def test_kernel_parzen():
def w(k):
w = np.empty(k + 1)
for i in range(k + 1):
z = i / (k + 1)
if z > 0.5:
w[i] = 2 * (1 - z)**3
else:
w[i] = 1 - 6 * z**2 + 6 * z**3
return w
assert_equal(w(0), kernel_weight_parzen(0))
assert_equal(w(1), kernel_weight_parzen(1))
assert_equal(w(10), kernel_weight_parzen(10))
def test_kernel_weight_direct(weight_data, center, debias):
bandwidth = 12
wm = KernelWeightMatrix(center,
debias,
kernel="parzen",
bandwidth=bandwidth)
x, z, eps, sigma = weight_data
weights = wm.weight_matrix(x, z, eps, sigma=sigma)
k = len(z)
ze = [z[i] * eps[:, i:i + 1] for i in range(k)]
ze = np.concatenate(ze, 1)
if center:
ze = ze - ze.mean(0)
nobs = ze.shape[0]
direct = ze.T @ ze / nobs
w = kernel_weight_parzen(bandwidth)
for i in range(1, bandwidth + 1):
op = ze[:-i].T @ ze[i:] / nobs
direct += w[i] * (op + op.T)
if debias:
df = [vx.shape[1] * np.ones(vz.shape[1]) for vx, vz in zip(x, z)]
df = np.concatenate(df)[:, None]
df = np.sqrt(df)
adj = nobs / (nobs - df @ df.T)
direct *= adj
assert_allclose(weights, direct)
def test_kernel_direct(cov_data, debias):
x, z, eps, sigma = cov_data
k = len(x)
bandwidth = 12
cov = KernelCovariance(x, eps, sigma, sigma, gls=False, debiased=debias,
kernel='parzen', bandwidth=bandwidth)
assert cov.bandwidth == 12
xe = [x[i] * eps[:, i:i + 1] for i in range(k)]
xe = np.concatenate(xe, 1)
w = kernel_weight_parzen(12)
nobs = xe.shape[0]
xeex = np.zeros((xe.shape[1], xe.shape[1]))
xeex += xe.T @ xe / nobs
for i in range(1, bandwidth + 1):
op = xe[:-i].T @ xe[i:] / nobs
xeex += w[i] * (op + op.T)
xpxi = _xpxi(x)
direct = xpxi @ xeex @ xpxi / nobs
direct = (direct + direct.T) / 2
if debias:
df = [x[i].shape[1] * np.ones(x[i].shape[1]) for i in range(k)]
df = np.concatenate(df)[:, None]
adj = nobs / (nobs - np.sqrt(df) @ np.sqrt(df).T)
direct *= adj
direct = (direct + direct.T) / 2
assert_allclose(np.diag(direct), np.diag(cov.cov))
s = np.sqrt(np.diag(direct))[:, None]
r_direct = direct / (s @ s.T)
s = np.sqrt(np.diag(cov.cov))[:, None]
c_direct = direct / (s @ s.T)
assert_allclose(r_direct, c_direct, atol=1e-5)
