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estimate area under a curve defined by non-negative real-valued continuous function within a continuous interval using monte-carlo (#1785)

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* estimate area under a curve defined by non-negative real-valued continuous function within a continuous interval using monte-carlo

* run black; update comments

* Use f”strings” and drop unnecessary returns

Co-authored-by: Christian Clauss <cclauss@me.com>
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108
maths/monte_carlo.py
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@ -1,9 +1,10 @@
"""
@author: MatteoRaso
"""
from numpy import pi, sqrt
from math import pi, sqrt
from random import uniform
from statistics import mean
from typing import Callable
def pi_estimator(iterations: int):
@ -18,7 +19,7 @@ def pi_estimator(iterations: int):
6. Print the estimated and numpy value of pi
"""
# A local function to see if a dot lands in the circle.
def in_circle(x: float, y: float) -> bool:
def is_in_circle(x: float, y: float) -> bool:
distance_from_centre = sqrt((x ** 2) + (y ** 2))
# Our circle has a radius of 1, so a distance
# greater than 1 would land outside the circle.
@ -26,50 +27,99 @@ def pi_estimator(iterations: int):
# The proportion of guesses that landed in the circle
proportion = mean(
int(in_circle(uniform(-1.0, 1.0), uniform(-1.0, 1.0))) for _ in range(iterations)
int(is_in_circle(uniform(-1.0, 1.0), uniform(-1.0, 1.0)))
for _ in range(iterations)
)
# The ratio of the area for circle to square is pi/4.
pi_estimate = proportion * 4
print("The estimated value of pi is ", pi_estimate)
print("The numpy value of pi is ", pi)
print("The total error is ", abs(pi - pi_estimate))
print(f"The estimated value of pi is {pi_estimate}")
print(f"The numpy value of pi is {pi}")
print(f"The total error is {abs(pi - pi_estimate)}")
def area_under_line_estimator(iterations: int,
min_value: float=0.0,
max_value: float=1.0) -> float:
def area_under_curve_estimator(
iterations: int,
function_to_integrate: Callable[[float], float],
min_value: float = 0.0,
max_value: float = 1.0,
) -> float:
"""
An implementation of the Monte Carlo method to find area under
y = x where x lies between min_value to max_value
1. Let x be a uniformly distributed random variable between min_value to max_value
2. Expected value of x = (integration of x from min_value to max_value) / (max_value - min_value)
3. Finding expected value of x:
a single variable non-negative real-valued continuous function,
say f(x), where x lies within a continuous bounded interval,
say [min_value, max_value], where min_value and max_value are
finite numbers
1. Let x be a uniformly distributed random variable between min_value to
max_value
2. Expected value of f(x) =
(integrate f(x) from min_value to max_value)/(max_value - min_value)
3. Finding expected value of f(x):
a. Repeatedly draw x from uniform distribution
b. Expected value = average of those values
4. Actual value = (max_value^2 - min_value^2) / 2
b. Evaluate f(x) at each of the drawn x values
c. Expected value = average of the function evaluations
4. Estimated value of integral = Expected value * (max_value - min_value)
5. Returns estimated value
"""
return mean(uniform(min_value, max_value) for _ in range(iterations)) * (max_value - min_value)
return mean(
function_to_integrate(uniform(min_value, max_value)) for _ in range(iterations)
) * (max_value - min_value)
def area_under_line_estimator_check(iterations: int,
min_value: float=0.0,
max_value: float=1.0) -> None:
def area_under_line_estimator_check(
iterations: int, min_value: float = 0.0, max_value: float = 1.0
) -> None:
"""
Checks estimation error for area_under_line_estimator func
1. Calls "area_under_line_estimator" function
Checks estimation error for area_under_curve_estimator function
for f(x) = x where x lies within min_value to max_value
1. Calls "area_under_curve_estimator" function
2. Compares with the expected value
3. Prints estimated, expected and error value
"""
estimated_value = area_under_line_estimator(iterations, min_value, max_value)
expected_value = (max_value*max_value - min_value*min_value) / 2
def identity_function(x: float) -> float:
"""
Represents identity function
>>> [function_to_integrate(x) for x in [-2.0, -1.0, 0.0, 1.0, 2.0]]
[-2.0, -1.0, 0.0, 1.0, 2.0]
"""
return x
estimated_value = area_under_curve_estimator(
iterations, identity_function, min_value, max_value
)
expected_value = (max_value * max_value - min_value * min_value) / 2
print("******************")
print("Estimating area under y=x where x varies from ",min_value, " to ",max_value)
print("Estimated value is ", estimated_value)
print("Expected value is ", expected_value)
print("Total error is ", abs(estimated_value - expected_value))
print(f"Estimating area under y=x where x varies from {min_value} to {max_value}")
print(f"Estimated value is {estimated_value}")
print(f"Expected value is {expected_value}")
print(f"Total error is {abs(estimated_value - expected_value)}")
print("******************")
def pi_estimator_using_area_under_curve(iterations: int) -> None:
"""
Area under curve y = sqrt(4 - x^2) where x lies in 0 to 2 is equal to pi
"""
def function_to_integrate(x: float) -> float:
"""
Represents semi-circle with radius 2
>>> [function_to_integrate(x) for x in [-2.0, 0.0, 2.0]]
[0.0, 2.0, 0.0]
"""
return sqrt(4.0 - x * x)
estimated_value = area_under_curve_estimator(
iterations, function_to_integrate, 0.0, 2.0
)
print("******************")
print("Estimating pi using area_under_curve_estimator")
print(f"Estimated value is {estimated_value}")
print(f"Expected value is {pi}")
print(f"Total error is {abs(estimated_value - pi)}")
print("******************")