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FBackGainsCalculator.py
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FBackGainsCalculator.py
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from GainsCalculator import *
class FBackGainsCalculator(GainsCalculator):
"""
This class is a subclass of 'GainsCalculator'. Whereas the parent class
is for analyzing DAGs without feedback loops, this class can handle
feedback loops.
This class implements formulae derived in my book "Bayesuvius", in the
chapter entitled "LDEN diagrams with feedback loops". In that chapter,
I consider, for a graph with feedback loops,
the matrix A with entries A_{i,j}=\alpha_{ i|j}= inslice arrow gains
the matrix B with entries B_{i,j}=\beta_{i|j}= feedback arrow gains
I show that A and B satisfy a system of 2 linear equations with two
unknowns A,B.
Let CM_info = the single-time covariance matrices C^n, C^{n+1} with
entries C^t_{i,i}=<x^t_i, x^t_j> at times t=n and t=n+1, and the 2-times
covariance matrix cov2times_n with entries cov2times_n_{i,j}= <x^{ n}_i,
x^{n+1}_j>
To solve that system of 2 equations in (A, B), this class first solves
one equation for A in terms of B and CMinfo, thus obtaining A(B,
CMinfo). Then it substitutes A(B, CMinfo) into the remaining equation to
obtain B(CMinfo). Finally, it substitutes B(CMinfo) into A(B, CMinfo) to
get A(B(CMinfo), CMinfo).
self.alpha_list_with_betas and self.alpha_mat_with_betas are used to store
A(B, CM_info).
self.beta_list and self.beta_mat are used to store B(CM_info).
self.alpha_list and self.alpha_mat are used to store A(B(CM_info),
CM_info).
The value of random variable x at time n will be denoted by x^{[ n]}. We
will also use the notation
\Delta x^{[n]} = x^{[n+1]}- x^{[n]}
Set "delta=False" if you want 2-times correlations < x_i^{[n]},
x_j^{[n+1]}> in the final result to be expressed as themselves. Set
"delta=True" (recommended) if you want 2-times correlations < x_i^{[
n]}, x_j^{[n+1]}> in the final result to be replaced by 2 terms,
using the identity
< x_i^{[n]}, x_j^{[n+1]}>= < x_i^{[n]}, x_j^{[n]}> + < x_i^{[n]},
\Delta x_j^{[n]}>
Attributes
----------
# alpha_list and alpha_mat are inherited from parent class
alpha_list_with_betas: list[sp.Eq]
alpha_mat_with_betas: sp.Matrix
beta_list: list[sp.Eq]
beta_mat: sp.Matrix
delta: bool
"""
def __init__(self, graph, delta=True):
"""
Constructor
Parameters
----------
graph: FBackGraph
delta: bool
"""
GainsCalculator.__init__(self, graph)
self.delta = delta
# self.alpha_list and self.alpha_mat are inherited from parent class.
self.alpha_list_with_betas = None
self.alpha_mat_with_betas = None
self.beta_list = None
self.beta_mat = None
def calculate_gains(self, cov_mat_list_in=None, mat_K=None, time="n"):
"""
This method overrides the parent method. It calls the parent method
within itself. It fills in
self.alpha_list and self.alpha_mat
self.alpha_list_with_betas and self.alpha_mat_with_betas
self.beta_list and self.beta_mat
The inputs to cov_mat_list_in and mat_K do not matter as these
variables are reassigned internally.
Parameters
----------
cov_mat_list_in: list[sp.Matrix]
mat_K: sp.Matrix
time: None or str or int
Returns
-------
None
"""
dim = self.graph.num_nds
if time == "n":
time0 = "n"
time1 = "n_plus_one"
elif isinstance(time, int):
time0 = time
time1 = time + 1
else:
assert False
if cov_mat_list_in is None:
cov_mat0 = cov_sb_mat(dim, time=time0)
cov2times = cov2times_sb_mat(dim, time=time0)
cov_mat1 = cov_sb_mat(dim, time=time1)
d_cov2times = cov2times_sb_mat(dim, time=time0, delta=True)
else:
cov_mat0, cov2times, cov_mat1 = cov_mat_list_in
d_cov2times = cov2times - cov_mat0
mat_B = set_to_zero_fback_gains_without_arrows(self.graph,
beta_sb_mat(dim))
mat_K = mat_B * cov_mat0
calc = GainsCalculator(self.graph)
calc.calculate_gains(cov_mat_in=cov_mat1, mat_K=mat_K, time=time1)
self.alpha_mat_with_betas = deepcopy(calc.alpha_mat)
self.alpha_list_with_betas = deepcopy(calc.alpha_list)
self.alpha_mat = None
self.alpha_list = None
self.calculate_betas(cov_mat0, cov2times, d_cov2times, time=time0)
self.calculate_alphas()
def calculate_betas(self, cov_mat0, cov2times, d_cov2times, time):
"""
This method fills in self.beta_list and self.beta_mat. It's an
internal method called by calculate_gains().
Parameters
----------
cov_mat0: sp.Matrix
cov2times: sp.Matrix
d_cov2times: sp.Matrix
time: None or str or int
Returns
-------
None
"""
dim = self.graph.num_nds
mat_B = set_to_zero_fback_gains_without_arrows(self.graph,
beta_sb_mat(dim))
eq_list = eq_list0 = eq_list1 = None
eq_mat = eq_mat0 = eq_mat1 = None
unknowns = unknowns0 = unknowns1 = None
if not self.delta:
eq_mat = (sp.eye(dim) - self.alpha_mat_with_betas) * \
cov2times.T - \
mat_B * cov_mat0
else:
# delta method consists in solving
# b= A x
# by solving two systems of linear equations:
# b_0 = A x_0 solved for x_0
# b_1 = A x_1 solved for x_1
# so
# b= b_0 + b_1 = A(x_0 + x_1)
eq_mat0 = (sp.eye(dim) - self.alpha_mat_with_betas) * \
cov_mat0.T - mat_B * cov_mat0
eq_mat1 = (sp.eye(dim) - self.alpha_mat_with_betas) * \
d_cov2times.T - mat_B * cov_mat0
if not self.delta:
unknowns = []
eq_list = []
else:
unknowns0 = []
unknowns1 = []
eq_list0 = []
eq_list1 = []
for row, col in product(range(dim), range(dim)):
row_nd = self.graph.ord_nodes[row]
col_nd = self.graph.ord_nodes[col]
if not self.delta:
eq_list.append(eq_mat[row, col])
else:
eq_list0.append(eq_mat0[row, col])
eq_list1.append(eq_mat1[row, col])
if (col_nd, row_nd) in self.graph.fback_arrows:
beta_str = "beta_" + str(row) + "_L_" + str(col)
if not self.delta:
unknowns.append(sp.Symbol(beta_str))
else:
unknowns0.append(sp.Symbol(beta_str))
unknowns1.append(sp.Symbol(beta_str))
else:
if time == "n":
xtra_str = "n"
elif isinstance(time, int):
xtra_str = "n" + str(time)
else:
assert False
cov2times_str = "cov2times_" + xtra_str +\
"_" + str(col) + "_" + str(row)
if not self.delta:
unknowns.append(sp.Symbol(cov2times_str))
else:
# do nothing for unknowns0
unknowns1.append(sp.Symbol("d_" + cov2times_str))
# the comma does what is called sequence unpacking.
# draws out item from single item list
if not self.delta:
sol_list, = linsolve(eq_list, unknowns)
# print(str(sol_list))
sol_list = sp.factor(sol_list)
else:
sol_list0, = linsolve(eq_list0, unknowns0)
sol_list1, = linsolve(eq_list1, unknowns1)
sol_list = []
unknowns = unknowns1
for i in range(len(sol_list1)):
if i < len(unknowns0):
sol_list.append(sol_list0[i] + sol_list1[i])
else:
sol_list.append(sol_list1[i])
self.beta_list = []
self.beta_mat = sp.zeros(dim)
for i in range(len(sol_list)):
self.beta_list.append(sp.Eq(unknowns[i], sol_list[i]))
left_str = str(unknowns[i])
if left_str[0:3] == 'beta':
# print("kkkll", left_str)
row_str, col_str = left_str[4:].split("_L_")
self.beta_mat[int(row_str), int(col_str)] = sol_list[i]
def calculate_alphas(self):
"""
This method fills in self.alpha_list and self.alpha_mat. It's an
internal method called by calculate_gains().
Returns
-------
None
"""
dim = self.graph.num_nds
self.alpha_mat = deepcopy(self.alpha_mat_with_betas)
self.alpha_list = deepcopy(self.alpha_list_with_betas)
for row, col in product(range(dim), range(dim)):
beta_str = "beta_" + str(row) + "_L_" + str(col)
self.alpha_mat = sp.simplify(
self.alpha_mat.subs(sp.Symbol(beta_str),
self.beta_mat[row, col]))
for i in range(len(self.alpha_list)):
self.alpha_list[i] = sp.simplify(
self.alpha_list[i].subs(sp.Symbol(beta_str),
self.beta_mat[row, col]))
# print("ccvvf", self.alpha_mat)
def print_alpha_list_with_betas(self, verbose=False, time="n"):
"""
This method prints the info in self.alpha_list_with_betas. It does
this by calling latexify:print_list_sb().
Parameters
----------
verbose: bool
time: None or str or int
Returns
-------
sp.Symbol
"""
return print_list_sb(self.alpha_list_with_betas,
self.graph,
verbose=verbose,
time=time)
def print_beta_list(self, verbose=False, time="n"):
"""
This method prints the info in self.beta_list. It does this by
calling latexify:print_list_sb().
Parameters
----------
verbose: bool
time: None or str or int
Returns
-------
sp.Symbol
"""
return print_list_sb(self.beta_list,
self.graph,
verbose=verbose,
time=time)
def print_alpha_list(self, verbose=False, time="n"):
"""
This method prints the info in self.alpha_list. It does this by
calling latexify:print_list_sb().
Parameters
----------
verbose: bool
time: None or str or int
Returns
-------
sp.Symbol
"""
return print_list_sb(self.alpha_list,
self.graph,
verbose=verbose,
time=time)
if __name__ == "__main__":
def main():
path = 'dot_atlas/fback-2node.dot'
graph = FBackGraph(path)
cal = FBackGainsCalculator(graph)
cal.calculate_gains()
cal.print_alpha_list_with_betas(verbose=True)
cal.print_beta_list(verbose=True)
cal.print_alpha_list(verbose=True)
main()