sigma = 0.1
A = [[phi_1, phi_2, phi_3, phi_4],
[1, 0, 0, 0],
[0, 1, 0, 0],
[0, 0, 1, 0]]
C = [sigma, 0, 0, 0]
G = [1, 0, 0, 0]
T = 30
ar = LSS(A, C, G)
ymin, ymax = -0.8, 1.25
fig, ax = plt.subplots(figsize=(8,4))
ax.set_xlim(ymin, ymax)
ax.set_xlabel(r'$y_t$', fontsize=16)
x, y = ar.replicate(T=T, num_reps=500000)
mu_x, mu_y, Sigma_x, Sigma_y = ar.moments(T=T, mu_0=np.ones(4))
f_y = norm(loc=float(mu_y), scale=float(np.sqrt(Sigma_y)))
y = y.flatten()
ax.hist(y, bins=50, normed=True, alpha=0.4)
ygrid = np.linspace(ymin, ymax, 150)
ax.plot(ygrid, f_y.pdf(ygrid), 'k-', lw=2, alpha=0.8, label='true density')
ax.legend()
plt.show()
