def domain(self):
from sympy import Interval, Range, oo, CartesianSpace
shape = self.shape
if self.is_integer:
if self.is_positive:
interval = Range(1, oo)
elif self.is_nonnegative:
interval = Range(0, oo)
elif self.is_negative:
interval = Range(-oo, 0)
elif self.is_nonpositive:
interval = Range(-oo, 1)
else:
interval = Range(-oo, oo)
elif self.is_extended_real:
if self.is_positive:
interval = Interval(0, oo, left_open=True)
elif self.is_nonnegative:
interval = Interval(0, oo)
elif self.is_negative:
interval = Interval(-oo, 0, right_open=True)
elif self.is_nonpositive:
interval = Interval(-oo, 0)
else:
interval = Interval(-oo, oo)
else:
interval = S.Complexes
return CartesianSpace(interval, *shape)
def test_arithmetic_sums():
assert summation(1, (n, a, b)) == b - a + 1
assert Sum(S.NaN, (n, a, b)) is S.NaN
assert Sum(x, (n, a, a)).doit() == x
assert Sum(x, (x, a, a)).doit() == a
assert Sum(x, (n, 1, a)).doit() == a*x
assert Sum(x, (x, Range(1, 11))).doit() == 55
assert Sum(x, (x, Range(1, 11, 2))).doit() == 25
assert Sum(x, (x, Range(1, 10, 2))) == Sum(x, (x, Range(9, 0, -2)))
lo, hi = 1, 2
s1 = Sum(n, (n, lo, hi))
s2 = Sum(n, (n, hi, lo))
assert s1 != s2
assert s1.doit() == 3 and s2.doit() == 0
lo, hi = x, x + 1
s1 = Sum(n, (n, lo, hi))
s2 = Sum(n, (n, hi, lo))
assert s1 != s2
assert s1.doit() == 2*x + 1 and s2.doit() == 0
assert Sum(Integral(x, (x, 1, y)) + x, (x, 1, 2)).doit() == \
y**2 + 2
assert summation(1, (n, 1, 10)) == 10
assert summation(2*n, (n, 0, 10**10)) == 100000000010000000000
assert summation(4*n*m, (n, a, 1), (m, 1, d)).expand() == \
2*d + 2*d**2 + a*d + a*d**2 - d*a**2 - a**2*d**2
assert summation(cos(n), (n, -2, 1)) == cos(-2) + cos(-1) + cos(0) + cos(1)
assert summation(cos(n), (n, x, x + 2)) == cos(x) + cos(x + 1) + cos(x + 2)
assert isinstance(summation(cos(n), (n, x, x + S.Half)), Sum)
assert summation(k, (k, 0, oo)) == oo
assert summation(k, (k, Range(1, 11))) == 55
def test_issue_12533():
d = IndexedBase('d')
assert IndexedBase(range(5)) == Range(0, 5, 1)
assert d[0].subs(Symbol("d"), range(5)) == 0
assert d[0].subs(d, range(5)) == 0
assert d[1].subs(d, range(5)) == 1
assert Indexed(Range(5), 2) == 2
def generate_solver(self):
""" Generates solver of the wave equation in C code """
declarations = self.declarate_variables_same_type(['t', 'x', 'y'],
integer)
t, x, y, z = symbols('t, x, y, z', integer=True)
dt, h_x, h_y, h_z = symbols('dt, h_x, h_y, h_z', real=True)
time_m, time_M, x_m, x_M, y_m, y_M, z_m, z_M = symbols(
'time_m, time_M, x_m, x_M, y_m, y_M, z_m, z_M', integer=True)
u = Element('u', [3, 21, 21])
u = Declaration(u)
u.variable.type = real
body = self.get_stencil_code()
if self.dimensions == 3:
for_z = For(z, Range(z_m, z_M, 1), [body])
for_y = For(y, Range(y_m, y_M, 1), [for_z])
elif self.dimensions == 2:
for_y = For(y, Range(y_m, y_M, 1), [body])
for_x = For(x, Range(x_m, x_M, 1), [for_y])
for_time = For(t, Range(time_m, time_M, 1), [for_x])
code = CodeBlock(*declarations, for_time)
solver_prototype = FunctionPrototype(
integer, 'Solver',
[dt, h_x, h_y, u, time_M, time_m, x_M, x_m, y_M, y_m])
solver = FunctionDefinition.from_FunctionPrototype(
solver_prototype, code)
return ccode(solver)
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 set(self):
if not isinstance(self.n, Integer):
i = Symbol('i', integer=True, positive=True)
return Product(Intersection(S.Naturals0, Interval(0, self.n//i)),
(i, 1, self.n))
prod_set = Range(0, self.n + 1)
for i in range(2, self.n + 1):
prod_set *= Range(0, self.n//i + 1)
return prod_set
def test_Range():
# Only works in Python 3 where range returns a range type
if PY3:
builtin_range = range
else:
builtin_range = xrange
assert sympify(builtin_range(10)) == Range(10)
assert _sympify(builtin_range(10)) == Range(10)
def is_lower(self):
"""Check if matrix is a lower triangular matrix. True can be returned
even if the matrix is not square.
Examples
========
>>> from sympy import Matrix
>>> m = Matrix(2, 2, [1, 0, 0, 1])
>>> m
Matrix([
[1, 0],
[0, 1]])
>>> m.is_lower
True
>>> m = Matrix(4, 3, [0, 0, 0, 2, 0, 0, 1, 4 , 0, 6, 6, 5])
>>> m
Matrix([
[0, 0, 0],
[2, 0, 0],
[1, 4, 0],
[6, 6, 5]])
>>> m.is_lower
True
>>> from sympy.abc import x, y
>>> m = Matrix(2, 2, [x**2 + y, y**2 + x, 0, x + y])
>>> m
Matrix([
[x**2 + y, x + y**2],
[ 0, x + y]])
>>> m.is_lower
False
See Also
========
is_upper
is_diagonal
is_lower_hessenberg
"""
from sympy import Range, Min
i = self.generate_var(domain=Range(0, Min(self.rows, self.cols - 1)))
j = i.generate_var(free_symbols=self.free_symbols,
domain=Range(i + 1, self.cols))
assert i < j
return self[i, j] == 0
def is_upper(self):
"""Check if matrix is an upper triangular matrix. True can be returned
even if the matrix is not square.
Examples
========
>>> from sympy import Matrix
>>> m = Matrix(2, 2, [1, 0, 0, 1])
>>> m
Matrix([
[1, 0],
[0, 1]])
>>> m.is_upper
True
>>> m = Matrix(4, 3, [5, 1, 9, 0, 4 , 6, 0, 0, 5, 0, 0, 0])
>>> m
Matrix([
[5, 1, 9],
[0, 4, 6],
[0, 0, 5],
[0, 0, 0]])
>>> m.is_upper
True
>>> m = Matrix(2, 3, [4, 2, 5, 6, 1, 1])
>>> m
Matrix([
[4, 2, 5],
[6, 1, 1]])
>>> m.is_upper
False
See Also
========
is_lower
is_diagonal
is_upper_hessenberg
"""
from sympy import Range, Min
j = self.generate_var(domain=Range(0, Min(self.cols, self.rows - 1)))
i = j.generate_var(free_symbols=self.free_symbols,
domain=Range(j + 1, self.rows))
assert i > j
return self[i, j] == 0
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 domain_nonzero(self, x):
from sympy import Range, oo
n, k = self.args
p = k.as_poly(x)
if p.degree() == 1:
alpha = p.coeff_monomial(x)
beta = p.coeff_monomial(S.One)
if alpha.is_number:
if alpha == S.One:
# consider binomial(2*n, n), x = n
if not n._has(x):
return Range(-beta, n - beta + 1)
elif alpha == S.NegativeOne:
return Range(beta - n, beta + 1)
return Range(-oo, oo)
def test_issue_17256():
from sympy import Symbol, Range, Sum
x = Symbol('x')
s1 = Sum(x + 1, (x, 1, 9))
s2 = Sum(x + 1, (x, Range(1, 10)))
a = Symbol('a')
r1 = s1.xreplace({x: a})
r2 = s2.xreplace({x: a})
r1.doit() == r2.doit()
s1 = Sum(x + 1, (x, 0, 9))
s2 = Sum(x + 1, (x, Range(10)))
a = Symbol('a')
r1 = s1.xreplace({x: a})
r2 = s2.xreplace({x: a})
assert r1 == r2
def readIn(path, upToColumn, changeOrigin):
if changeOrigin:
xOrigin = 56502
yOrigin = 3754100
zOrigin = -100
else:
xOrigin = 0
yOrigin = 0
zOrigin = 0
with open(path, 'rb') as csvfile:
reader = csv.reader(csvfile, delimiter=' ', quotechar='|')
data = []
for row in reader:
data.append(row)
MAT = np.matrix(data, dtype=float_)
safe = 0
while (safe < 100):
try:
MAT = np.delete(MAT, upToColumn, 1)
except:
print "outa while loop"
break
safe += 1
for i in Range((MAT.shape[0])):
MAT[i, 0] = MAT[i, 0] + xOrigin
MAT[i, 1] = MAT[i, 1] + yOrigin
MAT[i, 2] = MAT[i, 2] + zOrigin
# pprint (MAT)
print MAT
return MAT
def test_fcode_For():
x, y = symbols('x y')
f = For(x, Range(0, 10, 2), [Assignment(y, x * y)])
sol = fcode(f)
assert sol == (" do x = 0, 10, 2\n"
" y = x*y\n"
" end do")
def BandPart(x, *limits):
(num_lower, ), (num_upper, ) = limits
m, n = x.shape
excludes = num_lower.free_symbols | num_upper.free_symbols
i = x.generate_var(excludes=excludes, var='i', integer=True)
excludes.add(i)
j = x.generate_var(excludes=excludes, var='j', integer=True)
return x * Lamda[j:n, i:m](Bool(
Contains(i - j, Range(-num_upper, num_lower + 1))))
def test_union_iter():
# Use Range because it is ordered
u = Union(Range(3), Range(5), Range(3), evaluate=False)
# Round robin
assert list(u) == [0, 0, 0, 1, 1, 1, 2, 2, 2, 3, 4]
from timescale.timescale import Timescale_new as tsc
from sympy import (Set, S, Rational, Lambda, Symbol, Range, oo, Interval,
latex)
from sympy.sets.sets import FiniteSet, Interval
from sympy.sets.fancysets import ImageSet
t = Symbol('t')
T = Symbol(r'\mathbb{T}')
# Iterables
print(latex(T))
TS0 = tsc(S.EmptySet, latex(T) + '=' + latex(S.EmptySet))
TS1 = tsc(Range(-2, 1), latex(T) + '=' + latex(Range(-2, 1)))
TS2 = tsc(Range(-oo, 5, 5), latex(T) + '=' + latex(Range(-oo, 5, 5)))
TS3 = tsc(Range(0, oo, 7), latex(T) + '=' + latex(Range(0, oo, 7)))
TS4 = tsc(S.Naturals, latex(T) + '=' + latex(S.Naturals))
TS5 = tsc(S.Naturals0, latex(T) + '=' + latex(S.Naturals0))
TS6 = tsc(S.Integers, latex(T) + '=' + latex(S.Integers))
TS7 = tsc(ImageSet(Lambda(t, -t), S.Naturals),
latex(T) + '=' + latex(ImageSet(Lambda(t, -t), S.Naturals)))
# TODO: Create an instance of rational numbers in some subset like Interval(0,1s).
# TODO: Recorrer las fracciones de la siguiente manera:
# https://mathoverflow.net/questions/200656/is-there-a-natural-bijection-from-mathbbn-to-mathbbq
# Non-interables
TS9 = tsc(S.Reals, latex(T) + '=' + latex(S.Reals)) # Means Interval(-oo,+oo).
TS10 = tsc(S.UniversalSet, latex(T) + '=' + latex(S.UniversalSet))
# print(list(TS1.ts))
# TODO: Implementar raise
from sympy import Symbol, pprint
from sympy.abc import i, j, k
from sympy import Range
n = Symbol('n', integer=True)
r = Range(n, n + 20, 3)
r.inf
pprint(r)
def test_latex_Range():
assert latex(Range(1, 51)) == \
r'\left\{1, 2, \ldots, 50\right\}'
assert latex(Range(1, 4)) == r'\left\{1, 2, 3\right\}'
def test_Range():
# Only works in Python 3 where range returns a range type
assert sympify(range(10)) == Range(10)
assert _sympify(range(10)) == Range(10)
def test_For():
f = For(n, Range(0, 3),
(Assignment(A[n, 0], x + n), aug_assign(x, '+', y)))
f = For(n, (1, 2, 3, 4, 5), (Assignment(A[n, 0], x + n), ))
assert f.func(*f.args) == f
raises(TypeError, lambda: For(n, x, (x + y, )))
def Z(self):
z = 0
for i in Range(1, round(self.k / self.R)):
z += (1 / i)
z += log(self.R / self.delta)
return 1 + z * self.R / self.k
def set(self):
return set(list(Range(1, self.k + 1)))
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
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 domain_nonzero(self, x):
from sympy import Interval, Range, oo
if x.is_integer:
return Range(-oo, oo)
else:
return Interval(-oo, oo)
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 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
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
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([[]])
assert X.communication_classes() == []
assert X.canonical_form() == ([], Matrix([[]]))
assert X.decompose() == ([], Matrix([[]]), Matrix([[]]), 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 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 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(0.739072, 6)
assert Float(P(Eq(Y[500], Y[240]), Eq(Y[120], 1)),
14) == Float(1 - P(Ne(Y[500], Y[240]), Eq(Y[120], 1)), 14)
assert Float(P(Gt(Y[350], Y[100]), Eq(Y[75], 2)),
14) == Float(1 - P(Le(Y[350], Y[100]), Eq(Y[75], 2)), 14)
assert Float(P(Lt(Y[400], Y[210]), Eq(Y[161], 0)),
14) == Float(1 - P(Ge(Y[400], Y[210]), Eq(Y[161], 0)), 14)
def set(self):
return Range(0, S.Infinity)**len(self.p)
def set(self):
prod_set = Range(0, self.n//1 + 1)
for i in range(2, self.n + 1):
prod_set *= Range(0, self.n//i + 1)
return prod_set
def set(self):
return Range(0, self.n)**len(self.p)
from sympy import Symbol, pprint
from sympy.abc import i, j, k
from sympy import Range
n = Symbol('n', integer=True)
r = Range(n, n * n, 3)
r.inf
pprint(r)
