from demos.setup import np, plt from compecon.quad import qnwlogn from compecon.tools import nodeunif from mpl_toolkits.mplot3d import Axes3D from matplotlib import cm # Univariate Taylor approximation x = np.linspace(-1, 1, 100) y = (x + 1) * np.exp(2 * x) y1 = 1 + 3 * x y2 = 1 + 3 * x + 8 * x**2 plt.figure(figsize=[6, 6]) plt.plot(x, y, 'k', linewidth=3, label='Function') plt.plot(x, y1, 'b', linewidth=3, label='1st order approximation') plt.plot(x, y2, 'r', linewidth=3, label='2nd order approximation') plt.legend() plt.xticks([-1, 0, 1]) plt.show() ## Bivariate Taylor approximation nplot = [101, 101] a = [0, -1] b = [2, 1] x1, x2 = nodeunif(nplot, a, b) x1.shape = nplot x2.shape = nplot y = np.exp(x2) * x1**2
return p * (alpha[0] + alpha[1] * a + alpha[2] * a ** 2 )
def transition(p, x, i, j, in_, e):
return pbar + gamma * (p - pbar) + e
model = DPmodel(basis, profit, transition,
# i=['a={}'.format(a+1) for a in range(A)],
i=[a + 1 for a in range(A)],
j=['keep', 'replace'],
discount=delta, e=e, w=w, h=h)
# SOLUTION
S = model.solve()
pr = np.linspace(pmin, pmax, 10 * n)
# Plot Action-Contingent Value Functions
pp = demo.qplot('unit profit', 'value_j', 'i',
data=S,
main='Action-Contingent Value Functions',
xlab='Net Unit Profit',
ylab='Value')
print(pp)
'''
frm = '{:21} {:6.3f} {:8.1e} {:7.6f}'
prt = lambda d, t, n, x: print(frm.format(d, t, n, *x))
print('{:21} {:^6} {:^8} {:^7}\n{}'.format('Algorithm','Time','Norm','x','-' * 51));
prt('Newton minmax', t1, n1, x1)
prt('Newton semismooth', t2, n2, x2)
# ### Plot results
# Here we use the methods *ssmooth* and *minmax* from class **MCP** to compute the semi-smooth and minimax transformations.
# In[7]:
fig = plt.figure()
original = {'label':'Original', 'alpha':0.5, 'color':'gray'}
x = np.linspace(-0.5, 2.5, 500)
ax1 = fig.add_subplot(121, title='Difficult NCP', aspect=1,
xlabel='x', xlim=[-0.5, 2.5], ylim=[-1, 1.5])
ax1.axhline(ls='--', color='gray')
ax1.plot(x, billups(x)[0], **original)
ax1.plot(x, Billups.ssmooth(x), label='Semismooth')
ax1.plot(x, Billups.minmax(x), label='Minmax')
ax1.legend(loc='best')
x = np.linspace(-0.03, 0.03, 500)
ax2 = fig.add_subplot(122, title='Difficult NCP Magnified', aspect=1,
xlabel='x', xlim = [-.035, .035], ylim=[ -.01, .06])
ax2.axhline(ls='--', color='gray')
ax2.plot(x, Billups.original(x), **original)
ax2.plot(x, Billups.ssmooth(x), label='Semismooth')
from compecon.tools import gridmake, getindex
## DEMDDP07 Renewable resource model
# Model Parameters
delta = 0.9 # discount factor
alpha = 4.0 # growth function parameter
beta = 1.0 # growth function parameter
gamma = 0.5 # demand function parameter
cost = 0.2 # unit cost of harvest
# State Space
smin = 0 # minimum state
smax = 8 # maximum state
n = 200 # number of states
S = np.linspace(smin, smax, n) # vector of states
# Action Space
xmin = 0 # minimum action
xmax = 6 # maximum action
m = 100 # number of actions
X = np.linspace(xmin, xmax, m) # vector of actions
# Reward Function
f = np.full((m, n), -np.inf)
for k in range(m):
f[k, S >= X[k]] = (X[k] ** (1 - gamma)) / (1 - gamma) - cost * X[k]
# State Transition Function
from compecon.tools import gridmake, getindex
## DEMDDP07 Renewable resource model
# Model Parameters
delta = 0.9 # discount factor
alpha = 4.0 # growth function parameter
beta = 1.0 # growth function parameter
gamma = 0.5 # demand function parameter
cost = 0.2 # unit cost of harvest
# State Space
smin = 0 # minimum state
smax = 8 # maximum state
n = 200 # number of states
S = np.linspace(smin, smax, n) # vector of states
# Action Space
xmin = 0 # minimum action
xmax = 6 # maximum action
m = 100 # number of actions
X = np.linspace(xmin, xmax, m) # vector of actions
# Reward Function
f = np.full((m, n), -np.inf)
for k in range(m):
f[k, S >= X[k]] = (X[k]**(1 - gamma)) / (1 - gamma) - cost * X[k]
# State Transition Function
g = np.zeros_like(f)
from compecon.tools import nodeunif
from mpl_toolkits.mplot3d import Axes3D
from matplotlib import cm
""" Approximating using the CompEcon toolbox """
'''Univariate approximation'''
# Define function and derivative
f1 = lambda x: np.exp(-2 * x)
d1 = lambda x: -2 * np.exp(-2 * x)
# Fit approximant
n, a, b = 10, -1, 1
f1fit = BasisChebyshev(n, a, b, f=f1)
# Graph approximation error for function and derivative
axopts = {'xlabel': 'x', 'ylabel': 'Error', 'xticks': [-1, 0, 1]}
x = np.linspace(a, b, 1001)
fig = plt.figure(figsize=[12, 6])
ax1 = fig.add_subplot(121, title='Function approximation error', **axopts)
ax1.axhline(linestyle='--', color='gray', linewidth=2)
ax1.plot(f1fit.nodes, np.zeros_like(f1fit.nodes), 'ro', markersize=12)
ax1.plot(x, f1fit(x) - f1(x))
ax2 = fig.add_subplot(122, title='Derivative approximation error', **axopts)
ax2.plot(x, np.zeros_like(x), '--', color='gray', linewidth=2)
ax2.plot(f1fit.nodes, np.zeros_like(f1fit.nodes), 'ro', markersize=12)
ax2.plot(x, f1fit(x, 1) - d1(x))
''' Bivariate Interpolation '''
# Define function
f2 = lambda x: np.cos(x[0]) / np.exp(x[1])
def resid(c, tnodes, T, n, F, r, k, eta, s0):
F.c = np.reshape(c[:], (2, n))
(p, s), d = F(tnodes, [[0, 1]])
d[0] -= (r * p + k)
d[1] += p ** -eta
(p_0, p_T), (s_0, s_T) = F([0, T])
return np.r_[d.flatten(), s_0 - s0, s_T]
storage = NLP(resid, F.c.flatten(), tnodes, T, n, F, r, k, eta, s0)
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')
import warnings
warnings.simplefilter('ignore')
""" Uniform-node and Chebyshev-node polynomial approximation of Runge's function
and compute condition numbers of associated interpolation matrices
"""
# Runge function
runge = lambda x: 1 / (1 + 25 * x**2)
# Set points of approximation interval
a, b = -1, 1
# Construct plotting grid
nplot = 1001
x = np.linspace(a, b, nplot)
y = runge(x)
# Plot Runge's Function
fig1 = plt.figure(figsize=[6, 9])
ax1 = fig1.add_subplot(211,
title="Runge's Function",
xlabel='',
ylabel='y',
xticks=[])
ax1.plot(x, y)
ax1.text(-0.8, 0.8, r'$y = \frac{1}{1+25x^2}$', fontsize=18)
# Initialize data matrices
n = np.arange(3, 33, 2)
nn = n.size
frm = '{:21} {:6.3f} {:8.1e} {:7.6f}'
prt = lambda d, t, n, x: print(frm.format(d, t, n, *x))
print('{:21} {:^6} {:^8} {:^7}\n{}'.format('Algorithm', 'Time', 'Norm',
'x', '-' * 51))
prt('Newton minmax', t1, n1, x1)
prt('Newton semismooth', t2, n2, x2)
# ### Plot results
# Here we use the methods *ssmooth* and *minmax* from class **MCP** to compute the semi-smooth and minimax transformations.
# In[7]:
fig = plt.figure()
original = {'label': 'Original', 'alpha': 0.5, 'color': 'gray'}
x = np.linspace(-0.5, 2.5, 500)
ax1 = fig.add_subplot(121,
title='Difficult NCP',
aspect=1,
xlabel='x',
xlim=[-0.5, 2.5],
ylim=[-1, 1.5])
ax1.axhline(ls='--', color='gray')
ax1.plot(x, billups(x)[0], **original)
ax1.plot(x, Billups.ssmooth(x), label='Semismooth')
ax1.plot(x, Billups.minmax(x), label='Minmax')
ax1.legend(loc='best')
x = np.linspace(-0.03, 0.03, 500)
ax2 = fig.add_subplot(122,
""" Approximating using the CompEcon toolbox """
'''Univariate approximation'''
# Define function and derivative
f1 = lambda x: np.exp(-2 * x)
d1 = lambda x: -2 * np.exp(-2 * x)
# Fit approximant
n, a, b = 10, -1, 1
f1fit = BasisChebyshev(n, a, b, f=f1)
# Graph approximation error for function and derivative
axopts = {'xlabel': 'x', 'ylabel': 'Error', 'xticks': [-1, 0, 1]}
x = np.linspace(a, b, 1001)
fig = plt.figure(figsize=[12, 6])
ax1 = fig.add_subplot(121, title='Function approximation error', **axopts)
ax1.axhline(linestyle='--', color='gray', linewidth=2)
ax1.plot(f1fit.nodes, np.zeros_like(f1fit.nodes), 'ro', markersize=12)
ax1.plot(x, f1fit(x) - f1(x))
ax2 = fig.add_subplot(122, title='Derivative approximation error', **axopts)
ax2.plot(x, np.zeros_like(x), '--', color='gray', linewidth=2)
ax2.plot(f1fit.nodes, np.zeros_like(f1fit.nodes), 'ro', markersize=12)
ax2.plot(x, f1fit(x, 1) - d1(x))
''' Bivariate Interpolation '''
# Define function
# Set points of approximation interval # In[3]: a, b = -1, 1 # Construct plotting grid # In[4]: nplot = 1001 x = np.linspace(a, b, nplot) y = runge(x) # Plot Runge's Function # Initialize data matrices # In[5]: n = np.arange(3, 33, 2) nn = n.size errunif, errcheb = (np.zeros([nn, nplot]) for k in range(2)) nrmunif, nrmcheb, conunif, concheb = (np.zeros(nn) for k in range(4))
'\nThe approximate second derivative of exp(-x) at x=0 is {:.15f}'.format(
d2))
print(
"The 'exact' second derivative of exp(-x) at x=0 is {:.15f}".format(1))
# ... and one may even evaluate the approximant's definite integral between the left endpoint a and x:
int = F(x, -1)
print('\nThe approximate integral of exp(-x) between x=-1 and x=0 is {:.15f}'.
format(int))
print("The 'exact' integral of exp(-x) between x=-1 and x=0 is {:.15f}".
format(np.exp(1) - 1))
# One may evaluate the accuracy of the Chebychev polynomial approximant by
# computing the approximation error on a highly refined grid of points:
ngrid = 5001 # number of grid nodes
xgrid = np.linspace(a, b, ngrid) # generate refined grid for plotting
yapp = F(xgrid) # approximant values at grid nodes
yact = f(xgrid) # actual function values at grid points
demo.figure('Chebychev Approximation Error for exp(-x)', 'x', 'Error')
plt.plot(xgrid, yapp - yact)
plt.plot(xgrid, np.zeros(ngrid), 'k--', linewidth=2)
# The plot indicates that an order 10 Chebychev approximation scheme, produces approximation errors
# no bigger in magnitude than 6x10^-10. The approximation error exhibits the "Chebychev equioscillation
# property", oscilating relatively uniformly throughout the approximation domain.
#
# This commonly occurs when function being approximated is very smooth, as is the case here but should not
# be expected when the function is not smooth. Further notice how the approximation error is exactly 0 at the
# approximation nodes --- which is true by contruction.
print('\nCompute function norm')
print('\tnorm 1 = {:6.4f}, norm 2 = {:6.4f}'.format(q1, q2))
# Compute Function Metrics
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])
def resid(c, tnodes, T, n, F, r, k, eta, s0):
F.c = np.reshape(c[:], (2, n))
(p, s), d = F(tnodes, [[0, 1]])
d[0] -= (r * p + k)
d[1] += p**-eta
(p_0, p_T), (s_0, s_T) = F([0, T])
return np.r_[d.flatten(), s_0 - s0, s_T]
storage = NLP(resid, F.c.flatten(), tnodes, T, n, F, r, k, eta, s0)
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()