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plots.py
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plots.py
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"""
Author:
Oliver Sheridan-Methven, October 2020.
Description:
The various plots for the article.
"""
import plotting_configuration
import matplotlib.pylab as plt
from mpl_toolkits.axes_grid1.inset_locator import inset_axes, InsetPosition, mark_inset
import numpy as np
import pandas as pd
from scipy.stats import norm, ncx2
from scipy.integrate import quad as integrate
from approximate_random_variables.approximate_gaussian_distribution import construct_piecewise_constant_approximation, construct_symmetric_piecewise_polynomial_approximation, rademacher_approximation
from approximate_random_variables.approximate_non_central_chi_squared import construct_inverse_non_central_chi_squared_interpolated_polynomial_approximation
from mpmath import mp, mpf
from timeit import default_timer as timer
from functools import wraps
from progressbar import progressbar
import json
def time_function(func):
""" A decorator to time a function. """
@wraps(func)
def wrapper(*args, **kwargs):
start_time = timer()
results = func(*args, **kwargs)
elapsed_time = timer() - start_time
return results, elapsed_time
return wrapper
mp.dps = 50
norm_inv = norm.ppf
def plot_piecewise_constant_approximation(savefig=False, plot_from_json=True):
if plot_from_json:
with open('piecewise_constant_gaussian_approximation.json', "r") as input_file:
results = json.load(input_file)
u, exact, approximation = results['uniforms'], results['exact'], results['approximate']
N, w, x, y = results['samples'], results['samples_uniform'], results['samples_exact'], results['samples_approx']
else:
u = np.linspace(0, 1, 1000)[1:-1]
norm_inv_approx = construct_piecewise_constant_approximation(norm.ppf, 8)
exact = norm_inv(u)
approximation = norm_inv_approx(u)
# For the QQ-plot
N = 1000
w = np.random.random(N)
x = norm_inv(w)
norm_inv_approx = construct_piecewise_constant_approximation(norm.ppf, 8)
y = norm_inv_approx(w)
plt.clf()
ax1 = plt.gca()
ax1.plot(u, exact, 'k--', label=r'$\Phi^{-1}(x)$')
ax1.plot(u, approximation, 'k,', label=r'__nolegend__')
ax1.plot([], [], 'k-', label=r'$Q(x)$')
ax1.set_xlabel(r"$x$")
ax1.set_xticks([0, 1])
ax1.set_yticks([-3, 0, 3])
ax1.set_ylim(-3, 3)
ax1.legend(frameon=False)
# The qq-plot
qlim = 3
bbox = ax1.get_window_extent()
ratio = bbox.width / bbox.height
size=43
ax2 = inset_axes(ax1, width=str(size / ratio) + '%', height=str(size) + '%', loc="lower right", borderpad=0.5)
ax2.set_box_aspect(1)
ax2.cla()
ax2.plot(x, x, '--', color='grey')
ax2.plot(x, y, 'k.', ms=0.7)
ax2.set_xlim(-qlim, qlim)
ax2.set_ylim(-qlim, qlim)
ax2.set_xticks([])
ax2.set_yticks([-qlim, 0, qlim])
if savefig:
plt.savefig('piecewise_constant_gaussian_approximation.pdf', format='pdf', bbox_inches='tight', transparent=True)
if not plot_from_json:
with open('piecewise_constant_gaussian_approximation.json', "w") as output_file:
output_file.write(json.dumps({'uniforms': u.tolist(),
'exact': norm_inv(u).tolist(),
'approximate': norm_inv_approx(u).tolist(),
'samples': N,
'samples_uniform': w.tolist(),
'samples_exact': x.tolist(),
'samples_approx': y.tolist()}, indent=4))
def plot_piecewise_constant_error(savefig=False, plot_from_json=True):
if plot_from_json:
with open('piecewise_constant_gaussian_approximation_error.json', "r") as input_file:
results = json.load(input_file)
for p in results:
for data_type in ['data', 'bound']:
results[p][data_type] = {float(k): float(v) for k, v in results[p][data_type].items()}
else:
results = {1 << i: {'data': {}, 'bound': {}} for i in range(1, 4)}
for n in [1 << i for i in range(11)]:
norm_inv_approx = construct_piecewise_constant_approximation(norm.ppf, n)
discontinuities = np.linspace(0, 1, n + 1)
for p in results:
p_norm = integrate(lambda u: (norm_inv(u) - norm_inv_approx(u)) ** p, 0, 1, points=discontinuities, limit=50 + 10 * n)[0] ** (1.0 / p)
results[p]['data'][n] = p_norm
for p in results:
n, p_norm = zip(*results[p]['data'].items())
q = np.linspace(2, np.log2(n[-1]), 100) # For the analytic bound from Giles
x = 2.0 ** q
y = 2.0 ** (-q / p) * q ** -0.5
y = y / y[-1] * p_norm[-1] # Rescaled
results[p]['bound'] = {k: v for k, v in zip(x, y)}
plt.clf()
markers = (i for i in {'o', 's', 'v'})
for p in results:
n, p_norm = zip(*results[p]['data'].items())
n_bound, p_bound = zip(*results[p]['bound'].items())
marker = next(markers)
plt.plot(n, p_norm, 'k{}'.format(marker), label=r'$p = {}$'.format(p))
plt.plot(n, p_norm, 'k:', label=r'__nolegend__')
plt.plot(n_bound, p_bound, 'k--', label='__nolegend__')
plt.plot([], [], 'k--', label=r'$O(2^{-q/p} q^{-1/2})$')
plt.yscale('log')
plt.xscale('log')
plt.ylabel(r'$\lVert Z - \widetilde{Z}\rVert_p$')
plt.xlabel('Intervals')
plt.legend(frameon=False, handlelength=1, borderaxespad=0)
if savefig:
plt.savefig('piecewise_constant_gaussian_approximation_error.pdf', format='pdf', bbox_inches='tight', transparent=True)
if not plot_from_json:
with open('piecewise_constant_gaussian_approximation_error.json', "w") as output_file:
output_file.write(json.dumps(results, indent=4))
def plot_piecewise_linear_gaussian_approximation(savefig=False, plot_from_json=True):
if plot_from_json:
with open('piecewise_linear_gaussian_approximation.json', "r") as input_file:
results = json.load(input_file)
u, exact, approximate = results['uniforms'], results['exact'], results['approximate']
N, w, x, y = results['samples'], results['samples_uniform'], results['samples_exact'], results['samples_approx']
else:
u = np.linspace(0, 1, 1000)[1:-1]
norm_inv_approx = construct_symmetric_piecewise_polynomial_approximation(norm.ppf, n_intervals=5, polynomial_order=1)
exact, approximate = norm_inv(u), norm_inv_approx(u)
# For the QQ-plot
N = 1000
w = np.random.random(N)
x = norm_inv(w)
y = norm_inv_approx(w)
plt.clf()
ax1 = plt.gca()
ax1.plot(u, exact, 'k--', label=r'$\Phi^{-1}(x)$')
ax1.plot(u, approximate, 'k,', label=r'__nolegend__')
ax1.plot([], [], 'k-', label=r'$D(x)$')
ax1.set_xlabel(r"$x$")
ax1.set_xticks([0, 1])
ax1.set_yticks([-3, 0, 3])
ax1.set_ylim(-3, 3)
ax1.legend(frameon=False)
# The qq-plot
qlim = 3
bbox = ax1.get_window_extent()
ratio = bbox.width / bbox.height
size=43
ax2 = inset_axes(ax1, width=str(size / ratio) + '%', height=str(size) + '%', loc="lower right", borderpad=0.5)
ax2.set_box_aspect(1)
ax2.cla()
ax2.plot(x, x, '--', color='grey')
ax2.plot(x, y, 'k.', ms=0.7)
ax2.set_xlim(-qlim, qlim)
ax2.set_ylim(-qlim, qlim)
ax2.set_xticks([])
ax2.set_yticks([-qlim, 0, qlim])
if savefig:
plt.savefig('piecewise_linear_gaussian_approximation.pdf', format='pdf', bbox_inches='tight', transparent=True)
if not plot_from_json:
with open('piecewise_linear_gaussian_approximation.json', "w") as output_file:
output_file.write(json.dumps({'uniforms': u.tolist(),
'exact': norm_inv(u).tolist(),
'approximate': norm_inv_approx(u).tolist(),
'samples': N,
'samples_uniform': w.tolist(),
'samples_exact': x.tolist(),
'samples_approx': y.tolist()}, indent=4))
def plot_piecewise_linear_gaussian_approximation_error(savefig=False, plot_from_json=True):
if plot_from_json:
with open('piecewise_linear_gaussian_approximation_error.json', "r") as input_file:
results = json.load(input_file)
results = {k: {int(x): float(y) for x, y in v.items()} for k, v in results.items()}
else:
polynomial_orders = range(6)
interval_sizes = [2, 4, 8, 16]
results = {s: {} for s in interval_sizes}
for n_intervals in results:
for polynomial_order in polynomial_orders:
approximate_inverse_gaussian_cdf = construct_symmetric_piecewise_polynomial_approximation(norm.ppf, n_intervals + 1, polynomial_order) # +1 as we have the 0 interval which is measure 0.
discontinuities = [0.5 ** (i + 2) for i in range(n_intervals)] # Makes the numerical integration involved in the RMSE easier.
rmse = integrate(lambda u: 2.0 * (norm.ppf(u) - approximate_inverse_gaussian_cdf(u)) ** 2, 0, 0.5, points=discontinuities)[0] ** 0.5
results[n_intervals][polynomial_order] = rmse
polynomial_orders = sorted(list(results.values())[0].keys())
plt.clf()
for n_intervals in results:
poly_orders, rmse = zip(*results[n_intervals].items())
plt.plot(poly_orders, rmse, 'ko:', label='__nolengend__')
plt.plot([], [], 'ko', label=n_intervals)
plt.gca().text(poly_orders[-1] + 0.2, rmse[-1], str(n_intervals), va='center')
plt.yscale('log')
plt.ylabel(r'$\lVert Z - \widetilde{Z}\rVert_2$')
plt.xlabel('Polynomial order')
plt.xlim(None, 5.7)
plt.ylim(1e-4, 1e0)
plt.xticks(polynomial_orders)
if savefig:
plt.savefig('piecewise_linear_gaussian_approximation_error.pdf', format='pdf', bbox_inches='tight', transparent=True)
if not plot_from_json:
with open('piecewise_linear_gaussian_approximation_error.json', "w") as output_file:
output_file.write(json.dumps(results, indent=4))
def produce_geometric_brownian_motion_paths(dt, method=None, approx=None):
"""
Perform path simulations of a geometric Brownian motion.
:param dt: Float. (Fraction of time).
:param method: Str.
:param approx: List.
:return: List. [x_fine_exact, x_coarse_exact, x_fine_approx, x_coarse_approx]
"""
assert isinstance(dt, float) and np.isfinite(dt) and dt > 0 and (1.0 / dt).is_integer()
assert isinstance(method, str) and method in ['euler_maruyama', 'milstein']
assert approx is not None
# The parameters.
x_0 = 1.0
mu = 0.05
sigma = 0.2
T = 1.0
dt = dt * T
t_fine = dt
t_coarse = 2 * dt
sqrt_t_fine = t_fine ** 0.5
w_coarse_exact = 0.0
w_coarse_approx = 0.0
x_fine_exact = x_0
x_coarse_exact = x_0
x_fine_approx = x_0
x_coarse_approx = x_0
n_fine = int(1.0 / dt)
update_coarse = False
x_0, mu, sigma, T, dt, t_fine, t_coarse, sqrt_t_fine, w_coarse_exact, w_coarse_approx = [mpf(i) for i in [x_0, mu, sigma, T, dt, t_fine, t_coarse, sqrt_t_fine, w_coarse_exact, w_coarse_approx]]
fabs = mp.fabs
path_update = None
if method == 'euler_maruyama':
path_update = lambda x, w, t: x + mu * x * t + sigma * x * w
elif method == 'milstein':
path_update = lambda x, w, t: x + mu * x * t + sigma * x * w + 0.5 * sigma * sigma * (w * w - t)
assert path_update is not None
for n in range(n_fine):
u = np.random.uniform()
z_exact = norm.ppf(u)
z_approx = approx(u)
z_approx = z_approx if isinstance(z_approx, float) else z_approx[0]
w_fine_exact = sqrt_t_fine * z_exact
w_fine_approx = sqrt_t_fine * z_approx
w_coarse_exact += w_fine_exact
w_coarse_approx += w_fine_approx
x_fine_exact = path_update(x_fine_exact, w_fine_exact, t_fine)
x_fine_approx = path_update(x_fine_approx, w_fine_approx, t_fine)
if update_coarse:
x_coarse_exact = path_update(x_coarse_exact, w_coarse_exact, t_coarse)
x_coarse_approx = path_update(x_coarse_approx, w_coarse_approx, t_coarse)
w_coarse_exact *= 0.0
w_coarse_approx *= 0.0
update_coarse = not update_coarse # We toggle to achieve pairwise summation.
assert not update_coarse # This should have been the last thing we did.
return [x_fine_exact, x_coarse_exact, x_fine_approx, x_coarse_approx]
def produce_cir_paths_with_only_gaussians(dt, approx=None, **kwargs):
"""
Perform path simulations of a geometric Brownian motion.
:param dt: Float. (Fraction of time).
:param method: Str.
:param approx: List.
:return: List. [x_fine_exact, x_coarse_exact, x_fine_approx, x_coarse_approx]
"""
assert isinstance(dt, float) and np.isfinite(dt) and dt > 0 and (1.0 / dt).is_integer()
assert approx is not None
# The parameters.
params = kwargs
kappa, theta, sigma = params['kappa'], params['theta'], params['sigma']
T = 1.0
x_0 = 1.0
dt = dt * T
sqrt_t = dt ** 0.5
c1 = 4.0 * kappa / (sigma ** 2 * (1.0 - np.exp(-kappa * dt)))
c2 = c1 * np.exp(-kappa * dt)
df = 4.0 * kappa * theta / (sigma ** 2)
dt = dt * T
t_fine = dt
t_coarse = 2 * dt
sqrt_t_fine = t_fine ** 0.5
w_coarse_exact = 0.0
w_coarse_approx = 0.0
x_fine_exact = x_0
x_coarse_exact = x_0
x_fine_approx = x_0
x_coarse_approx = x_0
n_fine = int(1.0 / dt)
update_coarse = False
# x_0, T, dt, t_fine, t_coarse, sqrt_t_fine, w_coarse_exact, w_coarse_approx = [mpf(i) for i in [x_0, T, dt, t_fine, t_coarse, sqrt_t_fine, w_coarse_exact, w_coarse_approx]]
# fabs = mp.fabs
path_update = lambda x, w, t: x + kappa * (theta - x) * t + sigma * np.sqrt(np.fabs(x)) * w
for n in range(n_fine):
u = np.random.uniform()
z_exact = norm.ppf(u)
z_approx = approx(u)
z_approx = z_approx if isinstance(z_approx, float) else z_approx[0]
w_fine_exact = sqrt_t_fine * z_exact
w_fine_approx = sqrt_t_fine * z_approx
w_coarse_exact += w_fine_exact
w_coarse_approx += w_fine_approx
x_fine_exact = path_update(x_fine_exact, w_fine_exact, t_fine)
x_fine_approx = path_update(x_fine_approx, w_fine_approx, t_fine)
if update_coarse:
x_coarse_exact = path_update(x_coarse_exact, w_coarse_exact, t_coarse)
x_coarse_approx = path_update(x_coarse_approx, w_coarse_approx, t_coarse)
w_coarse_exact *= 0.0
w_coarse_approx *= 0.0
update_coarse = not update_coarse # We toggle to achieve pairwise summation.
assert not update_coarse # This should have been the last thing we did.
return [x_fine_exact, x_coarse_exact, x_fine_approx, x_coarse_approx]
def plot_variance_reduction_geometric_brownian_motion(savefig=False, plot_from_json=True):
methods = ['euler_maruyama', 'milstein']
if plot_from_json:
results = {}
for method in methods:
with open('variance_reduction_{}_scheme.json'.format(method), "r") as input_file:
results[method] = json.load(input_file)
results[method] = {k: {float(x): y for x, y in v.items()} for k, v in results[method].items()}
else:
deltas = [2.0 ** -i for i in range(1, 7)]
inverse_norm = norm.ppf
piecewise_constant = construct_piecewise_constant_approximation(inverse_norm, n_intervals=1024)
piecewise_linear = construct_symmetric_piecewise_polynomial_approximation(inverse_norm, n_intervals=16, polynomial_order=1)
piecewise_cubic = construct_symmetric_piecewise_polynomial_approximation(inverse_norm, n_intervals=16, polynomial_order=3)
approximations = {'constant': piecewise_constant, 'linear': piecewise_linear, 'cubic': piecewise_cubic, 'rademacher': rademacher_approximation}
results = {method: {term: {} for term in ['original'] + list(approximations.keys())} for method in methods} # Store the values of delta and the associated data.
time_per_level = 2.0
paths_min = 64
for method in results:
for approx_name, approx in approximations.items():
for dt in deltas:
_, elapsed_time_per_path = time_function(produce_geometric_brownian_motion_paths)(dt, method, approx)
paths_required = int(time_per_level / elapsed_time_per_path)
if paths_required < paths_min:
print("More time required for {} and {} with dt={}".format(method, approx_name, dt))
break
originals, corrections = [[None for i in range(paths_required)] for j in range(2)]
for path in range(paths_required):
x_fine_exact, x_coarse_exact, x_fine_approx, x_coarse_approx = produce_geometric_brownian_motion_paths(dt, method, approx)
originals[path] = x_fine_exact - x_coarse_exact
corrections[path] = min((x_fine_exact - x_coarse_exact) - (x_fine_approx - x_coarse_approx), (x_fine_exact - x_fine_approx) - (x_coarse_exact - x_coarse_approx), sum([x_fine_exact, -x_coarse_exact, -x_fine_approx, x_coarse_approx])) # might need revising for near machine precision.
originals, corrections = [[j ** 2 for j in i] for i in [originals, corrections]]
for name, values in [['original', originals], [approx_name, corrections]]:
mean = np.mean(values)
std = np.std(values) / (len(values) ** 0.5)
[mean, std] = [float(i) for i in [mean, std]]
results[method][name][dt] = [mean, std]
markers = {'original': 'd', 'constant': 'o', 'linear': 'v', 'cubic': 's', 'rademacher': 'x'}
deltas = list(list(list(results.items())[0][1].items())[0][1].keys())
for method in results:
plt.clf()
for approx_name in results[method]:
x, y = zip(*results[method][approx_name].items())
y, y_std = list(zip(*y))
y_error = 1 * np.array(y_std)
plt.errorbar(x, y, y_error, None, 'k{}:'.format(markers[approx_name]))
plt.xscale('log', base=2)
plt.yscale('log', base=2)
plt.xlabel(r'Fine time increment $\delta^{\mathrm{f}}$')
plt.ylabel('Variance')
y_min_base_2 = 50
plt.ylim(2 ** -y_min_base_2, 2 ** -10)
plt.yticks([2 ** -i for i in range(10, y_min_base_2 + 1, 10)])
plt.xticks(deltas)
if savefig:
plt.savefig('variance_reduction_{}_scheme.pdf'.format(method), format='pdf', bbox_inches='tight', transparent=True)
if not plot_from_json:
with open('variance_reduction_{}_scheme.json'.format(method), "w") as output_file:
output_file.write(json.dumps(results[method], indent=4))
def plot_variance_reduction_cir_with_only_approx_gaussian_mlmc(savefig=False, plot_from_json=True):
if plot_from_json:
with open('variance_reduction_cir_with_only_approx_gaussian_mlmc.json', "r") as input_file:
results = json.load(input_file)
else:
deltas = [2.0 ** -i for i in range(1, 9)]
inverse_norm = norm.ppf
piecewise_constant = construct_piecewise_constant_approximation(inverse_norm, n_intervals=1024)
piecewise_linear = construct_symmetric_piecewise_polynomial_approximation(inverse_norm, n_intervals=16, polynomial_order=1)
approximations = {'constant': piecewise_constant, 'linear': piecewise_linear}
params = {'kappa': 0.5, 'theta': 1.0, 'sigma': 1.0}
results = {term: {} for term in ['original'] + list(approximations.keys())}
time_per_level = 60.0
paths_min = 64
for approx_name, approx in approximations.items():
for dt in deltas:
_, elapsed_time_per_path = time_function(produce_cir_paths_with_only_gaussians)(dt, approx, **params)
paths_required = int(time_per_level / elapsed_time_per_path)
if paths_required < paths_min:
print("More time required for {} with dt={}".format(approx_name, dt))
break
originals, corrections = [[None for i in range(paths_required)] for j in range(2)]
for path in range(paths_required):
x_fine_exact, x_coarse_exact, x_fine_approx, x_coarse_approx = produce_cir_paths_with_only_gaussians(dt, approx, **params)
originals[path] = x_fine_exact - x_coarse_exact
corrections[path] = min((x_fine_exact - x_coarse_exact) - (x_fine_approx - x_coarse_approx), (x_fine_exact - x_fine_approx) - (x_coarse_exact - x_coarse_approx), sum([x_fine_exact, -x_coarse_exact, -x_fine_approx, x_coarse_approx])) # might need revising for near machine precision.
originals, corrections = [[j ** 2 for j in i] for i in [originals, corrections]]
for name, values in [['original', originals], [approx_name, corrections]]:
mean = np.mean(values)
std = np.std(values) / (len(values) ** 0.5)
[mean, std] = [float(i) for i in [mean, std]]
results[name][dt] = [mean, std]
markers = {'original': 'd', 'constant': 'o', 'linear': 'v', 'cubic': 's', 'rademacher': 'x'}
deltas = list(list(results.items())[0][1].keys())
levels = [int(i) for i in np.log2(1.0/np.array([float(i) for i in deltas]))]
plt.clf()
ls = {'original': (None, None), 'constant': (15, 15), 'linear': (10, 3, 4, 3)}
leg = {'original': 'baseline', 'constant': 'constant', 'linear': 'dyadic'}
for approx_name in results:
x, y = zip(*results[approx_name].items())
x = [float(i) for i in x]
l = [int(i) for i in np.log2(1.0/np.array(x))]
y, y_std = list(zip(*y))
y_error = 1 * np.array(y_std)
plt.errorbar(l, y, y_error, None, 'ko-', dashes=ls[approx_name], label=leg[approx_name])
plt.yscale('log', base=10)
plt.xlabel(r'level $\ell$')
plt.ylabel('Variance')
plt.ylim(1e-8, 1e1)
plt.xlim(0, None)
plt.xticks(levels)
plt.legend(frameon=False, handlelength=4)
if savefig:
plt.savefig('variance_reduction_cir_with_only_approx_gaussian_mlmc.pdf', format='pdf', bbox_inches='tight', transparent=True)
if not plot_from_json:
with open('variance_reduction_cir_with_only_approx_gaussian_mlmc.json', "w") as output_file:
output_file.write(json.dumps(results, indent=4))
def inverse_non_central_chi_squared_abdel_aty(u, df, nc):
"""The approximation from Abdel-Aty, cf: https://en.wikipedia.org/wiki/Noncentral_chi-squared_distribution#Approximation_(including_for_quantiles)"""
k = df
l = nc
f = (k + l) ** 2 / (k + 2.0*l)
x = norm.ppf(u)
x *= np.sqrt(2.0/(9.0 * f))
x += 1.0 - 2.0/(9.0 * f)
x = x ** 3
x *= (k + l)
return x
def inverse_non_central_chi_squared_sankaran(u, df, nc):
"""The approximation from Sankaran, cf: https://en.wikipedia.org/wiki/Noncentral_chi-squared_distribution#Approximation_(including_for_quantiles)"""
k = df
l = nc
h = 1.0 - (2.0/3.0) * (k + l) * (k + 3.0 * l) / ((k + 2.0 * l)**2)
p = (k + 2.0*l) / ((k + l)**2)
m = (h - 1.0) * (1 - 3.0*h)
x = norm.ppf(u)
x *= h * np.sqrt(2.0 * p) * (1.0 + 0.5 * m * p)
x += 1.0 + h * p * (h - 1.0 + 0.5 * (2.0 - h) * m * p)
x = x ** (1.0/h)
x *= (k + l)
return x
def rmse_of_non_central_chi_squared_polynomial_approximations():
lambdas = [1, 5, 10, 50, 100, 200]
nus = [1, 5, 10, 50, 100]
poly_orders = [1, 3, 5]
n_intervals = 16
results = {poly_order: {nu: {} for nu in nus} for poly_order in poly_orders}
for poly_order in poly_orders:
for nu in nus:
ncx2_approx = construct_inverse_non_central_chi_squared_interpolated_polynomial_approximation(dof=nu, n_intervals=n_intervals + 1, polynomial_order=poly_order)
discontinuities = sorted([0.5 ** (i + 2) for i in range(n_intervals)] + [0.5] + [1.0 - 0.5 ** (i + 2) for i in range(n_intervals)])
for l in lambdas:
rmse = integrate(lambda u: (ncx2.ppf(u, df=nu, nc=l) - ncx2_approx(u, non_centrality=l)) ** 2, 0, 1, points=discontinuities, limit=50 + 10 * len(discontinuities))[0] ** 0.5
results[poly_order][nu][l] = rmse
for poly_order, result in results.items():
df = pd.DataFrame(result)
df.index = df.index.rename('lambda')
df.columns = df.columns.rename('nu')
print(poly_order, df.min().min(), df.max().max())
print(round(df, 3))
print('\n')
print(round(df, 3).apply(lambda x: ' & '.join([str(i) for i in list(x)]) + r' \\', axis=1))
print('\n' * 3)
# For the approximations by Abdel-Aty and Sankaran
approxes = {'Abdel-Aty': inverse_non_central_chi_squared_abdel_aty, 'Sankaran': inverse_non_central_chi_squared_sankaran}
results = {approx: {nu: {} for nu in nus} for approx in approxes.keys()}
for name in results.keys():
ncx2_approx = approxes[name]
for nu in progressbar(nus):
for l in lambdas:
limits=[50,10,1]
for limit in limits:
rmse = integrate(lambda u: (ncx2.ppf(u, df=nu, nc=l) - ncx2_approx(u, df=nu, nc=l)) ** 2, 0, 1, limit=limit)[0] ** 0.5
if not np.isnan(rmse):
break
results[name][nu][l] = rmse
for name, result in results.items():
df = pd.DataFrame(result)
df.index = df.index.rename('lambda')
df.columns = df.columns.rename('nu')
print(name, df.min().min(), df.max().max())
print(round(df, 3))
print('\n')
print(round(df, 3).apply(lambda x: ' & '.join([str(i) for i in list(x)]) + r' \\', axis=1))
print('\n' * 3)
def produce_cox_ingersoll_ross_paths_by_approx_euler_maruyama(dt, gaussian_approximations=None, **kwargs):
assert isinstance(dt, float) and np.isfinite(dt) and dt > 0 and (1.0 / dt).is_integer()
assert gaussian_approximations is not None
# The parameters.
params = kwargs
kappa, theta, sigma = params['kappa'], params['theta'], params['sigma']
T = 1.0
x_0 = 1.0
dt = dt * T
sqrt_t = dt ** 0.5
c1 = 4.0 * kappa / (sigma ** 2 * (1.0 - np.exp(-kappa * dt)))
c2 = c1 * np.exp(-kappa * dt)
df = 4.0 * kappa * theta / (sigma ** 2)
euler_maruyama_update = lambda x, w, t: x + kappa * (theta - x) * t + sigma * np.sqrt(np.fabs(x)) * w
x_euler_maruyama = x_0
x_approximations = [x_0] * len(gaussian_approximations)
n_increments = int(1.0 / dt)
for n in range(n_increments):
u = np.random.uniform()
z = norm.ppf(u)
z_approxes = [approx(u) for approx in gaussian_approximations]
dw = sqrt_t * z
dw_approxes = [sqrt_t * z in z_approxes]
x_euler_maruyama = euler_maruyama_update(x_euler_maruyama, dw, dt)
x_approximations = [euler_maruyama_update(x_euler_maruyama, dw_approx, dt) for dw_approx in dw_approxes]
return [x_euler_maruyama, *x_approximations]
def produce_cox_ingersoll_ross_paths(dt, approximations=None, full_path=False, **kwargs):
assert isinstance(dt, float) and np.isfinite(dt) and dt > 0 and (1.0 / dt).is_integer()
assert approximations is not None
# The parameters.
params = kwargs
kappa, theta, sigma = params['kappa'], params['theta'], params['sigma']
T = 1.0
x_0 = 1.0
dt = dt * T
sqrt_t = dt ** 0.5
c1 = 4.0 * kappa / (sigma ** 2 * (1.0 - np.exp(-kappa * dt)))
c2 = c1 * np.exp(-kappa * dt)
df = 4.0 * kappa * theta / (sigma ** 2)
euler_maruyama_update = lambda x, w, t: x + kappa * (theta - x) * t + sigma * np.sqrt(np.fabs(x)) * w
exact_update = lambda u, x: ncx2.ppf(u, df=df, nc=x * c2) / c1
approximate_update = lambda u, x, approx: approx(u, non_centrality=x * c2)[0] / c1
x_exact = x_0
x_euler_maruyama = x_0
x_approximations = [x_0] * len(approximations)
if full_path:
paths = []
paths.append([x_euler_maruyama, x_exact, *x_approximations])
n_increments = int(1.0 / dt)
for n in range(n_increments):
u = np.random.uniform()
z = norm.ppf(u)
dw = sqrt_t * z
x_euler_maruyama = euler_maruyama_update(x_euler_maruyama, dw, dt)
x_exact = exact_update(u, x_exact)
x_approximations = [approximate_update(u, x_approximate, approx) for approx, x_approximate in zip(approximations, x_approximations)]
if full_path:
paths.append([x_euler_maruyama, x_exact, *x_approximations])
if full_path:
x_euler_maruyama, x_exact, *x_approximations = list(zip(*paths))
return [x_euler_maruyama, x_exact, *x_approximations]
def plot_variance_reduction_cir_process(savefig=False, plot_from_json=True):
poly_orders = {'linear': 1, 'cubic': 3}
poly_markers = (i for i in ['s', 'd'])
markers = {**{'exact': 'o', 'euler_maruyama': 'v'}, **{k: next(poly_markers) for k in poly_orders}}
if plot_from_json:
with open('variance_reduction_cir_process.json', "r") as input_file:
results = json.load(input_file)
results = {k: {float(x): y for x, y in v.items()} for k, v in results.items()}
else:
deltas = [0.5 ** i for i in range(8)]
params = {'kappa': 0.5, 'theta': 1.0, 'sigma': 1.0}
nu = 4.0 * params['kappa'] * params['theta'] / (params['sigma'] ** 2)
approximations = [construct_inverse_non_central_chi_squared_interpolated_polynomial_approximation(dof=nu, polynomial_order=poly_order) for poly_order in [1, 3]]
results = {k: {} for k in ['exact', 'euler_maruyama'] + list(poly_orders.keys())}
time_per_level = 5.0
paths_min = 64
for dt in progressbar(deltas):
_, elapsed_time_per_path = time_function(produce_cox_ingersoll_ross_paths)(dt, approximations, **params)
paths_required = int(time_per_level / elapsed_time_per_path)
if paths_required < paths_min:
print("More time required for dt={}".format(dt))
break
exacts, euler_maruyamas, linears, cubics = [[None for i in range(paths_required)] for j in range(4)]
for path in range(paths_required):
x_euler_maruyama, x_exact, x_linear, x_cubic = produce_cox_ingersoll_ross_paths(dt, approximations, **params)
exacts[path] = x_exact
euler_maruyamas[path] = x_exact - x_euler_maruyama
linears[path] = x_exact - x_linear
cubics[path] = x_exact - x_cubic
exacts, euler_maruyamas, linears, cubics = [[j ** 2 for j in i] for i in [exacts, euler_maruyamas, linears, cubics]]
for name, values in {'exact': exacts, 'euler_maruyama': euler_maruyamas, 'linear': linears, 'cubic': cubics}.items():
mean = np.mean(values)
std = np.std(values) / (len(values) ** 0.5)
results[name][dt] = [mean, std]
plt.clf()
for name in results:
x, y = zip(*results[name].items())
y, y_std = list(zip(*y))
y_error = 1 * np.array(y_std)
plt.errorbar(x, y, y_error, None, 'k{}:'.format(markers[name]))
plt.xscale('log', base=2)
plt.yscale('log', base=2)
plt.xticks(x)
plt.ylim(2 ** -25, 2 ** 2)
plt.yticks([2 ** -i for i in range(0, 30, 5)])
plt.xlabel(r'Time increment $\delta$')
plt.ylabel('Variance')
if savefig:
plt.savefig('variance_reduction_cir_process.pdf', format='pdf', bbox_inches='tight', transparent=True)
if not plot_from_json:
with open('variance_reduction_cir_process.json', "w") as output_file:
output_file.write(json.dumps(results, indent=4))
def plot_variance_reduction_cir_process_asian_option(savefig=False, plot_from_json=True):
poly_orders = {'linear': 1, 'cubic': 3}
poly_markers = (i for i in ['s', 'd'])
markers = {**{'exact': 'o', 'euler_maruyama': 'v'}, **{k: next(poly_markers) for k in poly_orders}}
full_paths = True
if plot_from_json:
with open('variance_reduction_cir_process_asian_option.json', "r") as input_file:
results = json.load(input_file)
results = {k: {float(x): y for x, y in v.items()} for k, v in results.items()}
else:
deltas = [0.5 ** i for i in range(8)]
params = {'kappa': 0.5, 'theta': 1.0, 'sigma': 1.0}
nu = 4.0 * params['kappa'] * params['theta'] / (params['sigma'] ** 2)
approximations = [construct_inverse_non_central_chi_squared_interpolated_polynomial_approximation(dof=nu, polynomial_order=poly_order) for poly_order in [1, 3]]
results = {k: {} for k in ['exact', 'euler_maruyama'] + list(poly_orders.keys())}
time_per_level = 5.0
paths_min = 64
for dt in progressbar(deltas):
_, elapsed_time_per_path = time_function(produce_cox_ingersoll_ross_paths)(dt, approximations, full_paths, **params)
paths_required = int(time_per_level / elapsed_time_per_path)
if paths_required < paths_min:
print("More time required for dt={}".format(dt))
break
exacts, euler_maruyamas, linears, cubics = [[None for i in range(paths_required)] for j in range(4)]
for path in range(paths_required):
x_euler_maruyama, x_exact, x_linear, x_cubic = produce_cox_ingersoll_ross_paths(dt, approximations, full_paths, **params)
x_euler_maruyama, x_exact, x_linear, x_cubic = [np.mean(i) for i in [x_euler_maruyama, x_exact, x_linear, x_cubic]] # The arithmetic mean
exacts[path] = x_exact
euler_maruyamas[path] = x_exact - x_euler_maruyama
linears[path] = x_exact - x_linear
cubics[path] = x_exact - x_cubic
exacts, euler_maruyamas, linears, cubics = [[j ** 2 for j in i] for i in [exacts, euler_maruyamas, linears, cubics]]
for name, values in {'exact': exacts, 'euler_maruyama': euler_maruyamas, 'linear': linears, 'cubic': cubics}.items():
mean = np.mean(values)
std = np.std(values) / (len(values) ** 0.5)
results[name][dt] = [mean, std]
plt.clf()
for name in results:
x, y = zip(*results[name].items())
y, y_std = list(zip(*y))
y_error = 1 * np.array(y_std)
plt.errorbar(x, y, y_error, None, 'k{}:'.format(markers[name]))
plt.xscale('log', base=2)
plt.yscale('log', base=2)
plt.xticks(x)
plt.ylim(2 ** -27, 2 ** 2)
plt.yticks([2 ** -i for i in range(0, 30, 5)])
plt.xlabel(r'Time increment $\delta$')
plt.ylabel('Variance')
if savefig:
plt.savefig('variance_reduction_cir_process_asian_option.pdf', format='pdf', bbox_inches='tight', transparent=True)
if not plot_from_json:
with open('variance_reduction_cir_process_asian_option.json', "w") as output_file:
output_file.write(json.dumps(results, indent=4))
def plot_non_central_chi_squared_polynomial_approximation(savefig=False, plot_from_json=True):
if plot_from_json:
with open('non_central_chi_squared_linear_approximation.json', "r") as input_file:
results = json.load(input_file)
results = {k: {x: {float(u): w for u, w in y.items()} for x, y in v.items()} for k, v in results.items()}
else:
dof = 1.0
ncx2_approx = construct_inverse_non_central_chi_squared_interpolated_polynomial_approximation(dof, n_intervals=4 + 1)
u = np.concatenate([np.linspace(0.0, 1.0, 1000)[:-1], np.logspace(-10, -1, 100), 1.0 - np.logspace(-10, -1, 100)])
u.sort()
non_centralities = [1.0, 10.0, 20.0]
results = {non_centrality: {} for non_centrality in non_centralities}
for non_centrality in results:
exact, approximate = ncx2.ppf(u, df=dof, nc=non_centrality), ncx2_approx(u, non_centrality=non_centrality)
results[non_centrality]['exact'] = {x: y for x, y in zip(u, exact)}
results[non_centrality]['approximate'] = {x: y for x, y in zip(u, approximate)}
_u = u[1:-1] # The end points can be singular, so we avoid these.
abdel_aty = inverse_non_central_chi_squared_abdel_aty(_u, df=dof, nc=non_centrality)
sankaran = inverse_non_central_chi_squared_sankaran(_u, df=dof, nc=non_centrality)
results[non_centrality]['abdel_aty'] = {x: y for x, y in zip(_u, abdel_aty)}
results[non_centrality]['sankaran'] = {x: y for x, y in zip(_u, sankaran)}
plt.clf()
for non_centrality in results:
exact, approximate = results[non_centrality]['exact'], results[non_centrality]['approximate']
plt.plot(*zip(*exact.items()), 'k--')
plt.plot(*zip(*approximate.items()), 'k,')
abdel_aty, sankaran = results[non_centrality]['abdel_aty'], results[non_centrality]['sankaran']
# plt.plot(*zip(*abdel_aty.items()), 'r,')
# plt.plot(*zip(*abdel_aty.items()), 'b,')
plt.plot([], [], 'k--', label=r'$C^{-1}_{\nu}(x;\lambda)$')
plt.plot([], [], 'k-', label=r'$\widetilde{C}^{-1}_{\nu}(x;\lambda)$')
plt.ylim(0, 50)
plt.yticks([i for i in range(0, 51, 10)])
plt.xticks([0, 1])
plt.xlabel(r'$x$')
plt.legend(frameon=False)
if savefig:
plt.savefig('non_central_chi_squared_linear_approximation.pdf', format='pdf', bbox_inches='tight', transparent=True)
if not plot_from_json:
with open('non_central_chi_squared_linear_approximation.json', "w") as output_file:
output_file.write(json.dumps(results, indent=4))
def print_speed_up_and_efficiencies(variances_reductions, cost_reductions):
for V, c0 in zip(variances_reductions, cost_reductions):
c = 1.0 / c0
C = 1.0 + c
e = (1.0 + np.sqrt(V * C / c)) ** 2
s = c * e
m = np.sqrt(s / c)
M = np.sqrt(s * V / C)
print(1.0 / s, 100.0 / e, m, 1.0 / M)
def print_speed_up_and_efficiencies_non_central_chi_squared():
variances_reductions = [2 ** -15, 2 ** -25]
cost_reductions = [300.0, 300.0]
print_speed_up_and_efficiencies(variances_reductions, cost_reductions)
def print_speed_up_and_efficiencies_gaussian():
variances_reductions = [2 ** -1, 2 ** -13, 2 ** -14, 2 ** -25]
cost_reductions = [9.0, 6.0, 7.0, 5.0]
print_speed_up_and_efficiencies(variances_reductions, cost_reductions)
if __name__ == '__main__':
plot_params = dict(savefig=True, plot_from_json=True)
# plot_params = dict(savefig=True, plot_from_json=False)
# plot_params = dict(savefig=False, plot_from_json=True)
# plot_params = dict(savefig=False, plot_from_json=False)
plot_piecewise_constant_approximation(**plot_params)
plot_piecewise_constant_error(**plot_params)
plot_piecewise_linear_gaussian_approximation(**plot_params)
plot_piecewise_linear_gaussian_approximation_error(**plot_params)
plot_variance_reduction_geometric_brownian_motion(**plot_params)
plot_variance_reduction_cir_with_only_approx_gaussian_mlmc(**plot_params)
plot_variance_reduction_cir_process(**plot_params)
plot_variance_reduction_cir_process_asian_option(**plot_params)
plot_non_central_chi_squared_polynomial_approximation(**plot_params)
print_speed_up_and_efficiencies_gaussian()