X[2], 1) | Eq(X[2], 2), Eq(X[1], 1)) == Rational(2, 3)
assert P(Eq(X[2], 1) & Eq(X[2], 2), Eq(X[1], 1)) is S.Zero
assert P(Ne(X[2], 2), Eq(X[1], 1)) == Rational(1, 3)
a = X.state_space.args[0]
c = X.state_space.args[2]
assert (E(X[1] ** 2, Eq(X[0], 1)) - (a**2/3 + 2*c**2/3)).simplify() == 0
assert (variance(X[1], Eq(X[0], 1)) - (2*(-a/3 + c/3)**2/3 + (2*a/3 - 2*c/3)**2/3)).simplify() == 0
raises(ValueError, lambda: E(X[1], Eq(X[2], 1)))
from sympy import *
from sympy.stats import Normal, density, E, variance
var("tau nact pact dt std damping scale tau_threshold act_threshold", real=True)
var("dt std alpha scale time_constant", real=True, positive=True)
noise = Normal("noise", 0, std)
alpha = 1 - exp(-dt/time_constant)
#nact = alpha*pact + (1 - alpha)*(atan((tau - tau_threshold)*scale) + noise)
#dact = (nact - pact).simplify()
#pprint(dact)
#nact = pact + (-damping*pact + atan((tau - tau_threshold)*scale) + noise)*dt
dact = -alpha*pact + atan((tau - tau_threshold)*scale)*dt + noise*sqrt(dt)
print(dact)
pprint(density(dact))
m = E(dact)
v = variance(dact).simplify()
print(m)
print(v)
def test_DiscreteMarkovChain():
# pass only the name
X = DiscreteMarkovChain("X")
assert X.state_space == S.Reals
assert X.index_set == S.Naturals0
assert X.transition_probabilities == None
t = symbols('t', positive=True, integer=True)
assert isinstance(X[t], RandomIndexedSymbol)
assert E(X[0]) == Expectation(X[0])
raises(TypeError, lambda: DiscreteMarkovChain(1))
raises(NotImplementedError, lambda: X(t))
# pass name and state_space
Y = DiscreteMarkovChain("Y", [1, 2, 3])
assert Y.transition_probabilities == None
assert Y.state_space == FiniteSet(1, 2, 3)
assert P(Eq(Y[2], 1), Eq(Y[0], 2)) == Probability(Eq(Y[2], 1), Eq(Y[0], 2))
assert E(X[0]) == Expectation(X[0])
raises(TypeError, lambda: DiscreteMarkovChain("Y", dict((1, 1))))
# pass name, state_space and transition_probabilities
T = Matrix([[0.5, 0.2, 0.3], [0.2, 0.5, 0.3], [0.2, 0.3, 0.5]])
TS = MatrixSymbol('T', 3, 3)
Y = DiscreteMarkovChain("Y", [0, 1, 2], T)
YS = DiscreteMarkovChain("Y", [0, 1, 2], TS)
assert YS._transient2transient() == None
assert YS._transient2absorbing() == None
assert Y.joint_distribution(1, Y[2],
3) == JointDistribution(Y[1], Y[2], Y[3])
raises(ValueError, lambda: Y.joint_distribution(Y[1].symbol, Y[2].symbol))
assert P(Eq(Y[3], 2), Eq(Y[1], 1)).round(2) == Float(0.36, 2)
assert str(P(Eq(YS[3], 2), Eq(YS[1], 1))) == \
"T[0, 2]*T[1, 0] + T[1, 1]*T[1, 2] + T[1, 2]*T[2, 2]"
TO = Matrix([[0.25, 0.75, 0], [0, 0.25, 0.75], [0.75, 0, 0.25]])
assert P(Eq(Y[3], 2),
Eq(Y[1], 1) & TransitionMatrixOf(Y, TO)).round(3) == Float(
0.375, 3)
assert E(Y[3], evaluate=False) == Expectation(Y[3])
assert E(Y[3], Eq(Y[2], 1)).round(2) == Float(1.1, 3)
TSO = MatrixSymbol('T', 4, 4)
raises(
ValueError,
lambda: str(P(Eq(YS[3], 2),
Eq(YS[1], 1) & TransitionMatrixOf(YS, TSO))))
raises(TypeError,
lambda: DiscreteMarkovChain("Z", [0, 1, 2], symbols('M')))
raises(
ValueError,
lambda: DiscreteMarkovChain("Z", [0, 1, 2], MatrixSymbol('T', 3, 4)))
raises(IndexError, lambda: str(P(Eq(YS[3], 3), Eq(YS[1], 1))))
raises(ValueError, lambda: str(P(Eq(YS[1], 1), Eq(YS[2], 2))))
raises(ValueError, lambda: E(Y[3], Eq(Y[2], 6)))
raises(ValueError, lambda: E(Y[2], Eq(Y[3], 1)))
# extended tests for probability queries
TO1 = Matrix([[S(1) / 4, S(3) / 4, 0], [S(1) / 3,
S(1) / 3,
S(1) / 3], [0,
S(1) / 4,
S(3) / 4]])
assert P(
And(Eq(Y[2], 1), Eq(Y[1], 1), Eq(Y[0], 0)),
Eq(Probability(Eq(Y[0], 0)),
S(1) / 4) & TransitionMatrixOf(Y, TO1)) == S(1) / 16
assert P(And(Eq(Y[2], 1), Eq(Y[1], 1), Eq(Y[0], 0)), TransitionMatrixOf(Y, TO1)) == \
Probability(Eq(Y[0], 0))/4
assert P(
Lt(X[1], 2) & Gt(X[1], 0),
Eq(X[0], 2) & StochasticStateSpaceOf(X, [0, 1, 2])
& TransitionMatrixOf(X, TO1)) == S(1) / 4
assert P(
Ne(X[1], 2) & Ne(X[1], 1),
Eq(X[0], 2) & StochasticStateSpaceOf(X, [0, 1, 2])
& TransitionMatrixOf(X, TO1)) == S(0)
raises(
ValueError, lambda: str(
P(And(Eq(Y[2], 1), Eq(Y[1], 1), Eq(Y[0], 0)), Eq(Y[1], 1))))
# testing properties of Markov chain
TO2 = Matrix([[S(1), 0, 0], [S(1) / 3, S(1) / 3,
S(1) / 3], [0, S(1) / 4,
S(3) / 4]])
TO3 = Matrix([[S(1) / 4, S(3) / 4, 0], [S(1) / 3,
S(1) / 3,
S(1) / 3], [0,
S(1) / 4,
S(3) / 4]])
Y2 = DiscreteMarkovChain('Y', trans_probs=TO2)
Y3 = DiscreteMarkovChain('Y', trans_probs=TO3)
assert Y3._transient2absorbing() == None
raises(ValueError, lambda: Y3.fundamental_matrix())
assert Y2.is_absorbing_chain() == True
assert Y3.is_absorbing_chain() == False
TO4 = Matrix([[S(1) / 5, S(2) / 5, S(2) / 5],
[S(1) / 10, S(1) / 2, S(2) / 5],
[S(3) / 5, S(3) / 10, S(1) / 10]])
Y4 = DiscreteMarkovChain('Y', trans_probs=TO4)
w = ImmutableMatrix([[S(11) / 39, S(16) / 39, S(4) / 13]])
assert Y4.limiting_distribution == w
assert Y4.is_regular() == True
TS1 = MatrixSymbol('T', 3, 3)
Y5 = DiscreteMarkovChain('Y', trans_probs=TS1)
assert Y5.limiting_distribution(w, TO4).doit() == True
TO6 = Matrix([[S(1), 0, 0, 0, 0], [S(1) / 2, 0,
S(1) / 2, 0, 0],
[0, S(1) / 2, 0, S(1) / 2, 0], [0, 0,
S(1) / 2, 0,
S(1) / 2], [0, 0, 0, 0, 1]])
Y6 = DiscreteMarkovChain('Y', trans_probs=TO6)
assert Y6._transient2absorbing() == ImmutableMatrix([[S(1) / 2, 0], [0, 0],
[0, S(1) / 2]])
assert Y6._transient2transient() == ImmutableMatrix(
[[0, S(1) / 2, 0], [S(1) / 2, 0, S(1) / 2], [0, S(1) / 2, 0]])
assert Y6.fundamental_matrix() == ImmutableMatrix(
[[S(3) / 2, S(1), S(1) / 2], [S(1), S(2), S(1)],
[S(1) / 2, S(1), S(3) / 2]])
assert Y6.absorbing_probabilites() == ImmutableMatrix(
[[S(3) / 4, S(1) / 4], [S(1) / 2, S(1) / 2], [S(1) / 4,
S(3) / 4]])
# testing miscellaneous queries
T = Matrix([[S(1) / 2, S(1) / 4, S(1) / 4], [S(1) / 3, 0,
S(2) / 3],
[S(1) / 2, S(1) / 2, 0]])
X = DiscreteMarkovChain('X', [0, 1, 2], T)
assert P(
Eq(X[1], 2) & Eq(X[2], 1) & Eq(X[3], 0),
Eq(P(Eq(X[1], 0)),
S(1) / 4) & Eq(P(Eq(X[1], 1)),
S(1) / 4)) == S(1) / 12
assert P(Eq(X[2], 1) | Eq(X[2], 2), Eq(X[1], 1)) == S(2) / 3
assert P(Eq(X[2], 1) & Eq(X[2], 2), Eq(X[1], 1)) == S(0)
assert P(Ne(X[2], 2), Eq(X[1], 1)) == S(1) / 3
assert E(X[1]**2, Eq(X[0], 1)) == S(8) / 3
assert variance(X[1], Eq(X[0], 1)) == S(8) / 9
raises(ValueError, lambda: E(X[1], Eq(X[2], 1)))
def test_dice():
# TODO: Make iid method!
X, Y, Z = Die('X', 6), Die('Y', 6), Die('Z', 6)
a, b, t, p = symbols('a b t p')
assert E(X) == 3 + S.Half
assert variance(X) == Rational(35, 12)
assert E(X + Y) == 7
assert E(X + X) == 7
assert E(a * X + b) == a * E(X) + b
assert variance(X + Y) == variance(X) + variance(Y) == cmoment(X + Y, 2)
assert variance(X + X) == 4 * variance(X) == cmoment(X + X, 2)
assert cmoment(X, 0) == 1
assert cmoment(4 * X, 3) == 64 * cmoment(X, 3)
assert covariance(X, Y) is S.Zero
assert covariance(X, X + Y) == variance(X)
assert density(Eq(cos(X * S.Pi), 1))[True] == S.Half
assert correlation(X, Y) == 0
assert correlation(X, Y) == correlation(Y, X)
assert smoment(X + Y, 3) == skewness(X + Y)
assert smoment(X + Y, 4) == kurtosis(X + Y)
assert smoment(X, 0) == 1
assert P(X > 3) == S.Half
assert P(2 * X > 6) == S.Half
assert P(X > Y) == Rational(5, 12)
assert P(Eq(X, Y)) == P(Eq(X, 1))
assert E(X, X > 3) == 5 == moment(X, 1, 0, X > 3)
assert E(X, Y > 3) == E(X) == moment(X, 1, 0, Y > 3)
assert E(X + Y, Eq(X, Y)) == E(2 * X)
assert moment(X, 0) == 1
assert moment(5 * X, 2) == 25 * moment(X, 2)
assert quantile(X)(p) == Piecewise((nan, (p > 1) | (p < 0)),\
(S.One, p <= Rational(1, 6)), (S(2), p <= Rational(1, 3)), (S(3), p <= S.Half),\
(S(4), p <= Rational(2, 3)), (S(5), p <= Rational(5, 6)), (S(6), p <= 1))
assert P(X > 3, X > 3) is S.One
assert P(X > Y, Eq(Y, 6)) is S.Zero
assert P(Eq(X + Y, 12)) == Rational(1, 36)
assert P(Eq(X + Y, 12), Eq(X, 6)) == Rational(1, 6)
assert density(X + Y) == density(Y + Z) != density(X + X)
d = density(2 * X + Y**Z)
assert d[S(22)] == Rational(1, 108) and d[S(4100)] == Rational(
1, 216) and S(3130) not in d
assert pspace(X).domain.as_boolean() == Or(
*[Eq(X.symbol, i) for i in [1, 2, 3, 4, 5, 6]])
assert where(X > 3).set == FiniteSet(4, 5, 6)
assert characteristic_function(X)(t) == exp(6 * I * t) / 6 + exp(
5 * I * t) / 6 + exp(4 * I * t) / 6 + exp(3 * I * t) / 6 + exp(
2 * I * t) / 6 + exp(I * t) / 6
assert moment_generating_function(X)(
t) == exp(6 * t) / 6 + exp(5 * t) / 6 + exp(4 * t) / 6 + exp(
3 * t) / 6 + exp(2 * t) / 6 + exp(t) / 6
assert median(X) == FiniteSet(3, 4)
D = Die('D', 7)
assert median(D) == FiniteSet(4)
# Bayes test for die
BayesTest(X > 3, X + Y < 5)
BayesTest(Eq(X - Y, Z), Z > Y)
BayesTest(X > 3, X > 2)
# arg test for die
raises(ValueError, lambda: Die('X', -1)) # issue 8105: negative sides.
raises(ValueError, lambda: Die('X', 0))
raises(ValueError, lambda: Die('X', 1.5)) # issue 8103: non integer sides.
# symbolic test for die
n, k = symbols('n, k', positive=True)
D = Die('D', n)
dens = density(D).dict
assert dens == Density(DieDistribution(n))
assert set(dens.subs(n, 4).doit().keys()) == set([1, 2, 3, 4])
assert set(dens.subs(n, 4).doit().values()) == set([Rational(1, 4)])
k = Dummy('k', integer=True)
assert E(D).dummy_eq(Sum(Piecewise((k / n, k <= n), (0, True)), (k, 1, n)))
assert variance(D).subs(n, 6).doit() == Rational(35, 12)
ki = Dummy('ki')
cumuf = cdf(D)(k)
assert cumuf.dummy_eq(
Sum(Piecewise((1 / n, (ki >= 1) & (ki <= n)), (0, True)), (ki, 1, k)))
assert cumuf.subs({n: 6, k: 2}).doit() == Rational(1, 3)
t = Dummy('t')
cf = characteristic_function(D)(t)
assert cf.dummy_eq(
Sum(Piecewise((exp(ki * I * t) / n, (ki >= 1) & (ki <= n)), (0, True)),
(ki, 1, n)))
assert cf.subs(
n,
3).doit() == exp(3 * I * t) / 3 + exp(2 * I * t) / 3 + exp(I * t) / 3
mgf = moment_generating_function(D)(t)
assert mgf.dummy_eq(
Sum(Piecewise((exp(ki * t) / n, (ki >= 1) & (ki <= n)), (0, True)),
(ki, 1, n)))
assert mgf.subs(n,
3).doit() == exp(3 * t) / 3 + exp(2 * t) / 3 + exp(t) / 3
def test_zeta():
s = S(5)
x = Zeta('x', s)
assert E(x) == zeta(s - 1) / zeta(s)
assert simplify(
variance(x)) == (zeta(s) * zeta(s - 2) - zeta(s - 1)**2) / zeta(s)**2
def test_literal_probability():
X = Normal('X', 2, 3)
Y = Normal('Y', 3, 4)
Z = Poisson('Z', 4)
W = Poisson('W', 3)
x = symbols('x', real=True)
y, w, z = symbols('y, w, z')
assert Probability(X > 0).evaluate_integral() == probability(X > 0)
assert Probability(X > x).evaluate_integral() == probability(X > x)
assert Probability(X > 0).rewrite(Integral).doit() == probability(X > 0)
assert Probability(X > x).rewrite(Integral).doit() == probability(X > x)
assert Expectation(X).evaluate_integral() == expectation(X)
assert Expectation(X).rewrite(Integral).doit() == expectation(X)
assert Expectation(X**2).evaluate_integral() == expectation(X**2)
assert Expectation(x * X).args == (x * X, )
assert Expectation(x * X).doit() == x * Expectation(X)
assert Expectation(2 * X + 3 * Y + z * X * Y).doit(
) == 2 * Expectation(X) + 3 * Expectation(Y) + z * Expectation(X * Y)
assert Expectation(2 * X + 3 * Y + z * X * Y).args == (2 * X + 3 * Y +
z * X * Y, )
assert Expectation(sin(X)) == Expectation(sin(X)).doit()
assert Expectation(
2 * x * sin(X) * Y + y * X**2 +
z * X * Y).doit() == 2 * x * Expectation(sin(X) * Y) + y * Expectation(
X**2) + z * Expectation(X * Y)
assert Variance(w).args == (w, )
assert Variance(w).doit() == 0
assert Variance(X).evaluate_integral() == Variance(X).rewrite(
Integral).doit() == variance(X)
assert Variance(X + z).args == (X + z, )
assert Variance(X + z).doit() == Variance(X)
assert Variance(X * Y).args == (Mul(X, Y), )
assert type(Variance(X * Y)) == Variance
assert Variance(z * X).doit() == z**2 * Variance(X)
assert Variance(
X + Y).doit() == Variance(X) + Variance(Y) + 2 * Covariance(X, Y)
assert Variance(X + Y + Z +
W).doit() == (Variance(X) + Variance(Y) + Variance(Z) +
Variance(W) + 2 * Covariance(X, Y) +
2 * Covariance(X, Z) + 2 * Covariance(X, W) +
2 * Covariance(Y, Z) + 2 * Covariance(Y, W) +
2 * Covariance(W, Z))
assert Variance(X**2).evaluate_integral() == variance(X**2)
assert Variance(X**2) == Variance(X**2)
assert Variance(x * X**2).doit() == x**2 * Variance(X**2)
assert Variance(sin(X)).args == (sin(X), )
assert Variance(sin(X)).doit() == Variance(sin(X))
assert Variance(x * sin(X)).doit() == x**2 * Variance(sin(X))
assert Covariance(w, z).args == (w, z)
assert Covariance(w, z).doit() == 0
assert Covariance(X, w).doit() == 0
assert Covariance(w, X).doit() == 0
assert Covariance(X, Y).args == (X, Y)
assert type(Covariance(X, Y)) == Covariance
assert Covariance(z * X + 3, Y).doit() == z * Covariance(X, Y)
assert Covariance(X, X).args == (X, X)
assert Covariance(X, X).doit() == Variance(X)
assert Covariance(z * X + 3, w * Y + 4).doit() == w * z * Covariance(X, Y)
assert Covariance(X, Y) == Covariance(Y, X)
assert Covariance(X + Y, Z + W).doit() == Covariance(W, X) + Covariance(
W, Y) + Covariance(X, Z) + Covariance(Y, Z)
assert Covariance(
x * X + y * Y, z * Z +
w * W).doit() == (x * w * Covariance(W, X) + w * y * Covariance(W, Y) +
x * z * Covariance(X, Z) + y * z * Covariance(Y, Z))
assert Covariance(x * X**2 + y * sin(Y), z * Y * Z**2 +
w * W).doit() == (w * x * Covariance(W, X**2) +
w * y * Covariance(sin(Y), W) +
x * z * Covariance(Y * Z**2, X**2) +
y * z * Covariance(Y * Z**2, sin(Y)))
assert Covariance(X, X**2).doit() == Covariance(X, X**2)
assert Covariance(X, sin(X)).doit() == Covariance(sin(X), X)
assert Covariance(X**2, sin(X) * Y).doit() == Covariance(sin(X) * Y, X**2)
def _eval_rewrite_as_Integral(self, arg, condition=None, **kwargs):
return variance(self.args[0], self._condition, evaluate=False)
def test_DiscreteMarkovChain():
# pass only the name
X = DiscreteMarkovChain("X")
assert isinstance(X.state_space, Range)
assert isinstance(X.index_of, Range)
assert not X._is_numeric
assert X.index_set == S.Naturals0
assert isinstance(X.transition_probabilities, MatrixSymbol)
t = symbols('t', positive=True, integer=True)
assert isinstance(X[t], RandomIndexedSymbol)
assert E(X[0]) == Expectation(X[0])
raises(TypeError, lambda: DiscreteMarkovChain(1))
raises(NotImplementedError, lambda: X(t))
raises(ValueError, lambda: sample_stochastic_process(t))
raises(ValueError, lambda: next(sample_stochastic_process(X)))
# pass name and state_space
# any hashable object should be a valid state
# states should be valid as a tuple/set/list/Tuple/Range
sym = symbols('a', real=True)
state_spaces = [(1, 2, 3), [Str('Hello'), sym, DiscreteMarkovChain],
Tuple(1, exp(sym), Str('World'), sympify=False),
Range(-1, 7, 2)]
chains = [
DiscreteMarkovChain("Y", state_spaces[0]),
DiscreteMarkovChain("Y", state_spaces[1]),
DiscreteMarkovChain("Y", state_spaces[2])
]
for i, Y in enumerate(chains):
assert isinstance(Y.transition_probabilities, MatrixSymbol)
assert Y.state_space == Tuple(*state_spaces[i])
assert Y.number_of_states == 3
assert not Y._is_numeric # because no transition matrix is provided
assert Y.index_of[state_spaces[i][0]] == 0
assert Y.index_of[state_spaces[i][1]] == 1
assert Y.index_of[state_spaces[i][2]] == 2
with ignore_warnings(
UserWarning): # TODO: Restore tests once warnings are removed
assert P(Eq(Y[2], 1), Eq(Y[0], 2),
evaluate=False) == Probability(Eq(Y[2], 1), Eq(Y[0], 2))
assert E(Y[0]) == Expectation(Y[0])
raises(ValueError, lambda: next(sample_stochastic_process(Y)))
raises(TypeError, lambda: DiscreteMarkovChain("Y", dict((1, 1))))
Y = DiscreteMarkovChain("Y", Range(1, t, 2))
assert Y.number_of_states == ceiling((t - 1) / 2)
raises(NotImplementedError, lambda: Y.index_of)
# pass name and transition_probabilities
chains = [
DiscreteMarkovChain("Y", trans_probs=Matrix([[]])),
DiscreteMarkovChain("Y", trans_probs=Matrix([[0, 1], [1, 0]])),
DiscreteMarkovChain("Y",
trans_probs=Matrix([[pi, 1 - pi], [sym, 1 - sym]]))
]
for Z in chains:
assert Z.number_of_states == Z.transition_probabilities.shape[0]
assert isinstance(Z.transition_probabilities, ImmutableDenseMatrix)
assert isinstance(Z.state_space, Tuple)
assert Z._is_numeric
# pass name, state_space and transition_probabilities
T = Matrix([[0.5, 0.2, 0.3], [0.2, 0.5, 0.3], [0.2, 0.3, 0.5]])
TS = MatrixSymbol('T', 3, 3)
Y = DiscreteMarkovChain("Y", [0, 1, 2], T)
YS = DiscreteMarkovChain("Y", ['One', 'Two', 3], TS)
assert YS._transient2transient() == None
assert YS._transient2absorbing() == None
assert Y.joint_distribution(1, Y[2],
3) == JointDistribution(Y[1], Y[2], Y[3])
raises(ValueError, lambda: Y.joint_distribution(Y[1].symbol, Y[2].symbol))
assert P(Eq(Y[3], 2), Eq(Y[1], 1)).round(2) == Float(0.36, 2)
assert (P(Eq(YS[3], 2), Eq(YS[1], 1)) -
(TS[0, 2] * TS[1, 0] + TS[1, 1] * TS[1, 2] +
TS[1, 2] * TS[2, 2])).simplify() == 0
assert P(Eq(YS[1], 1), Eq(YS[2], 2)) == Probability(Eq(YS[1], 1))
assert P(Eq(YS[3], 3), Eq(YS[1], 1)) is S.Zero
TO = Matrix([[0.25, 0.75, 0], [0, 0.25, 0.75], [0.75, 0, 0.25]])
assert P(Eq(Y[3], 2),
Eq(Y[1], 1) & TransitionMatrixOf(Y, TO)).round(3) == Float(
0.375, 3)
with ignore_warnings(
UserWarning): ### TODO: Restore tests once warnings are removed
assert E(Y[3], evaluate=False) == Expectation(Y[3])
assert E(Y[3], Eq(Y[2], 1)).round(2) == Float(1.1, 3)
TSO = MatrixSymbol('T', 4, 4)
raises(
ValueError,
lambda: str(P(Eq(YS[3], 2),
Eq(YS[1], 1) & TransitionMatrixOf(YS, TSO))))
raises(TypeError,
lambda: DiscreteMarkovChain("Z", [0, 1, 2], symbols('M')))
raises(
ValueError,
lambda: DiscreteMarkovChain("Z", [0, 1, 2], MatrixSymbol('T', 3, 4)))
raises(ValueError, lambda: E(Y[3], Eq(Y[2], 6)))
raises(ValueError, lambda: E(Y[2], Eq(Y[3], 1)))
# extended tests for probability queries
TO1 = Matrix([[Rational(1, 4), Rational(3, 4), 0],
[Rational(1, 3),
Rational(1, 3),
Rational(1, 3)], [0, Rational(1, 4),
Rational(3, 4)]])
assert P(
And(Eq(Y[2], 1), Eq(Y[1], 1), Eq(Y[0], 0)),
Eq(Probability(Eq(Y[0], 0)), Rational(1, 4))
& TransitionMatrixOf(Y, TO1)) == Rational(1, 16)
assert P(And(Eq(Y[2], 1), Eq(Y[1], 1), Eq(Y[0], 0)), TransitionMatrixOf(Y, TO1)) == \
Probability(Eq(Y[0], 0))/4
assert P(
Lt(X[1], 2) & Gt(X[1], 0),
Eq(X[0], 2) & StochasticStateSpaceOf(X, [0, 1, 2])
& TransitionMatrixOf(X, TO1)) == Rational(1, 4)
assert P(
Lt(X[1], 2) & Gt(X[1], 0),
Eq(X[0], 2) & StochasticStateSpaceOf(X, [None, 'None', 1])
& TransitionMatrixOf(X, TO1)) == Rational(1, 4)
assert P(
Ne(X[1], 2) & Ne(X[1], 1),
Eq(X[0], 2) & StochasticStateSpaceOf(X, [0, 1, 2])
& TransitionMatrixOf(X, TO1)) is S.Zero
assert P(
Ne(X[1], 2) & Ne(X[1], 1),
Eq(X[0], 2) & StochasticStateSpaceOf(X, [None, 'None', 1])
& TransitionMatrixOf(X, TO1)) is S.Zero
assert P(And(Eq(Y[2], 1), Eq(Y[1], 1), Eq(Y[0], 0)),
Eq(Y[1], 1)) == 0.1 * Probability(Eq(Y[0], 0))
# testing properties of Markov chain
TO2 = Matrix([[S.One, 0, 0],
[Rational(1, 3),
Rational(1, 3),
Rational(1, 3)], [0, Rational(1, 4),
Rational(3, 4)]])
TO3 = Matrix([[Rational(1, 4), Rational(3, 4), 0],
[Rational(1, 3),
Rational(1, 3),
Rational(1, 3)], [0, Rational(1, 4),
Rational(3, 4)]])
Y2 = DiscreteMarkovChain('Y', trans_probs=TO2)
Y3 = DiscreteMarkovChain('Y', trans_probs=TO3)
assert Y3._transient2absorbing() == None
raises(ValueError, lambda: Y3.fundamental_matrix())
assert Y2.is_absorbing_chain() == True
assert Y3.is_absorbing_chain() == False
TO4 = Matrix([[Rational(1, 5),
Rational(2, 5),
Rational(2, 5)], [Rational(1, 10), S.Half,
Rational(2, 5)],
[Rational(3, 5),
Rational(3, 10),
Rational(1, 10)]])
Y4 = DiscreteMarkovChain('Y', trans_probs=TO4)
w = ImmutableMatrix([[Rational(11, 39),
Rational(16, 39),
Rational(4, 13)]])
assert Y4.limiting_distribution == w
assert Y4.is_regular() == True
TS1 = MatrixSymbol('T', 3, 3)
Y5 = DiscreteMarkovChain('Y', trans_probs=TS1)
assert Y5.limiting_distribution(w, TO4).doit() == True
TO6 = Matrix([[S.One, 0, 0, 0, 0], [S.Half, 0, S.Half, 0, 0],
[0, S.Half, 0, S.Half, 0], [0, 0, S.Half, 0, S.Half],
[0, 0, 0, 0, 1]])
Y6 = DiscreteMarkovChain('Y', trans_probs=TO6)
assert Y6._transient2absorbing() == ImmutableMatrix([[S.Half, 0], [0, 0],
[0, S.Half]])
assert Y6._transient2transient() == ImmutableMatrix([[0, S.Half, 0],
[S.Half, 0, S.Half],
[0, S.Half, 0]])
assert Y6.fundamental_matrix() == ImmutableMatrix(
[[Rational(3, 2), S.One, S.Half], [S.One, S(2), S.One],
[S.Half, S.One, Rational(3, 2)]])
assert Y6.absorbing_probabilities() == ImmutableMatrix(
[[Rational(3, 4), Rational(1, 4)], [S.Half, S.Half],
[Rational(1, 4), Rational(3, 4)]])
# testing miscellaneous queries
T = Matrix([[S.Half, Rational(1, 4),
Rational(1, 4)], [Rational(1, 3), 0,
Rational(2, 3)], [S.Half, S.Half, 0]])
X = DiscreteMarkovChain('X', [0, 1, 2], T)
assert P(
Eq(X[1], 2) & Eq(X[2], 1) & Eq(X[3], 0),
Eq(P(Eq(X[1], 0)), Rational(1, 4))
& Eq(P(Eq(X[1], 1)), Rational(1, 4))) == Rational(1, 12)
assert P(Eq(X[2], 1) | Eq(X[2], 2), Eq(X[1], 1)) == Rational(2, 3)
assert P(Eq(X[2], 1) & Eq(X[2], 2), Eq(X[1], 1)) is S.Zero
assert P(Ne(X[2], 2), Eq(X[1], 1)) == Rational(1, 3)
assert E(X[1]**2, Eq(X[0], 1)) == Rational(8, 3)
assert variance(X[1], Eq(X[0], 1)) == Rational(8, 9)
raises(ValueError, lambda: E(X[1], Eq(X[2], 1)))
raises(ValueError, lambda: DiscreteMarkovChain('X', [0, 1], T))
# testing miscellaneous queries with different state space
X = DiscreteMarkovChain('X', ['A', 'B', 'C'], T)
assert P(
Eq(X[1], 2) & Eq(X[2], 1) & Eq(X[3], 0),
Eq(P(Eq(X[1], 0)), Rational(1, 4))
& Eq(P(Eq(X[1], 1)), Rational(1, 4))) == Rational(1, 12)
assert P(Eq(X[2], 1) | Eq(X[2], 2), Eq(X[1], 1)) == Rational(2, 3)
assert P(Eq(X[2], 1) & Eq(X[2], 2), Eq(X[1], 1)) is S.Zero
assert P(Ne(X[2], 2), Eq(X[1], 1)) == Rational(1, 3)
a = X.state_space.args[0]
c = X.state_space.args[2]
assert (E(X[1]**2, Eq(X[0], 1)) -
(a**2 / 3 + 2 * c**2 / 3)).simplify() == 0
assert (variance(X[1], Eq(X[0], 1)) -
(2 * (-a / 3 + c / 3)**2 / 3 +
(2 * a / 3 - 2 * c / 3)**2 / 3)).simplify() == 0
raises(ValueError, lambda: E(X[1], Eq(X[2], 1)))
def test_ContinuousMarkovChain():
T1 = Matrix([[S(-2), S(2), S.Zero], [S.Zero, S.NegativeOne, S.One],
[Rational(3, 2), Rational(3, 2),
S(-3)]])
C1 = ContinuousMarkovChain('C', [0, 1, 2], T1)
assert C1.limiting_distribution() == ImmutableMatrix(
[[Rational(3, 19), Rational(12, 19),
Rational(4, 19)]])
T2 = Matrix([[-S.One, S.One, S.Zero], [S.One, -S.One, S.Zero],
[S.Zero, S.One, -S.One]])
C2 = ContinuousMarkovChain('C', [0, 1, 2], T2)
A, t = C2.generator_matrix, symbols('t', positive=True)
assert C2.transition_probabilities(A)(t) == Matrix(
[[S.Half + exp(-2 * t) / 2, S.Half - exp(-2 * t) / 2, 0],
[S.Half - exp(-2 * t) / 2, S.Half + exp(-2 * t) / 2, 0],
[
S.Half - exp(-t) + exp(-2 * t) / 2, S.Half - exp(-2 * t) / 2,
exp(-t)
]])
with ignore_warnings(
UserWarning): ### TODO: Restore tests once warnings are removed
assert P(Eq(C2(1), 1), Eq(C2(0), 1),
evaluate=False) == Probability(Eq(C2(1), 1), Eq(C2(0), 1))
assert P(Eq(C2(1), 1), Eq(C2(0), 1)) == exp(-2) / 2 + S.Half
assert P(
Eq(C2(1), 0) & Eq(C2(2), 1) & Eq(C2(3), 1),
Eq(P(Eq(C2(1), 0)),
S.Half)) == (Rational(1, 4) - exp(-2) / 4) * (exp(-2) / 2 + S.Half)
assert P(
Not(Eq(C2(1), 0) & Eq(C2(2), 1) & Eq(C2(3), 2)) |
(Eq(C2(1), 0) & Eq(C2(2), 1) & Eq(C2(3), 2)),
Eq(P(Eq(C2(1), 0)), Rational(1, 4))
& Eq(P(Eq(C2(1), 1)), Rational(1, 4))) is S.One
assert E(C2(Rational(3, 2)),
Eq(C2(0), 2)) == -exp(-3) / 2 + 2 * exp(Rational(-3, 2)) + S.Half
assert variance(C2(Rational(3, 2)), Eq(
C2(0),
1)) == ((S.Half - exp(-3) / 2)**2 * (exp(-3) / 2 + S.Half) +
(Rational(-1, 2) - exp(-3) / 2)**2 * (S.Half - exp(-3) / 2))
raises(KeyError, lambda: P(Eq(C2(1), 0), Eq(P(Eq(C2(1), 1)), S.Half)))
assert P(Eq(C2(1), 0), Eq(P(Eq(C2(5), 1)),
S.Half)) == Probability(Eq(C2(1), 0))
TS1 = MatrixSymbol('G', 3, 3)
CS1 = ContinuousMarkovChain('C', [0, 1, 2], TS1)
A = CS1.generator_matrix
assert CS1.transition_probabilities(A)(t) == exp(t * A)
C3 = ContinuousMarkovChain(
'C', [Symbol('0'), Symbol('1'), Symbol('2')], T2)
assert P(Eq(C3(1), 1), Eq(C3(0), 1)) == exp(-2) / 2 + S.Half
assert P(Eq(C3(1), Symbol('1')), Eq(C3(0),
Symbol('1'))) == exp(-2) / 2 + S.Half
#test probability queries
G = Matrix([[-S(1), Rational(1, 10),
Rational(9, 10)], [Rational(2, 5), -S(1),
Rational(3, 5)],
[Rational(1, 2), Rational(1, 2), -S(1)]])
C = ContinuousMarkovChain('C', state_space=[0, 1, 2], gen_mat=G)
assert P(Eq(C(7.385), C(3.19)), Eq(C(0.862),
0)).round(5) == Float(0.35469, 5)
assert P(Gt(C(98.715), C(19.807)), Eq(C(11.314),
2)).round(5) == Float(0.32452, 5)
assert P(Le(C(5.9), C(10.112)), Eq(C(4), 1)).round(6) == Float(0.675214, 6)
assert Float(P(Eq(C(7.32), C(2.91)), Eq(C(2.63), 1)),
14) == Float(1 - P(Ne(C(7.32), C(2.91)), Eq(C(2.63), 1)), 14)
assert Float(P(Gt(C(3.36), C(1.101)), Eq(C(0.8), 2)),
14) == Float(1 - P(Le(C(3.36), C(1.101)), Eq(C(0.8), 2)), 14)
assert Float(P(Lt(C(4.9), C(2.79)), Eq(C(1.61), 0)),
14) == Float(1 - P(Ge(C(4.9), C(2.79)), Eq(C(1.61), 0)), 14)
assert P(Eq(C(5.243), C(10.912)), Eq(C(2.174),
1)) == P(Eq(C(10.912), C(5.243)),
Eq(C(2.174), 1))
assert P(Gt(C(2.344), C(9.9)), Eq(C(1.102),
1)) == P(Lt(C(9.9), C(2.344)),
Eq(C(1.102), 1))
assert P(Ge(C(7.87), C(1.008)), Eq(C(0.153),
1)) == P(Le(C(1.008), C(7.87)),
Eq(C(0.153), 1))
#test symbolic queries
a, b, c, d = symbols('a b c d')
query = P(Eq(C(a), b), Eq(C(c), d))
assert query.subs({
a: 3.65,
b: 2,
c: 1.78,
d: 1
}).evalf().round(10) == P(Eq(C(3.65), 2), Eq(C(1.78), 1)).round(10)
query_gt = P(Gt(C(a), b), Eq(C(c), d))
query_le = P(Le(C(a), b), Eq(C(c), d))
assert query_gt.subs({
a: 13.2,
b: 0,
c: 3.29,
d: 2
}).evalf() + query_le.subs({
a: 13.2,
b: 0,
c: 3.29,
d: 2
}).evalf() == 1
query_ge = P(Ge(C(a), b), Eq(C(c), d))
query_lt = P(Lt(C(a), b), Eq(C(c), d))
assert query_ge.subs({
a: 7.43,
b: 1,
c: 1.45,
d: 0
}).evalf() + query_lt.subs({
a: 7.43,
b: 1,
c: 1.45,
d: 0
}).evalf() == 1
#test issue 20078
assert (2 * C(1) + 3 * C(1)).simplify() == 5 * C(1)
assert (2 * C(1) - 3 * C(1)).simplify() == -C(1)
assert (2 * (0.25 * C(1))).simplify() == 0.5 * C(1)
assert (2 * C(1) * 0.25 * C(1)).simplify() == 0.5 * C(1)**2
assert (C(1)**2 + C(1)**3).simplify() == (C(1) + 1) * C(1)**2
def test_DiscreteMarkovChain():
# pass only the name
X = DiscreteMarkovChain("X")
assert isinstance(X.state_space, Range)
assert X.index_set == S.Naturals0
assert isinstance(X.transition_probabilities, MatrixSymbol)
t = symbols('t', positive=True, integer=True)
assert isinstance(X[t], RandomIndexedSymbol)
assert E(X[0]) == Expectation(X[0])
raises(TypeError, lambda: DiscreteMarkovChain(1))
raises(NotImplementedError, lambda: X(t))
raises(NotImplementedError, lambda: X.communication_classes())
raises(NotImplementedError, lambda: X.canonical_form())
raises(NotImplementedError, lambda: X.decompose())
nz = Symbol('n', integer=True)
TZ = MatrixSymbol('M', nz, nz)
SZ = Range(nz)
YZ = DiscreteMarkovChain('Y', SZ, TZ)
assert P(Eq(YZ[2], 1), Eq(YZ[1], 0)) == TZ[0, 1]
raises(ValueError, lambda: sample_stochastic_process(t))
raises(ValueError, lambda: next(sample_stochastic_process(X)))
# pass name and state_space
# any hashable object should be a valid state
# states should be valid as a tuple/set/list/Tuple/Range
sym, rainy, cloudy, sunny = symbols('a Rainy Cloudy Sunny', real=True)
state_spaces = [(1, 2, 3), [Str('Hello'), sym, DiscreteMarkovChain],
Tuple(1, exp(sym), Str('World'), sympify=False),
Range(-1, 5, 2), [rainy, cloudy, sunny]]
chains = [
DiscreteMarkovChain("Y", state_space) for state_space in state_spaces
]
for i, Y in enumerate(chains):
assert isinstance(Y.transition_probabilities, MatrixSymbol)
assert Y.state_space == state_spaces[i] or Y.state_space == FiniteSet(
*state_spaces[i])
assert Y.number_of_states == 3
with ignore_warnings(
UserWarning): # TODO: Restore tests once warnings are removed
assert P(Eq(Y[2], 1), Eq(Y[0], 2),
evaluate=False) == Probability(Eq(Y[2], 1), Eq(Y[0], 2))
assert E(Y[0]) == Expectation(Y[0])
raises(ValueError, lambda: next(sample_stochastic_process(Y)))
raises(TypeError, lambda: DiscreteMarkovChain("Y", dict((1, 1))))
Y = DiscreteMarkovChain("Y", Range(1, t, 2))
assert Y.number_of_states == ceiling((t - 1) / 2)
# pass name and transition_probabilities
chains = [
DiscreteMarkovChain("Y", trans_probs=Matrix([[]])),
DiscreteMarkovChain("Y", trans_probs=Matrix([[0, 1], [1, 0]])),
DiscreteMarkovChain("Y",
trans_probs=Matrix([[pi, 1 - pi], [sym, 1 - sym]]))
]
for Z in chains:
assert Z.number_of_states == Z.transition_probabilities.shape[0]
assert isinstance(Z.transition_probabilities, ImmutableDenseMatrix)
# pass name, state_space and transition_probabilities
T = Matrix([[0.5, 0.2, 0.3], [0.2, 0.5, 0.3], [0.2, 0.3, 0.5]])
TS = MatrixSymbol('T', 3, 3)
Y = DiscreteMarkovChain("Y", [0, 1, 2], T)
YS = DiscreteMarkovChain("Y", ['One', 'Two', 3], TS)
assert Y.joint_distribution(1, Y[2],
3) == JointDistribution(Y[1], Y[2], Y[3])
raises(ValueError, lambda: Y.joint_distribution(Y[1].symbol, Y[2].symbol))
assert P(Eq(Y[3], 2), Eq(Y[1], 1)).round(2) == Float(0.36, 2)
assert (P(Eq(YS[3], 2), Eq(YS[1], 1)) -
(TS[0, 2] * TS[1, 0] + TS[1, 1] * TS[1, 2] +
TS[1, 2] * TS[2, 2])).simplify() == 0
assert P(Eq(YS[1], 1), Eq(YS[2], 2)) == Probability(Eq(YS[1], 1))
assert P(Eq(YS[3], 3), Eq(
YS[1],
1)) == TS[0, 2] * TS[1, 0] + TS[1, 1] * TS[1, 2] + TS[1, 2] * TS[2, 2]
TO = Matrix([[0.25, 0.75, 0], [0, 0.25, 0.75], [0.75, 0, 0.25]])
assert P(Eq(Y[3], 2),
Eq(Y[1], 1) & TransitionMatrixOf(Y, TO)).round(3) == Float(
0.375, 3)
with ignore_warnings(
UserWarning): ### TODO: Restore tests once warnings are removed
assert E(Y[3], evaluate=False) == Expectation(Y[3])
assert E(Y[3], Eq(Y[2], 1)).round(2) == Float(1.1, 3)
TSO = MatrixSymbol('T', 4, 4)
raises(
ValueError,
lambda: str(P(Eq(YS[3], 2),
Eq(YS[1], 1) & TransitionMatrixOf(YS, TSO))))
raises(TypeError,
lambda: DiscreteMarkovChain("Z", [0, 1, 2], symbols('M')))
raises(
ValueError,
lambda: DiscreteMarkovChain("Z", [0, 1, 2], MatrixSymbol('T', 3, 4)))
raises(ValueError, lambda: E(Y[3], Eq(Y[2], 6)))
raises(ValueError, lambda: E(Y[2], Eq(Y[3], 1)))
# extended tests for probability queries
TO1 = Matrix([[Rational(1, 4), Rational(3, 4), 0],
[Rational(1, 3),
Rational(1, 3),
Rational(1, 3)], [0, Rational(1, 4),
Rational(3, 4)]])
assert P(
And(Eq(Y[2], 1), Eq(Y[1], 1), Eq(Y[0], 0)),
Eq(Probability(Eq(Y[0], 0)), Rational(1, 4))
& TransitionMatrixOf(Y, TO1)) == Rational(1, 16)
assert P(And(Eq(Y[2], 1), Eq(Y[1], 1), Eq(Y[0], 0)), TransitionMatrixOf(Y, TO1)) == \
Probability(Eq(Y[0], 0))/4
assert P(
Lt(X[1], 2) & Gt(X[1], 0),
Eq(X[0], 2) & StochasticStateSpaceOf(X, [0, 1, 2])
& TransitionMatrixOf(X, TO1)) == Rational(1, 4)
assert P(
Lt(X[1], 2) & Gt(X[1], 0),
Eq(X[0], 2) & StochasticStateSpaceOf(X, [None, 'None', 1])
& TransitionMatrixOf(X, TO1)) == Rational(1, 4)
assert P(
Ne(X[1], 2) & Ne(X[1], 1),
Eq(X[0], 2) & StochasticStateSpaceOf(X, [0, 1, 2])
& TransitionMatrixOf(X, TO1)) is S.Zero
assert P(
Ne(X[1], 2) & Ne(X[1], 1),
Eq(X[0], 2) & StochasticStateSpaceOf(X, [None, 'None', 1])
& TransitionMatrixOf(X, TO1)) is S.Zero
assert P(And(Eq(Y[2], 1), Eq(Y[1], 1), Eq(Y[0], 0)),
Eq(Y[1], 1)) == 0.1 * Probability(Eq(Y[0], 0))
# testing properties of Markov chain
TO2 = Matrix([[S.One, 0, 0],
[Rational(1, 3),
Rational(1, 3),
Rational(1, 3)], [0, Rational(1, 4),
Rational(3, 4)]])
TO3 = Matrix([[Rational(1, 4), Rational(3, 4), 0],
[Rational(1, 3),
Rational(1, 3),
Rational(1, 3)], [0, Rational(1, 4),
Rational(3, 4)]])
Y2 = DiscreteMarkovChain('Y', trans_probs=TO2)
Y3 = DiscreteMarkovChain('Y', trans_probs=TO3)
assert Y3.fundamental_matrix() == ImmutableMatrix(
[[176, 81, -132], [36, 141, -52], [-44, -39, 208]]) / 125
assert Y2.is_absorbing_chain() == True
assert Y3.is_absorbing_chain() == False
assert Y2.canonical_form() == ([0, 1, 2], TO2)
assert Y3.canonical_form() == ([0, 1, 2], TO3)
assert Y2.decompose() == ([0, 1,
2], TO2[0:1, 0:1], TO2[1:3, 0:1], TO2[1:3, 1:3])
assert Y3.decompose() == ([0, 1, 2], TO3, Matrix(0, 3,
[]), Matrix(0, 0, []))
TO4 = Matrix([[Rational(1, 5),
Rational(2, 5),
Rational(2, 5)], [Rational(1, 10), S.Half,
Rational(2, 5)],
[Rational(3, 5),
Rational(3, 10),
Rational(1, 10)]])
Y4 = DiscreteMarkovChain('Y', trans_probs=TO4)
w = ImmutableMatrix([[Rational(11, 39),
Rational(16, 39),
Rational(4, 13)]])
assert Y4.limiting_distribution == w
assert Y4.is_regular() == True
assert Y4.is_ergodic() == True
TS1 = MatrixSymbol('T', 3, 3)
Y5 = DiscreteMarkovChain('Y', trans_probs=TS1)
assert Y5.limiting_distribution(w, TO4).doit() == True
assert Y5.stationary_distribution(condition_set=True).subs(
TS1, TO4).contains(w).doit() == S.true
TO6 = Matrix([[S.One, 0, 0, 0, 0], [S.Half, 0, S.Half, 0, 0],
[0, S.Half, 0, S.Half, 0], [0, 0, S.Half, 0, S.Half],
[0, 0, 0, 0, 1]])
Y6 = DiscreteMarkovChain('Y', trans_probs=TO6)
assert Y6.fundamental_matrix() == ImmutableMatrix(
[[Rational(3, 2), S.One, S.Half], [S.One, S(2), S.One],
[S.Half, S.One, Rational(3, 2)]])
assert Y6.absorbing_probabilities() == ImmutableMatrix(
[[Rational(3, 4), Rational(1, 4)], [S.Half, S.Half],
[Rational(1, 4), Rational(3, 4)]])
TO7 = Matrix([[Rational(1, 2),
Rational(1, 4),
Rational(1, 4)], [Rational(1, 2), 0,
Rational(1, 2)],
[Rational(1, 4),
Rational(1, 4),
Rational(1, 2)]])
Y7 = DiscreteMarkovChain('Y', trans_probs=TO7)
assert Y7.is_absorbing_chain() == False
assert Y7.fundamental_matrix() == ImmutableDenseMatrix(
[[Rational(86, 75),
Rational(1, 25),
Rational(-14, 75)],
[Rational(2, 25), Rational(21, 25),
Rational(2, 25)],
[Rational(-14, 75),
Rational(1, 25),
Rational(86, 75)]])
# test for zero-sized matrix functionality
X = DiscreteMarkovChain('X', trans_probs=Matrix([[]]))
assert X.number_of_states == 0
assert X.stationary_distribution() == Matrix([[]])
assert X.communication_classes() == []
assert X.canonical_form() == ([], Matrix([[]]))
assert X.decompose() == ([], Matrix([[]]), Matrix([[]]), Matrix([[]]))
assert X.is_regular() == False
assert X.is_ergodic() == False
# test communication_class
# see https://drive.google.com/drive/folders/1HbxLlwwn2b3U8Lj7eb_ASIUb5vYaNIjg?usp=sharing
# tutorial 2.pdf
TO7 = Matrix([[0, 5, 5, 0, 0], [0, 0, 0, 10, 0], [5, 0, 5, 0, 0],
[0, 10, 0, 0, 0], [0, 3, 0, 3, 4]]) / 10
Y7 = DiscreteMarkovChain('Y', trans_probs=TO7)
tuples = Y7.communication_classes()
classes, recurrence, periods = list(zip(*tuples))
assert classes == ([1, 3], [0, 2], [4])
assert recurrence == (True, False, False)
assert periods == (2, 1, 1)
TO8 = Matrix([[0, 0, 0, 10, 0, 0], [5, 0, 5, 0, 0, 0], [0, 4, 0, 0, 0, 6],
[10, 0, 0, 0, 0, 0], [0, 10, 0, 0, 0, 0], [0, 0, 0, 5, 5, 0]
]) / 10
Y8 = DiscreteMarkovChain('Y', trans_probs=TO8)
tuples = Y8.communication_classes()
classes, recurrence, periods = list(zip(*tuples))
assert classes == ([0, 3], [1, 2, 5, 4])
assert recurrence == (True, False)
assert periods == (2, 2)
TO9 = Matrix(
[[2, 0, 0, 3, 0, 0, 3, 2, 0, 0], [0, 10, 0, 0, 0, 0, 0, 0, 0, 0],
[0, 2, 2, 0, 0, 0, 0, 0, 3, 3], [0, 0, 0, 3, 0, 0, 6, 1, 0, 0],
[0, 0, 0, 0, 5, 5, 0, 0, 0, 0], [0, 0, 0, 0, 0, 10, 0, 0, 0, 0],
[4, 0, 0, 5, 0, 0, 1, 0, 0, 0], [2, 0, 0, 4, 0, 0, 2, 2, 0, 0],
[3, 0, 1, 0, 0, 0, 0, 0, 4, 2], [0, 0, 4, 0, 0, 0, 0, 0, 3, 3]]) / 10
Y9 = DiscreteMarkovChain('Y', trans_probs=TO9)
tuples = Y9.communication_classes()
classes, recurrence, periods = list(zip(*tuples))
assert classes == ([0, 3, 6, 7], [1], [2, 8, 9], [5], [4])
assert recurrence == (True, True, False, True, False)
assert periods == (1, 1, 1, 1, 1)
# test canonical form
# see https://www.dartmouth.edu/~chance/teaching_aids/books_articles/probability_book/Chapter11.pdf
# example 11.13
T = Matrix([[1, 0, 0, 0, 0], [S(1) / 2, 0, S(1) / 2, 0, 0],
[0, S(1) / 2, 0, S(1) / 2, 0], [0, 0,
S(1) / 2, 0,
S(1) / 2], [0, 0, 0, 0,
S(1)]])
DW = DiscreteMarkovChain('DW', [0, 1, 2, 3, 4], T)
states, A, B, C = DW.decompose()
assert states == [0, 4, 1, 2, 3]
assert A == Matrix([[1, 0], [0, 1]])
assert B == Matrix([[S(1) / 2, 0], [0, 0], [0, S(1) / 2]])
assert C == Matrix([[0, S(1) / 2, 0], [S(1) / 2, 0, S(1) / 2],
[0, S(1) / 2, 0]])
states, new_matrix = DW.canonical_form()
assert states == [0, 4, 1, 2, 3]
assert new_matrix == Matrix([[1, 0, 0, 0, 0], [0, 1, 0, 0, 0],
[S(1) / 2, 0, 0, S(1) / 2, 0],
[0, 0, S(1) / 2, 0,
S(1) / 2], [0, S(1) / 2, 0,
S(1) / 2, 0]])
# test regular and ergodic
# https://www.dartmouth.edu/~chance/teaching_aids/books_articles/probability_book/Chapter11.pdf
T = Matrix([[0, 4, 0, 0, 0], [1, 0, 3, 0, 0], [0, 2, 0, 2, 0],
[0, 0, 3, 0, 1], [0, 0, 0, 4, 0]]) / 4
X = DiscreteMarkovChain('X', trans_probs=T)
assert not X.is_regular()
assert X.is_ergodic()
T = Matrix([[0, 1], [1, 0]])
X = DiscreteMarkovChain('X', trans_probs=T)
assert not X.is_regular()
assert X.is_ergodic()
# http://www.math.wisc.edu/~valko/courses/331/MC2.pdf
T = Matrix([[2, 1, 1], [2, 0, 2], [1, 1, 2]]) / 4
X = DiscreteMarkovChain('X', trans_probs=T)
assert X.is_regular()
assert X.is_ergodic()
# https://docs.ufpr.br/~lucambio/CE222/1S2014/Kemeny-Snell1976.pdf
T = Matrix([[1, 1], [1, 1]]) / 2
X = DiscreteMarkovChain('X', trans_probs=T)
assert X.is_regular()
assert X.is_ergodic()
# test is_absorbing_chain
T = Matrix([[0, 1, 0], [1, 0, 0], [0, 0, 1]])
X = DiscreteMarkovChain('X', trans_probs=T)
assert not X.is_absorbing_chain()
# https://en.wikipedia.org/wiki/Absorbing_Markov_chain
T = Matrix([[1, 1, 0, 0], [0, 1, 1, 0], [1, 0, 0, 1], [0, 0, 0, 2]]) / 2
X = DiscreteMarkovChain('X', trans_probs=T)
assert X.is_absorbing_chain()
T = Matrix([[2, 0, 0, 0, 0], [1, 0, 1, 0, 0], [0, 1, 0, 1, 0],
[0, 0, 1, 0, 1], [0, 0, 0, 0, 2]]) / 2
X = DiscreteMarkovChain('X', trans_probs=T)
assert X.is_absorbing_chain()
# test custom state space
Y10 = DiscreteMarkovChain('Y', [1, 2, 3], TO2)
tuples = Y10.communication_classes()
classes, recurrence, periods = list(zip(*tuples))
assert classes == ([1], [2, 3])
assert recurrence == (True, False)
assert periods == (1, 1)
assert Y10.canonical_form() == ([1, 2, 3], TO2)
assert Y10.decompose() == ([1, 2, 3], TO2[0:1, 0:1], TO2[1:3,
0:1], TO2[1:3,
1:3])
# testing miscellaneous queries
T = Matrix([[S.Half, Rational(1, 4),
Rational(1, 4)], [Rational(1, 3), 0,
Rational(2, 3)], [S.Half, S.Half, 0]])
X = DiscreteMarkovChain('X', [0, 1, 2], T)
assert P(
Eq(X[1], 2) & Eq(X[2], 1) & Eq(X[3], 0),
Eq(P(Eq(X[1], 0)), Rational(1, 4))
& Eq(P(Eq(X[1], 1)), Rational(1, 4))) == Rational(1, 12)
assert P(Eq(X[2], 1) | Eq(X[2], 2), Eq(X[1], 1)) == Rational(2, 3)
assert P(Eq(X[2], 1) & Eq(X[2], 2), Eq(X[1], 1)) is S.Zero
assert P(Ne(X[2], 2), Eq(X[1], 1)) == Rational(1, 3)
assert E(X[1]**2, Eq(X[0], 1)) == Rational(8, 3)
assert variance(X[1], Eq(X[0], 1)) == Rational(8, 9)
raises(ValueError, lambda: E(X[1], Eq(X[2], 1)))
raises(ValueError, lambda: DiscreteMarkovChain('X', [0, 1], T))
# testing miscellaneous queries with different state space
X = DiscreteMarkovChain('X', ['A', 'B', 'C'], T)
assert P(
Eq(X[1], 2) & Eq(X[2], 1) & Eq(X[3], 0),
Eq(P(Eq(X[1], 0)), Rational(1, 4))
& Eq(P(Eq(X[1], 1)), Rational(1, 4))) == Rational(1, 12)
assert P(Eq(X[2], 1) | Eq(X[2], 2), Eq(X[1], 1)) == Rational(2, 3)
assert P(Eq(X[2], 1) & Eq(X[2], 2), Eq(X[1], 1)) is S.Zero
assert P(Ne(X[2], 2), Eq(X[1], 1)) == Rational(1, 3)
a = X.state_space.args[0]
c = X.state_space.args[2]
assert (E(X[1]**2, Eq(X[0], 1)) -
(a**2 / 3 + 2 * c**2 / 3)).simplify() == 0
assert (variance(X[1], Eq(X[0], 1)) -
(2 * (-a / 3 + c / 3)**2 / 3 +
(2 * a / 3 - 2 * c / 3)**2 / 3)).simplify() == 0
raises(ValueError, lambda: E(X[1], Eq(X[2], 1)))
#testing queries with multiple RandomIndexedSymbols
T = Matrix([[Rational(5, 10),
Rational(3, 10),
Rational(2, 10)],
[Rational(2, 10),
Rational(7, 10),
Rational(1, 10)],
[Rational(3, 10),
Rational(3, 10),
Rational(4, 10)]])
Y = DiscreteMarkovChain("Y", [0, 1, 2], T)
assert P(Eq(Y[7], Y[5]), Eq(Y[2], 0)).round(5) == Float(0.44428, 5)
assert P(Gt(Y[3], Y[1]), Eq(Y[0], 0)).round(2) == Float(0.36, 2)
assert P(Le(Y[5], Y[10]), Eq(Y[4], 2)).round(6) == Float(0.583120, 6)
assert Float(P(Eq(Y[10], Y[5]), Eq(Y[4], 1)),
14) == Float(1 - P(Ne(Y[10], Y[5]), Eq(Y[4], 1)), 14)
assert Float(P(Gt(Y[8], Y[9]), Eq(Y[3], 2)),
14) == Float(1 - P(Le(Y[8], Y[9]), Eq(Y[3], 2)), 14)
assert Float(P(Lt(Y[1], Y[4]), Eq(Y[0], 0)),
14) == Float(1 - P(Ge(Y[1], Y[4]), Eq(Y[0], 0)), 14)
assert P(Eq(Y[5], Y[10]), Eq(Y[2], 1)) == P(Eq(Y[10], Y[5]), Eq(Y[2], 1))
assert P(Gt(Y[1], Y[2]), Eq(Y[0], 1)) == P(Lt(Y[2], Y[1]), Eq(Y[0], 1))
assert P(Ge(Y[7], Y[6]), Eq(Y[4], 1)) == P(Le(Y[6], Y[7]), Eq(Y[4], 1))
#test symbolic queries
a, b, c, d = symbols('a b c d')
T = Matrix([[Rational(1, 10),
Rational(4, 10),
Rational(5, 10)],
[Rational(3, 10),
Rational(4, 10),
Rational(3, 10)],
[Rational(7, 10),
Rational(2, 10),
Rational(1, 10)]])
Y = DiscreteMarkovChain("Y", [0, 1, 2], T)
query = P(Eq(Y[a], b), Eq(Y[c], d))
assert query.subs({
a: 10,
b: 2,
c: 5,
d: 1
}).evalf().round(4) == P(Eq(Y[10], 2), Eq(Y[5], 1)).round(4)
assert query.subs({
a: 15,
b: 0,
c: 10,
d: 1
}).evalf().round(4) == P(Eq(Y[15], 0), Eq(Y[10], 1)).round(4)
query_gt = P(Gt(Y[a], b), Eq(Y[c], d))
query_le = P(Le(Y[a], b), Eq(Y[c], d))
assert query_gt.subs({
a: 5,
b: 2,
c: 1,
d: 0
}).evalf() + query_le.subs({
a: 5,
b: 2,
c: 1,
d: 0
}).evalf() == 1
query_ge = P(Ge(Y[a], b), Eq(Y[c], d))
query_lt = P(Lt(Y[a], b), Eq(Y[c], d))
assert query_ge.subs({
a: 4,
b: 1,
c: 0,
d: 2
}).evalf() + query_lt.subs({
a: 4,
b: 1,
c: 0,
d: 2
}).evalf() == 1
#test issue 20078
assert (2 * Y[1] + 3 * Y[1]).simplify() == 5 * Y[1]
assert (2 * Y[1] - 3 * Y[1]).simplify() == -Y[1]
assert (2 * (0.25 * Y[1])).simplify() == 0.5 * Y[1]
assert ((2 * Y[1]) * (0.25 * Y[1])).simplify() == 0.5 * Y[1]**2
assert (Y[1]**2 + Y[1]**3).simplify() == (Y[1] + 1) * Y[1]**2
def test_yule_simon():
rho = S(3)
x = YuleSimon('x', rho)
assert simplify(E(x)) == rho / (rho - 1)
assert simplify(variance(x)) == rho**2 / ((rho - 1)**2 * (rho - 2))
assert isinstance(E(x, evaluate=False), Sum)
def test_weibull():
a, b = symbols('a b', positive=True)
X = Weibull('x', a, b)
assert simplify(E(X)) == simplify(a * gamma(1 + 1 / b))
assert simplify(variance(X)) == simplify(a**2 * gamma(1 + 2 / b) - E(X)**2)
