mirror of
https://github.com/vale981/bachelor_thesis
synced 2025-03-10 04:46:40 -04:00
427 lines
15 KiB
Python
427 lines
15 KiB
Python
"""
|
|
Simple monte carlo integration implementation.
|
|
Author: Valentin Boettcher <hiro@protagon.space>
|
|
"""
|
|
import numpy as np
|
|
from scipy.optimize import minimize_scalar, root
|
|
from dataclasses import dataclass
|
|
|
|
|
|
def _process_interval(interval):
|
|
assert len(interval) == 2, 'An interval has two endpoints'
|
|
|
|
a, b = interval
|
|
if b < a:
|
|
a, b = b, a
|
|
|
|
return [a, b]
|
|
|
|
|
|
@dataclass
|
|
class IntegrationResult:
|
|
"""
|
|
A simple container class to hold the result, uncertainty and
|
|
sampling size of the naive monte-carlo integration.
|
|
"""
|
|
|
|
result: float
|
|
sigma: float
|
|
N: int
|
|
|
|
@property
|
|
def combined_result(self):
|
|
"""
|
|
Get the result and accuracy combined as tuple.
|
|
"""
|
|
|
|
return self.result, self.sigma
|
|
|
|
|
|
def integrate(f, interval, epsilon=.01,
|
|
seed=None, **kwargs) -> IntegrationResult:
|
|
"""Monte-Carlo integrates the function `f` over an interval.
|
|
|
|
:param f: function of one variable, kwargs are passed to it
|
|
:param tuple interval: a 2-tuple of numbers, specifiying the
|
|
integration range
|
|
:param epsilon: desired accuracy
|
|
:param seed: the seed for the rng, if not specified, the system
|
|
time is used
|
|
|
|
:returns: the integration result
|
|
|
|
:rtype: IntegrationResult
|
|
"""
|
|
|
|
interval = _process_interval(interval)
|
|
if seed:
|
|
np.random.seed(seed)
|
|
|
|
interval_length = (interval[1] - interval[0])
|
|
|
|
# guess the correct N
|
|
probe_points = np.random.uniform(interval[0], interval[1],
|
|
int(interval_length*10))
|
|
|
|
num_points = int((interval_length
|
|
* f(probe_points, **kwargs).std()/epsilon)**2*1.1 + 1)
|
|
|
|
# now we iterate until we hit the desired epsilon
|
|
while True:
|
|
points = np.random.uniform(interval[0], interval[1], num_points)
|
|
sample = f(points, **kwargs)
|
|
|
|
integral = np.sum(sample)/num_points*interval_length
|
|
|
|
# the deviation gets multiplied by the square root of the interval
|
|
# lenght, because it is the standard deviation of the integral.
|
|
sample_std = np.std(sample)*interval_length
|
|
deviation = sample_std*np.sqrt(1/(num_points - 1))
|
|
|
|
if deviation < epsilon:
|
|
return IntegrationResult(integral, deviation, num_points)
|
|
|
|
# then we refine our guess, the factor 1.1
|
|
num_points = int((sample_std/epsilon)**2*1.1)
|
|
|
|
|
|
def find_upper_bound(f, interval, **kwargs):
|
|
"""Find the upper bound of a function.
|
|
|
|
:param f: function of one scalar and some kwargs that are passed
|
|
on to it
|
|
:param interval: interval to look in
|
|
|
|
:returns: the upper bound of the function
|
|
:rtype: float
|
|
"""
|
|
|
|
upper_bound = minimize_scalar(lambda *args: -f(*args, **kwargs),
|
|
bounds=interval, method='bounded')
|
|
if upper_bound.success:
|
|
return -upper_bound.fun
|
|
else:
|
|
raise RuntimeError('Could not find an upper bound.')
|
|
|
|
|
|
def sample_unweighted(f, interval, upper_bound=None, seed=None,
|
|
chunk_size=100, report_efficiency=False, **kwargs):
|
|
"""Samples a distribution proportional to f by hit and miss.
|
|
Implemented as a generator.
|
|
|
|
:param f: function of one scalar to sample, should be positive,
|
|
superflous kwargs are passed to it
|
|
:param interval: the interval to sample from
|
|
:param upper_bound: an upper bound to the function, optional
|
|
:param seed: the seed for the rng, if not specified, the system
|
|
time is used
|
|
:param chunk_size: the size of the chunks of random numbers
|
|
allocated per unit interval
|
|
:yields: random nubers following the distribution of f
|
|
:rtype: float
|
|
"""
|
|
|
|
interval = _process_interval(interval)
|
|
interval_length = (interval[1] - interval[0])
|
|
|
|
if seed:
|
|
np.random.seed(seed)
|
|
|
|
upper_bound_fn, upper_bound_integral, upper_bound_integral_inverse = None, None, None
|
|
# i know....
|
|
|
|
if not upper_bound:
|
|
upper_bound_value = find_upper_bound(f, interval, **kwargs)
|
|
def upper_bound_fn(x): return upper_bound_value
|
|
def upper_bound_integral(x): return upper_bound_value*x
|
|
def upper_bound_integral_inverse(y): return y/upper_bound_value
|
|
|
|
elif len(upper_bound) == 2:
|
|
upper_bound_fn, upper_bound_integral =\
|
|
upper_bound
|
|
|
|
def upper_inv(points): # not for performance right now...
|
|
return np.array([root(lambda y: upper_bound_integral(y) - x, x0=0,
|
|
jac=upper_bound_fn).x for x in points]).T
|
|
|
|
upper_bound_integral_inverse = upper_inv
|
|
|
|
elif len(upper_bound) == 3:
|
|
upper_bound_fn, upper_bound_integral, upper_bound_integral_inverse =\
|
|
upper_bound
|
|
else:
|
|
raise ValueError(
|
|
'The upper bound must be `None` or a three element sequence!')
|
|
|
|
def allocate_random_chunk():
|
|
return np.random.uniform([upper_bound_integral(interval[0]), 0],
|
|
[upper_bound_integral(interval[1]), 1],
|
|
[int(chunk_size*interval_length), 2])
|
|
|
|
total_points = 0
|
|
total_accepted = 0
|
|
|
|
while True:
|
|
points = allocate_random_chunk()
|
|
points[:, 0] = upper_bound_integral_inverse(points[:, 0])
|
|
sample_points = points[:, 0][np.where(f(points[:, 0]) >
|
|
points[:, 1]*upper_bound_fn(points[:, 0]))]
|
|
|
|
if report_efficiency:
|
|
total_points += points.size
|
|
total_accepted += sample_points.size
|
|
|
|
for point in sample_points:
|
|
yield (point, total_accepted/total_points) \
|
|
if report_efficiency else point
|
|
|
|
|
|
@dataclass
|
|
class VegasIntegrationResult(IntegrationResult):
|
|
"""
|
|
A simple container class to hold the result, uncertainty and
|
|
sampling size of the naive monte-carlo integration.
|
|
|
|
The ascertained increment borders, as well as the total sample
|
|
size are available as well.
|
|
"""
|
|
|
|
increment_borders: np.ndarray
|
|
vegas_iterations: int
|
|
|
|
|
|
def integrate_vegas(f, interval, seed=None, num_increments=5,
|
|
target_epsilon=1e-3, epsilon=1e-2, alpha=1.5, acumulate=True,
|
|
**kwargs) -> VegasIntegrationResult:
|
|
"""Integrate the given function (in one dimension) with the vegas
|
|
algorithm to reduce variance. This implementation follows the
|
|
description given in JOURNAL OF COMPUTATIONAL 27, 192-203 (1978).
|
|
|
|
All iterations contribute to the final result.
|
|
|
|
:param f: function of one variable, kwargs are passed to it
|
|
:param tuple interval: a 2-tuple of numbers, specifiying the
|
|
integration range
|
|
:param seed: the seed for the rng, if not specified, the system
|
|
time is used
|
|
:param num_increments: the number increments in which to divide
|
|
the interval
|
|
:param point_density: the number of random points per unit
|
|
interval
|
|
:param epsilon: the breaking condition, if the magnitude of the
|
|
difference between the increment positions in subsequent
|
|
iterations does not change more then epsilon the computation
|
|
is considered to have converged
|
|
:param alpha: controls the the convergence speed, should be
|
|
between 1 and 2 (the lower the faster)
|
|
|
|
:returns: the intregal, the standard deviation, an array of
|
|
increment borders which can be used in subsequent
|
|
sampling
|
|
|
|
:rtype: tuple
|
|
"""
|
|
|
|
interval = _process_interval(interval)
|
|
interval_length = (interval[1] - interval[0])
|
|
|
|
if seed:
|
|
np.random.seed(seed)
|
|
|
|
# no clever logic is being used to define the vegas iteration
|
|
# sample density for the sake of simplicity
|
|
points_per_increment = int(100*interval_length/num_increments)
|
|
total_points = points_per_increment*num_increments
|
|
|
|
# start with equally sized intervals
|
|
interval_borders = np.linspace(*interval, num_increments + 1,
|
|
endpoint=True)
|
|
|
|
def evaluate_integrand(interval_borders, interval_lengths,
|
|
samples_per_increment):
|
|
intervals = np.array((interval_borders[:-1], interval_borders[1:]))
|
|
sample_points = np.random.uniform(*intervals,
|
|
(samples_per_increment,
|
|
num_increments)).T
|
|
|
|
weighted_f_values = f(sample_points, **kwargs) * \
|
|
interval_lengths[:, None]
|
|
|
|
# the mean here has absorbed the num_increments
|
|
integral_steps = weighted_f_values.mean(axis=1)
|
|
integral = integral_steps.sum()
|
|
variance = \
|
|
((f(sample_points, **kwargs).std(axis=1)
|
|
* interval_lengths)**2).sum() / (samples_per_increment - 1)
|
|
return integral, integral_steps, variance
|
|
|
|
K = num_increments*1000
|
|
increment_borders = interval_borders[1:-1] - interval_borders[0]
|
|
|
|
integrals = []
|
|
variances = []
|
|
|
|
vegas_iterations, integral, variance = 0, 0, 0
|
|
while True:
|
|
vegas_iterations += 1
|
|
interval_lengths = interval_borders[1:] - interval_borders[:-1]
|
|
integral, integral_steps, variance = \
|
|
evaluate_integrand(interval_borders,
|
|
interval_lengths, points_per_increment)
|
|
|
|
integrals.append(integral)
|
|
variances.append(variance)
|
|
|
|
# alpha controls the convergence speed
|
|
μ = np.abs(integral_steps)/integral
|
|
new_increments = (K*((μ - 1)/(np.log(μ)))**alpha).astype(int)
|
|
group_size = new_increments.sum()/num_increments
|
|
|
|
new_increment_borders = np.empty_like(increment_borders)
|
|
|
|
# this whole code does a very simple thing: it eats up
|
|
# sub-increments until it has `group_size` of them
|
|
i = 0 # position in increment count list
|
|
j = 0 # position in new_incerement_borders
|
|
# the number of sub-increments still available
|
|
rest = new_increments[0]
|
|
head = group_size # the number of sub-increments needed to
|
|
# fill one increment
|
|
current = 0 # the current position in the interval relative
|
|
# to its beginning
|
|
|
|
while i < num_increments and (j < (num_increments - 1)):
|
|
if new_increments[i] == 0:
|
|
i += 1
|
|
rest = new_increments[i]
|
|
|
|
current_increment_size = interval_lengths[i]/new_increments[i]
|
|
|
|
if head <= rest:
|
|
current += head*current_increment_size
|
|
new_increment_borders[j] = current
|
|
rest -= head
|
|
head = group_size
|
|
j += 1
|
|
|
|
else:
|
|
current += rest*current_increment_size
|
|
head -= rest
|
|
i += 1
|
|
rest = new_increments[i]
|
|
|
|
interval_borders[1:-1] = interval_borders[0] + increment_borders
|
|
if np.linalg.norm(increment_borders - new_increment_borders) < epsilon:
|
|
break
|
|
|
|
increment_borders = new_increment_borders
|
|
|
|
# brute force increase of the sample size
|
|
if np.sqrt(variance) >= target_epsilon:
|
|
while True:
|
|
integral, _, variance = \
|
|
evaluate_integrand(interval_borders,
|
|
interval_lengths,
|
|
points_per_increment)
|
|
|
|
integrals.append(integral)
|
|
variances.append(variance)
|
|
|
|
if np.sqrt(variance) <= target_epsilon:
|
|
break
|
|
|
|
points_per_increment += int(100 *
|
|
interval_length/num_increments)
|
|
|
|
if acumulate:
|
|
integrals = np.array(integrals)
|
|
variances = np.array(variances)
|
|
integral = np.sum(integrals**3/variances**2) \
|
|
/ np.sum(integrals**2/variances**2)
|
|
variance = 1/np.sqrt(np.sum(integrals**2/variances**2)) \
|
|
* integral
|
|
|
|
return VegasIntegrationResult(integral,
|
|
np.sqrt(variance),
|
|
points_per_increment*num_increments,
|
|
interval_borders,
|
|
vegas_iterations)
|
|
|
|
|
|
def sample_stratified(f, increment_borders, seed=None, chunk_size=100,
|
|
report_efficiency=False, **kwargs):
|
|
"""Samples a distribution proportional to f by hit and miss.
|
|
Implemented as a generator.
|
|
|
|
:param f: function of one scalar to sample, should be positive,
|
|
superflous kwargs are passed to it
|
|
:param interval: the interval to sample from
|
|
:param seed: the seed for the rng, if not specified, the system
|
|
time is used
|
|
:param chunk_size: the size of the chunks of random numbers
|
|
allocated per unit interval
|
|
:yields: random nubers following the distribution of f
|
|
"""
|
|
|
|
increment_count = increment_borders.size - 1
|
|
increment_lenghts = increment_borders[1:] - increment_borders[:-1]
|
|
weights = increment_count*increment_lenghts
|
|
increment_chunk = int(chunk_size/increment_count)
|
|
chunk_size = increment_chunk*increment_count
|
|
|
|
upper_bound = \
|
|
np.array([find_upper_bound(lambda x: f(x, **kwargs)*weight,
|
|
[left_border, right_border])
|
|
for weight, left_border, right_border
|
|
in zip(weights,
|
|
increment_borders[:-1],
|
|
increment_borders[1:])]).max()
|
|
|
|
total_samples = 0
|
|
total_accepted = 0
|
|
|
|
while True:
|
|
increment_samples = np.random.uniform(increment_borders[:-1],
|
|
increment_borders[1:],
|
|
[increment_chunk,
|
|
increment_count])
|
|
increment_y_samples = np.random.uniform(0, 1,
|
|
[increment_chunk,
|
|
increment_count])
|
|
f_weighted = f(increment_samples)*weights # numpy magic at work here
|
|
mask = f_weighted > increment_y_samples*upper_bound
|
|
|
|
if report_efficiency:
|
|
total_samples += chunk_size
|
|
total_accepted += np.count_nonzero(mask)
|
|
|
|
for point in increment_samples[mask]:
|
|
yield (point, total_accepted/total_samples) \
|
|
if report_efficiency else point
|
|
|
|
|
|
def sample_unweighted_array(num, *args, increment_borders=None,
|
|
report_efficiency=False, **kwargs):
|
|
"""Sample `num` elements from a distribution. The rest of the
|
|
arguments is analogous to `sample_unweighted`.
|
|
"""
|
|
|
|
sample_arr = np.empty(num)
|
|
if 'chunk_size' not in kwargs:
|
|
kwargs['chunk_size'] = num*10
|
|
samples = \
|
|
sample_unweighted(*args,
|
|
report_efficiency=report_efficiency,
|
|
**kwargs) \
|
|
if increment_borders is None else \
|
|
sample_stratified(*args,
|
|
increment_borders=increment_borders,
|
|
report_efficiency=report_efficiency, **kwargs)
|
|
|
|
for i, sample in zip(range(num), samples):
|
|
if report_efficiency:
|
|
sample_arr[i], _ = sample
|
|
else:
|
|
sample_arr[i] = sample
|
|
|
|
return (sample_arr, next(samples)[1]) if report_efficiency else sample_arr
|