def test_variance_prop_with_covar():
x, y, z = symbols('x y z')
phi, t = consts = symbols('phi t')
a = RandomSymbol(x)
var_x = Variance(a)
b = RandomSymbol(y)
var_y = Variance(b)
c = RandomSymbol(z)
var_z = Variance(c)
covar_x_y = Covariance(a, b)
covar_x_z = Covariance(a, c)
covar_y_z = Covariance(b, c)
cases = {
x + y: var_x + var_y + 2*covar_x_y,
a + y: var_x + var_y + 2*covar_x_y,
x + y + z: var_x + var_y + var_z + \
2*covar_x_y + 2*covar_x_z + 2*covar_y_z,
2*x: 4*var_x,
x*y: var_x*y**2 + var_y*x**2 + 2*covar_x_y/(x*y),
1/x: var_x/x**4,
exp(x): var_x*exp(2*x),
exp(2*x): 4*var_x*exp(4*x),
exp(-x*t): t**2*var_x*exp(-2*t*x),
}
for inp, out in cases.items():
obs = variance_prop(inp, consts=consts, include_covar=True)
assert out == obs
def covariance(X, Y, condition=None, **kwargs):
"""
Covariance of two random expressions
The expectation that the two variables will rise and fall together
Covariance(X,Y) = E( (X-E(X)) * (Y-E(Y)) )
Examples
========
>>> from sympy.stats import Exponential, covariance
>>> from sympy import Symbol
>>> rate = Symbol('lambda', positive=True, real=True, finite=True)
>>> X = Exponential('X', rate)
>>> Y = Exponential('Y', rate)
>>> covariance(X, X)
lambda**(-2)
>>> covariance(X, Y)
0
>>> covariance(X, Y + rate*X)
1/lambda
"""
if (is_random(X) and pspace(X) == PSpace()) or (is_random(Y)
and pspace(Y) == PSpace()):
from sympy.stats.symbolic_probability import Covariance
return Covariance(X, Y, condition)
return expectation((X - expectation(X, condition, **kwargs)) *
(Y - expectation(Y, condition, **kwargs)), condition,
**kwargs)
def expand(self, **hints):
arg = self.args[0]
condition = self._condition
if not is_random(arg):
return ZeroMatrix(*self.shape)
if isinstance(arg, RandomSymbol):
return self
elif isinstance(arg, Add):
rv = []
for a in arg.args:
if is_random(a):
rv.append(a)
variances = Add(
*map(lambda xv: Variance(xv, condition).expand(), rv))
map_to_covar = lambda x: 2 * Covariance(*x, condition=condition
).expand()
covariances = Add(
*map(map_to_covar, itertools.combinations(rv, 2)))
return variances + covariances
elif isinstance(arg, (Mul, MatMul)):
nonrv = []
rv = []
for a in arg.args:
if is_random(a):
rv.append(a)
else:
nonrv.append(a)
if len(rv) == 0:
return ZeroMatrix(*self.shape)
# Avoid possible infinite loops with MatMul:
if len(nonrv) == 0:
return self
# Variance of many multiple matrix products is not implemented:
if len(rv) > 1:
return self
return Mul.fromiter(nonrv) * Variance(
Mul.fromiter(rv),
condition) * (Mul.fromiter(nonrv)).transpose()
# this expression contains a RandomSymbol somehow:
return self
def variance_prop(expr, consts=(), include_covar=False):
r"""Symbolically propagates variance (`\sigma^2`) for expressions.
This is computed as as seen in [1]_.
Parameters
==========
expr : Expr
A sympy expression to compute the variance for.
consts : sequence of Symbols, optional
Represents symbols that are known constants in the expr,
and thus have zero variance. All symbols not in consts are
assumed to be variant.
include_covar : bool, optional
Flag for whether or not to include covariances, default=False.
Returns
=======
var_expr : Expr
An expression for the total variance of the expr.
The variance for the original symbols (e.g. x) are represented
via instance of the Variance symbol (e.g. Variance(x)).
Examples
========
>>> from sympy import symbols, exp
>>> from sympy.stats.error_prop import variance_prop
>>> x, y = symbols('x y')
>>> variance_prop(x + y)
Variance(x) + Variance(y)
>>> variance_prop(x * y)
x**2*Variance(y) + y**2*Variance(x)
>>> variance_prop(exp(2*x))
4*exp(4*x)*Variance(x)
References
==========
.. [1] https://en.wikipedia.org/wiki/Propagation_of_uncertainty
"""
args = expr.args
if len(args) == 0:
if expr in consts:
return S(0)
elif isinstance(expr, RandomSymbol):
return Variance(expr).doit()
elif isinstance(expr, Symbol):
return Variance(RandomSymbol(expr)).doit()
else:
return S(0)
nargs = len(args)
var_args = list(
map(variance_prop, args, repeat(consts, nargs),
repeat(include_covar, nargs)))
if isinstance(expr, Add):
var_expr = Add(*var_args)
if include_covar:
terms = [2 * Covariance(_arg0_or_var(x), _arg0_or_var(y)).doit() \
for x, y in combinations(var_args, 2)]
var_expr += Add(*terms)
elif isinstance(expr, Mul):
terms = [v / a**2 for a, v in zip(args, var_args)]
var_expr = simplify(expr**2 * Add(*terms))
if include_covar:
terms = [2*Covariance(_arg0_or_var(x), _arg0_or_var(y)).doit()/(a*b) \
for (a, b), (x, y) in zip(combinations(args, 2),
combinations(var_args, 2))]
var_expr += Add(*terms)
elif isinstance(expr, Pow):
b = args[1]
v = var_args[0] * (expr * b / args[0])**2
var_expr = simplify(v)
elif isinstance(expr, exp):
var_expr = simplify(var_args[0] * expr**2)
else:
# unknown how to proceed, return variance of whole expr.
var_expr = Variance(expr)
return var_expr
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)
