S = BasisSpline(n, a, b, f=ff)
L = BasisSpline(n, a, b, k=1, f=ff)
# Compute actual and fitted values on grid
y = ff(x) # actual
yc = C(x) # Chebychev approximant
ys = S(x) # cubic spline approximant
yl = L(x) # linear spline approximant
# Plot function approximations
plt.figure()
ymin = np.floor(y.min())
ymax = np.ceil(y.max())
xlim = [a, b]
ylim = [-0.2, 1.2] if ifunc == 2 else [ymin, ymax]
subfig(1, x, y, xlim, ylim, 'Function')
subfig(2, x, yc, xlim, ylim, 'Chebyshev')
subfig(3, x, ys, xlim, ylim, 'Cubic Spline')
subfig(4, x, yl, xlim, ylim, 'Linear Spline')
# Plot function approximation error
plt.figure()
plt.plot(x, np.c_[yc - y, ys - y], linewidth=3)
plt.xlabel('x')
plt.ylabel('Approximation Error')
plt.legend(['Chebychev Polynomial', 'Cubic Spline'])
# pltlegend('boxoff')
plt.show()
# Simulate Model
T = 20
nrep = 50000
sinit = np.full((1, nrep), smin)
data = growth.simulate(T, sinit)
# Plot Simulated State Path
subdata = data[data['_rep'] < 3][['time', 'wealth', '_rep']]
subdata.pivot(index='time', columns='_rep', values='wealth').plot(legend=False,
lw=1)
data.groupby('time')['wealth'].mean().plot(color='k', linestyle='--')
plt.title('Simulated and Expected Wealth')
plt.xlabel('Period')
plt.ylabel('Wealth')
# Plot Simulated Policy Path
subdata = data[data['_rep'] < 3][['time', 'investment', '_rep']]
subdata.pivot(index='time', columns='_rep',
values='investment').plot(legend=False, lw=1)
plt.plot(data.groupby('time')['investment'].mean(), 'k-')
plt.title('Simulated and Expected Investment')
plt.xlabel('Period')
plt.ylabel('Investment')
# Print Steady-State and Ergodic Moments
ff = '\t%15s = %5.2f'
D = data[data['time'] == T][['wealth', 'investment']]
plt.figure(figsize=[12, 6])
demo.subplot(1, 2, 1, 'Chebychev Collocation Residual\nand Approximation Error', 'Wealth', 'Residual/Error')
plt.plot(Wealth, np.c_[S.resid, v - vtrue], Wealth, np.zeros_like(Wealth), 'k--')
plt.legend(['Residual','Error'], loc='lower right')
# Plot Linear-Quadratic Approximation Error
demo.subplot(1, 2, 2, 'Linear-Quadratic Approximation Error', 'Wealth', 'Error')
plt.plot(Wealth, vlq - vtrue)
# Plot State and Policy Paths
opts = dict(spec='r*', offset=(-5, -5), fs=11, ha='right')
data[['Wealth', 'Investment']].plot()
plt.title('State and Policy Paths')
demo.annotate(T, sstar, 'steady-state wealth\n = %.2f' % sstar, **opts)
demo.annotate(T, kstar, 'steady-state investment\n = %.2f' % kstar, **opts)
plt.xlabel('Period')
plt.ylabel('Wealth/Investment')
plt.xlim([0, T + 0.5])
# Print Steady-State
print('\n\nSteady-State')
print(' Wealth = %5.4f' % sstar)
print(' Investment = %5.4f' % kstar)
plt.show()
## SIMULATION
# Simulate Model
T = 30
nrep = 100000
sinit = np.full((1, nrep), smin)
data = model.simulate(T, sinit, seed=945)
# Plot Simulated State Path
D = data[data['_rep'] < 3][['time', 'reservoir', '_rep']]
D.pivot(index='time', columns='_rep', values='reservoir').plot(legend=False, lw=1)
data.groupby('time')['reservoir'].mean().plot(color='k', linestyle='--')
plt.title('Simulated and Expected Reservoir Level')
plt.xlabel('Year')
plt.ylabel('Reservoir Level')
# Plot Simulated Policy Path
D = data[data['_rep'] < 3][['time', 'released', '_rep']]
D.pivot('time', '_rep', 'released').plot(legend=False, lw=1)
data.groupby('time')['released'].mean().plot(color='k', linestyle='--')
plt.title('Simulated and Expected Irrigation')
plt.xlabel('Year')
plt.ylabel('Irrigation')
# Print Steady-State and Ergodic Moments
ff = '\t%15s = %5.2f'
D = data[data['time'] == T]
## SIMULATION
# Simulate Model
T = 30
nrep = 100000
sinit = np.full((1, nrep), smin)
data = model.simulate(T, sinit, seed=945)
# Plot Simulated State Path
D = data[data['_rep'] < 3][['time', 'reservoir', '_rep']]
D.pivot(index='time', columns='_rep', values='reservoir').plot(legend=False,
lw=1)
data.groupby('time')['reservoir'].mean().plot(color='k', linestyle='--')
plt.title('Simulated and Expected Reservoir Level')
plt.xlabel('Year')
plt.ylabel('Reservoir Level')
# Plot Simulated Policy Path
D = data[data['_rep'] < 3][['time', 'released', '_rep']]
D.pivot('time', '_rep', 'released').plot(legend=False, lw=1)
data.groupby('time')['released'].mean().plot(color='k', linestyle='--')
plt.title('Simulated and Expected Irrigation')
plt.xlabel('Year')
plt.ylabel('Irrigation')
# Print Steady-State and Ergodic Moments
ff = '\t%15s = %5.2f'
D = data[data['time'] == T]
F.c = np.reshape(c, (2, n))
nplot = 501
t = np.linspace(0, T, nplot)
(p, s), (dp, ds) = F(t, [[0, 1]])
res_p = dp - r * p - k
res_s = ds + p ** -eta
plt.figure()
plt.subplot(2, 1, 1)
plt.plot(t, res_p)
plt.title('Residuals')
plt.ylabel('d(price) residual')
plt.subplot(2, 1, 2)
plt.plot(t, res_s)
plt.xlabel('time')
plt.ylabel('d(storage) residual')
plt.figure()
plt.subplot(2, 1, 1)
plt.plot(t, p)
plt.title('Solution')
plt.ylabel('Price')
plt.subplot(2, 1, 2)
plt.plot(t, s)
plt.xlabel('time')
plt.ylabel('Stock')
if abs(i - j) > 1:
AA[i,j] = 0
n = np.hstack((np.arange(50, 250, 50), np.arange(300, 1100, 100)))
ratio = np.empty(n.size)
for k in range(n.size):
A = AA[:n[k], :n[k]]
b = bb[:n[k]]
tt = tic()
for i in range(100):
x = solve(A, b)
toc1 = toc(tt)
S = csc_matrix(A)
tt = tic()
for i in range(100):
x = spsolve(S, b)
toc2 = toc(tt)
ratio[k] = toc2 / toc1
# Plot effort ratio
plt.figure(figsize=[6, 6])
plt.plot(n, ratio)
plt.xlabel('n')
plt.ylabel('Ratio')
plt.title('Ratio of Sparse Solve Time to Full Solve Time')
plt.show()
a, b = 0, 2
f = lambda x: x**3 + x**2 + 1
g = lambda x: x**3 + 2
p1, p2 = 1, 2
q1 = quad(lambda x: np.abs(f(x)-g(x)) ** p1, a, b)[0] ** (1 / p1)
q2 = quad(lambda x: np.abs(f(x)-g(x)) ** p2, a, b)[0] ** (1 / p2)
print('\nCompute function metrics')
print('\tnorm 1 = {:6.4f}, norm 2 = {:6.4f}'.format(q1, q2))
# Illustrate function metrics
x = np.linspace(a, b, 200)
plt.figure(figsize=[12, 4])
plt.subplot(1, 2, 1)
plt.plot([0, 2], [0, 0], 'k:', linewidth=4)
plt.plot(x, f(x) - g(x), 'b', linewidth=4, label='f - g')
plt.xlabel('x')
plt.ylabel('y')
plt.xticks([0, 1, 2])
plt.yticks([-1, 0, 1, 2, 3])
plt.title('f - g')
plt.subplot(1, 2, 2)
plt.plot(x, np.abs(f(x) - g(x)), 'b', linewidth=4, label='f - g')
plt.xlabel('x')
plt.ylabel('y')
plt.xticks([0, 1, 2])
plt.yticks([0, 1, 2, 3])
plt.title('|f - g|')
plt.show()
c = storage.broyden(print=True)
F.c = np.reshape(c, (2, n))
nplot = 501
t = np.linspace(0, T, nplot)
(p, s), (dp, ds) = F(t, [[0, 1]])
res_p = dp - r * p - k
res_s = ds + p**-eta
plt.figure()
plt.subplot(2, 1, 1)
plt.plot(t, res_p)
plt.title('Residuals')
plt.ylabel('d(price) residual')
plt.subplot(2, 1, 2)
plt.plot(t, res_s)
plt.xlabel('time')
plt.ylabel('d(storage) residual')
plt.figure()
plt.subplot(2, 1, 1)
plt.plot(t, p)
plt.title('Solution')
plt.ylabel('Price')
plt.subplot(2, 1, 2)
plt.plot(t, s)
plt.xlabel('time')
plt.ylabel('Stock')
plt.show()
