def test_contraction_permutation_mix():
Me = M.subs(k, 3).as_explicit()
Ne = N.subs(k, 3).as_explicit()
cg1 = CodegenArrayContraction(
CodegenArrayPermuteDims(CodegenArrayTensorProduct(M, N),
Permutation([0, 2, 1, 3])), (2, 3))
cg2 = CodegenArrayContraction(CodegenArrayTensorProduct(M, N), (1, 3))
assert cg1 == cg2
assert recognize_matrix_expression(cg2) == M * N.T
cge1 = tensorcontraction(
permutedims(tensorproduct(Me, Ne), Permutation([0, 2, 1, 3])), (2, 3))
cge2 = tensorcontraction(tensorproduct(Me, Ne), (1, 3))
assert cge1 == cge2
cg1 = CodegenArrayPermuteDims(CodegenArrayTensorProduct(M, N),
Permutation([0, 1, 3, 2]))
cg2 = CodegenArrayTensorProduct(
M, CodegenArrayPermuteDims(N, Permutation([1, 0])))
assert cg1 == cg2
assert recognize_matrix_expression(cg1) == CodegenArrayTensorProduct(
M, N.T)
assert recognize_matrix_expression(cg2) == CodegenArrayTensorProduct(
M, N.T)
cg1 = CodegenArrayContraction(
CodegenArrayPermuteDims(CodegenArrayTensorProduct(M, N, P, Q),
Permutation([0, 2, 3, 1, 4, 5, 7, 6])), (1, 2),
(3, 5))
cg2 = CodegenArrayContraction(
CodegenArrayTensorProduct(
M, N, P, CodegenArrayPermuteDims(Q, Permutation([1, 0]))), (1, 5),
(2, 3))
assert cg1 == cg2
assert recognize_matrix_expression(cg1) == CodegenArrayTensorProduct(
M * P.T * Trace(N), Q.T)
assert recognize_matrix_expression(cg2) == CodegenArrayTensorProduct(
M * P.T * Trace(N), Q.T)
cg1 = CodegenArrayContraction(
CodegenArrayPermuteDims(CodegenArrayTensorProduct(M, N, P, Q),
Permutation([1, 0, 4, 6, 2, 7, 5, 3])), (0, 1),
(2, 6), (3, 7))
cg2 = CodegenArrayPermuteDims(
CodegenArrayContraction(CodegenArrayTensorProduct(M, P, Q, N), (0, 1),
(2, 3), (4, 7)), [1, 0])
assert cg1 == cg2
cg1 = CodegenArrayContraction(
CodegenArrayPermuteDims(CodegenArrayTensorProduct(M, N, P, Q),
Permutation([1, 0, 4, 6, 7, 2, 5, 3])), (0, 1),
(2, 6), (3, 7))
cg2 = CodegenArrayPermuteDims(
CodegenArrayContraction(
CodegenArrayTensorProduct(CodegenArrayPermuteDims(M, [1, 0]), N, P,
Q), (0, 1), (3, 6), (4, 5)),
Permutation([1, 0]))
assert cg1 == cg2
def compute_Gamma(g_deriv, gUP):
"""Return Christoffel symbols
"""
g_derivT = Array([(g_deriv[:, :, i]).transpose() for i in range(4)])
gUgd = tensorproduct(gUP, g_deriv)
gUgdT = tensorproduct(gUP, g_derivT)
return 1 / 2 * (tensorcontraction(gUgd, (1, 3)) + tensorcontraction(
gUgdT, (1, 3)) - tensorcontraction(gUgd, (1, 2)))
def test_arrayexpr_contraction_permutation_mix():
Me = M.subs(k, 3).as_explicit()
Ne = N.subs(k, 3).as_explicit()
cg1 = ArrayContraction(PermuteDims(ArrayTensorProduct(M, N), Permutation([0, 2, 1, 3])), (2, 3))
cg2 = ArrayContraction(ArrayTensorProduct(M, N), (1, 3))
assert cg1 == cg2
cge1 = tensorcontraction(permutedims(tensorproduct(Me, Ne), Permutation([0, 2, 1, 3])), (2, 3))
cge2 = tensorcontraction(tensorproduct(Me, Ne), (1, 3))
assert cge1 == cge2
cg1 = PermuteDims(ArrayTensorProduct(M, N), Permutation([0, 1, 3, 2]))
cg2 = ArrayTensorProduct(M, PermuteDims(N, Permutation([1, 0])))
assert cg1 == cg2
cg1 = ArrayContraction(
PermuteDims(
ArrayTensorProduct(M, N, P, Q), Permutation([0, 2, 3, 1, 4, 5, 7, 6])),
(1, 2), (3, 5)
)
cg2 = ArrayContraction(
ArrayTensorProduct(M, N, P, PermuteDims(Q, Permutation([1, 0]))),
(1, 5), (2, 3)
)
assert cg1 == cg2
cg1 = ArrayContraction(
PermuteDims(
ArrayTensorProduct(M, N, P, Q), Permutation([1, 0, 4, 6, 2, 7, 5, 3])),
(0, 1), (2, 6), (3, 7)
)
cg2 = PermuteDims(
ArrayContraction(
ArrayTensorProduct(M, P, Q, N),
(0, 1), (2, 3), (4, 7)),
[1, 0]
)
assert cg1 == cg2
cg1 = ArrayContraction(
PermuteDims(
ArrayTensorProduct(M, N, P, Q), Permutation([1, 0, 4, 6, 7, 2, 5, 3])),
(0, 1), (2, 6), (3, 7)
)
cg2 = PermuteDims(
ArrayContraction(
ArrayTensorProduct(PermuteDims(M, [1, 0]), N, P, Q),
(0, 1), (3, 6), (4, 5)
),
Permutation([1, 0])
)
assert cg1 == cg2
def init_riemann(self):
""" Riemann tensor of the metric, which is a 4-index tensor. """
riemann = sp.MutableDenseNDimArray(np.zeros((self.dim,)*4)) # Inizializing 4-index tensor
dchr = sp.MutableDenseNDimArray(np.zeros((self.dim,)*4)) # Derivative of Christoffel symbols
if isinstance(self.chr, type(None)):
self.init_chr() # Initialize Christoffel symbols (if not already done)
for mu in range(self.dim):
dchr[:,:,:,mu] = sp.diff(self.chr, self.variables[mu])
for sigma in range(self.dim):
for rho in range(self.dim):
riemann[rho,sigma,:,:] = dchr[rho,:,sigma,:].transpose() - dchr[rho,:,sigma,:] \
+ sp.tensorcontraction(sp.tensorproduct(self.chr[rho,:,:], self.chr[:,:,sigma]),(1,2)) \
- (sp.tensorcontraction(sp.tensorproduct(self.chr[rho,:,:], self.chr[:,:,sigma]),(1,2))).transpose()
self.riemann = sp.simplify(riemann)
def from_riemann(cls, riemann, parent_metric=None):
"""
Get Ricci Tensor calculated from Riemann Tensor
Parameters
----------
riemann : ~einsteinpy.symbolic.riemann.RiemannCurvatureTensor
Riemann Tensor
parent_metric : ~einsteinpy.symbolic.metric.MetricTensor or None
Corresponding Metric for the Ricci Tensor.
None if it should inherit the Parent Metric of Riemann Tensor.
Defaults to None.
"""
if not riemann.config == "ulll":
riemann = riemann.change_config(newconfig="ulll",
metric=parent_metric)
if parent_metric is None:
parent_metric = riemann.parent_metric
return cls(
simplify_sympy_array(
sympy.tensorcontraction(riemann.tensor(), (0, 2))),
riemann.syms,
config="ll",
parent_metric=parent_metric,
)
def test_EinsteinTensor_trace_negetive_of_Ricci_Scalar_in_4D(metric):
# https://en.wikipedia.org/wiki/Einstein_tensor#Trace
G1 = EinsteinTensor.from_metric(metric)
G2 = G1.change_config("ul")
val1 = simplify(tensorcontraction(G2.tensor(), (0, 1)))
val2 = RicciScalar.from_metric(metric).expr
assert simplify(val1 + val2) == 0
def from_riccitensor(cls, riccitensor, parent_metric=None):
"""
Get Ricci Scalar calculated from Ricci Tensor
Parameters
----------
riccitensor: ~einsteinpy.symbolic.metric.RicciTensor
Ricci Tensor
parent_metric : ~einsteinpy.symbolic.metric.MetricTensor or None
Corresponding Metric for the Ricci Scalar.
Defaults to None.
"""
if not riccitensor.config == "ul":
riccitensor = riccitensor.change_config(newconfig="ul",
metric=parent_metric)
if parent_metric is None:
parent_metric = riccitensor.parent_metric
ricci_scalar = tensorcontraction(riccitensor.tensor(), (0, 1))
return cls(
simplify_sympy_array(ricci_scalar),
riccitensor.syms,
parent_metric=parent_metric,
)
def partial_trace(dm, n_qubits, subsystem):
# This is the same calc. as for the tf_qc/qc.trace
# First we find the last consecutive block to trace away
block_end = subsystem[-1]
block_start = block_end
for idx in reversed(subsystem):
if block_start - idx <= 1:
block_start = idx
else:
break
n_static1 = block_start # First part of static qubits
n_static2 = (n_qubits - 1) - block_end # Second part of static qubits
n_static = n_static1 + n_static2
n_qubits2trace = block_end - block_start + 1 # Qubits to trace away
new_shape = [
2**n_static1, 2**n_qubits2trace, 2**n_static2, 2**n_static1,
2**n_qubits2trace, 2**n_static2
]
flat_dm = sp.flatten(dm)
new_dm = sp.NDimArray(flat_dm, shape=new_shape)
trace_result = tensorcontraction(new_dm, (1, 4))
reshaped_result = trace_result.reshape(2**n_static, 2**n_static)
# We must now recursively to the same to the lest of the subsystems
idx_of_start = subsystem.index(block_start)
new_subsystem = subsystem[:idx_of_start]
return partial_trace(reshaped_result, n_static, new_subsystem) if len(
new_subsystem) > 0 else reshaped_result.tomatrix()
def tensordot(a, b):
s1 = a.shape
s2 = b.shape
assert s1[-1] == s2[0]
s3 = s1[:-1] + s2[1:]
k = len(s1) - 1
return tensorcontraction(tensorproduct(a, b), (k, k + 1))
def test_array_as_explicit_call():
assert ZeroArray(3, 2, 4).as_explicit() == ImmutableDenseNDimArray.zeros(
3, 2, 4)
assert OneArray(3, 2, 4).as_explicit() == ImmutableDenseNDimArray(
[1 for i in range(3 * 2 * 4)]).reshape(3, 2, 4)
k = Symbol("k")
X = ArraySymbol("X", k, 3, 2)
raises(ValueError, lambda: X.as_explicit())
raises(ValueError, lambda: ZeroArray(k, 2, 3).as_explicit())
raises(ValueError, lambda: OneArray(2, k, 2).as_explicit())
A = ArraySymbol("A", 3, 3)
B = ArraySymbol("B", 3, 3)
texpr = tensorproduct(A, B)
assert isinstance(texpr, ArrayTensorProduct)
assert texpr.as_explicit() == tensorproduct(A.as_explicit(),
B.as_explicit())
texpr = tensorcontraction(A, (0, 1))
assert isinstance(texpr, ArrayContraction)
assert texpr.as_explicit() == A[0, 0] + A[1, 1] + A[2, 2]
texpr = tensordiagonal(A, (0, 1))
assert isinstance(texpr, ArrayDiagonal)
assert texpr.as_explicit() == ImmutableDenseNDimArray(
[A[0, 0], A[1, 1], A[2, 2]])
texpr = permutedims(A, [1, 0])
assert isinstance(texpr, PermuteDims)
assert texpr.as_explicit() == permutedims(A.as_explicit(), [1, 0])
def geodesic_ncurve(surface: ParametricSurface, ic_uv, ic_uv_t, t1=5, dt=0.05):
from sympy import lambdify
from scipy.integrate import ode as sciode
import numpy as np
from sympy import symbols, Function, Array, tensorproduct, tensorcontraction
t = symbols('t', real=True)
u = Function(surface.sym(0), real=True)(t)
v = Function(surface.sym(1), real=True)(t)
second_term_tensor = tensorproduct(
surface.christoffel_symbol.tensor().subs(
{surface.sym(0):u, surface.sym(1):v}),
Array([u, v]).diff(t),
Array([u, v]).diff(t))
second_term_tensor = tensorcontraction(second_term_tensor, (1, 3), (2, 4))
u_t = Function(str(u)+'^{\prime}', real=True)(t)
v_t = Function(str(v)+'^{\prime}', real=True)(t)
lambdify_sympy = lambdify((u, u_t, v, v_t), [
u_t,
-second_term_tensor[0].subs({u.diff(t):u_t, v.diff(t):v_t}),
v_t,
-second_term_tensor[1].subs({u.diff(t):u_t, v.diff(t):v_t})])
x0, t0 = [ic_uv[0], ic_uv_t[0], ic_uv[1], ic_uv_t[1]], 0.0
scioder = sciode(lambda t,X: lambdify_sympy(*X)).set_integrator('vode', method='bdf')
scioder.set_initial_value(x0, t0)
num_of_t = int(t1 / dt); # num_of_t
u_arr = np.empty((num_of_t, 4)); u_arr[0] = x0
t_arr = np.arange(num_of_t) * dt
i = 0
while scioder.successful() and i < num_of_t-1:
i += 1
u_arr[i] = scioder.integrate(scioder.t+dt)
return t_arr, (u_arr[:, 0], u_arr[:, 2])
def test_weyl_contraction_1st_3rd_indices_zero():
mw1 = anti_de_sitter_metric()
w1 = WeylTensor.from_metric(mw1)
t1 = simplify_sympy_array(
sympy.tensorcontraction(w1.change_config("ulll").tensor(), (0, 2)))
t0 = sympy.Array(np.zeros(shape=t1.shape, dtype=int))
assert t1 == t0
def tensor_contractions(T, indeces):
"""If indeces is a tuple with two integers, then perform the contraction
between those two indeces. If indeces is a list of tuples, then apply
multiple contractions.
"""
if (type(indeces) is tuple):
ret = tensorcontraction(T, indeces)
elif (type(indeces) is list):
T = tensorcontraction(T, indeces[0])
for i in range(1, len(indeces)):
T = tensorcontraction(
T, find_indeces_for_contraction(indeces[i - 1], indeces[i]))
ret = T
else:
raise TypeError("indeces in tensor_contractions has to be a list of \
tuples or a tuple")
return ret
def compute_Upsilon(g_deriv, uUP):
"""Return Upsilon =
Upsilon_ alpha beta = g_beta gamma, alpha uUP^gamma
"""
gdU = tensorproduct(g_deriv, uUP)
return -Matrix(tensorcontraction(gdU, (1, 3)))
def LineElement():
print("Starting Test: Line Element...")
try:
t, x, y, z = sympy.symbols("t x y z")
dt, dx, dy, dz = sympy.symbols('dt dx dy dz')
eta = bt.GRMetric([t, x, y, z],
sympy.Matrix([[-1, 0, 0, 0], [0, 1, 0, 0],
[0, 0, 1, 0], [0, 0, 0, 1]]))
dx = bt.GRTensor([eta], sympy.Array([dt, dx, dy, dz]))
rhs_ = sympy.tensorcontraction(
sympy.tensorproduct(eta.lowered, dx.vals), (1, 2))
rhs = sympy.tensorcontraction(sympy.tensorproduct(rhs_, dx.vals),
(0, 1))
assert rhs + dt**2 - dx**2 - dy**2 - dz**2 == 0
print("Test: Line Element - Passed")
return 1
except:
print(rhs)
print(type(rhs))
print("Test: Line Element - Failed")
return 0
def lorentz_transform(self, transformation_matrix):
"""
Performs a Lorentz transform on the tensor.
Parameters
----------
transformation_matrix : ~sympy.tensor.array.dense_ndim_array.ImmutableDenseNDimArray or list
Sympy Array or multi-dimensional list containing Sympy Expressions
Returns
-------
~einsteinpy.symbolic.tensor.BaseRelativityTensor
lorentz transformed tensor(or vector)
"""
tm = sympy.Array(transformation_matrix)
t = self.tensor()
for i in range(self.order):
if self.config[i] == "u":
t = simplify(
tensorcontraction(tensorproduct(tm, t), (1, 2 + i)))
else:
t = simplify(
tensorcontraction(tensorproduct(tm, t), (0, 2 + i)))
tmp = np.array(t.tolist()).reshape(t.shape)
source, dest = list(range(len(t.shape))), list(range(len(t.shape)))
dest.pop(i)
dest.insert(0, i)
tmp = np.moveaxis(tmp, source, dest)
t = sympy.Array(tmp)
return BaseRelativityTensor(
t,
syms=self.syms,
config=self.config,
parent_metric=None,
variables=self.variables,
functions=self.functions,
)
def CovDeriv():
print("Starting Test: Covariant Derivative...")
x, y = sympy.symbols('x y')
test_metric = sympy.Matrix([[1, 0], [0, x**2]])
g = bt.GRMetric([x, y], metric=test_metric)
a, b = sympy.symbols('a b')
A = bt.GRTensor([a],
sympy.Array(
[x * sympy.cos(2 * y), -x * x * sympy.sin(2 * y)]))
A.raise_index(a, g)
Aab = bt.CovariantDerivative(b, [x, y], g, A)
# print(Aab.vals)
assert sympy.tensorcontraction(Aab.vals, (0, 1)) == 0
return 1
def tensor3_vector_product(T, v):
"""Implements a product of a rank-3 tensor (3D array) with a vector using
tensor product and tensor contraction.
Parameters
----------
T: sp.Array of dimensions n x m x k
v: sp.Array of dimensions k x 1
Returns
-------
A: sp.Array of dimensions n x m
Example
-------
>>>T = sp.Array([[[1, 4, 7, 10], [2, 5, 8, 11], [3, 6, 9, 12]],
[[13, 16, 19, 22], [14, 17, 20, 23], [15, 18, 21, 24]]])
⎡⎡1 4 7 10⎤ ⎡13 16 19 22⎤⎤
⎢⎢ ⎥ ⎢ ⎥⎥
⎢⎢2 5 8 11⎥ ⎢14 17 20 23⎥⎥
⎢⎢ ⎥ ⎢ ⎥⎥
⎣⎣3 6 9 12⎦ ⎣15 18 21 24⎦⎦
>>>v = sp.Array([1, 2, 3, 4]).reshape(4, 1)
⎡1⎤
⎢ ⎥
⎢2⎥
⎢ ⎥
⎢3⎥
⎢ ⎥
⎣4⎦
>>>tensor3_vector_product(T, v)
⎡⎡70⎤ ⎡190⎤⎤
⎢⎢ ⎥ ⎢ ⎥⎥
⎢⎢80⎥ ⎢200⎥⎥
⎢⎢ ⎥ ⎢ ⎥⎥
⎣⎣90⎦ ⎣210⎦⎦
"""
import sympy as sp
assert (T.rank() == 3)
# reshape v to ensure 1D vector so that contraction do not contain x 1
# dimension
v.reshape(v.shape[0], )
p = sp.tensorproduct(T, v)
return sp.tensorcontraction(p, (2, 3))
def tensor_product(tensor1, tensor2, i=None, j=None):
"""Tensor Product of ``tensor1`` and ``tensor2``
Parameters
----------
tensor1 : ~einsteinpy.symbolic.BaseRelativityTensor
tensor2 : ~einsteinpy.symbolic.BaseRelativityTensor
i : int, optional
contract ``i``th index of ``tensor1``
j : int, optional
contract ``j``th index of ``tensor2``
Returns
-------
~einsteinpy.symbolic.BaseRelativityTensor
tensor of appropriate rank
Raises
------
ValueError
Raised when ``i`` and ``j`` both indicate 'u' or 'l' indices
"""
product = tensorproduct(tensor1.arr, tensor2.arr)
if (i or j) is None:
newconfig = tensor1.config + tensor2.config
else:
if tensor1.config[i] == tensor2.config[j]:
raise ValueError(
"Index summation not allowed between %s and %s indices"
% (tensor1.config[i], tensor2.config[j])
)
product = simplify(tensorcontraction(product, (i, len(tensor1.config) + j)))
con = tensor1.config[:i] + tensor1.config[i + 1 :]
fig = tensor2.config[:j] + tensor2.config[j + 1 :]
newconfig = con + fig
return BaseRelativityTensor(
product,
syms=tensor1.syms,
config=newconfig,
parent_metric=tensor1.parent_metric,
variables=tensor1.variables,
functions=tensor1.functions,
)
def chain_config_change():
t = sympy.Array(tensor.tensor())
difflist = _difference_list(newconfig, tensor.config)
for i, action in enumerate(difflist):
if action == 0:
continue
else:
t = simplify(
tensorcontraction(tensorproduct(met_dict[action], t),
(1, 2 + i)))
# reshuffle the indices
dest = list(range(len(t.shape)))
dest.remove(0)
dest.insert(i, 0)
t = sympy.permutedims(t, dest)
return t
def test_covdiff():
"""
Test the covariant derivative of a tensor on the 2-sphere.
The contraction of the specified tensor after being raised should be 0,
because it is a divergenceless tensor.
"""
x, y = sympy.symbols('x y')
test_metric = sympy.Matrix([[1, 0], [0, x**2]])
g = bt.GRMetric([x, y], metric=test_metric)
a, b = sympy.symbols('a b')
A = bt.GRTensor([a],
sympy.Array(
[x * sympy.cos(2 * y), -x * x * sympy.sin(2 * y)]))
A.raise_index(a, g)
Aab = bt.CovariantDerivative(b, [x, y], g, A)
assert sympy.tensorcontraction(Aab.vals, (0, 1)) == 0
return 1
def chain_config_change():
t = sympy.Array(tensor.tensor())
difflist = _difference_list(newconfig, tensor.config)
for i, action in enumerate(difflist):
if action == 0:
continue
else:
t = simplify(
tensorcontraction(tensorproduct(met_dict[action], t),
(1, 2 + i)))
# reshuffle the indices
tmp = np.array(t.tolist()).reshape(t.shape)
source, dest = list(range(len(t.shape))), list(
range(len(t.shape)))
dest.pop(i)
dest.insert(0, i)
tmp = np.moveaxis(tmp, source, dest)
t = sympy.Array(tmp)
return t
def test_array_as_explicit_matrix_symbol():
A = MatrixSymbol("A", 3, 3)
B = MatrixSymbol("B", 3, 3)
texpr = tensorproduct(A, B)
assert isinstance(texpr, ArrayTensorProduct)
assert texpr.as_explicit() == tensorproduct(A.as_explicit(),
B.as_explicit())
texpr = tensorcontraction(A, (0, 1))
assert isinstance(texpr, ArrayContraction)
assert texpr.as_explicit() == A[0, 0] + A[1, 1] + A[2, 2]
texpr = tensordiagonal(A, (0, 1))
assert isinstance(texpr, ArrayDiagonal)
assert texpr.as_explicit() == ImmutableDenseNDimArray(
[A[0, 0], A[1, 1], A[2, 2]])
texpr = permutedims(A, [1, 0])
assert isinstance(texpr, PermuteDims)
assert texpr.as_explicit() == permutedims(A.as_explicit(), [1, 0])
def test_array_symbol_and_element():
A = ArraySymbol("A", 2)
A0 = ArrayElement(A, (0,))
A1 = ArrayElement(A, (1,))
assert A.as_explicit() == ImmutableDenseNDimArray([A0, A1])
A2 = tensorproduct(A, A)
assert A2.shape == (2, 2)
# TODO: not yet supported:
# assert A2.as_explicit() == Array([[A[0]*A[0], A[1]*A[0]], [A[0]*A[1], A[1]*A[1]]])
A3 = tensorcontraction(A2, (0, 1))
assert A3.shape == ()
# TODO: not yet supported:
# assert A3.as_explicit() == Array([])
A = ArraySymbol("A", 2, 3, 4)
Ae = A.as_explicit()
assert Ae == ImmutableDenseNDimArray(
[[[ArrayElement(A, (i, j, k)) for k in range(4)] for j in range(3)] for i in range(2)])
p = permutedims(A, Permutation(0, 2, 1))
assert isinstance(p, PermuteDims)
def tensor_contract(T, ind1, ind2):
"""Contract two indices of a tensor.
Arguments:
T (Tensor) -- Tensor to contract
ind1 (Symbol) -- First index to contract.
ind2 (Symbol) -- Second index to contract.
Return Type -- Tensor
"""
if ind1 * ind2 > 0:
raise AttributeError("Index 1 and 2 cannot be in the same state")
if ind1 not in T.indices or ind2 not in T.indices:
raise AttributeError(f"Index {ind1} or {ind2} not found in tensor T")
i1 = T.indices.index(ind1)
i2 = T.indices.index(ind2)
new_indices = T.indices
new_indices.remove(ind1)
new_indices.remove(ind2)
new_vals = sympy.tensorcontraction(T.vals, (i1, i2))
return objects.Tensor(new_indices, new_vals)
exec('W.append(w' + str(FromLayer) + str(i) + str(j) + ')')
W = sp.Array(W)
W = W.reshape(outputNeurons, inputNeurons)
return (W)
#_weights_between_layers_1_and_2____
W_l1tol2 = weights(1, 4, 4)
#_weights_between_layers_2_and_3____
W_l2tol3 = weights(2, 4, 4)
#_weights_between_layers_3_and_4____
W_l3tol4 = weights(3, 4, 2)
#_weights_between_layers_4_and_5____
W_l4tol5 = weights(4, 2, 1)
#______________________________________________________________________________
#_function_definition_and_input______________
f = lambda x: 1 / (1 + sp.exp(-x))
input = sp.Array([inp1, inp2, inp3, inp4])
#_____________________________________________
#_Outputs_____________________________________________________________________
o_l1 = [f(input[0]), f(input[1]), f(input[2]), f(input[3])]
o_l2 = sp.tensorcontraction(
sp.tensorproduct([o_l1], W_l1tol2)[0, :, :, :].applyfunc(f), (0, 1))
o_l3 = sp.tensorcontraction(
sp.tensorproduct([o_l2], W_l2tol3)[0, :, :, :].applyfunc(f), (0, 1))
o_l4 = sp.tensorcontraction(
sp.tensorproduct([o_l3], W_l3tol4)[0, :, :, :].applyfunc(f), (0, 1))
o_l5 = sp.tensorcontraction(
sp.tensorproduct([o_l4], W_l4tol5)[0, :, :, :].applyfunc(f), (0, 1))
#______________________________________________________________________________
def row_wise_div(tensor):
return sym.Matrix(
tensorcontraction(derive_by_array(tensor, [x, y]), (0, 2)))
def as_explicit(self):
return tensorcontraction(self.expr.as_explicit(), *self.contraction_indices)
W_l2tol3 = weights(2, 4, 4)
#_weights_between_layers_3_and_4____
W_l3tol4 = weights(3, 4, 2)
#_weights_between_layers_4_and_5____
W_l4tol5 = weights(4, 2, 1)
#______________________________________________________________________________
#_function_definition_and_input______________
f = lambda x: 1 / (1 + sp.exp(-x))
input = sp.Array([inp1, inp2, inp3, inp4])
#_____________________________________________
#_Outputs_____________________________________________________________________
o_l1 = sp.Array([[f(input[0]), f(input[1]), f(input[2]), f(input[3])]])
o_l2 = sp.Array([
sp.tensorcontraction(
sp.tensorproduct(o_l1, W_l1tol2)[0, :, :, :].applyfunc(f), (0, 1))
])
o_l3 = sp.Array([
sp.tensorcontraction(
sp.tensorproduct(o_l2, W_l2tol3)[0, :, :, :].applyfunc(f), (0, 1))
])
o_l4 = sp.Array([
sp.tensorcontraction(
sp.tensorproduct(o_l3, W_l3tol4)[0, :, :, :].applyfunc(f), (0, 1))
])
o_l5 = sp.Array([
sp.tensorcontraction(
sp.tensorproduct(o_l4, W_l4tol5)[0, :, :, :].applyfunc(f), (0, 1))
])
#______________________________________________________________________________
def SARIMACorrelation(trendparams: tuple = (0, 0, 0),
seasonalparams: tuple = (0, 0, 0, 1),
trendAR=None,
trendMA=None,
seasonAR=None,
seasonMA=None):
p, d, q = trendparams
P, D, Q, m = seasonalparams
print(f'SARIMA({p},{d},{q})({P},{D},{Q},{m})')
assert type(p) is int, 'Input parameter "p" is not an integer type.'
assert type(d) is int, 'Input parameter "d" is not an integer type.'
assert type(q) is int, 'Input parameter "q" is not an integer type.'
assert type(P) is int, 'Input parameter "P" is not an integer type.'
assert type(D) is int, 'Input parameter "D" is not an integer type.'
assert type(Q) is int, 'Input parameter "Q" is not an integer type.'
assert type(m) is int, 'Input parameter "m" is not an integer type.'
if trendAR:
assert len(
trendAR
) == p, f'The len(trendAR) must be {p}. Reset the parameters.'
else:
trendAR = [0] * p
if trendMA:
assert len(
trendMA
) == q, f'The len(trendMA) must be {q}. Reset the parameters.'
else:
trendMA = [0] * q
if seasonAR:
assert len(
seasonAR
) == P, f'The len(seasonAR) must be {P}. Reset the parameters.'
else:
seasonAR = [0] * P
if seasonMA:
assert len(
seasonMA
) == Q, f'The len(seasonMA) must be {Q}. Reset the parameters.'
else:
seasonMA = [0] * Q
Y_order = p + P * m + d + D * m
e_order = q + Q * m
# define Y, e
Y, e = sympy.symbols('Y_t, e_t')
I, J = sympy.symbols('i, j')
Y_ = {}
e_ = {}
Y_['t'] = Y
Y__ = [[Y_['t']]]
e_['t'] = e
e__ = [[e_['t']]]
for i in range(1, Y_order + 1):
Y_[f't-{i}'] = sympy.symbols(f'Y_t-{i}')
Y__.append([
Y_[f't-{i}'] * (I**i)
]) # Y__ = [ [Y_['t']], [Y_['t-1']], ..., [Y_['t-(p+P*m+q+Q*m)']] ]
for i in range(1, e_order + 1):
e_[f't-{i}'] = sympy.symbols(f'e_t-{i}')
e__.append([
e_[f't-{i}'] * (J**i)
]) # e__ = [ [e_['t']], [e_['t-1']], ..., [e_['t-(q+Q*m)']] ]
# define L
L = sympy.symbols('L')
S_Lag = L**m
T_Lag = L
S_Lag_Diff = (1 - L**m)**D
T_Lag_Diff = (1 - L)**d
# define coefficients : phis(T), Phis(S), thetas(T), Thetas(S)
T_phi = {}
T_phis = []
L_byT_phi = []
S_phi = {}
S_phis = []
L_byS_phi = []
T_theta = {}
T_thetas = []
L_byT_theta = []
S_theta = {}
S_thetas = []
L_byS_theta = []
for p_ in range(0, p + 1):
T_phi[p_] = sympy.symbols(f'phi_{p_}')
T_phis.append(
-T_phi[p_]) # T_phis = [T_phi[0], T_phi[1], ..., T_phi[p]]
L_byT_phi.append([T_Lag**p_
]) # L_byT_phi = [[L**0], [L**1], ..., [L**p]]
for P_ in range(0, P + 1):
S_phi[P_] = sympy.symbols(f'Phi_{P_}')
S_phis.append(
-S_phi[P_]) # S_phis = [S_phi[0], S_phi[1], ..., S_phi[P]]
L_byS_phi.append([
S_Lag**P_
]) # L_byS_phi = [[(L**m)**0], [(L**m)**1], ..., [(L**m)**P]]
for q_ in range(0, q + 1):
T_theta[q_] = sympy.symbols(f'theta_{q_}')
T_thetas.append(
T_theta[q_]
) # T_thetas = [T_theta[0], T_theta[1], ..., T_theta[q]]
L_byT_theta.append([T_Lag**q_
]) # L_byT_theta = [[L**0], [L**1], ..., [L**q]]
for Q_ in range(0, Q + 1):
S_theta[Q_] = sympy.symbols(f'Theta_{Q_}')
S_thetas.append(
S_theta[Q_]
) # S_thetas = [T_theta[0], T_theta[1], ..., T_theta[Q]]
L_byS_theta.append([
S_Lag**Q_
]) # L_byS_theta = [[(L**m)**0], [(L**m)**1], ..., [(L**m)**Q]]
T_phi_Lag = sympy.Matrix([T_phis]) * sympy.Matrix(L_byT_phi)
S_phi_Lag = sympy.Matrix([S_phis]) * sympy.Matrix(L_byS_phi)
T_theta_Lag = sympy.Matrix([T_thetas]) * sympy.Matrix(L_byT_theta)
S_theta_Lag = sympy.Matrix([S_thetas]) * sympy.Matrix(L_byS_theta)
Y_operator = (T_phi_Lag * S_phi_Lag * T_Lag_Diff * S_Lag_Diff).subs(
T_phi[0], -1).subs(S_phi[0], -1)[0]
e_operator = (T_theta_Lag * S_theta_Lag).subs(T_theta[0],
1).subs(S_theta[0], 1)[0]
Y_operation = sympy.collect(Y_operator.expand(), L)
e_operation = sympy.collect(e_operator.expand(), L)
Y_coeff = sympy.Poly(Y_operation, L).all_coeffs()[::-1]
e_coeff = sympy.Poly(e_operation, L).all_coeffs()[::-1]
Y_term = sympy.Matrix([Y_coeff]) * sympy.Matrix(Y__) # left-side
e_term = sympy.Matrix([e_coeff]) * sympy.Matrix(e__) # right-side
Time_Series = {}
Time_Series['Y_t(i,j)'] = sympy.Poly(Y - Y_term[0] + e_term[0], (I, J))
Time_Series['Y_t'] = Time_Series['Y_t(i,j)'].subs(I, 1).subs(J, 1)
for i in range(1, int(p + P * m + d + D * m) + 1):
Time_Series['Y_t'] = sympy.collect(Time_Series['Y_t'],
Y_[f't-{i}']).simplify()
for i in range(1, int(q + Q * m) + 1):
Time_Series['Y_t'] = sympy.collect(Time_Series['Y_t'],
e_[f't-{i}']).simplify()
print('* Time Series Equation(Analytic Form)')
sympy.pprint(Time_Series['Y_t'])
print()
Time_Series['Analytic_Coeff_of_Y'] = Time_Series['Y_t(i,j)'].subs(
J, 0).all_coeffs()[::-1]
Time_Series['Analytic_Coeff_of_e'] = Time_Series['Y_t(i,j)'].subs(
I, 0).all_coeffs()[::-1]
Time_Series['Numeric_Coeff_of_Y'] = Time_Series['Y_t(i,j)'].subs(
J, 0) - e_['t']
Time_Series['Numeric_Coeff_of_e'] = Time_Series['Y_t(i,j)'].subs(I, 0)
for i, (phi, Np) in enumerate(zip(list(T_phi.values())[1:], trendAR)):
Time_Series['Numeric_Coeff_of_Y'] = Time_Series[
'Numeric_Coeff_of_Y'].subs(phi, Np)
for i, (Phi, NP) in enumerate(zip(list(S_phi.values())[1:], seasonAR)):
Time_Series['Numeric_Coeff_of_Y'] = Time_Series[
'Numeric_Coeff_of_Y'].subs(Phi, NP)
for i, (theta, Nt) in enumerate(zip(list(T_theta.values())[1:], trendMA)):
Time_Series['Numeric_Coeff_of_e'] = Time_Series[
'Numeric_Coeff_of_e'].subs(theta, Nt)
for i, (Theta, NT) in enumerate(zip(list(S_theta.values())[1:], seasonMA)):
Time_Series['Numeric_Coeff_of_e'] = Time_Series[
'Numeric_Coeff_of_e'].subs(Theta, NT)
print('* Time Series Equation(Numeric Form)')
sympy.pprint((Time_Series['Numeric_Coeff_of_Y'] +
Time_Series['Numeric_Coeff_of_e']).subs(I, 1).subs(J, 1))
Time_Series['Numeric_Coeff_of_Y'] = sympy.Poly(
Time_Series['Numeric_Coeff_of_Y'], I).all_coeffs()[::-1]
Time_Series['Numeric_Coeff_of_e'] = sympy.Poly(
Time_Series['Numeric_Coeff_of_e'], J).all_coeffs()[::-1]
final_coeffs = [[], []]
print('\n* Y params')
print(f'- TAR({trendparams[0]}) phi : {trendAR}')
print(f'- TMA({trendparams[2]}) theta : {trendMA}')
for i, (A_coeff_Y, N_coeff_Y) in enumerate(
zip(Time_Series['Analytic_Coeff_of_Y'],
Time_Series['Numeric_Coeff_of_Y'])):
if i == 0:
pass
elif i != 0:
A_coeff_Y = A_coeff_Y.subs(Y_[f"t-{i}"], 1)
N_coeff_Y = N_coeff_Y.subs(Y_[f"t-{i}"], 1)
print(f't-{i} : {A_coeff_Y} > {round(N_coeff_Y, 5)}')
final_coeffs[0].append(N_coeff_Y)
print('\n* e params')
print(f'- SAR({seasonalparams[0]}) Phi : {seasonAR}')
print(f'- SMA({seasonalparams[2]}) Theta : {seasonMA}')
for i, (A_coeff_e, N_coeff_e) in enumerate(
zip(Time_Series['Analytic_Coeff_of_e'],
Time_Series['Numeric_Coeff_of_e'])):
if i == 0:
A_coeff_e = A_coeff_e.subs(e_[f"t"], 1)
N_coeff_e = N_coeff_e.subs(e_[f"t"], 1)
print(f't-{i} : {A_coeff_e} > {1}')
elif i != 0:
A_coeff_e = A_coeff_e.subs(e_[f"t-{i}"], 1)
N_coeff_e = N_coeff_e.subs(e_[f"t-{i}"], 1)
print(f't-{i} : {A_coeff_e} > {round(N_coeff_e, 5)}')
final_coeffs[1].append(N_coeff_e)
print('\n* Quasi-Convergence Factor')
try:
print(
'Y :',
sympy.tensorcontraction(
sympy.Array(final_coeffs[0]).applyfunc(lambda x: x**2), (0, )))
print(
'e :',
sympy.tensorcontraction(
sympy.Array(final_coeffs[1]).applyfunc(lambda x: x**2), (0, )))
except:
pass
_, axes = plt.subplots(5, 1, figsize=(12, 15))
ar_params = np.array(final_coeffs[0])
ma_params = np.array(final_coeffs[1])
ar, ma = np.r_[1, -ar_params], np.r_[1, ma_params]
y = smt.ArmaProcess(ar, ma).generate_sample(300, burnin=50)
axes[0].plot(y, 'o-')
axes[0].grid(True)
axes[1].stem(smt.ArmaProcess(ar, ma).acf(lags=40))
axes[1].set_xlim(-1, 41)
axes[1].set_ylim(-1.1, 1.1)
axes[1].set_title(
"Theoretical autocorrelation function of an SARIMA process")
axes[1].grid(True)
axes[2].stem(smt.ArmaProcess(ar, ma).pacf(lags=40))
axes[2].set_xlim(-1, 41)
axes[2].set_ylim(-1.1, 1.1)
axes[2].set_title(
"Theoretical partial autocorrelation function of an SARIMA process")
axes[2].grid(True)
smt.graphics.plot_acf(y, lags=40, ax=axes[3])
axes[3].set_xlim(-1, 41)
axes[3].set_ylim(-1.1, 1.1)
axes[3].set_title(
"Experimental autocorrelation function of an SARIMA process")
axes[3].grid(True)
smt.graphics.plot_pacf(y, lags=40, ax=axes[4])
axes[4].set_xlim(-1, 41)
axes[4].set_ylim(-1.1, 1.1)
axes[4].set_title(
"Experimental partial autocorrelation function of an SARIMA process")
axes[4].grid(True)
plt.tight_layout()
plt.show()
