def time_iteration(model,
dr0=None,
dprocess=None,
with_complementarities=True,
verbose=True,
grid={},
maxit=1000,
inner_maxit=10,
tol=1e-6,
hook=None,
details=False,
interp_method='cubic'):
'''
Finds a global solution for ``model`` using backward time-iteration.
This algorithm iterates on the residuals of the arbitrage equations
Parameters
----------
model : Model
model to be solved
verbose : boolean
if True, display iterations
dr0 : decision rule
initial guess for the decision rule
dprocess : DiscretizedProcess (model.exogenous.discretize())
discretized process to be used
with_complementarities : boolean (True)
if False, complementarity conditions are ignored
grid: grid options
overload the values set in `options:grid` section
maxit: maximum number of iterations
inner_maxit: maximum number of iteration for inner solver
tol: tolerance criterium for successive approximations
hook: Callable
function to be called within each iteration, useful for debugging purposes
Returns
-------
decision rule :
approximated solution
'''
from dolo import dprint
def vprint(t):
if verbose:
print(t)
if dprocess is None:
dprocess = model.exogenous.discretize()
n_ms = dprocess.n_nodes # number of exogenous states
n_mv = dprocess.n_inodes(
0) # this assume number of integration nodes is constant
x0 = model.calibration['controls']
parms = model.calibration['parameters']
n_x = len(x0)
n_s = len(model.symbols['states'])
endo_grid = model.get_endo_grid(**grid)
exo_grid = dprocess.grid
mdr = DecisionRule(exo_grid,
endo_grid,
dprocess=dprocess,
interp_method=interp_method)
grid = mdr.endo_grid.nodes
N = grid.shape[0]
controls_0 = numpy.zeros((n_ms, N, n_x))
if dr0 is None:
controls_0[:, :, :] = x0[None, None, :]
else:
if isinstance(dr0, AlgoResult):
dr0 = dr0.dr
try:
for i_m in range(n_ms):
controls_0[i_m, :, :] = dr0(i_m, grid)
except Exception:
for i_m in range(n_ms):
m = dprocess.node(i_m)
controls_0[i_m, :, :] = dr0(m, grid)
f = model.functions['arbitrage']
g = model.functions['transition']
if 'controls_lb' in model.functions and with_complementarities == True:
lb_fun = model.functions['controls_lb']
ub_fun = model.functions['controls_ub']
lb = numpy.zeros_like(controls_0) * numpy.nan
ub = numpy.zeros_like(controls_0) * numpy.nan
for i_m in range(n_ms):
m = dprocess.node(i_m)[None, :]
p = parms[None, :]
m = numpy.repeat(m, N, axis=0)
p = numpy.repeat(p, N, axis=0)
lb[i_m, :, :] = lb_fun(m, grid, p)
ub[i_m, :, :] = ub_fun(m, grid, p)
else:
with_complementarities = False
sh_c = controls_0.shape
controls_0 = controls_0.reshape((-1, n_x))
from dolo.numeric.optimize.newton import newton, SerialDifferentiableFunction
from dolo.numeric.optimize.ncpsolve import ncpsolve
err = 10
it = 0
if with_complementarities:
lb = lb.reshape((-1, n_x))
ub = ub.reshape((-1, n_x))
if verbose:
headline = '|{0:^4} | {1:10} | {2:8} | {3:8} | {4:3} |'.format(
'N', ' Error', 'Gain', 'Time', 'nit')
stars = '-' * len(headline)
print(stars)
print(headline)
print(stars)
import time
t1 = time.time()
err_0 = numpy.nan
verbit = (verbose == 'full')
while err > tol and it < maxit:
it += 1
t_start = time.time()
mdr.set_values(controls_0.reshape(sh_c))
fn = lambda x: residuals_simple(f, g, grid, x.reshape(sh_c), mdr,
dprocess, parms).reshape((-1, n_x))
dfn = SerialDifferentiableFunction(fn)
res = fn(controls_0)
if hook:
hook()
if with_complementarities:
[controls, nit] = ncpsolve(dfn,
lb,
ub,
controls_0,
verbose=verbit,
maxit=inner_maxit)
else:
[controls, nit] = newton(dfn,
controls_0,
verbose=verbit,
maxit=inner_maxit)
err = abs(controls - controls_0).max()
err_SA = err / err_0
err_0 = err
controls_0 = controls
t_finish = time.time()
elapsed = t_finish - t_start
if verbose:
print('|{0:4} | {1:10.3e} | {2:8.3f} | {3:8.3f} | {4:3} |'.format(
it, err, err_SA, elapsed, nit))
controls_0 = controls.reshape(sh_c)
mdr.set_values(controls_0)
t2 = time.time()
if verbose:
print(stars)
print("Elapsed: {} seconds.".format(t2 - t1))
print(stars)
if not details:
return mdr
return TimeIterationResult(
mdr,
it,
with_complementarities,
dprocess,
err < tol, # x_converged: bool
tol, # x_tol
err, #: float
None, # log: object # TimeIterationLog
None # trace: object #{Nothing,IterationTrace}
)
def evaluate_policy(model,
mdr,
tol=1e-8,
maxit=2000,
grid={},
verbose=True,
initial_guess=None,
hook=None,
integration_orders=None,
details=False,
interp_type='cubic'):
"""Compute value function corresponding to policy ``dr``
Parameters:
-----------
model:
"dtcscc" model. Must contain a 'value' function.
mdr:
decision rule to evaluate
Returns:
--------
decision rule:
value function (a function of the space similar to a decision rule
object)
"""
process = model.exogenous
dprocess = process.discretize()
n_ms = dprocess.n_nodes() # number of exogenous states
n_mv = dprocess.n_inodes(
0) # this assume number of integration nodes is constant
x0 = model.calibration['controls']
v0 = model.calibration['values']
parms = model.calibration['parameters']
n_x = len(x0)
n_v = len(v0)
n_s = len(model.symbols['states'])
endo_grid = model.get_grid(**grid)
exo_grid = dprocess.grid
if initial_guess is not None:
mdrv = initial_guess
else:
mdrv = DecisionRule(exo_grid, endo_grid, interp_type=interp_type)
grid = mdrv.endo_grid.nodes()
N = grid.shape[0]
if isinstance(mdr, np.ndarray):
controls = mdr
else:
controls = np.zeros((n_ms, N, n_x))
for i_m in range(n_ms):
controls[i_m, :, :] = mdr.eval_is(i_m, grid)
values_0 = np.zeros((n_ms, N, n_v))
if initial_guess is None:
for i_m in range(n_ms):
values_0[i_m, :, :] = v0[None, :]
else:
for i_m in range(n_ms):
values_0[i_m, :, :] = initial_guess.eval_is(i_m, grid)
val = model.functions['value']
g = model.functions['transition']
sh_v = values_0.shape
err = 10
inner_maxit = 50
it = 0
if verbose:
headline = '|{0:^4} | {1:10} | {2:8} | {3:8} |'.format(
'N', ' Error', 'Gain', 'Time')
stars = '-' * len(headline)
print(stars)
print(headline)
print(stars)
t1 = time.time()
err_0 = np.nan
verbit = (verbose == 'full')
while err > tol and it < maxit:
it += 1
t_start = time.time()
mdrv.set_values(values_0.reshape(sh_v))
values = update_value(val, g, grid, controls, values_0, mdr, mdrv,
dprocess, parms).reshape((-1, n_v))
err = abs(values.reshape(sh_v) - values_0).max()
err_SA = err / err_0
err_0 = err
values_0 = values.reshape(sh_v)
t_finish = time.time()
elapsed = t_finish - t_start
if verbose:
print('|{0:4} | {1:10.3e} | {2:8.3f} | {3:8.3f} |'.format(
it, err, err_SA, elapsed))
# values_0 = values.reshape(sh_v)
t2 = time.time()
if verbose:
print(stars)
print("Elapsed: {} seconds.".format(t2 - t1))
print(stars)
if not details:
return mdrv
else:
return EvaluationResult(mdrv, it, tol, err)
def improved_time_iteration(
model,
method="jac",
dr0=None,
dprocess=None,
interp_method="cubic",
mu=2,
maxbsteps=10,
verbose=False,
tol=1e-8,
smaxit=500,
maxit=1000,
complementarities=True,
compute_radius=False,
invmethod="iti",
details=True,
):
def vprint(*args, **kwargs):
if verbose:
print(*args, **kwargs)
itprint = IterationsPrinter(
("N", int),
("f_x", float),
("d_x", float),
("Time_residuals", float),
("Time_inversion", float),
("Time_search", float),
("Lambda_0", float),
("N_invert", int),
("N_search", int),
verbose=verbose,
)
itprint.print_header("Start Improved Time Iterations.")
f = model.functions["arbitrage"]
g = model.functions["transition"]
x_lb = model.functions["arbitrage_lb"]
x_ub = model.functions["arbitrage_ub"]
parms = model.calibration["parameters"]
if dprocess is not None:
from dolo.numeric.grids import ProductGrid
endo_grid = model.domain.discretize()
grid = ProductGrid(dprocess.grid, endo_grid, names=["exo", "endo"])
else:
grid, dprocess = model.discretize()
endo_grid = grid["endo"]
exo_grid = grid["exo"]
n_m = max(dprocess.n_nodes, 1)
n_s = len(model.symbols["states"])
if interp_method in ("cubic", "linear"):
ddr = DecisionRule(
exo_grid, endo_grid, dprocess=dprocess, interp_method=interp_method
)
ddr_filt = DecisionRule(
exo_grid, endo_grid, dprocess=dprocess, interp_method=interp_method
)
else:
raise Exception("Unsupported interpolation method.")
# s = ddr.endo_grid
s = endo_grid.nodes
N = s.shape[0]
n_x = len(model.symbols["controls"])
x0 = (
model.calibration["controls"][
None,
None,
]
.repeat(n_m, axis=0)
.repeat(N, axis=1)
)
if dr0 is not None:
for i_m in range(n_m):
x0[i_m, :, :] = dr0.eval_is(i_m, s)
ddr.set_values(x0)
steps = 0.5 ** numpy.arange(maxbsteps)
lb = x0.copy()
ub = x0.copy()
for i_m in range(n_m):
m = dprocess.node(i_m)
lb[i_m, :] = x_lb(m, s, parms)
ub[i_m, :] = x_ub(m, s, parms)
x = x0
# both affect the precision
ddr.set_values(x)
## memory allocation
n_im = dprocess.n_inodes(0) # we assume it is constant for now
jres = numpy.zeros((n_m, n_im, N, n_x, n_x))
S_ij = numpy.zeros((n_m, n_im, N, n_s))
for it in range(maxit):
jres[...] = 0.0
S_ij[...] = 0.0
t1 = time.time()
# compute derivatives and residuals:
# res: residuals
# dres: derivatives w.r.t. x
# jres: derivatives w.r.t. ~x
# fut_S: future states
ddr.set_values(x)
#
# ub[ub>100000] = 100000
# lb[lb<-100000] = -100000
#
# sh_x = x.shape
# rr =euler_residuals(f,g,s,x,ddr,dp,parms, diff=False, with_jres=False,set_dr=True)
# print(rr.shape)
#
# from iti.fb import smooth_
# jj = smooth_(rr, x, lb, ub)
#
# print("Errors with complementarities")
# print(abs(jj.max()))
#
# exit()
#
from dolo.numeric.optimize.newton import SerialDifferentiableFunction
sh_x = x.shape
ff = SerialDifferentiableFunction(
lambda u: euler_residuals(
f,
g,
s,
u.reshape(sh_x),
ddr,
dprocess,
parms,
diff=False,
with_jres=False,
set_dr=False,
).reshape((-1, sh_x[2]))
)
res, dres = ff(x.reshape((-1, sh_x[2])))
res = res.reshape(sh_x)
dres = dres.reshape((sh_x[0], sh_x[1], sh_x[2], sh_x[2]))
junk, jres, fut_S = euler_residuals(
f,
g,
s,
x,
ddr,
dprocess,
parms,
diff=False,
with_jres=True,
set_dr=False,
jres=jres,
S_ij=S_ij,
)
# if there are complementerities, we modify derivatives
if complementarities:
res, dres, jres = smooth(res, dres, jres, x - lb)
res[...] *= -1
dres[...] *= -1
jres[...] *= -1
res, dres, jres = smooth(res, dres, jres, ub - x, pos=-1.0)
res[...] *= -1
dres[...] *= -1
jres[...] *= -1
err_0 = abs(res).max()
# premultiply by A
jres[...] *= -1.0
for i_m in range(n_m):
for j_m in range(n_im):
M = jres[i_m, j_m, :, :, :]
X = dres[i_m, :, :, :].copy()
sol = solve_tensor(X, M)
t2 = time.time()
# new version
if invmethod == "gmres":
ddx = solve_gu(dres.copy(), res.copy())
L = Operator(jres, fut_S, ddr_filt)
n0 = L.counter
L.addid = True
ttol = err_0 / 100
sol = scipy.sparse.linalg.gmres(
L, ddx.ravel(), tol=ttol
) # , maxiter=1, restart=smaxit)
lam0 = 0.01
nn = L.counter - n0
tot = sol[0].reshape(ddx.shape)
else:
# compute inversion
tot, nn, lam0 = invert_jac(
res,
dres,
jres,
fut_S,
ddr_filt,
tol=tol,
maxit=smaxit,
verbose=(verbose == "full"),
)
# lam, lam_max, lambdas = radius_jac(res,dres,jres,fut_S,tol=tol,maxit=1000,verbose=(verbose=='full'),filt=ddr_filt)
# backsteps
t3 = time.time()
for i_bckstps, lam in enumerate(steps):
new_x = x - tot * lam
new_err = euler_residuals(
f, g, s, new_x, ddr, dprocess, parms, diff=False, set_dr=True
)
if complementarities:
new_err = smooth_nodiff(new_err, new_x - lb)
new_err = smooth_nodiff(-new_err, ub - new_x)
new_err = abs(new_err).max()
if new_err < err_0:
break
err_2 = abs(tot).max()
t4 = time.time()
itprint.print_iteration(
N=it,
f_x=err_0,
d_x=err_2,
Time_residuals=t2 - t1,
Time_inversion=t3 - t2,
Time_search=t4 - t3,
Lambda_0=lam0,
N_invert=nn,
N_search=i_bckstps,
)
if err_0 < tol:
break
x = new_x
ddr.set_values(x)
itprint.print_finished()
# if compute_radius:
# return ddx,L
# lam, lam_max, lambdas = radius_jac(res,dres,jres,fut_S,ddr_filt,tol=tol,maxit=smaxit,verbose=(verbose=='full'))
# return ddr, lam, lam_max, lambdas
# else:
if not details:
return ddr
else:
ddx = solve_gu(dres.copy(), res.copy())
L = Operator(jres, fut_S, ddr_filt)
if compute_radius:
lam = scipy.sparse.linalg.eigs(L, k=1, return_eigenvectors=False)
lam = abs(lam[0])
else:
lam = np.nan
# lam, lam_max, lambdas = radius_jac(res,dres,jres,fut_S,ddr_filt,tol=tol,maxit=smaxit,verbose=(verbose=='full'))
return ImprovedTimeIterationResult(
ddr, it, err_0, err_2, err_0 < tol, complementarities, lam, None, L
)
def simulate(
model: Model,
dr: DecisionRule,
*,
process=None,
N=1,
T=40,
s0=None,
i0=None,
m0=None,
driving_process=None,
seed=42,
stochastic=True,
):
"""Simulate a model using the specified decision rule.
Parameters
----------
model: Model
dr: decision rule
process:
s0: ndarray
initial state where all simulations start
driving_process: ndarray
realization of exogenous driving process (drawn randomly if None)
N: int
number of simulations
T: int
horizon for the simulations
seed: int
used to initialize the random number generator. Use it to replicate
exact same results among simulations
discard: boolean (False)
if True, then all simulations containing at least one non finite value
are discarded
Returns
-------
xarray.DataArray:
returns a ``T x N x n_v`` array where ``n_v``
is the number of variables.
"""
if isinstance(dr, AlgoResult):
dr = dr.dr
calib = model.calibration
parms = numpy.array(calib["parameters"])
if s0 is None:
s0 = calib["states"]
n_x = len(model.symbols["controls"])
n_s = len(model.symbols["states"])
s_simul = numpy.zeros((T, N, n_s))
x_simul = numpy.zeros((T, N, n_x))
s_simul[0, :, :] = s0[None, :]
# are we simulating a markov chain or a continuous process ?
if driving_process is not None:
if len(driving_process.shape) == 3:
m_simul = driving_process
sim_type = "continuous"
if m0 is None:
m0 = model.calibration["exogenous"]
x_simul[0, :, :] = dr.eval_ms(m0[None, :], s0[None, :])[0, :]
elif len(driving_process.shape) == 2:
i_simul = driving_process
nodes = dr.exo_grid.nodes
m_simul = nodes[i_simul]
# inds = i_simul.ravel()
# m_simul = np.reshape( np.concatenate( [nodes[i,:][None,:] for i in inds.ravel()], axis=0 ), inds.shape + (-1,) )
sim_type = "discrete"
x_simul[0, :, :] = dr.eval_is(i0, s0[None, :])[0, :]
else:
raise Exception("Incorrect specification of driving values.")
m0 = m_simul[0, :, :]
else:
from dolo.numeric.processes import DiscreteProcess
if process is None:
if hasattr(dr, "dprocess") and hasattr(dr.dprocess, "simulate"):
process = dr.dprocess
else:
process = model.exogenous
# detect type of simulation
if not isinstance(process, DiscreteProcess):
sim_type = "continuous"
else:
sim_type = "discrete"
if sim_type == "discrete":
if i0 is None:
i0 = 0
dp = process
m_simul = dp.simulate(N, T, i0=i0, stochastic=stochastic)
i_simul = find_index(m_simul, dp.values)
m0 = dp.node(i0)
x0 = dr.eval_is(i0, s0[None, :])[0, :]
else:
m_simul = process.simulate(N, T, m0=m0, stochastic=stochastic)
if isinstance(m_simul, xr.DataArray):
m_simul = m_simul.data
sim_type = "continuous"
if m0 is None:
m0 = model.calibration["exogenous"]
x0 = dr.eval_ms(m0[None, :], s0[None, :])[0, :]
x_simul[0, :, :] = x0[None, :]
f = model.functions["arbitrage"]
g = model.functions["transition"]
numpy.random.seed(seed)
mp = m0
for i in range(T):
m = m_simul[i, :, :]
s = s_simul[i, :, :]
if sim_type == "discrete":
i_m = i_simul[i, :]
xx = [
dr.eval_is(i_m[ii], s[ii, :][None, :])[0, :] for ii in range(s.shape[0])
]
x = np.row_stack(xx)
else:
x = dr.eval_ms(m, s)
x_simul[i, :, :] = x
ss = g(mp, s, x, m, parms)
if i < T - 1:
s_simul[i + 1, :, :] = ss
mp = m
if "auxiliary" not in model.functions: # TODO: find a better test than this
l = [s_simul, x_simul]
varnames = model.symbols["states"] + model.symbols["controls"]
else:
aux = model.functions["auxiliary"]
a_simul = aux(
m_simul.reshape((N * T, -1)),
s_simul.reshape((N * T, -1)),
x_simul.reshape((N * T, -1)),
parms,
)
a_simul = a_simul.reshape(T, N, -1)
l = [m_simul, s_simul, x_simul, a_simul]
varnames = (
model.symbols["exogenous"]
+ model.symbols["states"]
+ model.symbols["controls"]
+ model.symbols["auxiliaries"]
)
simul = numpy.concatenate(l, axis=2)
if sim_type == "discrete":
varnames = ["_i_m"] + varnames
simul = np.concatenate([i_simul[:, :, None], simul], axis=2)
data = xr.DataArray(
simul,
dims=["T", "N", "V"],
coords={"T": range(T), "N": range(N), "V": varnames},
)
return data
def value_iteration(model,
grid={},
tol=1e-6,
maxit=500,
maxit_howard=20,
verbose=False):
"""
Solve for the value function and associated Markov decision rule by iterating over
the value function.
Parameters:
-----------
model :
"dtmscc" model. Must contain a 'felicity' function.
grid :
grid options
dr :
decision rule to evaluate
Returns:
--------
mdr : Markov decision rule
The solved decision rule/policy function
mdrv: decision rule
The solved value function
"""
transition = model.functions['transition']
felicity = model.functions['felicity']
controls_lb = model.functions['controls_lb']
controls_ub = model.functions['controls_ub']
parms = model.calibration['parameters']
discount = model.calibration['beta']
x0 = model.calibration['controls']
m0 = model.calibration['exogenous']
s0 = model.calibration['states']
r0 = felicity(m0, s0, x0, parms)
process = model.exogenous
dprocess = process.discretize()
n_ms = dprocess.n_nodes() # number of exogenous states
n_mv = dprocess.n_inodes(
0) # this assume number of integration nodes is constant
endo_grid = model.get_grid(**grid)
exo_grid = dprocess.grid
mdrv = DecisionRule(exo_grid, endo_grid)
grid = mdrv.endo_grid.nodes()
N = grid.shape[0]
n_x = len(x0)
mdr = constant_policy(model)
controls_0 = np.zeros((n_ms, N, n_x))
for i_ms in range(n_ms):
controls_0[i_ms, :, :] = mdr.eval_is(i_ms, grid)
values_0 = np.zeros((n_ms, N, 1))
# for i_ms in range(n_ms):
# values_0[i_ms, :, :] = mdrv(i_ms, grid)
mdr = DecisionRule(exo_grid, endo_grid)
# mdr.set_values(controls_0)
# THIRD: value function iterations until convergence
it = 0
err_v = 100
err_v_0 = 0
gain_v = 0.0
err_x = 100
err_x_0 = 0
tol_x = 1e-5
tol_v = 1e-7
itprint = IterationsPrinter(
('N', int), ('Error_V', float), ('Gain_V', float), ('Error_x', float),
('Gain_x', float), ('Eval_n', int), ('Time', float),
verbose=verbose)
itprint.print_header('Start value function iterations.')
while (it < maxit) and (err_v > tol or err_x > tol_x):
t_start = time.time()
it += 1
mdr.set_values(controls_0)
if it > 2:
ev = evaluate_policy(model,
mdr,
initial_guess=mdrv,
verbose=False,
details=True)
else:
ev = evaluate_policy(model, mdr, verbose=False, details=True)
mdrv = ev.solution
for i_ms in range(n_ms):
values_0[i_ms, :, :] = mdrv.eval_is(i_ms, grid)
values = values_0.copy()
controls = controls_0.copy()
for i_m in range(n_ms):
m = dprocess.node(i_m)
for n in range(N):
s = grid[n, :]
x = controls[i_m, n, :]
lb = controls_lb(m, s, parms)
ub = controls_ub(m, s, parms)
bnds = [e for e in zip(lb, ub)]
def valfun(xx):
return -choice_value(transition, felicity, i_m, s, xx,
mdrv, dprocess, parms, discount)[0]
res = scipy.optimize.minimize(valfun, x, bounds=bnds)
controls[i_m, n, :] = res.x
values[i_m, n, 0] = -valfun(x)
# compute error, update value and dr
err_x = abs(controls - controls_0).max()
err_v = abs(values - values_0).max()
t_end = time.time()
elapsed = t_end - t_start
values_0 = values
controls_0 = controls
gain_x = err_x / err_x_0
gain_v = err_v / err_v_0
err_x_0 = err_x
err_v_0 = err_v
itprint.print_iteration(N=it,
Error_V=err_v,
Gain_V=gain_v,
Error_x=err_x,
Gain_x=gain_x,
Eval_n=ev.iterations,
Time=elapsed)
itprint.print_finished()
mdr = DecisionRule(exo_grid, endo_grid)
mdr.set_values(controls)
mdrv.set_values(values_0)
return mdr, mdrv
def evaluate_policy(
model,
mdr,
tol=1e-8,
maxit=2000,
grid={},
verbose=True,
dr0=None,
hook=None,
integration_orders=None,
details=False,
interp_method="cubic",
):
"""Compute value function corresponding to policy ``dr``
Parameters:
-----------
model:
"dtcscc" model. Must contain a 'value' function.
mdr:
decision rule to evaluate
Returns:
--------
decision rule:
value function (a function of the space similar to a decision rule
object)
"""
process = model.exogenous
grid, dprocess = model.discretize()
endo_grid = grid["endo"]
exo_grid = grid["exo"]
n_ms = dprocess.n_nodes # number of exogenous states
n_mv = dprocess.n_inodes(
0) # this assume number of integration nodes is constant
x0 = model.calibration["controls"]
v0 = model.calibration["values"]
parms = model.calibration["parameters"]
n_x = len(x0)
n_v = len(v0)
n_s = len(model.symbols["states"])
if dr0 is not None:
mdrv = dr0
else:
mdrv = DecisionRule(exo_grid, endo_grid, interp_method=interp_method)
s = mdrv.endo_grid.nodes
N = s.shape[0]
if isinstance(mdr, np.ndarray):
controls = mdr
else:
controls = np.zeros((n_ms, N, n_x))
for i_m in range(n_ms):
controls[i_m, :, :] = mdr.eval_is(i_m, s)
values_0 = np.zeros((n_ms, N, n_v))
if dr0 is None:
for i_m in range(n_ms):
values_0[i_m, :, :] = v0[None, :]
else:
for i_m in range(n_ms):
values_0[i_m, :, :] = dr0.eval_is(i_m, s)
val = model.functions["value"]
g = model.functions["transition"]
sh_v = values_0.shape
err = 10
inner_maxit = 50
it = 0
if verbose:
headline = "|{0:^4} | {1:10} | {2:8} | {3:8} |".format(
"N", " Error", "Gain", "Time")
stars = "-" * len(headline)
print(stars)
print(headline)
print(stars)
t1 = time.time()
err_0 = np.nan
verbit = verbose == "full"
while err > tol and it < maxit:
it += 1
t_start = time.time()
mdrv.set_values(values_0.reshape(sh_v))
values = update_value(val, g, s, controls, values_0, mdr, mdrv,
dprocess, parms).reshape((-1, n_v))
err = abs(values.reshape(sh_v) - values_0).max()
err_SA = err / err_0
err_0 = err
values_0 = values.reshape(sh_v)
t_finish = time.time()
elapsed = t_finish - t_start
if verbose:
print("|{0:4} | {1:10.3e} | {2:8.3f} | {3:8.3f} |".format(
it, err, err_SA, elapsed))
# values_0 = values.reshape(sh_v)
t2 = time.time()
if verbose:
print(stars)
print("Elapsed: {} seconds.".format(t2 - t1))
print(stars)
if not details:
return mdrv
else:
return EvaluationResult(mdrv, it, tol, err)
def time_iteration(model,
initial_guess=None,
with_complementarities=True,
verbose=True,
grid={},
maxit=1000,
inner_maxit=10,
tol=1e-6,
hook=None):
'''
Finds a global solution for ``model`` using backward time-iteration.
This algorithm iterates on the residuals of the arbitrage equations
Parameters
----------
model : NumericModel
"dtmscc" model to be solved
verbose : boolean
if True, display iterations
initial_dr : decision rule
initial guess for the decision rule
with_complementarities : boolean (True)
if False, complementarity conditions are ignored
grid: grid options
maxit: maximum number of iterations
inner_maxit: maximum number of iteration for inner solver
tol: tolerance criterium for successive approximations
Returns
-------
decision rule :
approximated solution
'''
from dolo import dprint
def vprint(t):
if verbose:
print(t)
process = model.exogenous
dprocess = process.discretize()
n_ms = dprocess.n_nodes() # number of exogenous states
n_mv = dprocess.n_inodes(
0) # this assume number of integration nodes is constant
x0 = model.calibration['controls']
parms = model.calibration['parameters']
n_x = len(x0)
n_s = len(model.symbols['states'])
endo_grid = model.get_grid(**grid)
interp_type = 'cubic'
exo_grid = dprocess.grid
mdr = DecisionRule(exo_grid, endo_grid)
grid = mdr.endo_grid.nodes()
N = grid.shape[0]
controls_0 = numpy.zeros((n_ms, N, n_x))
if initial_guess is None:
controls_0[:, :, :] = x0[None, None, :]
else:
for i_m in range(n_ms):
controls_0[i_m, :, :] = initial_guess(i_m, grid)
f = model.functions['arbitrage']
g = model.functions['transition']
if 'controls_lb' in model.functions and with_complementarities == True:
lb_fun = model.functions['controls_lb']
ub_fun = model.functions['controls_ub']
lb = numpy.zeros_like(controls_0) * numpy.nan
ub = numpy.zeros_like(controls_0) * numpy.nan
for i_m in range(n_ms):
m = dprocess.node(i_m)[None, :]
p = parms[None, :]
m = numpy.repeat(m, N, axis=0)
p = numpy.repeat(p, N, axis=0)
lb[i_m, :, :] = lb_fun(m, grid, p)
ub[i_m, :, :] = ub_fun(m, grid, p)
else:
with_complementarities = False
sh_c = controls_0.shape
controls_0 = controls_0.reshape((-1, n_x))
from dolo.numeric.optimize.newton import newton, SerialDifferentiableFunction
from dolo.numeric.optimize.ncpsolve import ncpsolve
err = 10
it = 0
if with_complementarities:
vprint("Solving WITH complementarities.")
lb = lb.reshape((-1, n_x))
ub = ub.reshape((-1, n_x))
if verbose:
headline = '|{0:^4} | {1:10} | {2:8} | {3:8} | {4:3} |'.format(
'N', ' Error', 'Gain', 'Time', 'nit')
stars = '-' * len(headline)
print(stars)
print(headline)
print(stars)
import time
t1 = time.time()
err_0 = numpy.nan
verbit = (verbose == 'full')
while err > tol and it < maxit:
it += 1
t_start = time.time()
mdr.set_values(controls_0.reshape(sh_c))
fn = lambda x: residuals_simple(f, g, grid, x.reshape(sh_c), mdr,
dprocess, parms).reshape((-1, n_x))
dfn = SerialDifferentiableFunction(fn)
res = fn(controls_0)
if hook:
hook()
if with_complementarities:
[controls, nit] = ncpsolve(dfn,
lb,
ub,
controls_0,
verbose=verbit,
maxit=inner_maxit)
else:
[controls, nit] = newton(dfn,
controls_0,
verbose=verbit,
maxit=inner_maxit)
err = abs(controls - controls_0).max()
err_SA = err / err_0
err_0 = err
controls_0 = controls
t_finish = time.time()
elapsed = t_finish - t_start
if verbose:
print('|{0:4} | {1:10.3e} | {2:8.3f} | {3:8.3f} | {4:3} |'.format(
it, err, err_SA, elapsed, nit))
controls_0 = controls.reshape(sh_c)
t2 = time.time()
if verbose:
print(stars)
print("Elapsed: {} seconds.".format(t2 - t1))
print(stars)
return mdr
def evaluate_policy(model,
mdr,
tol=1e-8,
maxit=2000,
grid={},
verbose=True,
initial_guess=None,
hook=None,
integration_orders=None,
details=False,
interp_type='cubic'):
"""Compute value function corresponding to policy ``dr``
Parameters:
-----------
model:
"dtcscc" model. Must contain a 'value' function.
mdr:
decision rule to evaluate
Returns:
--------
decision rule:
value function (a function of the space similar to a decision rule
object)
"""
process = model.exogenous
dprocess = process.discretize()
n_ms = dprocess.n_nodes() # number of exogenous states
n_mv = dprocess.n_inodes(
0) # this assume number of integration nodes is constant
x0 = model.calibration['controls']
v0 = model.calibration['values']
parms = model.calibration['parameters']
n_x = len(x0)
n_v = len(v0)
n_s = len(model.symbols['states'])
endo_grid = model.get_grid(**grid)
exo_grid = dprocess.grid
if initial_guess is not None:
mdrv = initial_guess
else:
mdrv = DecisionRule(exo_grid, endo_grid, interp_type=interp_type)
grid = mdrv.endo_grid.nodes()
N = grid.shape[0]
if isinstance(mdr, np.ndarray):
controls = mdr
else:
controls = np.zeros((n_ms, N, n_x))
for i_m in range(n_ms):
controls[i_m, :, :] = mdr.eval_is(i_m, grid)
values_0 = np.zeros((n_ms, N, n_v))
if initial_guess is None:
for i_m in range(n_ms):
values_0[i_m, :, :] = v0[None, :]
else:
for i_m in range(n_ms):
values_0[i_m, :, :] = initial_guess.eval_is(i_m, grid)
val = model.functions['value']
g = model.functions['transition']
sh_v = values_0.shape
err = 10
inner_maxit = 50
it = 0
if verbose:
headline = '|{0:^4} | {1:10} | {2:8} | {3:8} |'.format(
'N', ' Error', 'Gain', 'Time')
stars = '-' * len(headline)
print(stars)
print(headline)
print(stars)
t1 = time.time()
err_0 = np.nan
verbit = (verbose == 'full')
while err > tol and it < maxit:
it += 1
t_start = time.time()
mdrv.set_values(values_0.reshape(sh_v))
values = update_value(val, g, grid, controls, values_0, mdr, mdrv,
dprocess, parms).reshape((-1, n_v))
err = abs(values.reshape(sh_v) - values_0).max()
err_SA = err / err_0
err_0 = err
values_0 = values.reshape(sh_v)
t_finish = time.time()
elapsed = t_finish - t_start
if verbose:
print('|{0:4} | {1:10.3e} | {2:8.3f} | {3:8.3f} |'.format(
it, err, err_SA, elapsed))
# values_0 = values.reshape(sh_v)
t2 = time.time()
if verbose:
print(stars)
print("Elapsed: {} seconds.".format(t2 - t1))
print(stars)
if not details:
return mdrv
else:
return EvaluationResult(mdrv, it, tol, err)
def value_iteration(
model: Model,
*,
verbose: bool = False, #
details: bool = True, #
tol=1e-6,
maxit=500,
maxit_howard=20,
) -> ValueIterationResult:
"""
Solve for the value function and associated Markov decision rule by iterating over
the value function.
Parameters:
-----------
model :
model to be solved
dr :
decision rule to evaluate
Returns:
--------
mdr : Markov decision rule
The solved decision rule/policy function
mdrv: decision rule
The solved value function
"""
transition = model.functions["transition"]
felicity = model.functions["felicity"]
controls_lb = model.functions["controls_lb"]
controls_ub = model.functions["controls_ub"]
parms = model.calibration["parameters"]
discount = model.calibration["beta"]
x0 = model.calibration["controls"]
m0 = model.calibration["exogenous"]
s0 = model.calibration["states"]
r0 = felicity(m0, s0, x0, parms)
process = model.exogenous
grid, dprocess = model.discretize()
endo_grid = grid["endo"]
exo_grid = grid["exo"]
n_ms = dprocess.n_nodes # number of exogenous states
n_mv = dprocess.n_inodes(
0) # this assume number of integration nodes is constant
mdrv = DecisionRule(exo_grid, endo_grid)
s = mdrv.endo_grid.nodes
N = s.shape[0]
n_x = len(x0)
mdr = constant_policy(model)
controls_0 = np.zeros((n_ms, N, n_x))
for i_ms in range(n_ms):
controls_0[i_ms, :, :] = mdr.eval_is(i_ms, s)
values_0 = np.zeros((n_ms, N, 1))
# for i_ms in range(n_ms):
# values_0[i_ms, :, :] = mdrv(i_ms, grid)
mdr = DecisionRule(exo_grid, endo_grid)
# mdr.set_values(controls_0)
# THIRD: value function iterations until convergence
it = 0
err_v = 100
err_v_0 = 0
gain_v = 0.0
err_x = 100
err_x_0 = 0
tol_x = 1e-5
tol_v = 1e-7
itprint = IterationsPrinter(
("N", int),
("Error_V", float),
("Gain_V", float),
("Error_x", float),
("Gain_x", float),
("Eval_n", int),
("Time", float),
verbose=verbose,
)
itprint.print_header("Start value function iterations.")
while (it < maxit) and (err_v > tol or err_x > tol_x):
t_start = time.time()
it += 1
mdr.set_values(controls_0)
if it > 2:
ev = evaluate_policy(model,
mdr,
dr0=mdrv,
verbose=False,
details=True)
else:
ev = evaluate_policy(model, mdr, verbose=False, details=True)
mdrv = ev.solution
for i_ms in range(n_ms):
values_0[i_ms, :, :] = mdrv.eval_is(i_ms, s)
values = values_0.copy()
controls = controls_0.copy()
for i_m in range(n_ms):
m = dprocess.node(i_m)
for n in range(N):
s_ = s[n, :]
x = controls[i_m, n, :]
lb = controls_lb(m, s_, parms)
ub = controls_ub(m, s_, parms)
bnds = [e for e in zip(lb, ub)]
def valfun(xx):
return -choice_value(
transition,
felicity,
i_m,
s_,
xx,
mdrv,
dprocess,
parms,
discount,
)[0]
res = scipy.optimize.minimize(valfun, x, bounds=bnds)
controls[i_m, n, :] = res.x
values[i_m, n, 0] = -valfun(x)
# compute error, update value and dr
err_x = abs(controls - controls_0).max()
err_v = abs(values - values_0).max()
t_end = time.time()
elapsed = t_end - t_start
values_0 = values
controls_0 = controls
gain_x = err_x / err_x_0
gain_v = err_v / err_v_0
err_x_0 = err_x
err_v_0 = err_v
itprint.print_iteration(
N=it,
Error_V=err_v,
Gain_V=gain_v,
Error_x=err_x,
Gain_x=gain_x,
Eval_n=ev.iterations,
Time=elapsed,
)
itprint.print_finished()
mdr = DecisionRule(exo_grid, endo_grid)
mdr.set_values(controls)
mdrv.set_values(values_0)
if not details:
return mdr, mdrv
else:
return ValueIterationResult(
mdr, #:AbstractDecisionRule
mdrv, #:AbstractDecisionRule
it, #:Int
dprocess, #:AbstractDiscretizedProcess
err_x < tol_x, #:Bool
tol_x, #:Float64
err_x, #:Float64
err_v < tol_v, #:Bool
tol_v, #:Float64
err_v, #:Float64
None, # log: #:ValueIterationLog
None, # trace: #:Union{Nothing,IterationTrace
)
def value_iteration(model,
grid={},
tol=1e-6,
maxit=500,
maxit_howard=20,
verbose=False,
details=True):
"""
Solve for the value function and associated Markov decision rule by iterating over
the value function.
Parameters:
-----------
model :
"dtmscc" model. Must contain a 'felicity' function.
grid :
grid options
dr :
decision rule to evaluate
Returns:
--------
mdr : Markov decision rule
The solved decision rule/policy function
mdrv: decision rule
The solved value function
"""
transition = model.functions['transition']
felicity = model.functions['felicity']
controls_lb = model.functions['controls_lb']
controls_ub = model.functions['controls_ub']
parms = model.calibration['parameters']
discount = model.calibration['beta']
x0 = model.calibration['controls']
m0 = model.calibration['exogenous']
s0 = model.calibration['states']
r0 = felicity(m0, s0, x0, parms)
process = model.exogenous
dprocess = process.discretize()
n_ms = dprocess.n_nodes() # number of exogenous states
n_mv = dprocess.n_inodes(
0) # this assume number of integration nodes is constant
endo_grid = model.get_grid(**grid)
exo_grid = dprocess.grid
mdrv = DecisionRule(exo_grid, endo_grid)
grid = mdrv.endo_grid.nodes()
N = grid.shape[0]
n_x = len(x0)
mdr = constant_policy(model)
controls_0 = np.zeros((n_ms, N, n_x))
for i_ms in range(n_ms):
controls_0[i_ms, :, :] = mdr.eval_is(i_ms, grid)
values_0 = np.zeros((n_ms, N, 1))
# for i_ms in range(n_ms):
# values_0[i_ms, :, :] = mdrv(i_ms, grid)
mdr = DecisionRule(exo_grid, endo_grid)
# mdr.set_values(controls_0)
# THIRD: value function iterations until convergence
it = 0
err_v = 100
err_v_0 = 0
gain_v = 0.0
err_x = 100
err_x_0 = 0
tol_x = 1e-5
tol_v = 1e-7
itprint = IterationsPrinter(
('N', int), ('Error_V', float), ('Gain_V', float), ('Error_x', float),
('Gain_x', float), ('Eval_n', int), ('Time', float),
verbose=verbose)
itprint.print_header('Start value function iterations.')
while (it < maxit) and (err_v > tol or err_x > tol_x):
t_start = time.time()
it += 1
mdr.set_values(controls_0)
if it > 2:
ev = evaluate_policy(
model, mdr, initial_guess=mdrv, verbose=False, details=True)
else:
ev = evaluate_policy(model, mdr, verbose=False, details=True)
mdrv = ev.solution
for i_ms in range(n_ms):
values_0[i_ms, :, :] = mdrv.eval_is(i_ms, grid)
values = values_0.copy()
controls = controls_0.copy()
for i_m in range(n_ms):
m = dprocess.node(i_m)
for n in range(N):
s = grid[n, :]
x = controls[i_m, n, :]
lb = controls_lb(m, s, parms)
ub = controls_ub(m, s, parms)
bnds = [e for e in zip(lb, ub)]
def valfun(xx):
return -choice_value(transition, felicity, i_m, s, xx,
mdrv, dprocess, parms, discount)[0]
res = scipy.optimize.minimize(valfun, x, bounds=bnds)
controls[i_m, n, :] = res.x
values[i_m, n, 0] = -valfun(x)
# compute error, update value and dr
err_x = abs(controls - controls_0).max()
err_v = abs(values - values_0).max()
t_end = time.time()
elapsed = t_end - t_start
values_0 = values
controls_0 = controls
gain_x = err_x / err_x_0
gain_v = err_v / err_v_0
err_x_0 = err_x
err_v_0 = err_v
itprint.print_iteration(
N=it,
Error_V=err_v,
Gain_V=gain_v,
Error_x=err_x,
Gain_x=gain_x,
Eval_n=ev.iterations,
Time=elapsed)
itprint.print_finished()
mdr = DecisionRule(exo_grid, endo_grid)
mdr.set_values(controls)
mdrv.set_values(values_0)
if not details:
return mdr, mdrv
else:
return ValueIterationResult(
mdr, #:AbstractDecisionRule
mdrv, #:AbstractDecisionRule
it, #:Int
dprocess, #:AbstractDiscretizedProcess
err_x<tol_x, #:Bool
tol_x, #:Float64
err_x, #:Float64
err_v<tol_v, #:Bool
tol_v, #:Float64
err_v, #:Float64
None, #log: #:ValueIterationLog
None #trace: #:Union{Nothing,IterationTrace
)
def improved_time_iteration(model,
method='jac',
initial_dr=None,
interp_type='spline',
mu=2,
maxbsteps=10,
verbose=False,
tol=1e-8,
smaxit=500,
maxit=1000,
complementarities=True,
compute_radius=False,
invmethod='gmres',
details=True):
def vprint(*args, **kwargs):
if verbose:
print(*args, **kwargs)
itprint = IterationsPrinter(
('N', int), ('f_x', float), ('d_x', float), ('Time_residuals', float),
('Time_inversion', float), ('Time_search', float), ('Lambda_0', float),
('N_invert', int), ('N_search', int),
verbose=verbose)
itprint.print_header('Start Improved Time Iterations.')
f = model.functions['arbitrage']
g = model.functions['transition']
x_lb = model.functions['controls_lb']
x_ub = model.functions['controls_ub']
parms = model.calibration['parameters']
dp = model.exogenous.discretize()
n_m = max(dp.n_nodes(), 1)
n_s = len(model.symbols['states'])
grid = model.get_grid()
if interp_type == 'spline':
ddr = DecisionRule(dp.grid, grid)
ddr_filt = DecisionRule(dp.grid, grid)
elif interp_type == 'smolyak':
ddr = SmolyakDecisionRule(n_m, grid.min, grid.max, mu)
ddr_filt = SmolyakDecisionRule(n_m, grid.min, grid.max, mu)
derivative_type = 'numerical'
# s = ddr.endo_grid
s = grid.nodes()
N = s.shape[0]
n_x = len(model.symbols['controls'])
x0 = model.calibration['controls'][None,
None, ].repeat(n_m,
axis=0).repeat(N, axis=1)
if initial_dr is not None:
for i_m in range(n_m):
x0[i_m, :, :] = initial_dr.eval_is(i_m, s)
ddr.set_values(x0)
steps = 0.5**numpy.arange(maxbsteps)
lb = x0.copy()
ub = x0.copy()
for i_m in range(n_m):
m = dp.node(i_m)
lb[i_m, :] = x_lb(m, s, parms)
ub[i_m, :] = x_ub(m, s, parms)
x = x0
# both affect the precision
ddr.set_values(x)
## memory allocation
n_im = dp.n_inodes(0) # we assume it is constant for now
jres = numpy.zeros((n_m, n_im, N, n_x, n_x))
S_ij = numpy.zeros((n_m, n_im, N, n_s))
for it in range(maxit):
jres[...] = 0.0
S_ij[...] = 0.0
t1 = time.time()
# compute derivatives and residuals:
# res: residuals
# dres: derivatives w.r.t. x
# jres: derivatives w.r.t. ~x
# fut_S: future states
ddr.set_values(x)
#
# ub[ub>100000] = 100000
# lb[lb<-100000] = -100000
#
# sh_x = x.shape
# rr =euler_residuals(f,g,s,x,ddr,dp,parms, diff=False, with_jres=False,set_dr=True)
# print(rr.shape)
#
# from iti.fb import smooth_
# jj = smooth_(rr, x, lb, ub)
#
# print("Errors with complementarities")
# print(abs(jj.max()))
#
# exit()
#
from dolo.numeric.optimize.newton import SerialDifferentiableFunction
sh_x = x.shape
ff = SerialDifferentiableFunction(
lambda u: euler_residuals(f,
g,
s,
u.reshape(sh_x),
ddr,
dp,
parms,
diff=False,
with_jres=False,
set_dr=False).reshape((-1, sh_x[2])))
res, dres = ff(x.reshape((-1, sh_x[2])))
res = res.reshape(sh_x)
dres = dres.reshape((sh_x[0], sh_x[1], sh_x[2], sh_x[2]))
junk, jres, fut_S = euler_residuals(f,
g,
s,
x,
ddr,
dp,
parms,
diff=False,
with_jres=True,
set_dr=False,
jres=jres,
S_ij=S_ij)
# if there are complementerities, we modify derivatives
if complementarities:
res, dres, jres = smooth(res, dres, jres, x - lb)
res[...] *= -1
dres[...] *= -1
jres[...] *= -1
res, dres, jres = smooth(res, dres, jres, ub - x, pos=-1.0)
res[...] *= -1
dres[...] *= -1
jres[...] *= -1
err_0 = (abs(res).max())
# premultiply by A
jres[...] *= -1.0
for i_m in range(n_m):
for j_m in range(n_im):
M = jres[i_m, j_m, :, :, :]
X = dres[i_m, :, :, :].copy()
sol = solve_tensor(X, M)
t2 = time.time()
# new version
if invmethod == 'gmres':
import scipy.sparse.linalg
ddx = solve_gu(dres.copy(), res.copy())
L = Operator(jres, fut_S, ddr_filt)
n0 = L.counter
L.addid = True
ttol = err_0 / 100
sol = scipy.sparse.linalg.gmres(
L, ddx.ravel(), tol=ttol) #, maxiter=1, restart=smaxit)
lam0 = 0.01
nn = L.counter - n0
tot = sol[0].reshape(ddx.shape)
else:
# compute inversion
tot, nn, lam0 = invert_jac(res,
dres,
jres,
fut_S,
ddr_filt,
tol=tol,
maxit=smaxit,
verbose=(verbose == 'full'))
# lam, lam_max, lambdas = radius_jac(res,dres,jres,fut_S,tol=tol,maxit=1000,verbose=(verbose=='full'),filt=ddr_filt)
# backsteps
t3 = time.time()
for i_bckstps, lam in enumerate(steps):
new_x = x - tot * lam
new_err = euler_residuals(f,
g,
s,
new_x,
ddr,
dp,
parms,
diff=False,
set_dr=True)
if complementarities:
new_err = smooth_nodiff(new_err, new_x - lb)
new_err = smooth_nodiff(-new_err, ub - new_x)
new_err = abs(new_err).max()
if new_err < err_0:
break
err_2 = abs(tot).max()
t4 = time.time()
itprint.print_iteration(N=it,
f_x=err_0,
d_x=err_2,
Time_residuals=t2 - t1,
Time_inversion=t3 - t2,
Time_search=t4 - t3,
Lambda_0=lam0,
N_invert=nn,
N_search=i_bckstps)
if err_0 < tol:
break
x = new_x
ddr.set_values(x)
itprint.print_finished()
# if compute_radius:
# return ddx,L
# lam, lam_max, lambdas = radius_jac(res,dres,jres,fut_S,ddr_filt,tol=tol,maxit=smaxit,verbose=(verbose=='full'))
# return ddr, lam, lam_max, lambdas
# else:
if not details:
return ddr
else:
ddx = solve_gu(dres.copy(), res.copy())
L = Operator(jres, fut_S, ddr_filt)
lam = scipy.sparse.linalg.eigs(L, k=1, return_eigenvectors=False)
lam = abs(lam[0])
# lam, lam_max, lambdas = radius_jac(res,dres,jres,fut_S,ddr_filt,tol=tol,maxit=smaxit,verbose=(verbose=='full'))
return ImprovedTimeIterationResult(ddr, it, err_0, err_2, err_0 < tol,
complementarities, lam, None, L)
def egm(
model: Model,
dr0: DecisionRule = None,
verbose: bool = False,
details: bool = True,
a_grid=None,
η_tol=1e-6,
maxit=1000,
grid=None,
dp=None,
):
"""
a_grid: (numpy-array) vector of points used to discretize poststates; must be increasing
"""
assert len(model.symbols["states"]) == 1
assert (len(model.symbols["controls"]) == 1
) # we probably don't need this restriction
from dolo.numeric.processes import IIDProcess
iid_process = isinstance(model.exogenous, IIDProcess)
def vprint(t):
if verbose:
print(t)
p = model.calibration["parameters"]
if grid is None and dp is None:
grid, dp = model.discretize()
s = grid["endo"].nodes
funs = model.__original_gufunctions__
h = funs["expectation"]
gt = funs["half_transition"]
τ = funs["direct_response_egm"]
aτ = funs["reverse_state"]
lb = funs["arbitrage_lb"]
ub = funs["arbitrage_ub"]
if dr0 is None:
x0 = model.calibration["controls"]
dr0 = lambda i, s: x0[None, :].repeat(s.shape[0], axis=0)
n_m = dp.n_nodes
n_x = len(model.symbols["controls"])
if a_grid is None:
raise Exception("You must supply a grid for the post-states.")
assert a_grid.ndim == 1
a = a_grid[:, None]
N_a = a.shape[0]
N = s.shape[0]
n_h = len(model.symbols["expectations"])
xa = np.zeros((n_m, N_a, n_x))
sa = np.zeros((n_m, N_a, 1))
xa0 = np.zeros((n_m, N_a, n_x))
sa0 = np.zeros((n_m, N_a, 1))
z = np.zeros((n_m, N_a, n_h))
if verbose:
headline = "|{0:^4} | {1:10} |".format("N", " Error")
stars = "-" * len(headline)
print(stars)
print(headline)
print(stars)
for it in range(0, maxit):
if it == 0:
drfut = dr0
else:
def drfut(i, ss):
if iid_process:
i = 0
m = dp.node(i)
l_ = lb(m, ss, p)
u_ = ub(m, ss, p)
x = eval_linear((sa0[i, :, 0], ), xa0[i, :, 0], ss)[:, None]
x = np.minimum(x, u_)
x = np.maximum(x, l_)
return x
z[:, :, :] = 0
for i_m in range(n_m):
m = dp.node(i_m)
for i_M in range(dp.n_inodes(i_m)):
w = dp.iweight(i_m, i_M)
M = dp.inode(i_m, i_M)
S = gt(m, a, M, p)
print(it, i_m, i_M)
X = drfut(i_M, S)
z[i_m, :, :] += w * h(M, S, X, p)
xa[i_m, :, :] = τ(m, a, z[i_m, :, :], p)
sa[i_m, :, :] = aτ(m, a, xa[i_m, :, :], p)
if it > 1:
η = abs(xa - xa0).max() + abs(sa - sa0).max()
else:
η = 1
vprint("|{0:4} | {1:10.3e} |".format(it, η))
if η < η_tol:
break
sa0[...] = sa
xa0[...] = xa
# resample the result on the standard grid
endo_grid = grid["endo"]
exo_grid = grid["exo"]
mdr = DecisionRule(exo_grid, endo_grid, dprocess=dp, interp_method="cubic")
mdr.set_values(
np.concatenate([drfut(i, s)[None, :, :] for i in range(n_m)], axis=0))
sol = EGMResult(mdr, it, dp, (η < η_tol), η_tol, η)
return sol
def time_iteration(
model: Model,
*, #
dr0: DecisionRule = None, #
verbose: bool = True, #
details: bool = True, #
ignore_constraints: bool = False, #
trace: bool = False, #
dprocess=None,
maxit=1000,
inner_maxit=10,
tol=1e-6,
hook=None,
interp_method="cubic",
# obsolete
with_complementarities=None,
) -> TimeIterationResult:
"""Finds a global solution for ``model`` using backward time-iteration.
This algorithm iterates on the residuals of the arbitrage equations
Parameters
----------
model : Model
model to be solved
verbose : bool
if True, display iterations
dr0 : decision rule
initial guess for the decision rule
with_complementarities : bool (True)
if False, complementarity conditions are ignored
maxit: maximum number of iterations
inner_maxit: maximum number of iteration for inner solver
tol: tolerance criterium for successive approximations
hook: Callable
function to be called within each iteration, useful for debugging purposes
Returns
-------
decision rule :
approximated solution
"""
# deal with obsolete options
if with_complementarities is not None:
# TODO warn
pass
else:
with_complementarities = not ignore_constraints
if trace:
trace_details = []
else:
trace_details = None
from dolo import dprint
def vprint(t):
if verbose:
print(t)
grid, dprocess_ = model.discretize()
if dprocess is None:
dprocess = dprocess_
n_ms = dprocess.n_nodes # number of exogenous states
n_mv = dprocess.n_inodes(
0) # this assume number of integration nodes is constant
x0 = model.calibration["controls"]
parms = model.calibration["parameters"]
n_x = len(x0)
n_s = len(model.symbols["states"])
endo_grid = grid["endo"]
exo_grid = grid["exo"]
mdr = DecisionRule(exo_grid,
endo_grid,
dprocess=dprocess,
interp_method=interp_method)
s = mdr.endo_grid.nodes
N = s.shape[0]
controls_0 = numpy.zeros((n_ms, N, n_x))
if dr0 is None:
controls_0[:, :, :] = x0[None, None, :]
else:
if isinstance(dr0, AlgoResult):
dr0 = dr0.dr
try:
for i_m in range(n_ms):
controls_0[i_m, :, :] = dr0(i_m, s)
except Exception:
for i_m in range(n_ms):
m = dprocess.node(i_m)
controls_0[i_m, :, :] = dr0(m, s)
f = model.functions["arbitrage"]
g = model.functions["transition"]
if "arbitrage_lb" in model.functions and with_complementarities == True:
lb_fun = model.functions["arbitrage_lb"]
ub_fun = model.functions["arbitrage_ub"]
lb = numpy.zeros_like(controls_0) * numpy.nan
ub = numpy.zeros_like(controls_0) * numpy.nan
for i_m in range(n_ms):
m = dprocess.node(i_m)[None, :]
p = parms[None, :]
m = numpy.repeat(m, N, axis=0)
p = numpy.repeat(p, N, axis=0)
lb[i_m, :, :] = lb_fun(m, s, p)
ub[i_m, :, :] = ub_fun(m, s, p)
else:
with_complementarities = False
sh_c = controls_0.shape
controls_0 = controls_0.reshape((-1, n_x))
from dolo.numeric.optimize.newton import newton, SerialDifferentiableFunction
from dolo.numeric.optimize.ncpsolve import ncpsolve
err = 10
it = 0
if with_complementarities:
lb = lb.reshape((-1, n_x))
ub = ub.reshape((-1, n_x))
itprint = IterationsPrinter(
("N", int),
("Error", float),
("Gain", float),
("Time", float),
("nit", int),
verbose=verbose,
)
itprint.print_header("Start Time Iterations.")
import time
t1 = time.time()
err_0 = numpy.nan
verbit = verbose == "full"
while err > tol and it < maxit:
it += 1
t_start = time.time()
mdr.set_values(controls_0.reshape(sh_c))
if trace:
trace_details.append({"dr": copy.deepcopy(mdr)})
fn = lambda x: residuals_simple(f, g, s, x.reshape(sh_c), mdr,
dprocess, parms).reshape((-1, n_x))
dfn = SerialDifferentiableFunction(fn)
res = fn(controls_0)
if hook:
hook()
if with_complementarities:
[controls, nit] = ncpsolve(dfn,
lb,
ub,
controls_0,
verbose=verbit,
maxit=inner_maxit)
else:
[controls, nit] = newton(dfn,
controls_0,
verbose=verbit,
maxit=inner_maxit)
err = abs(controls - controls_0).max()
err_SA = err / err_0
err_0 = err
controls_0 = controls
t_finish = time.time()
elapsed = t_finish - t_start
itprint.print_iteration(N=it,
Error=err_0,
Gain=err_SA,
Time=elapsed,
nit=nit),
controls_0 = controls.reshape(sh_c)
mdr.set_values(controls_0)
if trace:
trace_details.append({"dr": copy.deepcopy(mdr)})
itprint.print_finished()
if not details:
return mdr
return TimeIterationResult(
mdr,
it,
with_complementarities,
dprocess,
err < tol, # x_converged: bool
tol, # x_tol
err, #: float
None, # log: object # TimeIterationLog
trace_details, # trace: object #{Nothing,IterationTrace}
)
def time_iteration(model, initial_guess=None, dprocess=None, with_complementarities=True,
verbose=True, grid={},
maxit=1000, inner_maxit=10, tol=1e-6, hook=None, details=False):
'''
Finds a global solution for ``model`` using backward time-iteration.
This algorithm iterates on the residuals of the arbitrage equations
Parameters
----------
model : Model
model to be solved
verbose : boolean
if True, display iterations
initial_guess : decision rule
initial guess for the decision rule
dprocess : DiscretizedProcess (model.exogenous.discretize())
discretized process to be used
with_complementarities : boolean (True)
if False, complementarity conditions are ignored
grid: grid options
overload the values set in `options:grid` section
maxit: maximum number of iterations
inner_maxit: maximum number of iteration for inner solver
tol: tolerance criterium for successive approximations
hook: Callable
function to be called within each iteration, useful for debugging purposes
Returns
-------
decision rule :
approximated solution
'''
from dolo import dprint
def vprint(t):
if verbose:
print(t)
if dprocess is None:
dprocess = model.exogenous.discretize()
n_ms = dprocess.n_nodes() # number of exogenous states
n_mv = dprocess.n_inodes(0) # this assume number of integration nodes is constant
x0 = model.calibration['controls']
parms = model.calibration['parameters']
n_x = len(x0)
n_s = len(model.symbols['states'])
endo_grid = model.get_grid(**grid)
exo_grid = dprocess.grid
mdr = DecisionRule(exo_grid, endo_grid)
grid = mdr.endo_grid.nodes()
N = grid.shape[0]
controls_0 = numpy.zeros((n_ms, N, n_x))
if initial_guess is None:
controls_0[:, :, :] = x0[None,None,:]
else:
if isinstance(initial_guess, AlgoResult):
initial_guess = initial_guess.dr
try:
for i_m in range(n_ms):
controls_0[i_m, :, :] = initial_guess(i_m, grid)
except Exception:
for i_m in range(n_ms):
m = dprocess.node(i_m)
controls_0[i_m, :, :] = initial_guess(m, grid)
f = model.functions['arbitrage']
g = model.functions['transition']
if 'controls_lb' in model.functions and with_complementarities==True:
lb_fun = model.functions['controls_lb']
ub_fun = model.functions['controls_ub']
lb = numpy.zeros_like(controls_0)*numpy.nan
ub = numpy.zeros_like(controls_0)*numpy.nan
for i_m in range(n_ms):
m = dprocess.node(i_m)[None,:]
p = parms[None,:]
m = numpy.repeat(m, N, axis=0)
p = numpy.repeat(p, N, axis=0)
lb[i_m,:,:] = lb_fun(m, grid, p)
ub[i_m,:,:] = ub_fun(m, grid, p)
else:
with_complementarities = False
sh_c = controls_0.shape
controls_0 = controls_0.reshape( (-1,n_x) )
from dolo.numeric.optimize.newton import newton, SerialDifferentiableFunction
from dolo.numeric.optimize.ncpsolve import ncpsolve
err = 10
it = 0
if with_complementarities:
vprint("Solving WITH complementarities.")
lb = lb.reshape((-1,n_x))
ub = ub.reshape((-1,n_x))
if verbose:
headline = '|{0:^4} | {1:10} | {2:8} | {3:8} | {4:3} |'.format( 'N',' Error', 'Gain','Time', 'nit' )
stars = '-'*len(headline)
print(stars)
print(headline)
print(stars)
import time
t1 = time.time()
err_0 = numpy.nan
verbit = (verbose == 'full')
while err>tol and it<maxit:
it += 1
t_start = time.time()
mdr.set_values(controls_0.reshape(sh_c))
fn = lambda x: residuals_simple(f, g, grid, x.reshape(sh_c), mdr, dprocess, parms).reshape((-1,n_x))
dfn = SerialDifferentiableFunction(fn)
res = fn(controls_0)
if hook:
hook()
if with_complementarities:
[controls,nit] = ncpsolve(dfn, lb, ub, controls_0, verbose=verbit, maxit=inner_maxit)
else:
[controls, nit] = newton(dfn, controls_0, verbose=verbit, maxit=inner_maxit)
err = abs(controls-controls_0).max()
err_SA = err/err_0
err_0 = err
controls_0 = controls
t_finish = time.time()
elapsed = t_finish - t_start
if verbose:
print('|{0:4} | {1:10.3e} | {2:8.3f} | {3:8.3f} | {4:3} |'.format( it, err, err_SA, elapsed, nit ))
controls_0 = controls.reshape(sh_c)
t2 = time.time()
if verbose:
print(stars)
print("Elapsed: {} seconds.".format(t2-t1))
print(stars)
if not details:
return mdr
return TimeIterationResult(
mdr,
it,
with_complementarities,
dprocess,
err<tol, # x_converged: bool
tol, # x_tol
err, #: float
None, # log: object # TimeIterationLog
None # trace: object #{Nothing,IterationTrace}
)