phi = lognorm(a_sigma)
def p(x, y):
d = s * x**alpha
return phi.pdf((y - (1 - delta) * x) / d) / d
# other data
n_a, n_b, n_y = 50, (5, 5), 20
a = np.random.rand(n_a) + 0.01
b = np.random.rand(*n_b) + 0.01
y = np.linspace(0, 10, 20)
lae_a = LAE(p, a)
lae_b = LAE(p, b)
def test_x_flattened():
"lae: is x flattened and reshaped"
# should have a trailing singleton dimension
assert_equal(lae_b.X.shape[-1], 1)
assert_equal(lae_a.X.shape[-1], 1)
def test_x_2d():
"lae: is x 2d"
assert_equal(lae_a.X.ndim, 2)
assert_equal(lae_b.X.ndim, 2)
Both x and y must be strictly positive.
"""
d = s * x**alpha
return phi.pdf((y - (1 - delta) * x) / d) / d
n = 10000 # Number of observations at each date t
T = 30 # Compute density of k_t at 1,...,T+1
# == Generate matrix s.t. t-th column is n observations of k_t == #
k = np.empty((n, T))
A = phi.rvs((n, T))
k[:, 0] = psi_0.rvs(n) # Draw first column from initial distribution
for t in range(T - 1):
k[:, t + 1] = s * A[:, t] * k[:, t]**alpha + (1 - delta) * k[:, t]
# == Generate T instances of LAE using this data, one for each date t == #
laes = [LAE(p, k[:, t]) for t in range(T)]
# == Plot == #
fig, ax = plt.subplots()
ygrid = np.linspace(0.01, 4.0, 200)
greys = [str(g) for g in np.linspace(0.0, 0.8, T)]
greys.reverse()
for psi, g in zip(laes, greys):
ax.plot(ygrid, psi(ygrid), color=g, lw=2, alpha=0.6)
ax.set_xlabel('capital')
title = r'Density of $k_1$ (lighter) to $k_T$ (darker) for $T={}$'
ax.set_title(title.format(T))
plt.show()
def psi_star(y):
return 2 * norm.pdf(y) * norm.cdf(delta * y)
def p(x, y):
return phi.pdf((y - theta * np.abs(x)) / d) / d
#generate n random number from normal distribution
Z = phi.rvs(n)
X = np.empty(n)
for t in range(n - 1):
X[t + 1] = theta * np.abs(X[t]) + d * Z[t]
psi_est = LAE(p, X)
k_est = gaussian_kde(X)
fig, ax = plt.subplots(figsize=(10, 7))
#generate 200 numbers from -3 to 3
ys = np.linspace(-3, 3, 200)
ax.plot(ys, psi_star(ys), "b-", lw=2, alpha=0.6, label="true")
ax.plot(ys, psi_est(ys), "g-", lw=2, alpha=0.6, label="look ahead estimate")
ax.plot(ys, k_est(ys), "k-", lw=2, alpha=0.6, label="kernel based estimate")
ax.legend(loc="upper left")
#.show()