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utils.py
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utils.py
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import numpy as np
import numba
from . import polynomial_matrix as pm
from numpy.random import uniform, randn
from numpy import sqrt
from itertools import combinations
def polar_to_rectangle(modulus, argument):
return modulus * np.exp(1j*argument)
def rectangle_to_polar(z):
return np.abs(z), np.angle(z)
def make_real_diag(x, cpx_pairs):
""" an array of roots, including n_pairs
of complex roots. the first lx-2*npairs are
real roots. the last are modulus /angle pairs
of roots.
Example:
make_real_diag([1, 2, 2, pi/4, 3, pi/3], 2)
Out[13]:
array([[ 1. , 0. , 0. , 0. , 0. ,
0. ],
[ 0. , 2. , 0. , 0. , 0. ,
0. ],
[ 0. , 0. , 1.41421356, 1.41421356, 0. ,
0. ],
[ 0. , 0. , -1.41421356, 1.41421356, 0. ,
0. ],
[ 0. , 0. , 0. , 0. , 1.5 ,
2.59807621],
[ 0. , 0. , 0. , 0. , -2.59807621,
1.5 ]])
"""
lx = len(x)
ret = np.zeros((lx, lx), dtype=float)
if (lx > 2*cpx_pairs):
np.fill_diagonal(
ret[:(lx-2*cpx_pairs), :(lx-2*cpx_pairs)], x[:(lx-2*cpx_pairs)])
if (cpx_pairs != 0):
for i in range(cpx_pairs):
root = polar_to_rectangle(
modulus=x[lx-2*cpx_pairs+2*i],
argument=x[lx-2*cpx_pairs+2*i+1])
ret[lx-2*cpx_pairs+2*i:lx-2*cpx_pairs+2*i+2,
lx-2*cpx_pairs+2*i:lx-2*cpx_pairs+2*i+2] =\
np.array([root.real, root.imag,
-root.imag, root.real]).reshape(2, 2)
return ret
def random_orthogonal(k):
"""Generate a random orthogonal matrix of size (k, k)
real matrix based on the paper
How to generate random matrices from the classical compact groups
example:
O = random_orthogonal(3)
print O
[[ 0.25452591 -0.92275001 0.28939416]
[ 0.96115429 0.27441539 0.0296417 ]
[-0.10676609 0.27060785 0.95675096]]
np.dot(O, O.T)
array([[ 1.00000000e+00, -1.21314803e-16, -2.59623113e-18],
[-1.21314803e-16, 1.00000000e+00, 3.20641969e-17],
[-2.59623113e-18, 3.20641969e-17, 1.00000000e+00]])
"""
z = randn(k, k) / sqrt(2.)
q, r = np.linalg.qr(z)
d = np.diagonal(r)
ph = d / np.abs(d)
q = np.multiply(q, ph, q)
return q
def random_innovation_series(D, OO, n):
"""
generate a random innovation series
with covariance matrix OO . D OO.T.
D is an array, diagonal
OO is orthogonal
Return M of size n times k
with M
"""
e = randn(n, D.shape[0])
e = e - np.mean(e, axis=0)[None, :]
return np.dot(
e * sqrt(D)[None, :], OO.T)
def gen_stable_model_p_2(Psi, k):
"""Psi = (2, d_2, 1, d_1)
Condition: d_2 + d_1 <= k
We do this by a generalization of SVD:
A random diagonal positive vector of
size d_2: Sigma_2
A random diagonal positve matrix of
size d_1: Sigma_1
d_1 + d_2 Random orthogonal vector of size k
break to matrices U_2,0 and U_1,0 of size
(k, d_2) and (k, d_1)
Another unrelated set of random orthogonal
vector of size d_1 + d_2 break to matrices
of V_2,0 and V_1, 0 of sizes
(k, d_2) and (k, d_1)
Another set of random orthogonal matrix of size 2*d_2
(so some could be zeros) so we can form
vector U_2,1 and V_2,1 such that
U_{2, 1} Sigma_2 V_{2, 1).T = 0
We form H_{i, j} = U_{i, j} Sigma_i^{1/2}
We adjust values of Sigma to make sure the system is stable
"""
low_bound = .1
high_bound = .97
uv21_size = .2
U0 = random_orthogonal(k)
V0 = random_orthogonal(k)
UV_21 = random_orthogonal(k) * uv21_size
U1 = random_orthogonal(k)
V1 = random_orthogonal(k)
d_2 = Psi[0][1]
d_1 = Psi[1][1]
u_tmp_21 = UV_21[:, :d_2]
v_tmp_21 = np.zeros((d_2, k), dtype=float)
if 2*d_2 <= k:
v_tmp_21[:, :] = UV_21[d_2:2*d_2, :]
else:
v_tmp_21[(k-d_2):, :] = UV_21[d_2:, :]
mm_degree = pm.calc_McMillan_degree(Psi)
H = np.zeros((k, mm_degree))
G = np.zeros((mm_degree, k))
F = pm.calc_Jordan_matrix(Psi, 0)
stable = False
max_search = 100
cnt = 0
while (not stable) and cnt < max_search:
sqrt_Sigma_2 = uniform(low_bound, high_bound, d_2)
sqrt_Sigma_1 = uniform(low_bound, high_bound, d_1)
H[:, :d_2] = U0[:, :d_2] * sqrt_Sigma_2[None, :]
H[:, d_2:2*d_2] = u_tmp_21 / sqrt_Sigma_2[None, :]
H[:, 2*d_2:] = U1[:, :d_1] * sqrt_Sigma_1[None, :]
G[:d_2, :] = V0[:d_2, :] * sqrt_Sigma_2[:, None]
G[d_2:2*d_2, :] = v_tmp_21 / sqrt_Sigma_2[:, None]
G[2*d_2:, :] = V1[:d_1, :] * sqrt_Sigma_1[:, None]
Phi = pm.state_to_Phi(H, F, G, Psi)
stable, roots, dd = pm.check_stable(Phi, 2)
cnt = cnt + 1
return stable, H, G, F, Phi
@numba.jit
def _calc_var_sim(Phi_arr, e):
"""Note that Polynomial matrix
is in high to low order.
Phi is usually is in low to high order
"""
n = e.shape[0]
p = Phi_arr.shape[2]
ret = np.zeros_like(e)
for j in range(0, n):
for i in range(min(j, p)):
ret[j, :] += np.dot(
Phi_arr[:, :, p-i-1], ret[j-i-1, :])
ret[j, :] += e[j, :]
return ret
@numba.jit
def calc_residual(Y, Phi_arr):
n = Y.shape[0]
k = Y.shape[1]
p = Phi_arr.shape[2]
e = np.zeros_like(Y)
Y_ex = np.zeros((n+p, k))
Y_ex[-n:, :] = Y[:, :]
e[-n:, :] = Y[:, :]
for i in range(p):
e[-n:, :] -= np.dot(
Y_ex[p-i-1:p-i-1+n, :], Phi_arr[:, :, i].T)
return e
def VAR_sim(Phi, n, D=None, OO=None):
"""Generate a random stable series
based on Phi, where the innovation series
have covariance matrix OO D OO.T
"""
k = Phi.shape[0]
if D is None:
D = np.eye(k)
if OO is None:
OO = np.eye(k)
e = random_innovation_series(D, OO, n)
Phi_arr = Phi.PolynomialMatrix_to_3darray()
ret = _calc_var_sim(Phi_arr, e)
return ret, e
def list_all_psi_hat(m, p):
""" List all possible Psi. The funny formula below is just stars and bars map
"""
off = (1,)+(p-1)*(0,)
for c in combinations(range(m+p-1), p):
yield [b-a-1+o for a, b, o in zip((-1,)+c, c+(m+p-1,), off)]
def psi_hat_to_psi(Psi_hat):
p = len(Psi_hat)
return [(p-i, Psi_hat[i]) for i in range(p)]
def psi_counts(m, p):
import scipy
return int(scipy.special.comb(p+m-1, p))
if __name__ == '__main__':
pass