def test1():
'''
Runs the spectral factorization routine on a few trivial examples
and plots the results.
'''
#Specify zeros of the filter which produces the PSD
z1 = 0.5
z2 = -0.5 + 0.75j
z3 = 0.99 + 0.01j
H1_z, P1_z = [z1], [z1, 1/z1]
H2_z, P2_z = [z1, z2, z2.conjugate()], [z1, 1/z1, z2, 1/z2,
z2.conjugate(),
1/z2.conjugate()]
H3_z, P3_z = [z3, z3.conjugate()], [z3, 1/z3, z3.conjugate(),
1/z3.conjugate()]
for H_z, P_z in [(H1_z, P1_z), (H2_z, P2_z), (H3_z, P3_z)]:
P = poly(H_z)
P = polymul(P[::-1], P) #Creates a valid PSD polynomial
#Pass in only the causal portion
Q, s = spectral_factorization(P[len(P)/2:])
Q_zeros = roots(Q)
fig = plt.figure()
ax = fig.add_subplot(1,1,1)
uc = patches.Circle((0, 0), radius = 1, fill = False,
color = 'black', ls = 'dashed')
ax.add_patch(uc)
plt.xlim([-3, 3])
plt.ylim([-3, 3])
plt.plot([z.real for z in P_z], [z.imag for z in P_z],
'b+', markersize = 12, mew = 3,
label = 'Roots of $P(z)$')
plt.plot([z.real for z in Q_zeros], [z.imag for z in Q_zeros],
'rx', markersize = 12, mew = 3,
label = 'Roots of $\sigma Q(z)$')
plt.legend()
plt.gca().set_aspect('equal')
plt.title('Spectral Factorization')
plt.xlabel('Real Part')
plt.ylabel('Imaginary Part')
plt.show()
return
def test2():
'''
Performs a number of randomized tests with visualizations
'''
N = 10 #Number of tests
p = 7 #Number of roots
np.random.seed(1)
for i in range(N):
K = stats.expon.rvs(scale = 5)
b, H = random_arma(p, 0, k = K, p_radius = 1) #Random filter
H = b[0]*H
P = polymul(H[::-1], H) #Creates a valid PSD polynomial
#Pass in only the causal portion
Q, s = spectral_factorization(P[len(P)/2:])
Q_zeros = roots(Q)
P_zeros = roots(P)
fig = plt.figure()
ax = fig.add_subplot(1,1,1)
uc = patches.Circle((0, 0), radius = 1, fill = False,
color = 'black', ls = 'dashed')
ax.add_patch(uc)
plt.xlim([-3, 3])
plt.ylim([-3, 3])
plt.plot([z.real for z in P_zeros], [z.imag for z in P_zeros],
'b+', markersize = 12, mew = 3,
label = 'Roots of $P(z)$')
plt.plot([z.real for z in Q_zeros], [z.imag for z in Q_zeros],
'rx', markersize = 12, mew = 3,
label = 'Roots of $\sigma Q(z)$')
plt.legend()
plt.title('Spectral Factorization')
plt.xlabel('Real Part')
plt.ylabel('Imaginary Part')
plt.gca().set_aspect('equal')
plt.show()
return
def test3():
'''
Performs randomized tests with less visualization.
'''
N = 100 #Number of tests
p = [3, 7, 11, 16, 21, 35, 70, 141, 211] #Number of roots
np.random.seed(1)
#Tolerances for our checks
tols = [1./(10**i) for i in range(1, 10)]
factorization_fails = np.array([0]*len(p))
for p_index, num_roots in enumerate(p):
if prog_bar:
pbar = ProgressBar(widgets = ('p = %d ' % num_roots,
Bar(), Percentage(),
ETA()), maxval = N).start()
num_fails = np.array([0]*len(tols))
for trial_num in range(N):
if prog_bar:
pbar.update(trial_num)
K = stats.expon.rvs(scale = 5) #gain term
b, H = random_arma(num_roots, 0, k = K, p_radius = 1) #Random filter
H = b*H
#Creates a valid PSD polynomial
#Only the causal portion is needed
P = polymul(H[::-1], H)
try:
Q, s = spectral_factorization(P[len(P)/2:])
except FactorizationError:
factorization_fails[p_index] += 1
num_fails += 1
continue
Q_zeros = np.sort(roots(Q))
P_minphase_zeros = np.sort(roots(H))
for (atol_i, atol) in enumerate(tols):
if not (np.allclose(Q_zeros, P_minphase_zeros,
rtol = 0, atol = atol) and
np.isclose(K, s, rtol = 0, atol = atol)):
num_fails[atol_i] += 1
try:
total_fails = np.vstack((total_fails, num_fails))
except NameError:
total_fails = num_fails
if prog_bar:
pbar.finish()
total_fails = total_fails / float(N) #Make everything a ratio
if tab:
print 'Fraction of time (over %d trials) accuracy not within... '\
'(oo indicates a failure to return a valid factorization)' % N
headers = ['p', 'oo'] + tols
table = []
for i, pi in enumerate(p):
row_i = [pi] + [factorization_fails[i]] + list(total_fails[i])
table.append(row_i)
print tabulate(table, headers = headers)
results_file = open('test_results.txt', 'w')
results_file.write('Fraction of time (over %d trials) accuracy '\
'not within...\n(oo indicates a failure to '\
'return a valid factorization)\n' % N)
results_file.write(tabulate(table, headers = headers))
results_file.write('\n')
results_file.close()
return
